Frequency functions, monotonicity formulas, and the thin ...

Frequency functions, monotonicity formulas, and thethin obstacle problem

Donatella Danielli - Purdue University

IMA - University of MinnesotaMarch 4, 2013

Donatella Danielli - Purdue University The Thin Obstacle ProblemIMA - University of Minnesota March 4, 2013 1

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Thank you for the invitation!


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In this talk we will present an overview of the parabolic Signorini (or thinobstacle) problem.

In particular, we will focus on a new approach to this classical problem,based on monotonicity properties for a family of so-called frequencyfunctions, first introduced by Almgren.


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The Classical Obstacle Problem

The obstacle problem is a classic motivating example in the mathematicalstudy of variational inequalities and free boundary problems.

The problem consists in finding the equilibrium position of an elasticmembrane whose boundary is held fixed, and which is constrained to lieabove a given obstacle. Applications include the study of fluid filtration inporous media, constrained heating, elasto-plasticity, optimal control, andfinancial mathematics.


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The Classical Obstacle Problem

The obstacle problem is a classic motivating example in the mathematicalstudy of variational inequalities and free boundary problems.

The problem consists in finding the equilibrium position of an elasticmembrane whose boundary is held fixed, and which is constrained to lieabove a given obstacle. Applications include the study of fluid filtration inporous media, constrained heating, elasto-plasticity, optimal control, andfinancial mathematics.


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Figure : Example of a one-dimensional obstacle problem


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Mathematically, the obstacle problem consists of studying the properties ofminimizers of the Dirichlet integral

J(u) =

∫D|∇u|2 dx

in a domain D ⊂ Rn, among all configurations u(x) (representing thevertical displacement of the membrane) with prescribed boundary valuesu|∂D = f (x), and constrained to remain above the obstacle ϕ(x).

The solution breaks down into a region where the solution is equal to theobstacle function, known as the coincidence set, and a region where thesolution is above the obstacle. The interface between the two regions isthe so-called free boundary.


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Mathematically, the obstacle problem consists of studying the properties ofminimizers of the Dirichlet integral

J(u) =

∫D|∇u|2 dx

in a domain D ⊂ Rn, among all configurations u(x) (representing thevertical displacement of the membrane) with prescribed boundary valuesu|∂D = f (x), and constrained to remain above the obstacle ϕ(x).

The solution breaks down into a region where the solution is equal to theobstacle function, known as the coincidence set, and a region where thesolution is above the obstacle. The interface between the two regions isthe so-called free boundary.


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The obstacle problem can be reformulated as a standard problem invariational inequalities on Hilbert spaces. In fact, solving the obstacleproblem is equivalent to seeking a function

u ∈ K = v ∈W 1,2(D) | v |∂D = f (x), v ≥ ϕ

such that ∫D∇u · ∇(v − u) dx ≥ 0 for all v ∈ K .

Variational arguments show that the solution to the obstacle problem isharmonic away from the contact set, and that it is superharmonic on thecontact set. Hence, the solution is a superharmonic function.


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The study of the classical obstacle problem, initiated in the 60’s with thepioneering works of G. Stampacchia, H. Lewy, J. L. Lions, has led tobeautiful and deep developments in calculus of variations and geometricpartial differential equations. The crowning achievement has been thedevelopment, due to L. Caffarelli, of the theory of free boundaries.


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Optimal regularity of the solution The solution to the obstacle problemis C 1,1 (i.e. it has bounded second derivatives) when the obstacle itself hassuch regularity. In general, the second derivatives of the solutions arediscontinuous across the free boundary.

The free boundary The free boundary is characterized as a Holdercontinuous surface except at certain singular points, which are eitherisolated or contained on a C 1 manifold.


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Optimal regularity of the solution The solution to the obstacle problemis C 1,1 (i.e. it has bounded second derivatives) when the obstacle itself hassuch regularity. In general, the second derivatives of the solutions arediscontinuous across the free boundary.

The free boundary The free boundary is characterized as a Holdercontinuous surface except at certain singular points, which are eitherisolated or contained on a C 1 manifold.


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Background: The time-independent thin obstacle problem

Let Ω be a domain in Rn and M a smooth (n − 1)-dimensional manifoldin Rn that divides Ω into two parts: Ω+ and Ω−. For given functionsϕ :M→ R and g : ∂Ω→ R satisfying g > ϕ on M∩ ∂Ω, consider theproblem of minimizing the Dirichlet integral

DΩ(u) =

∫Ω|∇u|2dx

on the closed convex set

K = u ∈W 1,2(Ω) : u = g on ∂Ω, u ≥ ϕ on M∩ Ω.

This problem is known as the lower dimensional, or thin obstacleproblem. The thin obstacle is the function ϕ.


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Figure : Graphs of Re(x1 + i x2)3/2 and Re(x1 + i x2)6


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When u is constrained to stay above an obstacle ϕ which is assigned inthe whole domain Ω, i.e. when M = Ω, then we obtain the classicalobstacle problem.

Whereas the latter is by now well-understood, the thin obstacle problemstill presents considerable challenges and only recently there has beensome significant progress on it.


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When M and ϕ are smooth, it has been proved by Caffarelli (1979) thatthe minimizer u in the thin obstacle problem is of class C 1,α

loc (Ω± ∪M).The function u satisfies

∆u = 0 in Ω \M = Ω+ ∪ Ω−,

but in general u does not need to be harmonic across M. Instead, on M,one has the following complementary conditions

u − ϕ ≥ 0, ∂ν+u + ∂ν−u ≥ 0, (u − ϕ)(∂ν+u + ∂ν−u) = 0,

where ν± are the outer unit normals to Ω± on M.


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One of the main goals in the study of this problem is to understand theproperties of the coincidence set

Λ(u) := x ∈M : u = ϕ

and its boundary (in the relative topology of M)

Γ(u) := ∂MΛ(u),

i.e., the free boundary.

Simplifying assumptions:1. Vanishing thin obstacle ϕ.2. The manifold M is a flat portion of the boundary of the relevantdomain: M=Rn−1 × 0. In this case the thin obstacle problem is knownas the Signorini problem.


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One of the main goals in the study of this problem is to understand theproperties of the coincidence set

Λ(u) := x ∈M : u = ϕ

and its boundary (in the relative topology of M)

Γ(u) := ∂MΛ(u),

i.e., the free boundary.

Simplifying assumptions:1. Vanishing thin obstacle ϕ.2. The manifold M is a flat portion of the boundary of the relevantdomain: M=Rn−1 × 0. In this case the thin obstacle problem is knownas the Signorini problem.


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This problem, which arises in linear elasticity, was first proposed bySignorini in 1959. Signorini called it the “problem with ambiguousboundary conditions”. The existence and uniqueness of solutions wasproved by Fichera in 1963. It was Fichera who renamed it as “Signoriniproblem”.

In the original formulation, it consists of finding the elastic equilibriumconfiguration of an anisotropic non-homogeneous elastic body, resting on arigid frictionless surface and subject only to its mass forces.


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Figure : What will be the equilibrium configuration of an elastic body resting on arigid frictionless plane?


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Other applications include optimal control of temperature across a surface,in the modeling of semipermeable membranes where some salineconcentration can flow through the membrane only in one direction, andfinancial math (when the random variation of underlying asset changes ina discontinuous fashion, as a Levi process).


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Normalization

Since we are interested in properties of minimizers near free boundarypoints, after translation, rotation and scaling arguments we may consider afunction u defined in the upper half-ball B+

1 := B1 ∩ Rn+ satisfying

∆u = 0 in B+1 (0.1)

u ≥ 0, −∂xnu ≥ 0, u ∂xnu = 0 on B ′1 (0.2)

0 ∈ Γ(u) = ∂Λ(u) := ∂(x ′, 0) ∈ B ′1 | u(x ′, 0) = 0, (0.3)

where Λ(u) is the coincidence set and the boundary is in the relativetopology of B ′1. Here B ′1 := B1 ∩ (Rn−1 × 0).We denote by S the class of solutions of the normalized Signorini problem(0.1)–(0.3).


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What is the meaning of the boundary condition

u ≥ 0, −∂xnu ≥ 0, u ∂xnu = 0 on B ′1?

In the original formulation of the problem, it was requested that thesolution satisfied one of the two sets of conditions

u = 0, ∂xnu ≥ 0 (0.4)

oru > 0, ∂xnu = 0 (0.5)

The set of conditions (0.4) applies to points of the boundary of the bodywhich do not leave the contact set in the equilibrium configuration. Theset of conditions (0.5) applies instead to points of the boundary of thebody which leave that set in the equilibrium configuration.


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Recent Developments

Athanasopoulos-Caffarelli (2006): Optimal C 1,1/2 interior regularitywith flat M and ϕ = 0.

Athanasopoulos-Caffarelli-Salsa (2008): Fine regularity properties ofthe free boundary. Namely, the set of regular free boundary points islocally a C 1-manifold of dimension n − 2.Crucial tool: Monotonicity of Almgren’s Frequency Function

N(r , u) =r∫Br|∇u|2∫

∂Bru2

The name comes from fact that if u is a harmonic function in B1,homogeneous of degree κ, then N(r , u) = κ. When ϕ = 0 and M isflat, then a free boundary point is called regular if at such point thefrequency attains its least possible value N(0+, u) = 3/2.


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In the particular case Ω = Rn−1 × (0,∞) and M = Rn−1 × 0, theSignorini problem can be interpreted as an obstacle problem for thefractional Laplacian on Rn−1:

u − ϕ ≥ 0, (−∆x ′)su ≥ 0, (u − ϕ)(−∆x ′)

su = 0,

with s = 1/2.

Silvestre (2007): Almost optimal regularity of solutions, namelyu ∈ C 1,α(Rn−1) for any α < s, 0 < s < 1.Caffarelli-Salsa-Silvestre (2008): Optimal regularity C 1,s(Rn−1), freeboundary regularity.

Interesting aspect: In the above results, the thin obstacle ϕ is allowedto be nonzero, thanks to a suitable generalization of Almgren’smonotonicity of the frequency.

Garofalo-Petrosyan (2009): Structure of the singular set of solutionsto the thin obstacle problem by construction of two one-parameterfamilies of monotonicity formulas (of Weiss and Monneau type).


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Similarly to the classical obstacle problem, in the lower dimensionalobstacle problem the analysis of the free boundary revolves around thebehavior of the so-called blowups.In the Signorini problem Athanasopoulos, Caffarelli and Salsa consideredthe rescalings

ur (x) :=u(rx)(

1rn−1

∫∂Br

u2)1/2

,

and studied the limits as r → 0+, known as the blowups.

Generally the blowups might be different over different subsequencesr = rj → 0+.

The following result provides a tool to control the rescalings.


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Similarly to the classical obstacle problem, in the lower dimensionalobstacle problem the analysis of the free boundary revolves around thebehavior of the so-called blowups.In the Signorini problem Athanasopoulos, Caffarelli and Salsa consideredthe rescalings

ur (x) :=u(rx)(

1rn−1

∫∂Br

u2)1/2

,

and studied the limits as r → 0+, known as the blowups.

Generally the blowups might be different over different subsequencesr = rj → 0+.

The following result provides a tool to control the rescalings.


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Monotonicity of the Frequency

Theorem 1

Let u ∈ S, then the function

N(r , u) :=r∫Br|∇u|2∫

∂Bru2

is monotone increasing in r for 0 < r < 1. Moreover, N(r , u) ≡ κ for0 < r < 1 iff u is homogeneous of order κ in B1, i.e.

x · ∇u − κu = 0 in B1.

When u is a harmonic function this is a classical result of Almgren (1979).


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The Blowups

It follows easily from the monotonicity formula that, for r ≤ 1∫B1

|∇ur |2 = N(1, ur ) = N(r , u) ≤ N(1, u),

where in the last inequality we have used the monotonicity of thefrequency N(r , u) claimed in the previous theorem. The above inequality,and the C 1,α

loc estimates of Caffarelli, imply that there exists a nonzerofunction u0 ∈W 1,2(B1), called a blowup of u at the origin, such that for asubsequence r = rj → 0+

urj → u0 in W 1,2(B1)

urj → u0 in L2(∂B1)

urj → u0 in C 1loc(B ′1 ∪ B±1 )


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The monotonicity of the frequency easily implies the following

Proposition 2 (Homogeneity of blowups)

Let u ∈ S and denote by u0 any blowup of u as described above. Then,u0 ∈ S and it is a homogeneous function of degree κ = N(0+, u).

The following result was proved in part by Luis Silvestre in his Ph.D.Dissertation , and in part by Caffarelli, Salsa and Silvestre

Lemma 1 (Minimal homogeneity)

Let u ∈ S. Then

N(0+, u) ≥ 2− 1

2=

3

2.

Moreover, one has either

N(0+, u) = 2− 1

2or N(0+, u) ≥ 2.


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The monotonicity of the frequency easily implies the following

Proposition 2 (Homogeneity of blowups)

Let u ∈ S and denote by u0 any blowup of u as described above. Then,u0 ∈ S and it is a homogeneous function of degree κ = N(0+, u).

The following result was proved in part by Luis Silvestre in his Ph.D.Dissertation , and in part by Caffarelli, Salsa and Silvestre

Lemma 1 (Minimal homogeneity)

Let u ∈ S. Then

N(0+, u) ≥ 2− 1

2=

3

2.

Moreover, one has either

N(0+, u) = 2− 1

2or N(0+, u) ≥ 2.


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Statement of the Parabolic Signorini Problem

Given a domain Ω in Rxn with a sufficiently regular boundary ∂Ω, arelatively open subset M⊂ ∂Ω, S = ∂Ω \M, consider the problem

∆v − ∂tv = 0 in ΩT := Ω× (0,T ] (0.6)

v ≥ ϕ, ∂νv ≥ 0, (v − ϕ)∂νv = 0 on MT :=M× (0,T ], (0.7)

v = g on ST := S × (0,T ] (0.8)

v(·, 0) = ϕ0 on Ω0 := Ω× 0 (0.9)

where ∂ν is the outer normal derivative on ∂Ω and ϕ :MT → R,ϕ0 : Ω0 → R, g : ST → R are prescribed functions satisfying thecompatibility conditions ϕ0 ≥ ϕ on M×0, g ≥ ϕ on ∂S × (0,T ],g = ϕ on S × 0. The condition (0.7) is known as the Signoriniboundary condition and the problem (0.6)–(0.9) as the Signorini problemfor the heat equation.


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As a weak (or generalized) solution of the Signorini problem weunderstand the solution v ∈ K of the variational inequality∫

ΩT

∇v∇(w − v) + ∂tv(w − v) ≥ 0 for every w ∈ K , v(·, 0) = ϕ0,

where K = w ∈W 1,02 (ΩT ) | w ≥ ϕ on MT , w = g on ST.


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Known Results

Regularity of the solution∇v ∈ Hα,α/2, 0 < α < 1, on compact subsets of ΩT ∪MT

• Athanasopoulos (1982)• Uraltseva (1985)• Arkhipova-Uraltseva (1996)

Poon’s Monotonicity Formula Poon (1996): If u is a solution ofthe heat equation in a unit strip, the parabolic frequency

Nu(r) =r 2∫Rn |∇u|2(x ,−r 2)ρ(x ,−r 2)dx∫Rn u(x ,−r 2)2ρ(x ,−r 2)dx

is monotone in r ∈ (0, 1). Here ρ denotes the backward heat kernelon S∞ = Rn × (−∞, 0], i.e.

ρ(x , t) = (−4πt)−n2 e

x2

4t .


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Known Results

Regularity of the solution∇v ∈ Hα,α/2, 0 < α < 1, on compact subsets of ΩT ∪MT

• Athanasopoulos (1982)• Uraltseva (1985)• Arkhipova-Uraltseva (1996)

Poon’s Monotonicity Formula Poon (1996): If u is a solution ofthe heat equation in a unit strip, the parabolic frequency

Nu(r) =r 2∫Rn |∇u|2(x ,−r 2)ρ(x ,−r 2)dx∫Rn u(x ,−r 2)2ρ(x ,−r 2)dx

is monotone in r ∈ (0, 1). Here ρ denotes the backward heat kernelon S∞ = Rn × (−∞, 0], i.e.

ρ(x , t) = (−4πt)−n2 e

x2

4t .


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Notations

Rn = x = (x1, x2, . . . , xn) : xi ∈ R (Euclidean space)

Rn+ = Rn ∩ xn > 0 (positive halfspace)

Rn− = Rn ∩ xn < 0 (negative halfspace)

Rn−1 ∼ Rn−1 × 0 ⊂ Rn

x ′ = (x1, x2, . . . , xn−1),

x ′′ = (x1, x2, . . . , xn−2)

x = (x ′, xn), x ′ = (x ′′, xn−1)


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For x0 ∈ Rn−1, t0 ∈ R we let

Br (x0) = x ∈ Rn | |x | < r (Euclidean ball)

B±r (x0) = Br (x0) ∩ Rn± (Euclidean halfball)

B ′r (x0) = Br (x) ∩ Rn−1 (“thin” ball)

Qr (x0, t0) = Br (x0)× (t0 − r 2, t0] (parabolic cylinder)

Q ′r (x0, t0) = B ′r (x0)× (t0 − r 2, t0] (“thin” parabolic cylinder)

Q±r (x0, t0) = B±r (x0)× (t0 − r 2, t0] (parabolic halfcylinders)

Qr (x0, t0) = Br (x0)× [t0, t0 + r 2) (upper parabolic cylinder)

Qr (x0, t0) = Br (x0)× (t0 − r 2, t0 + r 2) (full parabolic cylinder)

Sr = Rn × (−r 2, 0] (parabolic strip)

S±r = Rn± × (−r 2, 0] (parabolic halfstrip)

S ′r = Rn−1 × (−r 2, 0] (“thin” parabolic strip)


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Solutions in Half-Cylinders

The following is joint work with N. Garofalo, A. Petrosyan, and T. To.

Definition 3

The class Sϕ(Q+1 ) consists of functions v ∈W 2,1

2 (Q+1 ), with

∇v ∈ Hα,α/2(Q+1 ∪ Q ′1) for some 0 < α < 1, satisfying

∆v − ∂tv = 0 in Q+1

v − ϕ ≥ 0, −∂xnv ≥ 0, (v − ϕ)∂xnv = 0 on Q ′1,

and(0, 0) ∈ Γ(v) = ∂(x ′, t) ∈ Q ′1 | v(x ′, 0, t) > ϕ(x ′, t).


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Reduction to Vanishing Obstacle

The difference v(x , t)− ϕ(x ′, t) satisfies the Signorini conditions on Q ′1with zero obstacle, but at an expense of solving a nonhomogeneous heatequation instead of the homogeneous one.This difference may be extended to the strip S+

1 = Rn+ × (−1, 0] by

multiplying it by a suitable cutoff function ψ.The resulting function will satisfy

∆u − ∂tu = f (x , t) in S+1 ,

with

f (x , t) = −ψ(x)[∆′ϕ− ∂tϕ] + [v(x , t)− ϕ(x ′, t)]∆ψ + 2∇v∇ψ.

For smooth enough ϕ, the function f is bounded in S+1 !


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Solutions in Half-Strips

A function u is in the class Sf (S+1 ), for f ∈ L∞(S+

1 ), if u ∈W 2,12 (S+

1 ),∇u ∈ Hα,α/2(S+

1 ∪ S ′1), u has a compact support and solves

∆u − ∂tu = f in S+1 ,

u ≥ 0, −∂xnu ≥ 0, u∂xnu = 0 on S ′1,

and(0, 0) ∈ Γ(u) = ∂(x ′, t) ∈ S ′1 : u(x ′, 0, t) > 0.


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Generalized Monotonicity Formula

Problem Poon’s monotonicity formula requires the function u to becaloric in an entire strip, and it is not immediately applicable to caloricfunctions in the unit cylinder Q1.

(Partial) Solution Extend the function u, caloric in Q1, to the entire stripS1 by multiplying it by a spatial cutoff function ψ, supported in B1:

v(x , t) = u(x , t)ψ(x).

New Problem v is not caloric in S1 ⇒ Nv not monotone.

Hope Nv is “close” to being monotone.


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v(x , t) = u(x , t)ψ(x).




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v(x , t) = u(x , t)ψ(x).




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v(x , t) = u(x , t)ψ(x).




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Introduce quantities

hu(t)=

∫Rn

+

u(x , t)2ρ(x , t)dx

iu(t)= −t

∫Rn

+

|∇u(x , t)|2ρ(x , t)dx ,

for any function u on S+1 for which they make sense. Poon’s parabolic

frequency function is given by

Nu(r) =iu(−r 2)

hu(−r 2).

There are many substantial technical difficulties involved in working withthis function directly. To overcome such difficulties, consider averagedversions of hu and iu:

Hu(r)=1

r 2

∫ 0

−r2

hu(t)dt =1

r 2

∫S+r

u(x , t)2ρ(x , t)dxdt

Iu(r)=1

r 2

∫ 0

−r2

iu(t)dt =1

r 2

∫S+r

|t||∇u(x , t)|2ρ(x , t)dxdt


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The New Generalized Monotonicity Formula

Theorem 4

Let δ > 0. Then there exists C > 0, depending only on δ and n, such thatthe function

Φu(r) =1

2reCr

δ d

drlog maxHu(r), r 4−2δ+

3

2(eCr

δ − 1)

is nondecreasing for r ∈ (0, 1).

Remark 5

Hu(r) > r 4−2δ ⇒ Φu(r) ∼ 12 rH ′u(r)/Hu(r) = 2Nu, when f = 0.

The truncation of Hu(r) with r 4−2δ controls the error terms caused by theright-hand-side f .


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The New Generalized Monotonicity Formula

Theorem 4

Let δ > 0. Then there exists C > 0, depending only on δ and n, such thatthe function

Φu(r) =1

2reCr

δ d


3

2(eCr

δ − 1)

is nondecreasing for r ∈ (0, 1).

Remark 5

Hu(r) > r 4−2δ ⇒ Φu(r) ∼ 12 rH ′u(r)/Hu(r) = 2Nu, when f = 0.

The truncation of Hu(r) with r 4−2δ controls the error terms caused by theright-hand-side f .


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With the previous theorem in hands, we can study the existence and thehomogeneity properties of the blow-up’s.Since the function

Φu(r) =1

2reCr

δ d


3

2(eCr

δ − 1)

is nondecreasing for r ∈ (0, 1), the limit

κ := Φu(0+) = limr→0+

Φu(r)

exists.


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We have the following basic result concerning the values of κ.

Lemma 2

Let u ∈ Sf (S+1 ) and satisfy the conditions of the monotonicity theorem,

and let κ be as above. Then

κ ≤ 2− δ.

Moreover, if κ < 2− δ, then there exists ru > 0 such that Hu(r) ≥ r 4−2δ

for 0 < r ≤ ru. In particular,

κ =1

2lim

r→0+

rH ′u(r)

Hu(r)= 2 lim

r→0+

Iu(r)

Hu(r).


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Similarly to what was done in the elliptic case by Athanasopoulos,Caffarelli and Salsa, we introduce the parabolic rescalings.

Definition: For u ∈ Sf (S+1 ) and r > 0 define the rescalings

ur (x , t) :=u(rx , r 2t)

Hu(r)1/2, (x , t) ∈ S+

1/r = Rn+ × (−1/r 2, 0].

Our main result shows that, unless we are in the borderline case κ = 2− δ,we can study the blow-up’s of u at the origin.


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Existence and Homogeneity of Blowups

It is easy to see that ur solves the nonhomogeneous Signorini problem

∆ur − ∂tur = fr (x , t) in S+1/r

ur ≥ 0, −∂xnur ≥ 0, ur∂xnur = 0 on S ′1/r

with

fr (x , t) =r 2f (rx , r 2t)

Hu(r)1/2.


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Theorem 6

There is a subsequence rj → 0+ and a function u0 inS+∞ = Rn

+ × (−∞, 0] such that∫S+R

(|urj − u0|2 + |t||∇(urj − u0)|2)ρ→ 0.

We call any such u0 a blowup of u at the origin.

u0 is a nonzero global solution of the Signorini problem:

∆u0 − ∂tu0 = 0 in S+∞

u0 ≥ 0, −∂xnu0 ≥ 0, u0∂xnu0 = 0 on S ′∞

in the sense that it solves the Signorini problem in every Q+R .

u0 is parabolically homogeneous of degree κ:

u0(λx , λ2t) = λκu0(x , t), (x , t) ∈ S+∞, λ > 0


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Homogeneous global solutions of homogeneity 1 < κ < 2:

Let u be a nonzero κ-parabolically homogeneous solution of the Signoriniproblem in S+

∞ = Rn+ × (−∞, 0] with 1 < κ < 2. Then κ = 3/2 and

u(x , t) = C Re(x ′ · e + ixn)3/2+ in S+

∞

for some tangential direction e ∈ ∂B ′1.


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Optimal Regularity

Theorem 7

Let v ∈ Sfϕ(Q+

1 ) and f ∈ L∞(Q+1 ). Then ∇v ∈ H

12, 1

4 (Q+1/2 ∪ Q ′1/2) with

‖∇v‖H

12 ,

14 (Q+

1/2∪Q′

1/2)≤ Cn

(‖v‖

W 1,0∞ (Q+

1 )+ ‖f ‖L∞(Q+

1 ) + ‖ϕ‖H2,1(Q′1)

).

Regularity is optimal! (compare with elliptic case)


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The Free Boundary: The Regular Set

Analysis of homogeneous global solutions ⇒ κ = Φu(0+) ≥ 32 .

Definition 8

Let v ∈ Sϕ(Q+1 ) with ϕ ∈ H`,`/2(Q ′1), ` ≥ 2. We say that a free boundary

point (x0, t0) is regular ⇔ it has a minimal homogeneity κ = 3/2. R(v)denotes the set of regular free boundary points.

Theorem 9

Let v ∈ Sϕ(Q+1 ) with ϕ ∈ H`,`/2(Q ′1), ` ≥ 3 and such that (0, 0) ∈ R(v).

Then there exist δ = δv > 0 such that

Γ(v) ∩ Q ′δ = R(v) ∩ Q ′δ = xn−1 = g(x ′′, t) | (x ′′, t) ∈ Q ′′δ .

after a possible rotation in Rn−1, where g ∈ H1,1/2(Q ′′δ ) (i.e., g is aparabolically Lipschitz function).


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Thank you for your attention!


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Frequency functions, monotonicity formulas, and the thin ...

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