Power concave functions and Borell-Brascamp-Lieb inequalities SALANI PAOLO Università di Firenze IMA - University of Minneapolis, May 1 st , 2015 Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 1 / 34
Power concave functions and Borell-Brascamp-Liebinequalities
SALANI PAOLOUniversità di Firenze
IMA - University of Minneapolis, May 1st, 2015
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 1 / 34
Notations: p-means of non-negative numbers
Let p ∈ [−∞,+∞] and µ ∈ (0,1). Given two real numbers a > 0 and b > 0,the quantity
Mp(a,b;µ) =
maxa,b p = +∞((1− µ)ap + µbp)
1p for p 6= −∞,0,+∞
a1−µbµ p = 0mina,b p = −∞.
is the (µ-weighted) p-mean of a and b.
For a,b ≥ 0, we set Mp(a,b;µ) = 0 if ab = 0 and p ∈ R.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 2 / 34
Power concave functions
Let Ω be a convex set in Rn and p ∈ [−∞,∞]. A nonnegative function udefined in Ω is said p -concave if
u((1− λ)x + λy) ≥Mp(u(x),u(y);λ)
for all x , y ∈ Ω and λ ∈ (0,1). In the cases p = 0 and p = −∞, u is also saidlog-concave and quasi-concave in Ω, respectively.
In other words, a non-negative function u, with convex support Ω, isp-concave if:- it is a non-negative constant in Ω, for p = +∞;- up is concave in Ω, for p > 0;- log u is concave in Ω, for p = 0;- up is convex in Ω, for p < 0;- it is quasi-concave, i.e. all of its superlevel sets are convex, for p = −∞.For p = 1 corresponds to usual concavity.From Jensen’s inequality it follows that if u is p -concave, then u is q -concavefor every q ≤ p.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 3 / 34
Power concave functions
Let Ω be a convex set in Rn and p ∈ [−∞,∞]. A nonnegative function udefined in Ω is said p -concave if
u((1− λ)x + λy) ≥Mp(u(x),u(y);λ)
for all x , y ∈ Ω and λ ∈ (0,1). In the cases p = 0 and p = −∞, u is also saidlog-concave and quasi-concave in Ω, respectively.
In other words, a non-negative function u, with convex support Ω, isp-concave if:- it is a non-negative constant in Ω, for p = +∞;- up is concave in Ω, for p > 0;- log u is concave in Ω, for p = 0;- up is convex in Ω, for p < 0;- it is quasi-concave, i.e. all of its superlevel sets are convex, for p = −∞.For p = 1 corresponds to usual concavity.From Jensen’s inequality it follows that if u is p -concave, then u is q -concavefor every q ≤ p.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 3 / 34
The Borell-Brascamp-Lieb inequality
Let 0 < λ < 1,− 1n ≤ p ≤ ∞. Let u0,u1,h be nonnegative integrable functions
defined on Rn, satisfying
h((1− λ)x + λy) ≥ Mp(u0(x),u1(y), λ)
for all x ∈ Rn. Then∫Rn
h(x) dx ≥ Mq
(∫Rn
u0(x) dx ,∫Rn
u1(x) dx , λ)
where
q =
1/n p = +∞
ppn + 1
p ∈ (−1/n,+∞)
−∞ p = −1/n.
Henstock-Macbeath (1953), Dinghas (1957)
Borell (1975), Brascamp-Lieb (1976)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 4 / 34
The Borell-Brascamp-Lieb inequality
Let 0 < λ < 1,− 1n ≤ p ≤ ∞. Let u0,u1,h be nonnegative integrable functions
defined on Rn, satisfying
h((1− λ)x + λy) ≥ Mp(u0(x),u1(y), λ)
for all x ∈ Rn. Then∫Rn
h(x) dx ≥ Mq
(∫Rn
u0(x) dx ,∫Rn
u1(x) dx , λ)
where
q =
1/n p = +∞
ppn + 1
p ∈ (−1/n,+∞)
−∞ p = −1/n.
Henstock-Macbeath (1953), Dinghas (1957)
Borell (1975), Brascamp-Lieb (1976)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 4 / 34
The case p = 0
Prékopa-Leindler inequalityLet 0 < λ < 1 and let u0,u1 and h be nonnegative integrable functions on Rn
satisfyingh((1− λ)x + λy) ≥ u0(x)1−λu1(y)λ,
for all x , y ∈ Rn. Then∫Rn
h(x) dx ≥(∫
Rnu0(x) dx
)1−λ(∫Rn
u1(x) dx)λ
.
Prékopa, (1971 & 1973), Leindler (1972), Brascamp-Lieb (1975)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 5 / 34
Analysis versus Geometry
PL is a functional version of the Brunn-Minkowski inequality (in fact, the samecould be said for BBL for any p).
The Brunn-Minkowski inequalityK0,K1 measurable sets, λ ∈ [0,1], Kλ = (1− λ)K0 + λK1 and + is theMinkowski addition, then
|Kλ| ≥ M1/n (|K0|, |K1|;λ) =[(1− λ) |K0|
1n + λ |K1|
1n
]n(0.1)
provided Kλ is measurable as well.
Multiplicative form of BM|Kλ| ≥ |K0|1−λ|K1|λ
The BM inequality has strong and unexpected relations with many other
fundamental analytic and geometric inequalities.For references and a nice presentation, see R. J. Gardner (2002)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 6 / 34
Analysis versus Geometry
PL is a functional version of the Brunn-Minkowski inequality (in fact, the samecould be said for BBL for any p).
The Brunn-Minkowski inequalityK0,K1 measurable sets, λ ∈ [0,1], Kλ = (1− λ)K0 + λK1 and + is theMinkowski addition, then
|Kλ| ≥ M1/n (|K0|, |K1|;λ) =[(1− λ) |K0|
1n + λ |K1|
1n
]n(0.1)
provided Kλ is measurable as well.
Multiplicative form of BM|Kλ| ≥ |K0|1−λ|K1|λ
The BM inequality has strong and unexpected relations with many other
fundamental analytic and geometric inequalities.For references and a nice presentation, see R. J. Gardner (2002)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 6 / 34
Analysis versus Geometry
PL is a functional version of the Brunn-Minkowski inequality (in fact, the samecould be said for BBL for any p).
The Brunn-Minkowski inequalityK0,K1 measurable sets, λ ∈ [0,1], Kλ = (1− λ)K0 + λK1 and + is theMinkowski addition, then
|Kλ| ≥ M1/n (|K0|, |K1|;λ) =[(1− λ) |K0|
1n + λ |K1|
1n
]n(0.1)
provided Kλ is measurable as well.
Multiplicative form of BM|Kλ| ≥ |K0|1−λ|K1|λ
The BM inequality has strong and unexpected relations with many other
fundamental analytic and geometric inequalities.
For references and a nice presentation, see R. J. Gardner (2002)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 6 / 34
Analysis versus Geometry
PL is a functional version of the Brunn-Minkowski inequality (in fact, the samecould be said for BBL for any p).
The Brunn-Minkowski inequalityK0,K1 measurable sets, λ ∈ [0,1], Kλ = (1− λ)K0 + λK1 and + is theMinkowski addition, then
|Kλ| ≥ M1/n (|K0|, |K1|;λ) =[(1− λ) |K0|
1n + λ |K1|
1n
]n(0.1)
provided Kλ is measurable as well.
Multiplicative form of BM|Kλ| ≥ |K0|1−λ|K1|λ
The BM inequality has strong and unexpected relations with many other
fundamental analytic and geometric inequalities.For references and a nice presentation, see R. J. Gardner (2002)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 6 / 34
Rigidity
What happens if equality is achieved in one of the above mentionedinequalities?
BMEquality holds in BM if and only if K0 and K1 are convex and homothetic.
Equality conditions in BBL - Dubuc (1977)Equality holds in BBL for some p ∈ [−1/n,∞) if and only if
h is p-concave
and there exist suitable A,B,m,n > 0 and x1, xλ ∈ Rn such that
u0(x) = A h(mx + x1) , u1(x) = B h(nx + xλ) .
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 7 / 34
Rigidity
What happens if equality is achieved in one of the above mentionedinequalities?
BMEquality holds in BM if and only if K0 and K1 are convex and homothetic.
Equality conditions in BBL - Dubuc (1977)Equality holds in BBL for some p ∈ [−1/n,∞) if and only if
h is p-concave
and there exist suitable A,B,m,n > 0 and x1, xλ ∈ Rn such that
u0(x) = A h(mx + x1) , u1(x) = B h(nx + xλ) .
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 7 / 34
Rigidity
What happens if equality is achieved in one of the above mentionedinequalities?
BMEquality holds in BM if and only if K0 and K1 are convex and homothetic.
Equality conditions in BBL - Dubuc (1977)Equality holds in BBL for some p ∈ [−1/n,∞) if and only if
h is p-concave
and there exist suitable A,B,m,n > 0 and x1, xλ ∈ Rn such that
u0(x) = A h(mx + x1) , u1(x) = B h(nx + xλ) .
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 7 / 34
Stability/ImprovementsWhat happens if we are close to equality in one of the above mentionedinequalities?
Are the involved sets/functions close to be homothetic?In otherwords: is it possible to improve the above mentioned inequality in terms ofsome distance from the "rigid situation"?
There are stability/quantitative results for the Brunn-Minkowski inequality forconvex sets. See for instance: Diskant (1973), Groemer (1988), Schneider(1993), Figalli-Maggi-Pratelli (2009-2010), Figalli-Jerison preprint (2014).Not a complete list of course!
There are only three results about PL to my knowledge!
[1] K. M. Ball, K. J. Böröczky, Stability of the Prékopa-Leindler inequality,Mathematika, 56 (2010), no. 2, 339-356.
[2] K. M. Ball, K. J. Böröczky, Stability of some versions of thePrékopa-Leindler inequality, Monatsh. Math. 163 (2011), no. 1, 1-14.
[3] D. Bucur and I. Fragalà, Lower bounds for the Prékopa-Leindler deficit bysome distances modulo translations, J. Convex Anal. 21 (2014), no. 1,289-305.
(All for log-concave functions)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 8 / 34
Stability/ImprovementsWhat happens if we are close to equality in one of the above mentionedinequalities?Are the involved sets/functions close to be homothetic?
In otherwords: is it possible to improve the above mentioned inequality in terms ofsome distance from the "rigid situation"?
There are stability/quantitative results for the Brunn-Minkowski inequality forconvex sets. See for instance: Diskant (1973), Groemer (1988), Schneider(1993), Figalli-Maggi-Pratelli (2009-2010), Figalli-Jerison preprint (2014).Not a complete list of course!
There are only three results about PL to my knowledge!
[1] K. M. Ball, K. J. Böröczky, Stability of the Prékopa-Leindler inequality,Mathematika, 56 (2010), no. 2, 339-356.
[2] K. M. Ball, K. J. Böröczky, Stability of some versions of thePrékopa-Leindler inequality, Monatsh. Math. 163 (2011), no. 1, 1-14.
[3] D. Bucur and I. Fragalà, Lower bounds for the Prékopa-Leindler deficit bysome distances modulo translations, J. Convex Anal. 21 (2014), no. 1,289-305.
(All for log-concave functions)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 8 / 34
Stability/ImprovementsWhat happens if we are close to equality in one of the above mentionedinequalities?Are the involved sets/functions close to be homothetic?In otherwords: is it possible to improve the above mentioned inequality in terms ofsome distance from the "rigid situation"?
There are stability/quantitative results for the Brunn-Minkowski inequality forconvex sets. See for instance: Diskant (1973), Groemer (1988), Schneider(1993), Figalli-Maggi-Pratelli (2009-2010), Figalli-Jerison preprint (2014).Not a complete list of course!
There are only three results about PL to my knowledge!
[1] K. M. Ball, K. J. Böröczky, Stability of the Prékopa-Leindler inequality,Mathematika, 56 (2010), no. 2, 339-356.
[2] K. M. Ball, K. J. Böröczky, Stability of some versions of thePrékopa-Leindler inequality, Monatsh. Math. 163 (2011), no. 1, 1-14.
[3] D. Bucur and I. Fragalà, Lower bounds for the Prékopa-Leindler deficit bysome distances modulo translations, J. Convex Anal. 21 (2014), no. 1,289-305.
(All for log-concave functions)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 8 / 34
Stability/ImprovementsWhat happens if we are close to equality in one of the above mentionedinequalities?Are the involved sets/functions close to be homothetic?In otherwords: is it possible to improve the above mentioned inequality in terms ofsome distance from the "rigid situation"?
There are stability/quantitative results for the Brunn-Minkowski inequality forconvex sets.
See for instance: Diskant (1973), Groemer (1988), Schneider(1993), Figalli-Maggi-Pratelli (2009-2010), Figalli-Jerison preprint (2014).Not a complete list of course!
There are only three results about PL to my knowledge!
[1] K. M. Ball, K. J. Böröczky, Stability of the Prékopa-Leindler inequality,Mathematika, 56 (2010), no. 2, 339-356.
[2] K. M. Ball, K. J. Böröczky, Stability of some versions of thePrékopa-Leindler inequality, Monatsh. Math. 163 (2011), no. 1, 1-14.
[3] D. Bucur and I. Fragalà, Lower bounds for the Prékopa-Leindler deficit bysome distances modulo translations, J. Convex Anal. 21 (2014), no. 1,289-305.
(All for log-concave functions)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 8 / 34
Stability/ImprovementsWhat happens if we are close to equality in one of the above mentionedinequalities?Are the involved sets/functions close to be homothetic?In otherwords: is it possible to improve the above mentioned inequality in terms ofsome distance from the "rigid situation"?
There are stability/quantitative results for the Brunn-Minkowski inequality forconvex sets. See for instance: Diskant (1973), Groemer (1988), Schneider(1993), Figalli-Maggi-Pratelli (2009-2010), Figalli-Jerison preprint (2014).Not a complete list of course!
There are only three results about PL to my knowledge!
[1] K. M. Ball, K. J. Böröczky, Stability of the Prékopa-Leindler inequality,Mathematika, 56 (2010), no. 2, 339-356.
[2] K. M. Ball, K. J. Böröczky, Stability of some versions of thePrékopa-Leindler inequality, Monatsh. Math. 163 (2011), no. 1, 1-14.
[3] D. Bucur and I. Fragalà, Lower bounds for the Prékopa-Leindler deficit bysome distances modulo translations, J. Convex Anal. 21 (2014), no. 1,289-305.
(All for log-concave functions)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 8 / 34
Stability/ImprovementsWhat happens if we are close to equality in one of the above mentionedinequalities?Are the involved sets/functions close to be homothetic?In otherwords: is it possible to improve the above mentioned inequality in terms ofsome distance from the "rigid situation"?
There are stability/quantitative results for the Brunn-Minkowski inequality forconvex sets. See for instance: Diskant (1973), Groemer (1988), Schneider(1993), Figalli-Maggi-Pratelli (2009-2010), Figalli-Jerison preprint (2014).Not a complete list of course!
There are only three results about PL to my knowledge!
[1] K. M. Ball, K. J. Böröczky, Stability of the Prékopa-Leindler inequality,Mathematika, 56 (2010), no. 2, 339-356.
[2] K. M. Ball, K. J. Böröczky, Stability of some versions of thePrékopa-Leindler inequality, Monatsh. Math. 163 (2011), no. 1, 1-14.
[3] D. Bucur and I. Fragalà, Lower bounds for the Prékopa-Leindler deficit bysome distances modulo translations, J. Convex Anal. 21 (2014), no. 1,289-305.
(All for log-concave functions)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 8 / 34
Stability/ImprovementsWhat happens if we are close to equality in one of the above mentionedinequalities?Are the involved sets/functions close to be homothetic?In otherwords: is it possible to improve the above mentioned inequality in terms ofsome distance from the "rigid situation"?
There are stability/quantitative results for the Brunn-Minkowski inequality forconvex sets. See for instance: Diskant (1973), Groemer (1988), Schneider(1993), Figalli-Maggi-Pratelli (2009-2010), Figalli-Jerison preprint (2014).Not a complete list of course!
There are only three results about PL to my knowledge!
[1] K. M. Ball, K. J. Böröczky, Stability of the Prékopa-Leindler inequality,Mathematika, 56 (2010), no. 2, 339-356.
[2] K. M. Ball, K. J. Böröczky, Stability of some versions of thePrékopa-Leindler inequality, Monatsh. Math. 163 (2011), no. 1, 1-14.
[3] D. Bucur and I. Fragalà, Lower bounds for the Prékopa-Leindler deficit bysome distances modulo translations, J. Convex Anal. 21 (2014), no. 1,289-305.
(All for log-concave functions)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 8 / 34
Stability/ImprovementsWhat happens if we are close to equality in one of the above mentionedinequalities?Are the involved sets/functions close to be homothetic?In otherwords: is it possible to improve the above mentioned inequality in terms ofsome distance from the "rigid situation"?
There are stability/quantitative results for the Brunn-Minkowski inequality forconvex sets. See for instance: Diskant (1973), Groemer (1988), Schneider(1993), Figalli-Maggi-Pratelli (2009-2010), Figalli-Jerison preprint (2014).Not a complete list of course!
There are only three results about PL to my knowledge!
[1] K. M. Ball, K. J. Böröczky, Stability of the Prékopa-Leindler inequality,Mathematika, 56 (2010), no. 2, 339-356.
[2] K. M. Ball, K. J. Böröczky, Stability of some versions of thePrékopa-Leindler inequality, Monatsh. Math. 163 (2011), no. 1, 1-14.
[3] D. Bucur and I. Fragalà, Lower bounds for the Prékopa-Leindler deficit bysome distances modulo translations, J. Convex Anal. 21 (2014), no. 1,289-305.
(All for log-concave functions)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 8 / 34
Stability/ImprovementsWhat happens if we are close to equality in one of the above mentionedinequalities?Are the involved sets/functions close to be homothetic?In otherwords: is it possible to improve the above mentioned inequality in terms ofsome distance from the "rigid situation"?
There are stability/quantitative results for the Brunn-Minkowski inequality forconvex sets. See for instance: Diskant (1973), Groemer (1988), Schneider(1993), Figalli-Maggi-Pratelli (2009-2010), Figalli-Jerison preprint (2014).Not a complete list of course!
There are only three results about PL to my knowledge!
[1] K. M. Ball, K. J. Böröczky, Stability of the Prékopa-Leindler inequality,Mathematika, 56 (2010), no. 2, 339-356.
[2] K. M. Ball, K. J. Böröczky, Stability of some versions of thePrékopa-Leindler inequality, Monatsh. Math. 163 (2011), no. 1, 1-14.
[3] D. Bucur and I. Fragalà, Lower bounds for the Prékopa-Leindler deficit bysome distances modulo translations, J. Convex Anal. 21 (2014), no. 1,289-305.
(All for log-concave functions)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 8 / 34
Stability/ImprovementsWhat happens if we are close to equality in one of the above mentionedinequalities?Are the involved sets/functions close to be homothetic?In otherwords: is it possible to improve the above mentioned inequality in terms ofsome distance from the "rigid situation"?
There are stability/quantitative results for the Brunn-Minkowski inequality forconvex sets. See for instance: Diskant (1973), Groemer (1988), Schneider(1993), Figalli-Maggi-Pratelli (2009-2010), Figalli-Jerison preprint (2014).Not a complete list of course!
There are only three results about PL to my knowledge!
[1] K. M. Ball, K. J. Böröczky, Stability of the Prékopa-Leindler inequality,Mathematika, 56 (2010), no. 2, 339-356.
[2] K. M. Ball, K. J. Böröczky, Stability of some versions of thePrékopa-Leindler inequality, Monatsh. Math. 163 (2011), no. 1, 1-14.
[3] D. Bucur and I. Fragalà, Lower bounds for the Prékopa-Leindler deficit bysome distances modulo translations, J. Convex Anal. 21 (2014), no. 1,289-305.
(All for log-concave functions)Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 8 / 34
Stability/Improvements of PL for log-concave functions
The Ball-Böröczky [1] result is for log-concave functions in dimension 1 and itis written as a stability result: if
∫R hdx ≤ (1 + ε)
√∫R u0dx
∫R u1dx , then there
exist a > 0 and b ∈ Rn such that∫Rn|a(−1)i
ui (x + (−1)ib)− h(x)|dx ≤ γ ε1/6| log ε|2/3∫Rn
h dx .
This is extended to dimension n > 1 in [2], but only for log-concave evenfunctions.
Bucur-Fragalà [3] use the 1-dimensional result by Ball-Böröczky to write aquantitative version of the PL for log-concave functions in terms of some (alittle bit involved) distance between u0 and u1, that is∫
Rnh dx ≥
[1 + Ψλ,n(dn(u0,u1))
](∫Rn
u0dx)1−λ(∫
Rnu1dx
)λwhere dn measure the distance of u0 and u1 from coinciding up to anhomothety.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 9 / 34
Stability/Improvements of PL for log-concave functions
The Ball-Böröczky [1] result is for log-concave functions in dimension 1 and itis written as a stability result: if
∫R hdx ≤ (1 + ε)
√∫R u0dx
∫R u1dx , then there
exist a > 0 and b ∈ Rn such that∫Rn|a(−1)i
ui (x + (−1)ib)− h(x)|dx ≤ γ ε1/6| log ε|2/3∫Rn
h dx .
This is extended to dimension n > 1 in [2], but only for log-concave evenfunctions.
Bucur-Fragalà [3] use the 1-dimensional result by Ball-Böröczky to write aquantitative version of the PL for log-concave functions in terms of some (alittle bit involved) distance between u0 and u1, that is∫
Rnh dx ≥
[1 + Ψλ,n(dn(u0,u1))
](∫Rn
u0dx)1−λ(∫
Rnu1dx
)λwhere dn measure the distance of u0 and u1 from coinciding up to anhomothety.Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 9 / 34
Quantitative BBL for p-concave functions with p > 0
Joint work with D. Ghilli (Università di Padova).
Let H denotes the Hausdorff distance between sets in Rn, we set
H0(K ,L) = H(τ0K , τ1L), (0.2)
where τ1, τ0 are two homotheties (i.e. translation plus dilation) such that|τ0K | = |τ1L| = 1 and such that the centroids of τ0K and τ1L coincide.
Theorem 1 (Ghilli-S. 2015)Let p > 0 and assume that u0 and u1 are L1 p-concave functions, with convexcompact supports Ω0 and Ω1 respectively. Then, if H0(Ω0,Ω1) is smallenough, it holds∫
Ωλ
h(x) dx ≥M pnp+1
(I0, I1, λ)[1 + β H0(Ω0,Ω1)
(n+1)(p+1)p
](0.3)
where β is a constant depending on n, λ, p,M pnp+1
(I0, I1, λ) and thediameters and the measures of Ω0 and Ω1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 10 / 34
Quantitative BBL for p-concave functions with p > 0
Joint work with D. Ghilli (Università di Padova).Let H denotes the Hausdorff distance between sets in Rn, we set
H0(K ,L) = H(τ0K , τ1L), (0.2)
where τ1, τ0 are two homotheties (i.e. translation plus dilation) such that|τ0K | = |τ1L| = 1 and such that the centroids of τ0K and τ1L coincide.
Theorem 1 (Ghilli-S. 2015)Let p > 0 and assume that u0 and u1 are L1 p-concave functions, with convexcompact supports Ω0 and Ω1 respectively. Then, if H0(Ω0,Ω1) is smallenough, it holds∫
Ωλ
h(x) dx ≥M pnp+1
(I0, I1, λ)[1 + β H0(Ω0,Ω1)
(n+1)(p+1)p
](0.3)
where β is a constant depending on n, λ, p,M pnp+1
(I0, I1, λ) and thediameters and the measures of Ω0 and Ω1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 10 / 34
Quantitative BBL for p-concave functions with p > 0
Joint work with D. Ghilli (Università di Padova).Let H denotes the Hausdorff distance between sets in Rn, we set
H0(K ,L) = H(τ0K , τ1L), (0.2)
where τ1, τ0 are two homotheties (i.e. translation plus dilation) such that|τ0K | = |τ1L| = 1 and such that the centroids of τ0K and τ1L coincide.
Theorem 1 (Ghilli-S. 2015)Let p > 0 and assume that u0 and u1 are L1 p-concave functions, with convexcompact supports Ω0 and Ω1 respectively. Then, if H0(Ω0,Ω1) is smallenough, it holds∫
Ωλ
h(x) dx ≥M pnp+1
(I0, I1, λ)[1 + β H0(Ω0,Ω1)
(n+1)(p+1)p
](0.3)
where β is a constant depending on n, λ, p,M pnp+1
(I0, I1, λ) and thediameters and the measures of Ω0 and Ω1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 10 / 34
Quantitative BBL for p-concave functions with p > 0
Joint work with D. Ghilli (Università di Padova).
Let A denote the relative asymmetry of two sets, that is
A(K ,L) := infx∈Rn
|K ∆(x + λF )|
|K |, λ =
(|K ||L|
) 1n, (0.4)
where ∆ denotes the operation of symmetric difference, i.e.Ω ∆ B = (Ω \ B) ∪ (B \ Ω).
Theorem 2 (Ghilli-S. 2015)In the same assumptions and notation of Theorem 1, if A(Ω0,Ω1) is smallenough it holds∫
Ωλ
h(x) dx ≥M pnp+1
(I0, I1, λ)[1 + δ A(Ω0,Ω1)
2(p+1)p], (0.5)
where δ is a constant depending only on n, λ, p,M pnp+1
(I0, I1, λ) and on themeasures of Ω0 and Ω1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 11 / 34
Quantitative BBL for p-concave functions with p > 0
Joint work with D. Ghilli (Università di Padova).Let A denote the relative asymmetry of two sets, that is
A(K ,L) := infx∈Rn
|K ∆(x + λF )|
|K |, λ =
(|K ||L|
) 1n, (0.4)
where ∆ denotes the operation of symmetric difference, i.e.Ω ∆ B = (Ω \ B) ∪ (B \ Ω).
Theorem 2 (Ghilli-S. 2015)In the same assumptions and notation of Theorem 1, if A(Ω0,Ω1) is smallenough it holds∫
Ωλ
h(x) dx ≥M pnp+1
(I0, I1, λ)[1 + δ A(Ω0,Ω1)
2(p+1)p], (0.5)
where δ is a constant depending only on n, λ, p,M pnp+1
(I0, I1, λ) and on themeasures of Ω0 and Ω1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 11 / 34
Quantitative BBL for p-concave functions with p > 0
Joint work with D. Ghilli (Università di Padova).Let A denote the relative asymmetry of two sets, that is
A(K ,L) := infx∈Rn
|K ∆(x + λF )|
|K |, λ =
(|K ||L|
) 1n, (0.4)
where ∆ denotes the operation of symmetric difference, i.e.Ω ∆ B = (Ω \ B) ∪ (B \ Ω).
Theorem 2 (Ghilli-S. 2015)In the same assumptions and notation of Theorem 1, if A(Ω0,Ω1) is smallenough it holds∫
Ωλ
h(x) dx ≥M pnp+1
(I0, I1, λ)[1 + δ A(Ω0,Ω1)
2(p+1)p], (0.5)
where δ is a constant depending only on n, λ, p,M pnp+1
(I0, I1, λ) and on themeasures of Ω0 and Ω1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 11 / 34
Quantitative BBL for p-concave functions with p > 0Remarks. 0. Case p = 1 is easy! (In fact the same can be said forp = 1/k , k ∈ N)
1. In both theorems it is not necessary that all the involved functions arep-concave, it is just sufficient that h only is p-concave.
2. We in fact prove more than what stated above and the support sets Ω0 andΩ1 could be replaced by any couple of level sets of u0 and u1, suitably related.However, for the application we have in mind (quantitative BM inequalities forvariational functionals), we are mainly interested in the support sets.
3. The proof of both theorems essentially amounts to proving the followingand then applying existing quantitative results for the classical BM inequality.
Main theoremIf for some (small enough) ε > 0 it holds∫
Ωλ
h(x) dx ≤M pnp+1
(∫Ω0
u0(x) dx ,∫
Ω1
u1(x) dx ; λ
)+ ε, (0.6)
then|Ωλ| ≤ M 1
n(|Ω0|, |Ω1|, λ)
[1 + ηε
pp+1
]. (0.7)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 12 / 34
Quantitative BBL for p-concave functions with p > 0Remarks. 0. Case p = 1 is easy! (In fact the same can be said forp = 1/k , k ∈ N)
1. In both theorems it is not necessary that all the involved functions arep-concave, it is just sufficient that h only is p-concave.
2. We in fact prove more than what stated above and the support sets Ω0 andΩ1 could be replaced by any couple of level sets of u0 and u1, suitably related.However, for the application we have in mind (quantitative BM inequalities forvariational functionals), we are mainly interested in the support sets.
3. The proof of both theorems essentially amounts to proving the followingand then applying existing quantitative results for the classical BM inequality.
Main theoremIf for some (small enough) ε > 0 it holds∫
Ωλ
h(x) dx ≤M pnp+1
(∫Ω0
u0(x) dx ,∫
Ω1
u1(x) dx ; λ
)+ ε, (0.6)
then|Ωλ| ≤ M 1
n(|Ω0|, |Ω1|, λ)
[1 + ηε
pp+1
]. (0.7)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 12 / 34
Quantitative BBL for p-concave functions with p > 0Remarks. 0. Case p = 1 is easy! (In fact the same can be said forp = 1/k , k ∈ N)
1. In both theorems it is not necessary that all the involved functions arep-concave, it is just sufficient that h only is p-concave.
2. We in fact prove more than what stated above and the support sets Ω0 andΩ1 could be replaced by any couple of level sets of u0 and u1, suitably related.
However, for the application we have in mind (quantitative BM inequalities forvariational functionals), we are mainly interested in the support sets.
3. The proof of both theorems essentially amounts to proving the followingand then applying existing quantitative results for the classical BM inequality.
Main theoremIf for some (small enough) ε > 0 it holds∫
Ωλ
h(x) dx ≤M pnp+1
(∫Ω0
u0(x) dx ,∫
Ω1
u1(x) dx ; λ
)+ ε, (0.6)
then|Ωλ| ≤ M 1
n(|Ω0|, |Ω1|, λ)
[1 + ηε
pp+1
]. (0.7)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 12 / 34
Quantitative BBL for p-concave functions with p > 0Remarks. 0. Case p = 1 is easy! (In fact the same can be said forp = 1/k , k ∈ N)
1. In both theorems it is not necessary that all the involved functions arep-concave, it is just sufficient that h only is p-concave.
2. We in fact prove more than what stated above and the support sets Ω0 andΩ1 could be replaced by any couple of level sets of u0 and u1, suitably related.However, for the application we have in mind (quantitative BM inequalities forvariational functionals), we are mainly interested in the support sets.
3. The proof of both theorems essentially amounts to proving the followingand then applying existing quantitative results for the classical BM inequality.
Main theoremIf for some (small enough) ε > 0 it holds∫
Ωλ
h(x) dx ≤M pnp+1
(∫Ω0
u0(x) dx ,∫
Ω1
u1(x) dx ; λ
)+ ε, (0.6)
then|Ωλ| ≤ M 1
n(|Ω0|, |Ω1|, λ)
[1 + ηε
pp+1
]. (0.7)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 12 / 34
Quantitative BBL for p-concave functions with p > 0Remarks. 0. Case p = 1 is easy! (In fact the same can be said forp = 1/k , k ∈ N)
1. In both theorems it is not necessary that all the involved functions arep-concave, it is just sufficient that h only is p-concave.
2. We in fact prove more than what stated above and the support sets Ω0 andΩ1 could be replaced by any couple of level sets of u0 and u1, suitably related.However, for the application we have in mind (quantitative BM inequalities forvariational functionals), we are mainly interested in the support sets.
3. The proof of both theorems essentially amounts to proving the followingand then applying existing quantitative results for the classical BM inequality.
Main theoremIf for some (small enough) ε > 0 it holds∫
Ωλ
h(x) dx ≤M pnp+1
(∫Ω0
u0(x) dx ,∫
Ω1
u1(x) dx ; λ
)+ ε, (0.6)
then|Ωλ| ≤ M 1
n(|Ω0|, |Ω1|, λ)
[1 + ηε
pp+1
]. (0.7)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 12 / 34
Sketch of the proof
LetIi =
∫Ωi
ui dx i = 0,1 ,
Iλ =
∫Ωλ
h dx
andLi = max
Ωi
ui i = 0,1, Lλ = maxΩλ
h
Consider the distribution functions
µi (s) = |ui ≥ s| i = 0,1 , µλ(s) = |up,λ ≥ s|
Then
Ii =
∫ Li
0µi (s) ds i = 0,1, λ.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 13 / 34
Sketch of the proof
Notice that the assumption of BBL is equivalent to
h ≥Mp(s0, s1;λ) ⊇ (1− λ)u0 ≥ s0+ λu1 ≥ s1 (0.8)
for s0 ∈ [0,L0], s1 ∈ [0,L1].
Then, using the Brunn-Minkowski inequality, weget
µλ(Mp(s0, s1;λ)) ≥M 1n(µ0(s0), µ1(s1), λ). (0.9)
Now define the functions si : [0,1]→ [0,Li ] for i = 0,1 such that
si (t) :1Ii
∫ si (t)
0µi (s) ds = t for t ∈ [0,1] , (0.10)
and setsλ(t) =Mp(s0(t), s1(t), λ) t ∈ [0,1] .
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 14 / 34
Sketch of the proof
Notice that the assumption of BBL is equivalent to
h ≥Mp(s0, s1;λ) ⊇ (1− λ)u0 ≥ s0+ λu1 ≥ s1 (0.8)
for s0 ∈ [0,L0], s1 ∈ [0,L1]. Then, using the Brunn-Minkowski inequality, weget
µλ(Mp(s0, s1;λ)) ≥M 1n(µ0(s0), µ1(s1), λ). (0.9)
Now define the functions si : [0,1]→ [0,Li ] for i = 0,1 such that
si (t) :1Ii
∫ si (t)
0µi (s) ds = t for t ∈ [0,1] , (0.10)
and setsλ(t) =Mp(s0(t), s1(t), λ) t ∈ [0,1] .
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 14 / 34
Sketch of the proof
Notice that the assumption of BBL is equivalent to
h ≥Mp(s0, s1;λ) ⊇ (1− λ)u0 ≥ s0+ λu1 ≥ s1 (0.8)
for s0 ∈ [0,L0], s1 ∈ [0,L1]. Then, using the Brunn-Minkowski inequality, weget
µλ(Mp(s0, s1;λ)) ≥M 1n(µ0(s0), µ1(s1), λ). (0.9)
Now define the functions si : [0,1]→ [0,Li ] for i = 0,1 such that
si (t) :1Ii
∫ si (t)
0µi (s) ds = t for t ∈ [0,1] , (0.10)
and setsλ(t) =Mp(s0(t), s1(t), λ) t ∈ [0,1] .
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 14 / 34
Sketch of the proof
Thanks to (0.9), we get
µλ(sλ(t)) ≥M 1n(µ0(s0(t)), µ1(s1(t)), λ) t ∈ [0,1] (0.11)
Now, given any α ∈ (0,1), set
Fε = t ∈ [0,1] : µλ(sλ(t)) >M 1n(µ0(s0(t)), µ1(s1(t)), λ) + ε1−α (0.12)
andΓε = sλ(t) : t ∈ Fε . (0.13)
We want to find a bound of |Γε| in terms of ε and, playing with the integralsand using the assumption, it is actually possible and we find
|Γε| ≤ εα. (0.14)
Now choosing the right power α = pp+1 and using the p-concavity of h and the
Brunn-Minkowski inequality we get the conclusion.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 15 / 34
Sketch of the proof
Thanks to (0.9), we get
µλ(sλ(t)) ≥M 1n(µ0(s0(t)), µ1(s1(t)), λ) t ∈ [0,1] (0.11)
Now, given any α ∈ (0,1), set
Fε = t ∈ [0,1] : µλ(sλ(t)) >M 1n(µ0(s0(t)), µ1(s1(t)), λ) + ε1−α (0.12)
andΓε = sλ(t) : t ∈ Fε . (0.13)
We want to find a bound of |Γε| in terms of ε and, playing with the integralsand using the assumption, it is actually possible and we find
|Γε| ≤ εα. (0.14)
Now choosing the right power α = pp+1 and using the p-concavity of h and the
Brunn-Minkowski inequality we get the conclusion.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 15 / 34
Sketch of the proof
Thanks to (0.9), we get
µλ(sλ(t)) ≥M 1n(µ0(s0(t)), µ1(s1(t)), λ) t ∈ [0,1] (0.11)
Now, given any α ∈ (0,1), set
Fε = t ∈ [0,1] : µλ(sλ(t)) >M 1n(µ0(s0(t)), µ1(s1(t)), λ) + ε1−α (0.12)
andΓε = sλ(t) : t ∈ Fε . (0.13)
We want to find a bound of |Γε| in terms of ε and, playing with the integralsand using the assumption, it is actually possible and we find
|Γε| ≤ εα. (0.14)
Now choosing the right power α = pp+1 and using the p-concavity of h and the
Brunn-Minkowski inequality we get the conclusion.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 15 / 34
Examples of applications
1. Quantitative Brunn-Minkowksi inequalities for variational functional.
2. Quantitative Urysohn’s inequaities for the same functionals.
I will show in some detail the case of torsional rigidity.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 16 / 34
Brunn-Minkowksi inequalities for variationalfunctionals and PDEs
The leading idea is to find a way to compare solutions of different equations indifferent domains, I mean.... whiteboard —>
More explicitly, consider two sets Ω0 and Ω1 and a real number µ ∈ (0,1), anddenote by Ωµ the Minkowski convex combination (with coefficient µ) of Ω0 andΩ1, that is
Ωµ = (1− µ)Ω0 + µΩ1 = (1− µ)x0 + µ x1 : x0 ∈ Ω0, x1 ∈ Ω1 .
Correspondingly, let u0, u1 and uµ be the solutions of
(Pi )
Fi (x ,ui ,Dui ,D2ui ) = 0 in Ωi ,ui = 0 on ∂Ωi , i = 0,1, µui > 0 in Ωi ,
where Fi : Rn × R× Rn × Sn → R is a continuous (proper) degenerate ellipticoperator, i.e. decreasing with respect to u and increasing w.r.t. to the(Hessian) matrix variable A.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 17 / 34
Brunn-Minkowksi inequalities for variationalfunctionals and PDEs
The leading idea is to find a way to compare solutions of different equations indifferent domains, I mean.... whiteboard —>
More explicitly, consider two sets Ω0 and Ω1 and a real number µ ∈ (0,1), anddenote by Ωµ the Minkowski convex combination (with coefficient µ) of Ω0 andΩ1, that is
Ωµ = (1− µ)Ω0 + µΩ1 = (1− µ)x0 + µ x1 : x0 ∈ Ω0, x1 ∈ Ω1 .
Correspondingly, let u0, u1 and uµ be the solutions of
(Pi )
Fi (x ,ui ,Dui ,D2ui ) = 0 in Ωi ,ui = 0 on ∂Ωi , i = 0,1, µui > 0 in Ωi ,
where Fi : Rn × R× Rn × Sn → R is a continuous (proper) degenerate ellipticoperator, i.e. decreasing with respect to u and increasing w.r.t. to the(Hessian) matrix variable A.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 17 / 34
Brunn-Minkowksi inequalities for variationalfunctionals and PDEs
The leading idea is to find a way to compare solutions of different equations indifferent domains, I mean.... whiteboard —>
More explicitly, consider two sets Ω0 and Ω1 and a real number µ ∈ (0,1), anddenote by Ωµ the Minkowski convex combination (with coefficient µ) of Ω0 andΩ1, that is
Ωµ = (1− µ)Ω0 + µΩ1 = (1− µ)x0 + µ x1 : x0 ∈ Ω0, x1 ∈ Ω1 .
Correspondingly, let u0, u1 and uµ be the solutions of
(Pi )
Fi (x ,ui ,Dui ,D2ui ) = 0 in Ωi ,ui = 0 on ∂Ωi , i = 0,1, µui > 0 in Ωi ,
where Fi : Rn × R× Rn × Sn → R is a continuous (proper) degenerate ellipticoperator, i.e. decreasing with respect to u and increasing w.r.t. to the(Hessian) matrix variable A.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 17 / 34
Combinations of solutions in different sets
u0 sol. of (P0) in Ω0 u1 sol. of (P1) in Ω1
x0 ∈ Ω0 −→ u0(x0) x1 ∈ Ω1 −→ u1(x1)
uµ sol. of (Pµ) in Ωµ
x = (1− µ)x0 + µx1 ∈ Ωµ −→ uµ(x)
Question: are there suitable assumptions on the operators F0, F1 and Fµwhich permit to compare uµ with (a suitable convolution of) u0 and u1?Precisely, we want to find suitable conditions on the operators F0, F1, Fµwhich guarantee
uµ((1− µ)x0 + µx1) ≥ Mp(u0(x0),u1(x1);µ)
for every x0 ∈ Ω0, x1 ∈ Ω1, for some p ∈ R.
Equivalently
uµ(x) ≥ supMp(u0(x0),u1(x1);µ) : x0 ∈ Ω0, x1 ∈ Ω1, x = (1− µ)x0 + µx1
for every x ∈ Ωµ = (1− µ)Ω0 + µΩ1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 18 / 34
Combinations of solutions in different sets
u0 sol. of (P0) in Ω0 u1 sol. of (P1) in Ω1x0 ∈ Ω0 −→ u0(x0) x1 ∈ Ω1 −→ u1(x1)
uµ sol. of (Pµ) in Ωµ
x = (1− µ)x0 + µx1 ∈ Ωµ −→ uµ(x)
Question: are there suitable assumptions on the operators F0, F1 and Fµwhich permit to compare uµ with (a suitable convolution of) u0 and u1?Precisely, we want to find suitable conditions on the operators F0, F1, Fµwhich guarantee
uµ((1− µ)x0 + µx1) ≥ Mp(u0(x0),u1(x1);µ)
for every x0 ∈ Ω0, x1 ∈ Ω1, for some p ∈ R.
Equivalently
uµ(x) ≥ supMp(u0(x0),u1(x1);µ) : x0 ∈ Ω0, x1 ∈ Ω1, x = (1− µ)x0 + µx1
for every x ∈ Ωµ = (1− µ)Ω0 + µΩ1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 18 / 34
Combinations of solutions in different sets
u0 sol. of (P0) in Ω0 u1 sol. of (P1) in Ω1x0 ∈ Ω0 −→ u0(x0) x1 ∈ Ω1 −→ u1(x1)
uµ sol. of (Pµ) in Ωµ
x = (1− µ)x0 + µx1 ∈ Ωµ −→ uµ(x)
Question: are there suitable assumptions on the operators F0, F1 and Fµwhich permit to compare uµ with (a suitable convolution of) u0 and u1?Precisely, we want to find suitable conditions on the operators F0, F1, Fµwhich guarantee
uµ((1− µ)x0 + µx1) ≥ Mp(u0(x0),u1(x1);µ)
for every x0 ∈ Ω0, x1 ∈ Ω1, for some p ∈ R.
Equivalently
uµ(x) ≥ supMp(u0(x0),u1(x1);µ) : x0 ∈ Ω0, x1 ∈ Ω1, x = (1− µ)x0 + µx1
for every x ∈ Ωµ = (1− µ)Ω0 + µΩ1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 18 / 34
Combinations of solutions in different sets
u0 sol. of (P0) in Ω0 u1 sol. of (P1) in Ω1x0 ∈ Ω0 −→ u0(x0) x1 ∈ Ω1 −→ u1(x1)
uµ sol. of (Pµ) in Ωµ
x = (1− µ)x0 + µx1 ∈ Ωµ −→ uµ(x)
Question: are there suitable assumptions on the operators F0, F1 and Fµwhich permit to compare uµ with (a suitable convolution of) u0 and u1?Precisely, we want to find suitable conditions on the operators F0, F1, Fµwhich guarantee
uµ((1− µ)x0 + µx1) ≥ Mp(u0(x0),u1(x1);µ)
for every x0 ∈ Ω0, x1 ∈ Ω1, for some p ∈ R.
Equivalently
uµ(x) ≥ supMp(u0(x0),u1(x1);µ) : x0 ∈ Ω0, x1 ∈ Ω1, x = (1− µ)x0 + µx1
for every x ∈ Ωµ = (1− µ)Ω0 + µΩ1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 18 / 34
Combinations of solutions in different sets
u0 sol. of (P0) in Ω0 u1 sol. of (P1) in Ω1x0 ∈ Ω0 −→ u0(x0) x1 ∈ Ω1 −→ u1(x1)
uµ sol. of (Pµ) in Ωµ
x = (1− µ)x0 + µx1 ∈ Ωµ −→ uµ(x)
Question: are there suitable assumptions on the operators F0, F1 and Fµwhich permit to compare uµ with (a suitable convolution of) u0 and u1?
Precisely, we want to find suitable conditions on the operators F0, F1, Fµwhich guarantee
uµ((1− µ)x0 + µx1) ≥ Mp(u0(x0),u1(x1);µ)
for every x0 ∈ Ω0, x1 ∈ Ω1, for some p ∈ R.
Equivalently
uµ(x) ≥ supMp(u0(x0),u1(x1);µ) : x0 ∈ Ω0, x1 ∈ Ω1, x = (1− µ)x0 + µx1
for every x ∈ Ωµ = (1− µ)Ω0 + µΩ1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 18 / 34
Combinations of solutions in different sets
u0 sol. of (P0) in Ω0 u1 sol. of (P1) in Ω1x0 ∈ Ω0 −→ u0(x0) x1 ∈ Ω1 −→ u1(x1)
uµ sol. of (Pµ) in Ωµ
x = (1− µ)x0 + µx1 ∈ Ωµ −→ uµ(x)
Question: are there suitable assumptions on the operators F0, F1 and Fµwhich permit to compare uµ with (a suitable convolution of) u0 and u1?Precisely, we want to find suitable conditions on the operators F0, F1, Fµwhich guarantee
uµ((1− µ)x0 + µx1) ≥ Mp(u0(x0),u1(x1);µ)
for every x0 ∈ Ω0, x1 ∈ Ω1, for some p ∈ R.
Equivalently
uµ(x) ≥ supMp(u0(x0),u1(x1);µ) : x0 ∈ Ω0, x1 ∈ Ω1, x = (1− µ)x0 + µx1
for every x ∈ Ωµ = (1− µ)Ω0 + µΩ1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 18 / 34
Combinations of solutions in different sets
u0 sol. of (P0) in Ω0 u1 sol. of (P1) in Ω1x0 ∈ Ω0 −→ u0(x0) x1 ∈ Ω1 −→ u1(x1)
uµ sol. of (Pµ) in Ωµ
x = (1− µ)x0 + µx1 ∈ Ωµ −→ uµ(x)
Question: are there suitable assumptions on the operators F0, F1 and Fµwhich permit to compare uµ with (a suitable convolution of) u0 and u1?Precisely, we want to find suitable conditions on the operators F0, F1, Fµwhich guarantee
uµ((1− µ)x0 + µx1) ≥ Mp(u0(x0),u1(x1);µ)
for every x0 ∈ Ω0, x1 ∈ Ω1, for some p ∈ R.
Equivalently
uµ(x) ≥ supMp(u0(x0),u1(x1);µ) : x0 ∈ Ω0, x1 ∈ Ω1, x = (1− µ)x0 + µx1
for every x ∈ Ωµ = (1− µ)Ω0 + µΩ1.Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 18 / 34
The p-concave convolution
Let us define the function u∗p,µ : Ωµ → [0,+∞) as follows:
u∗p,µ(x) = supMp(u0(x0),u1(x1);µ) : x0 ∈ Ω0, x1 ∈ Ω1, x = (1− µ)x0 + µx1
and call it p-concave convolution of u0 and u1 (with weight µ).
When p = 1 it is the usual supremal convolution (from convex analysis) and,geometrically, it simply corresponds to the function whose graph is theMinkowski linear combination (in Rn+1) of the graphs of u0 and u1.
For p > 0 it corresponds to make the sup-conv (that is the Minkowskicombination of the graphs) of up
0 and up1 and then to raise to power 1/p.
For p = 0 it corresponds to exp(sup-conv of log u0 and log u1).
STRATEGY: if Fµ satisfies the comparison principle, to get the goal we havejust to prove that u∗p,µ is a viscosity subsolution of (Pµ).
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 19 / 34
The p-concave convolution
Let us define the function u∗p,µ : Ωµ → [0,+∞) as follows:
u∗p,µ(x) = supMp(u0(x0),u1(x1);µ) : x0 ∈ Ω0, x1 ∈ Ω1, x = (1− µ)x0 + µx1
and call it p-concave convolution of u0 and u1 (with weight µ).
When p = 1 it is the usual supremal convolution (from convex analysis) and,geometrically, it simply corresponds to the function whose graph is theMinkowski linear combination (in Rn+1) of the graphs of u0 and u1.
For p > 0 it corresponds to make the sup-conv (that is the Minkowskicombination of the graphs) of up
0 and up1 and then to raise to power 1/p.
For p = 0 it corresponds to exp(sup-conv of log u0 and log u1).
STRATEGY: if Fµ satisfies the comparison principle, to get the goal we havejust to prove that u∗p,µ is a viscosity subsolution of (Pµ).
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 19 / 34
The p-concave convolution
Let us define the function u∗p,µ : Ωµ → [0,+∞) as follows:
u∗p,µ(x) = supMp(u0(x0),u1(x1);µ) : x0 ∈ Ω0, x1 ∈ Ω1, x = (1− µ)x0 + µx1
and call it p-concave convolution of u0 and u1 (with weight µ).
When p = 1 it is the usual supremal convolution (from convex analysis) and,geometrically, it simply corresponds to the function whose graph is theMinkowski linear combination (in Rn+1) of the graphs of u0 and u1.
For p > 0 it corresponds to make the sup-conv (that is the Minkowskicombination of the graphs) of up
0 and up1 and then to raise to power 1/p.
For p = 0 it corresponds to exp(sup-conv of log u0 and log u1).
STRATEGY: if Fµ satisfies the comparison principle, to get the goal we havejust to prove that u∗p,µ is a viscosity subsolution of (Pµ).
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 19 / 34
The p-concave convolution
Let us define the function u∗p,µ : Ωµ → [0,+∞) as follows:
u∗p,µ(x) = supMp(u0(x0),u1(x1);µ) : x0 ∈ Ω0, x1 ∈ Ω1, x = (1− µ)x0 + µx1
and call it p-concave convolution of u0 and u1 (with weight µ).
When p = 1 it is the usual supremal convolution (from convex analysis) and,geometrically, it simply corresponds to the function whose graph is theMinkowski linear combination (in Rn+1) of the graphs of u0 and u1.
For p > 0 it corresponds to make the sup-conv (that is the Minkowskicombination of the graphs) of up
0 and up1 and then to raise to power 1/p.
For p = 0 it corresponds to exp(sup-conv of log u0 and log u1).
STRATEGY: if Fµ satisfies the comparison principle, to get the goal we havejust to prove that u∗p,µ is a viscosity subsolution of (Pµ).
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 19 / 34
The p-concave convolution
Let us define the function u∗p,µ : Ωµ → [0,+∞) as follows:
u∗p,µ(x) = supMp(u0(x0),u1(x1);µ) : x0 ∈ Ω0, x1 ∈ Ω1, x = (1− µ)x0 + µx1
and call it p-concave convolution of u0 and u1 (with weight µ).
When p = 1 it is the usual supremal convolution (from convex analysis) and,geometrically, it simply corresponds to the function whose graph is theMinkowski linear combination (in Rn+1) of the graphs of u0 and u1.
For p > 0 it corresponds to make the sup-conv (that is the Minkowskicombination of the graphs) of up
0 and up1 and then to raise to power 1/p.
For p = 0 it corresponds to exp(sup-conv of log u0 and log u1).
STRATEGY: if Fµ satisfies the comparison principle, to get the goal we havejust to prove that u∗p,µ is a viscosity subsolution of (Pµ).
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 19 / 34
Assumptions on the operators
For i = 0,1, µ and for a given p ≥ 0, for every fixed θ ∈ Rn we defineG(θ)
i,p : Ωi × (0,+∞)× Sn → R as follows:
G(θ)i,p (x , t ,A) = Fi (x , t
1p , t
1p−1θ, t
1p−3A) for p > 0 , (0.15)
G(θ)i,0 (x , t ,A) = Fi (x ,et ,etθ,etA) . (0.16)
Assumption (Aµ,p). Let µ ∈ (0,1) and p ≥ 0. We say that F0,F1,Fµ satisfythe assumption (Aµ,p) if, for every fixed θ ∈ Rn, the following holds:
G(θ)µ,p((1− µ)x0 + µx1, (1− µ)t0 + µt1, (1− µ)A0 + µA1
)≥
minG(θ)0,p(x0, t0,A0); G(θ)
1,p(x1, t1,A1)
for every x0 ∈ Ω0, x1 ∈ Ω1, t0, t1 > 0 and A0,A1 ∈ Sn.
If F0 = F1 = Fµ, we are simply requiring the operator Gθp to be quasi-concave,
i.e. with convex superlevel sets.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 20 / 34
Assumptions on the operators
For i = 0,1, µ and for a given p ≥ 0, for every fixed θ ∈ Rn we defineG(θ)
i,p : Ωi × (0,+∞)× Sn → R as follows:
G(θ)i,p (x , t ,A) = Fi (x , t
1p , t
1p−1θ, t
1p−3A) for p > 0 , (0.15)
G(θ)i,0 (x , t ,A) = Fi (x ,et ,etθ,etA) . (0.16)
Assumption (Aµ,p). Let µ ∈ (0,1) and p ≥ 0. We say that F0,F1,Fµ satisfythe assumption (Aµ,p) if, for every fixed θ ∈ Rn, the following holds:
G(θ)µ,p((1− µ)x0 + µx1, (1− µ)t0 + µt1, (1− µ)A0 + µA1
)≥
minG(θ)0,p(x0, t0,A0); G(θ)
1,p(x1, t1,A1)
for every x0 ∈ Ω0, x1 ∈ Ω1, t0, t1 > 0 and A0,A1 ∈ Sn.
If F0 = F1 = Fµ, we are simply requiring the operator Gθp to be quasi-concave,
i.e. with convex superlevel sets.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 20 / 34
BM for solutions of elliptic equations
Theorem 3 [S. (2014) - Ann. Inst. H.Poincaré in press]Let µ ∈ (0,1), Ωi an open bounded convex set and ui a classical solution of(Pi ) for i = 0,1. Assume that F0,F1,Fµ satisfy the assumption (Aµ,p) for somep ∈ [0,1). If p > 0, assume furthermore that for i = 0,1 it holds
lim infy→x
∂ui (y)
∂ν> 0 (0.17)
for every x ∈ ∂Ωi , where ν is any inward direction of Ωi at x .Then u∗p,µ is a viscosity subsolution of (Pµ).
Corollary (1)In the same assumption of the previous theorem, if Fµ satisfies a ComparisonPrinciple and uµ is a viscosity solution of (Pµ), then
uµ((1− µ)x0 + µ x1) ≥ Mp(u0(x0),u1(x1);µ) (0.18)
for every x0 ∈ Ω0, x1 ∈ Ω1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 21 / 34
BM for solutions of elliptic equations
Theorem 3 [S. (2014) - Ann. Inst. H.Poincaré in press]Let µ ∈ (0,1), Ωi an open bounded convex set and ui a classical solution of(Pi ) for i = 0,1. Assume that F0,F1,Fµ satisfy the assumption (Aµ,p) for somep ∈ [0,1). If p > 0, assume furthermore that for i = 0,1 it holds
lim infy→x
∂ui (y)
∂ν> 0 (0.17)
for every x ∈ ∂Ωi , where ν is any inward direction of Ωi at x .Then u∗p,µ is a viscosity subsolution of (Pµ).
Corollary (1)In the same assumption of the previous theorem, if Fµ satisfies a ComparisonPrinciple and uµ is a viscosity solution of (Pµ), then
uµ((1− µ)x0 + µ x1) ≥ Mp(u0(x0),u1(x1);µ) (0.18)
for every x0 ∈ Ω0, x1 ∈ Ω1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 21 / 34
Comparison
By a combination of the previous result with the BBL inequality, we cancompare the Lr norms of the involved functions:
Corollary (2)With the same assumptions and notation of the previous corollary, for everyr ∈ (0,+∞] we have
‖uµ‖Lr (Ωµ) ≥ Mq(‖u0‖Lr (Ω0), ‖u1‖Lr (Ω1);µ) , (0.19)
where
q =
pr
np+r for r ∈ (0,+∞)
p for r = +∞ .
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 22 / 34
A simple Example
For instance, let u0 and u1 be the solutions of the following problems∆u0 + f0(x) = 0 in Ω0u0 = 0 on ∂Ω0
and ∆u1 + f1(x) = 0 in Ω1
u1 = 0 on ∂Ω1 .
Then take µ ∈ (0,1) and set
Ω = (1− µ)Ω0 + µΩ1 ,
Now let uµ be the solution of ∆uµ + fµ(x) = 0 in Ω
uµ = 0 on ∂Ω.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 23 / 34
A simple exampleThen assumptio (Aµ,p) for p=1/3 reads
fµ((1− µ)x0 + µ x1) ≥ (1− µ)f0(x0) + µ f1(x1) (0.20)
The main theorem tells that we can estimate uµ in terms of u0 and u1;precisely it holds
uµ((1− µ)x0 + µ x1) ≥[(1− µ) 3
√u0(x0) + µ 3
√u1(x1)
]3
for every x0 ∈ Ω0, x1 ∈ Ω1.Moreover, by using the Borell-Brascamp-Lieb inequality we get
‖uµ‖Lr (Ωµ) ≥ Mq(‖u0‖Lr (Q), ‖u1‖Lr (B(0,1));µ)
for every r ∈ (0,+∞], where
q =
r
n+3r , r ∈ (0,+∞)
1/3 , r = +∞.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 24 / 34
ExamplesNotice in particular that, if f0 = f1 = fµ = f : Rn → [0,+∞) , condition (0.20)simply means f is concave. In this particular case, we can write the followingresult.
CorollaryLet f be a smooth nonnegative function defined in Rn. Let µ ∈ (0,1) and Ω0and Ω1 be convex subsets of Rn and denote by u0, u1 and uµ the solutions of ∆ui + f (x) = 0 in Ωi
ui = 0 on ∂Ωi
for i = 0,1, µ respectively, where Ωµ = (1− µ)Ω0 + µΩ1, as usual.Assume f is β-concave for some β ≥ 1, that is f β is concave.Then (0.18) holds with
p =β
1 + 2β.
In case f is a positive constant (β = +∞), the same conclusions follow withp = 1/2.
Similar results are obtained by substituting ∆u with a Finsler Laplacian ∆Huor with the Pucci’s operatorM−λ,Λ(D2u)
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 25 / 34
Consequences
This technique has several applications:
1. it can be used to prove concavity properties of solutions to elliptic andparabolic problems in convex domains;
2. it is possible to define a new kind of rearrangement (the mean-widthrearrangements) which apply to operators not in divergence form and permitsto obtain Talenti-like results for the associated equations;
3. it permits to prove Brunn-Minkowski type (and also Urysohn’s type)inequalities for many variational functionals.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 26 / 34
Consequences
This technique has several applications:
1. it can be used to prove concavity properties of solutions to elliptic andparabolic problems in convex domains;
2. it is possible to define a new kind of rearrangement (the mean-widthrearrangements) which apply to operators not in divergence form and permitsto obtain Talenti-like results for the associated equations;
3. it permits to prove Brunn-Minkowski type (and also Urysohn’s type)inequalities for many variational functionals.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 26 / 34
Consequences
This technique has several applications:
1. it can be used to prove concavity properties of solutions to elliptic andparabolic problems in convex domains;
2. it is possible to define a new kind of rearrangement (the mean-widthrearrangements) which apply to operators not in divergence form and permitsto obtain Talenti-like results for the associated equations;
3. it permits to prove Brunn-Minkowski type (and also Urysohn’s type)inequalities for many variational functionals.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 26 / 34
Consequences
This technique has several applications:
1. it can be used to prove concavity properties of solutions to elliptic andparabolic problems in convex domains;
2. it is possible to define a new kind of rearrangement (the mean-widthrearrangements) which apply to operators not in divergence form and permitsto obtain Talenti-like results for the associated equations;
3. it permits to prove Brunn-Minkowski type (and also Urysohn’s type)inequalities for many variational functionals.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 26 / 34
Brunn-Minkowski inequalities for variational functionals
When the involved equation is the Euler equation of some variationalfunctional, we can obtain a BM inequality for such a functional.
Probably the simplest case is that one of Torsional rigidity:
1τ(Ω)
= inf
∫Ω|∇u|2dx(∫Ω|u|)2 : u ∈W 1,2
0 (Ω),
∫Ω
|u|dx > 0
BM inequality for τ [Borell, 1985]
τ(Ωµ) ≥ M1/(n+2)(τ(Ω0), τ(Ω1);µ) =[(1− µ)τ(Ω0)1/(n+2) + µ τ(Ω1)1/(n+2)
]n+2
Equality holds if and only if Ω0 and Ω1 are homothetic [Colesanti 2005].
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 27 / 34
Brunn-Minkowski inequalities for variational functionals
When the involved equation is the Euler equation of some variationalfunctional, we can obtain a BM inequality for such a functional.Probably the simplest case is that one of Torsional rigidity:
1τ(Ω)
= inf
∫Ω|∇u|2dx(∫Ω|u|)2 : u ∈W 1,2
0 (Ω),
∫Ω
|u|dx > 0
BM inequality for τ [Borell, 1985]
τ(Ωµ) ≥ M1/(n+2)(τ(Ω0), τ(Ω1);µ) =[(1− µ)τ(Ω0)1/(n+2) + µ τ(Ω1)1/(n+2)
]n+2
Equality holds if and only if Ω0 and Ω1 are homothetic [Colesanti 2005].
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 27 / 34
Brunn-Minkowski inequalities for variational functionals
When the involved equation is the Euler equation of some variationalfunctional, we can obtain a BM inequality for such a functional.Probably the simplest case is that one of Torsional rigidity:
1τ(Ω)
= inf
∫Ω|∇u|2dx(∫Ω|u|)2 : u ∈W 1,2
0 (Ω),
∫Ω
|u|dx > 0
BM inequality for τ [Borell, 1985]
τ(Ωµ) ≥ M1/(n+2)(τ(Ω0), τ(Ω1);µ) =[(1− µ)τ(Ω0)1/(n+2) + µ τ(Ω1)1/(n+2)
]n+2
Equality holds if and only if Ω0 and Ω1 are homothetic [Colesanti 2005].
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 27 / 34
Brunn-Minkowski inequalities for variational functionals
When the involved equation is the Euler equation of some variationalfunctional, we can obtain a BM inequality for such a functional.Probably the simplest case is that one of Torsional rigidity:
1τ(Ω)
= inf
∫Ω|∇u|2dx(∫Ω|u|)2 : u ∈W 1,2
0 (Ω),
∫Ω
|u|dx > 0
BM inequality for τ [Borell, 1985]
τ(Ωµ) ≥ M1/(n+2)(τ(Ω0), τ(Ω1);µ) =[(1− µ)τ(Ω0)1/(n+2) + µ τ(Ω1)1/(n+2)
]n+2
Equality holds if and only if Ω0 and Ω1 are homothetic [Colesanti 2005].
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 27 / 34
BM for torsional rigidity
A proof of BM for τ with the above technique is quite simple.
Notice thatτ(Ωi ) =
∫Ωi
ui dx i = 0,1, µ
where ui is the solution of the torsion problem
(Pµ)
∆ui + 1 = 0 in Ωi ,i = 0,1, µ
ui = 0 on ∂Ωi .
Set
u∗1/2,µ(x) = sup[(1− µ)√
u0(x0) + µ√
u1(x1)]2 : (1− µ)x0 + µ x1 = x
By Theorem 3, u∗1/2,µ is a subsolution to the torsion problem in Ωµ.Then
uµ ≥ u∗1/2,µ
and we can use the BBL inequality to get the desired result, see Corollary 2.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 28 / 34
BM for torsional rigidity
A proof of BM for τ with the above technique is quite simple.Notice that
τ(Ωi ) =
∫Ωi
ui dx i = 0,1, µ
where ui is the solution of the torsion problem
(Pµ)
∆ui + 1 = 0 in Ωi ,i = 0,1, µ
ui = 0 on ∂Ωi .
Set
u∗1/2,µ(x) = sup[(1− µ)√
u0(x0) + µ√
u1(x1)]2 : (1− µ)x0 + µ x1 = x
By Theorem 3, u∗1/2,µ is a subsolution to the torsion problem in Ωµ.Then
uµ ≥ u∗1/2,µ
and we can use the BBL inequality to get the desired result, see Corollary 2.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 28 / 34
BM for torsional rigidity
A proof of BM for τ with the above technique is quite simple.Notice that
τ(Ωi ) =
∫Ωi
ui dx i = 0,1, µ
where ui is the solution of the torsion problem
(Pµ)
∆ui + 1 = 0 in Ωi ,i = 0,1, µ
ui = 0 on ∂Ωi .
Set
u∗1/2,µ(x) = sup[(1− µ)√
u0(x0) + µ√
u1(x1)]2 : (1− µ)x0 + µ x1 = x
By Theorem 3, u∗1/2,µ is a subsolution to the torsion problem in Ωµ.
Then
uµ ≥ u∗1/2,µ
and we can use the BBL inequality to get the desired result, see Corollary 2.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 28 / 34
BM for torsional rigidity
A proof of BM for τ with the above technique is quite simple.Notice that
τ(Ωi ) =
∫Ωi
ui dx i = 0,1, µ
where ui is the solution of the torsion problem
(Pµ)
∆ui + 1 = 0 in Ωi ,i = 0,1, µ
ui = 0 on ∂Ωi .
Set
u∗1/2,µ(x) = sup[(1− µ)√
u0(x0) + µ√
u1(x1)]2 : (1− µ)x0 + µ x1 = x
By Theorem 3, u∗1/2,µ is a subsolution to the torsion problem in Ωµ.Then
uµ ≥ u∗1/2,µ
and we can use the BBL inequality to get the desired result, see Corollary 2.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 28 / 34
Quantitative BM inequalities for τ
Now it’s easy to understand that it is possible to use the quantitative versionsof BBl to get corresponding quantitative versions of the BM inequality for τ .
Quantitative BM for τ [Ghilli-S. (2014)]Let Ω0 and Ω1 be convex bodies in Rn, then the following hold:
τ(Ωλ) ≥M 1n+2
(τ(Ω0), τ(Ω1), λ) + β H0(Ω0,Ω1)3(n+1) , (0.21)
τ(Ωλ) ≥M 1n+2
(τ(Ω0), τ(Ω1), λ) + δ A(Ω0,Ω1)6 , (0.22)
where β and δ are constants depending on n, λ,M pnp+1
(τ(Ω0, τ(Ω1), λ) andthe diameters and the measures of Ω0 and Ω1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 29 / 34
Quantitative BM inequalities for τ
Now it’s easy to understand that it is possible to use the quantitative versionsof BBl to get corresponding quantitative versions of the BM inequality for τ .
Quantitative BM for τ [Ghilli-S. (2014)]Let Ω0 and Ω1 be convex bodies in Rn, then the following hold:
τ(Ωλ) ≥M 1n+2
(τ(Ω0), τ(Ω1), λ) + β H0(Ω0,Ω1)3(n+1) , (0.21)
τ(Ωλ) ≥M 1n+2
(τ(Ω0), τ(Ω1), λ) + δ A(Ω0,Ω1)6 , (0.22)
where β and δ are constants depending on n, λ,M pnp+1
(τ(Ω0, τ(Ω1), λ) andthe diameters and the measures of Ω0 and Ω1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 29 / 34
Quantitative BM inequalities for τ
Now it’s easy to understand that it is possible to use the quantitative versionsof BBl to get corresponding quantitative versions of the BM inequality for τ .
Quantitative BM for τ [Ghilli-S. (2014)]Let Ω0 and Ω1 be convex bodies in Rn, then the following hold:
τ(Ωλ) ≥M 1n+2
(τ(Ω0), τ(Ω1), λ) + β H0(Ω0,Ω1)3(n+1) , (0.21)
τ(Ωλ) ≥M 1n+2
(τ(Ω0), τ(Ω1), λ) + δ A(Ω0,Ω1)6 , (0.22)
where β and δ are constants depending on n, λ,M pnp+1
(τ(Ω0, τ(Ω1), λ) andthe diameters and the measures of Ω0 and Ω1.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 29 / 34
Urysohn’s inequality for τ
Given a convex set Ω, we say that Ω]m is a rotation mean of Ω if there exist a
number m ∈ N and ρ1, . . . , ρm ∈ SO(n) such that
Ω]m =
1m
(ρ1Ω + · · ·+ ρmΩ) .
The following theorem is due to Hadwiger.
Theorem (Hadwiger)
Given an open bounded convex set Ω, there exists a sequence of rotationmeans of Ω converging in Hausdorff metric to a ball Ω] with diameter equal tothe mean width w(Ω) of Ω.
Notice that in the plane the mean width of a convex set coincides essentiallywith its perimeter. Precisely: w(Ω) = |∂Ω|/π. Then Ω] is a circle with thesame perimeter as Ω.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 30 / 34
Urysohn’s inequality for τ
Given a convex set Ω, we say that Ω]m is a rotation mean of Ω if there exist a
number m ∈ N and ρ1, . . . , ρm ∈ SO(n) such that
Ω]m =
1m
(ρ1Ω + · · ·+ ρmΩ) .
The following theorem is due to Hadwiger.
Theorem (Hadwiger)
Given an open bounded convex set Ω, there exists a sequence of rotationmeans of Ω converging in Hausdorff metric to a ball Ω] with diameter equal tothe mean width w(Ω) of Ω.
Notice that in the plane the mean width of a convex set coincides essentiallywith its perimeter. Precisely: w(Ω) = |∂Ω|/π. Then Ω] is a circle with thesame perimeter as Ω.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 30 / 34
Urysohn’s inequality for τ
Given a convex set Ω, we say that Ω]m is a rotation mean of Ω if there exist a
number m ∈ N and ρ1, . . . , ρm ∈ SO(n) such that
Ω]m =
1m
(ρ1Ω + · · ·+ ρmΩ) .
The following theorem is due to Hadwiger.
Theorem (Hadwiger)
Given an open bounded convex set Ω, there exists a sequence of rotationmeans of Ω converging in Hausdorff metric to a ball Ω] with diameter equal tothe mean width w(Ω) of Ω.
Notice that in the plane the mean width of a convex set coincides essentiallywith its perimeter. Precisely: w(Ω) = |∂Ω|/π. Then Ω] is a circle with thesame perimeter as Ω.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 30 / 34
Urysohn’s inequality for τ
By the BM inequality for torsional rigidity, we get
τ(Ω]m) ≥ τ(Ω) for evey m ,
then, passing to the limit, we obtain the following:
Urysohn’s ineq. for τ
τ(Ω) ≤ τ(Ω])
and = holds if and only if Ω = Ω].In other words: among sets with given mean width, the torsional rigidity ismaximized by balls.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 31 / 34
Urysohn’s inequality for τ
By the BM inequality for torsional rigidity, we get
τ(Ω]m) ≥ τ(Ω) for evey m ,
then, passing to the limit, we obtain the following:
Urysohn’s ineq. for τ
τ(Ω) ≤ τ(Ω])
and = holds if and only if Ω = Ω].In other words: among sets with given mean width, the torsional rigidity ismaximized by balls.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 31 / 34
Urysohn’s inequality for τ
By the BM inequality for torsional rigidity, we get
τ(Ω]m) ≥ τ(Ω) for evey m ,
then, passing to the limit, we obtain the following:
Urysohn’s ineq. for τ
τ(Ω) ≤ τ(Ω])
and = holds if and only if Ω = Ω].In other words: among sets with given mean width, the torsional rigidity ismaximized by balls.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 31 / 34
Quantitative Urysohn’s inequalities for τ
Let Ω be an open bounded convex set of Rn,n ≥ 2 with centroid in the origin.Let Ω] be the ball with the same mean-width of Ω with center in the origin.Then the following hold
τ(Ω]) ≥ τ(Ω)(
1 + µH3(n+1)), (0.23)
τ(Ω]) ≥ τ(Ω)(1 + νA6) , (0.24)
where H = H(Ω,Ω]) and A = maxA(Ω,Ωρ) : ρ rotation in Rn are smallenough, µ and ν are constants, the former depending on n, τ(Ω) and thediameter of Ω, the latter depending only on n and τ(Ω).
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 32 / 34
Possible extensions
Brunn-Minkowski type inequalities have been proved for several variationalfunctionals:
the first Dirichlet eigenvale of the Laplacian [Brascamp-Lieb,1976], Newton Capacity [Borell, 1983], p-capacity [Colesanti-S. 2003],Monge-Ampère eigenvalue [S., 2005], p-Laplacian eigenvalue[Colesanti-Cuoghi-S., 2006], the Bernoulli constant [Bianchini-S. 2009], theeigenvalue of Hessian equations [Lu-Ma-Xu, 2010] and [S., 2012], etc.
Then most of the above arguments, showed for the case of torsional rigidity,can be repeated for many other functionals.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 33 / 34
Possible extensions
Brunn-Minkowski type inequalities have been proved for several variationalfunctionals: the first Dirichlet eigenvale of the Laplacian [Brascamp-Lieb,1976], Newton Capacity [Borell, 1983], p-capacity [Colesanti-S. 2003],Monge-Ampère eigenvalue [S., 2005], p-Laplacian eigenvalue[Colesanti-Cuoghi-S., 2006], the Bernoulli constant [Bianchini-S. 2009], theeigenvalue of Hessian equations [Lu-Ma-Xu, 2010] and [S., 2012], etc.
Then most of the above arguments, showed for the case of torsional rigidity,can be repeated for many other functionals.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 33 / 34
Possible extensions
Brunn-Minkowski type inequalities have been proved for several variationalfunctionals: the first Dirichlet eigenvale of the Laplacian [Brascamp-Lieb,1976], Newton Capacity [Borell, 1983], p-capacity [Colesanti-S. 2003],Monge-Ampère eigenvalue [S., 2005], p-Laplacian eigenvalue[Colesanti-Cuoghi-S., 2006], the Bernoulli constant [Bianchini-S. 2009], theeigenvalue of Hessian equations [Lu-Ma-Xu, 2010] and [S., 2012], etc.
Then most of the above arguments, showed for the case of torsional rigidity,can be repeated for many other functionals.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 33 / 34
Possible extensions
Brunn-Minkowski type inequalities have been proved for several variationalfunctionals: the first Dirichlet eigenvale of the Laplacian [Brascamp-Lieb,1976], Newton Capacity [Borell, 1983], p-capacity [Colesanti-S. 2003],Monge-Ampère eigenvalue [S., 2005], p-Laplacian eigenvalue[Colesanti-Cuoghi-S., 2006], the Bernoulli constant [Bianchini-S. 2009], theeigenvalue of Hessian equations [Lu-Ma-Xu, 2010] and [S., 2012], etc.
Then most of the above arguments, showed for the case of torsional rigidity,can be repeated for many other functionals.
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 33 / 34
The end
THANKS!
Paolo Salani (DiMaI - Università di Firenze) Power concave and BBL IMA - Minneapolis 1/5/2015 34 / 34