Long-Time Behavior of the Mean Curvature Flow with ...acesar/docs/... · Long-Time Behavior 781 We are particularly interested in the asymptotic behavior as t→+ of solutions to

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Long-Time Behavior of the Mean Curvature Flow withPeriodic ForcingAnnalisa Cesaroni a & Matteo Novaga aa Dipartimento di Matematica Pura e Applicata, Università di Padova, Padova, ItalyAccepted author version posted online: 05 Feb 2013.Version of record first published: 10 Apr2013.

To cite this article: Annalisa Cesaroni & Matteo Novaga (2013): Long-Time Behavior of the Mean Curvature Flow with PeriodicForcing, Communications in Partial Differential Equations, 38:5, 780-801

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Communications in Partial Differential Equations, 38: 780–801, 2013Copyright © Taylor & Francis Group, LLCISSN 0360-5302 print/1532-4133 onlineDOI: 10.1080/03605302.2013.771508

Long-Time Behavior of theMean CurvatureFlow with Periodic Forcing

ANNALISA CESARONI AND MATTEO NOVAGA

Dipartimento di Matematica Pura e Applicata, Università di Padova,Padova, Italy

We consider the long-time behavior of the mean curvature flow in heterogeneousmedia with periodic fibrations, modeled as an additive driving force. Underappropriate assumptions on the forcing term, we show existence of generalizedtraveling waves with maximal speed of propagation, and we prove the convergenceof solutions to the forced mean curvature flow to these generalized waves.

Keywords Asymptotic behavior; Heterogeneous media; Mean curvature flow;Traveling waves.

Mathematics Subject Classification 35K55; 49L25; 53C44.

1. Introduction

We are interested in the long-time behavior of the mean curvature flow in aperiodic heterogeneous medium. The evolution law can be written as a forced meancurvature flow

v = � − gwhere v denotes the inward normal velocity of the evolving hypersurface, � its meancurvature (with the convention that � is positive on convex sets) and g is a periodicforcing term. In our model, we assume that the hypersurfaces are graphs withrespect to a fixed hyperplane and that the forcing term g does not depend on thevariable orthogonal to such hyperplane (fibered medium). Under these assumptionsthe evolving hypersurface coincides with the graph of the solution to the Cauchyproblemut =

√1+ �Du�2 div

(Du√

1+�Du�2

)+ g√1+ �Du�2 in �0�+��×�n

u�0� ·� = u0 in �n�(1)

Received October 12, 2011; Accepted January 13, 2013Address correspondence to Matteo Novaga, Dipartimento di Matematica Pura e

Applicata, Università di Padova, via Trieste 63, Padova 35121, Italy; E-mail: [email protected]

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Long-Time Behavior 781

We are particularly interested in the asymptotic behavior as t→ +� of solutionsto (1), where the initial data u0 and the forcing term g are assumed to be Lipschitzcontinuous and �n-periodic.

The expected result is that, under appropriate assumptions on g, there exists aunique constant c ∈ � and a periodic function � such that

u�t� y�− ct − ��y�→ 0� as t→ +�� uniformly in �n�

This is a result on the asymptotic stability of special solutions to (1), called travelingwave solutions, which are of the form �+ ct. The constant c and the function � arerespectively the propagation speed and the profile of the wave.

The first question we address in Section 3 of this paper is about existence oftraveling wave solutions to (1). We provide a construction of such solutions usinga variational approach developed in [23] (see also [24]). In particular, our solutionsare critical points of appropriate functionals, which are exponentially weightedarea functionals with a volume term, depending on the speed of propagation c.Exploiting this variational structure, we show existence of traveling waves underrather weak assumptions on the forcing term g, i.e.

∃A ⊆ �0� 1�n s.t.∫Ag�y�dy > Per�A��n�

where Per�A��n� is the periodic perimeter of A (see Section 2). Notice that, if∫�0�1�n g > 0, then the previous condition holds true by taking A = �0� 1�n.

As our solutions are in general not globally defined, we call them generalizedtraveling waves. In Propositions 3.7 and 3.10 we discuss the regularity of thesesolutions and of their support. Moreover, in Section 3.1 we list some strongerconditions on the forcing term, involving only the oscillation and the norm of g,under which we show existence of classical (i.e. globally defined) traveling waves (seeProposition 3.15).

We point out that the variational method selects the fastest traveling wavesfor (1) which are bounded above, in particular it is uniquely defined the speed ofpropagation c of such waves and it holds c ≥ ∫

�0�1�n g (see Corollary 3.2).We recall that the problem of existence of classical traveling waves for the

forced mean curvature flow has already been considered in the literature, underdifferent assumptions on the forcing term [9, 14, 19]. We also mention [22], wherethe authors construct V -shaped traveling waves in the whole space for a constantforcing term (see also [11, 20, 26] for similar results in the planar case). Theconstruction of the traveling fronts in these papers relies mainly on maximumprinciple type arguments, while we use here a variational approach.

The second question of interest is about the convergence, as t→ +�, ofthe solution to (1) to a traveling wave solution. We point out that the long-time behavior of solutions of parabolic problems using viscosity solutions typearguments has been extensively considered in the literature: see [7, 25] for the caseof semilinear and quasilinear parabolic problems in periodic environments, [13]where the author considers uniformly parabolic operators in bounded domains withNeumann boundary conditions, and [6] for the case of viscous Hamilton-Jacobiequations in bounded domains with Dirichlet boundary conditions. However, noneof these results applies to mean curvature type equations such as (1).

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782 Cesaroni and Novaga

In Section 4 we prove a convergence result under the assumption that thereexists a global traveling wave solution. In particular, in Corollary 4.9 we show thatthe solution u�t� y� to (1) satisfies

u�t� y�− ct→ ��y� in �1+��n�� as t→ +��

where �+ ct is the traveling wave, which in this case is unique up to an additiveconstant.

In the general case, we obtain a weaker convergence result. First, inProposition 4.6 we describe the asymptotic behavior as t→ +� of the maximum ofthe function u�t� ·�. Namely, letting Q �= �0� 1�n, we show that there exists a constantK > 0 such that

minQu0 + ct ≤ max

Qu�t� y� ≤ ct + K + log�1+ t�

c�

Then, in Theorem 4.7 we show that, along a sequence tn → +�,

u�tn� y�−maxQu�tn� ·� −→

{��y� locally in �1+��E�−� locally uniformly in Q\E

for all � ∈ �0� 1�, where �+ ct is a generalized traveling wave supported in E ⊂ Q,possibly depending on the sequence tn.

We point out that the proof of the convergence result, as well as the proofof existence of generalized waves, essentially uses variational methods, rather thanmaximum principle based arguments.

2. Notation and Preliminary Results

We refer to [2] for a general introduction to functions of bounded variation and setsof finite perimeter. Letting Q �= �0� 1�n, it is a classical result that any u ∈ BV�Q�admits a trace uQ on Q (see e.g. [2, Theorem 3.87]). Let 0Q �= Q ∩ y �

∏ni=1 yi =

0� and let � � 0Q→ Q be the function ��y� �= y +∑ni=1 i�y�ei, where i�y� = 1

if yi = 0 and i�y� = 0 otherwise. We consider the space BVper�Q� of functionswhich have periodic bounded variation in Q, where the periodic total variation ofu ∈ BV�Q� is defined as

�Du�per�Q� �= �Du��Q�+∫0Q

�uQ�y�− uQ��y��d�n−1�y�� (2)

The space BVper�Q� is the space BV�Q� endowed with the norm

�u�BVper�Q� �= �u�L1�Q� + �Du�per�Q��

Observe that BVper�Q� coincides with BV��n�, where �n �= �n/�n is the

n-dimensional torus. For every E ⊆ Q we define the periodic perimeter of E as

Per�E��n� �= �D�E�per�Q� (3)

where �E is the characteristic function of E. We recall the isoperimetric inequality [2]:

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Proposition 2.1. There exists Cn > 0 such that

Per�E��n� ≥ Cn�E� n−1n (4)

for all E ⊆ Q of finite perimeter and such that �E� ≤ 1/2.

Remark 2.2. Notice that C1 = 2.

In this paper we always make the following regularity assumption on the initialdatum and on the forcing term:

u0� g are Lipschitz continuous and �0� 1�n-periodic� (5)

Using the comparison principle [4] and (5), we get that there exists a uniquecontinuous solution u to (1) with periodic boundary conditions. Moreover, thissolution is locally Lipschitz continuous [12, 15] and hence smooth for all positivetimes, due to the regularity theory for parabolic problems.

Theorem 2.3. Under assumption (5), problem (1) admits a unique solution

u ∈ ��0�+��×Q� ∩�1+ �2 �2+��0� T�×Q�

for every � ∈ �0� 1� and T > 0, with periodic boundary conditions on Q. Moreover

ut ∈ L2��0�+��×Q� and Du�t� x� ∈ L��0� T�×Q� for every T > 0�

We need another condition on the forcing term g, in order to prove existenceof generalized traveling wave solutions to (1), namely we assume that

∃A ⊆ Q such that∫Ag�y�dy > Per�A��n�� (6)

Note that condition (6) implies maxQ g > 0, and is fullfilled for instance if∫Qg > 0.

Remark 2.4. In [5] (see also [10]) we considered a sort of complementary conditionto (6). Indeed it is proved that, if g has zero average and there exists � ∈ �0� 1� suchthat ∫

Ag�y�dy < �Per�A��n� ∀A ⊆ Q� (7)

then there exists a periodic stationary solution of (1).

We conclude this section by recalling a classical result about the regularityof hypersurfaces of prescribed bounded mean curvature [21, Theorem 4.1], [27,Theorem 1].

Theorem 2.5. Let K be a Caccioppoli set with bounded prescribed mean curvatureA�x� ∈ L�, x ∈ K. Then �k�K\�K� = 0 for every k > n− 8, and there exists � > 0,such that for every x ∈ �K we get that K ∩ B�x� �� = �K ∩ B�x� �� and K ∩ B�x� ��

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is a �1+� hypersurface for any � ∈ �0� 1�. Moreover, letting �Kn�n be a sequence ofCaccioppoli sets such that:

i) every Kn is a locally minimizer of the functional Per�V�+∫VAn�y�dy, with �An�� ≤

A independent of n,ii) Kn converges to K� locally in the L1-topology,

and letting xn ∈ Kn, with xn → x� as n→ +�, we have x� ∈ K�. If x� ∈ �K�,then xn ∈ �Kn for all n > n0, and the unit outward normal to �Kn at xn converges tothe unit outward normal to �K� at x�.

3. Existence and Regularity of Generalized Traveling Waves

We now show existence of special solutions to (1), which we call generalizedtraveling waves. They are solutions of the form ��x�+ ct, where the graph of � iscalled the profile of the traveling wave and c is called the traveling speed. Observethat to prove the existence of a traveling wave solution it is sufficient to determinec ∈ � such that the equation

−div(

D�√1+ �D��2

)= g�y�− c√

1+ �D��2 (8)

admits a �n-periodic solution � � �n → �. In the following we will show that itis always possible to define a unique traveling speed c for the problem under ourassumption (6) on the forcing, but in general, the previous equation does not admita global solution. We will prove that there exists a maximal set E ⊆ Q, which is asufficiently regular domain, and a function � � Q→ �−��+�� (which is defined upto additive constants) such that E = � > −��, � ∈ �2+��E� and solves

−div(

D�√1+ �D��2

)= g�y�− c√

1+ �D��2 � in E (9)

with the boundary conditions

��x�→ −� as dist�x� E\S�→ 0 (10)

where S ⊂ E is a closed set with � ��S� = 0 for all � > n− 8 (see Proposition 3.10).Moreover we will show that the solutions we construct satisfy also a strongerboundary condition, more natural in viscosity solutions theory, say

for every � ∈ �1per�Q�� − � achieves its minimum in E� (11)

First of all we note that the equation (8) can be interpreted, for any c > 0, as theEuler-Lagrange equation associated to the functional

Fc�� =∫Qec��y�

(√1+ �D��y��2 − g�y�

c

)dy � ∈ �1per�Q�� (12)

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Using the change of variable ��y� �= ec��y�c

, we can rewrite the functional Fc as

Fc�� = Gc�� =∫Q

√c2� 2�y�+ �D��y��2 − g�y��y�dy� (13)

which can extended as a lower semicontinuous functional on BVper�Q�, see [2]. UsingGc, we can extend the functional Fc to all measurable functions � � Q→ �−�� 0�such that ec��y� ∈ BVper�Q� (where we use the notation e−� = 0) by setting

Fc�� = Gc(ec��y�

c

)� (14)

In particular, for all such � the following representation formula holds (cfr. [18,Section 12]):

Fc�� = sup{∫Qec��y�

(div�′

c+ �n+1

)dy � ��′� �n+1� ∈ �1per�Q��n+1�� ′�2 + �2n+1 ≤ 1

}−∫Q

ec��y�

cg�y�dy (15)

which can be easily checked on smooth functions, and then extends by relaxationto all � such that ec��y� ∈ BVper�Q�.Proposition 3.1. Under the standing assumption (6) there exists a unique constantc > 0, with

∫Qg ≤ c ≤ maxQ g, such that

• if 0 < c < c, then infGc�� ∈ BVper�Q�� ≥ 0� = −�,• if c > c, then infGc�� ∈ BVper�Q�� ≥ 0� = 0, and Gc�� > 0 for every� �≡ 0,

• minGc�� ∈ BVper�Q�� ≥ 0� = 0, and there exists � �≡ 0 s.t. Gc�� = 0.Proof. As Gc is positively one-homogeneous, it follows that inf�∈BVper�Q��≥0Gc��can be either 0 or −�. By definition of Gc, if c > maxQ g, then Gc�� ≥ 0 for every� ≥ 0, so that inf�≥0Gc�� = 0.

Now we claim that under condition (6) there exists a function � such thatGc�� < 0 if c is sufficienlty small. Let A ⊆ Q be the set appearing in (6) and take� = �A. If A �= Q, then by condition (6) there exists k > 1 such that

Gc��A� = Per�A��n�+ c�A� −∫Ag < −�k− 1�Per�A��n�+ c�A��

By choosing 0 < c < �k− 1�Per�A��n�/�A�, we then obtain Gc��A� < 0, whichimplies inf�≥0Gc�� = −�. If A = Q, then

∫Qg > 0 by (6), so that

Gc��Q� < 0 for every 0 < c <∫Qg� (16)

For c > 0 we consider the constrained problem

inf{Gc�� ∈ BVper�Q�� ≥ 0�

∫Qg� = 1

}� (17)

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By the direct method of the Calculus of Variations, one can easily show that thisproblem admits a (possibly nonunique) minimizer �c [18]. We define the functionminimum value as

c ∈ �0�+�� → �c �= Gc��c�

and we claim that this function is continuous and strictly increasing. Notice that, byminimality of �c, we have∫

Qc�cdy ≤ Gc��c�+

∫Qg�cdy = �c + 1� (18)

The monotonicity of �c is due to the fact that c �→ Gc��c� is an increasing function.Indeed, for c1 < c2 we have

�c2 − �c1 = Gc2��c2�−Gc1��c1� ≥ Gc2��c2�−Gc1��c2�=∫Q

√c22�

2c2+ �D�c2 �2 −

√c21�

2c2+ �D�c2 �2

=∫Q

�c22 − c21�� 2c2√c22�

2c2+ �D�c2 �2 +

√c21�

2c2+ �D�c2 �2

> 0 �

Let us now show the continuity of c �→ �c. For c1 < c2 we get

��c2 − �c1 � = �c2 − �c1 = Gc2��c2�−Gc1��c1�≤ Gc2��c1�−Gc1��c1�

=∫Q

�c22 − c21�� 2c1√c22�

2c1+ �D�c1 �2 +

√c21�

2c1+ �D�c1 �2

≤∫Q

�c22 − c21�� 2c1�c1 + c2��c1

dy = �c2 − c1�∫Q�c1dy

≤ �c2 − c1��c1 + 1c1

(19)

where the last inequality follows from (18). Recalling that, for all c > 0,

�c + 1c

≤∫Q

√� 2 + �D� �

2

c2

for any fixed � ∈ BVper�Q� with � ≥ 0 and∫Qg� = 1, we have that the function

c �→ ��c + 1�/c is locally bounded on �0�+��, that is, for all c ∈ �m�M� there existsC > 0 such that ��c� ≤ C. Letting now c1� c2 ∈ �m�M�, from (19) it then follows

��c2 − �c1 � ≤ C�c1 − c2��

which shows the continuity of c �→ �c.

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Since this function is continuous and strictly increasing, it is possible to definec > 0 as the unique constant for which �c = Gc��c� = 0. From (16) it followsc ≥ ∫

Qg.

Observe that, due to the constraints, �c �≡ 0 and, due to the positiveone-homogeneity of Gc, k�c is also a minimizers of Gc for every k ≥ 0.

Finally, observe that necessarily if c > c and � �≡ 0, then Gc�� > 0. Onthe contrary, if Gc�� = 0 and � �≡ 0, then

∫Qg� = > 0. So −1� would be a

minimizer to (17), and �c = 0, for c > c, in contradiction with the monotonicity ofthe value function. �

Recalling (14), it is immediate to state the analogous result for the functional Fc.

Corollary 3.2. There exists a unique constant c > 0 with∫Qg ≤ c ≤ maxQ g such that

• if 0 < c < c, then infFc�� ec� ∈ BVper�Q�� = −�,• if c > c, then infFc�� ec� ∈ BVper�Q�� = 0, and Fc�� > 0 for all � �≡ −�,• there exists � � Q→ �−��+�� such that � �≡ −�, ec� ∈ BVper�Q� andFc�� = 0.

Remark 3.3. Notice that Proposition 3.1 and Corollary 3.2, assuring the existenceof generalized traveling waves solutions, requires only g ∈ L��Q�.

We now analyze the regularity of the minima of Fc (or equivalently of Gc).We first give a geometric representation of the functional Fc (cfr. [18,

Theorem 14.6]). Given c > 0 and � ⊂ Q×� we define a weighted perimeter

Perc��n ×�� = sup

{ ∫�ecz �div��y� z�+ c�n+1�y� z�� dy dz �

� ∈ �1per�Q×��n+1�� 2 ≤ 1}� (20)

Notice that, for all � ⊂ Q×� of locally finite perimeter we have

Perc��n ×�� =

∫∗�ecz d�n +

∫�ect∫0Q

��Q��y�− �Q��y��d�n−1�y�dt

where � is as in (2).

Proposition 3.4. Let � � Q→ �−��+�� be such that ec� ∈ BVper�Q�. Then

Fc�� = �c�� = Perc��n ×��−∫��

eczg�y�dy dz (21)

where �� = �y� z� ∈ Q×� � z < ��y�� is the epigraph of �.

Proof. By exploiting formula (15) and the definition of Perc in (20), it is possibleto check that Fc�� ≤ �c��. For the reverse inequality, we observe first of allthat (21) holds on smooth functions � ∈ �1per�Q� and then the inequality extends toall �’s by relaxation. For a similar argument see [18, Theorem 14.6]. �

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Lemma 3.5. Let � � Q→ �−��+�� be a non trivial minimizer of Fc, then theepigraph �� of � is a minimizer, under compact perturbations, of the functional �cdefined in (21).

Proof. We reason as in [18, Theorem 14.9]. Given F ⊂ Q×� such that∫Fecz dy dz < +�, we consider �F � Q→ �−��+�� be such that

ec�F �y�

c=∫ �F �y�−�

eczdz =∫Fy

eczdz for a.e. y ∈ Q�

where Fy �= z ∈ � � �y� z� ∈ F�. Observe that, by definition, ec�F ∈ BV�Q� and∫Feczg�y�dy dz =

∫Qec�F �y�

g�y�

cdy� (22)

Moreover, by definition of Perc, for all � = ��′� �n+1� ∈ �1per�Q��n+1� we have

Perc�F��n ×�� ≥

∫Fecz �div�′ + c�n+1� dy dz =

∫Qec�F

(div�′

c+ �n+1

)dy� (23)

By taking the supremum over all �’s in (23), and using the representationformula (15) and (22), we then get

�c�F� ≥ Fc��F� ≥ Fc�� = �c��

where the last equality follows from Proposition 3.4, thus proving the claim. �

Notice that if � is a minimizer of �c, then �+ �0� z� is also a minimizer for allz ∈ �, that is, the class of minimizers is invariant by vertical shifts. Reasoning as in[18, Proposition 5.14] (see also [1]) one can prove a density estimate for minimizersof �c.

Lemma 3.6. There exist constants � r0 > 0, depending only on n and �g��, such thatfor all minimizers � of �c, x ∈ � and r ∈ �0� r0� the following density estimate holds:

�� ∩ Br�x�� ≥ rn+1� (24)

Proposition 3.7. Let � � Q→ �−��+�� be a non trivial minimizer of Fc. Then �� =�� is a �

2+� hypersurface for all � < 1, out of a closed singular set S� ⊂ �� ofHausdorff dimension at most n− 7. Moreover, letting E� �= ��n ��\S�� the projectiononto �n of ��\S�, we have that1. E� is a open set and E� = int�E�� = int��n��,2. � ≡ −� a.e. on Q\E�,3. � ∈ �2+�loc �E�� for all � < 1,4. � solves (9) in E� with boundary conditions (10).

Finally, letting �̃ another minimizer of Fc, for every connected component Ei of E� thereexists ki ∈ � such that �̃ = �+ ki.

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Proof. By Lemma 3.5 �� is a minimizer of �c under compact perturbations.Classical results about regularity of minimal surfaces with prescribed curvature[1, 21] then imply that �� is �

2+� for all � < 1, out of a closed singular set S� ofHausdorff dimension at most n− 7.

Recalling that g is Lipschitz continuous and Perc��n ×�� < +�, we can

reason as in [18, p. 168 and Proposition 14.11] (see also [17]) to obtain that�n+1 �= 0 on ��\S�, where � = ��1� � � � � �n+1� denotes the exterior unit normal to ��.Reasoning as in [18, Theorem 14.13] it then follows E� = int�E�� = int��n�� and� ∈ �2+�loc �E��. From the density estimate (24) we can derive that � ≤ C for someC > 0, using the same argument as in Theorem 14.10 in [18]. So, this implies that �solves (9) in E� with boundary conditions (10).

To prove the last assertion we notice that, letting �̃ be another minimum of Fc,by convexity we have Fc��+ �1− ��̃� = 0 for every ∈ �0� 1�. By definition of Fcwe then get

0 = Fc(

�+ �1− ��̃

)= Fc��+ �1− �Fc

(�̃)

if and only if

�D�̃ = �̃D� on E� ∩ E�̃�

which implies the assertion. �

Remark 3.8. Integrating (9) on E� and using (10) we obtain

Per�E��n� = −

∫E�

div

(D�√

1+ �D��2)dy =

∫E�

(g�y�− c√

1+ �D��y��2)dy� (25)

which implies that E� has finite perimeter.

Corollary 3.9. Let � as in Proposition 3.7. Then � satisfies the boundaryconditions (11) on E�.

Proof. Let � ∈ �1per�Q�. By Proposition 3.7, minQ��− �� = minE��− ��. Assumeby contradiction that �− � attains its minimum at y0 ∈ E�. Without loss ofgenerality, we can assume that z0 �= ��y0� = ��y0� and that ��y�− ��y� > 0 forevery y �= y0. Again by Proposition 3.7, we have x0 �= �y0� z0� ∈ S�, where S� is thesingular set of ��.

Let us now blow-up the sets �� and the subgraph �� of � around x0 If we let

�s� �= x ∈ �n+1 � sx ∈ �� − x0��s� �= x ∈ �n+1 � sx ∈ �� − x0��

by standard arguments of the theory of minimal surfaces [18, Chapter 9], one canprove that along a subsequence si → 0, �si� converges to a half-space H ⊂ �n+1, and�si� converges to a minimal cone C. From the inclusion �� ⊆ �� it follows C ⊆ H ,

but this implies that C = H , thus leading to a contradiction since the cone C issingular. �

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We now define the maximal support E for minima of the functional Fc, andstudy the regularity of such set.

Proposition 3.10. There exists a set E = ∪ki=1Ei ⊆ Q, where Ei are connectedcomponents, such that the support of every minimum � of Fc is given by the union ofsome connected components of E. In particular, if E is connected, then there exists aunique nontrivial minimizer � of Fc, up to an additive constant.

Moreover, there exists a closed set S ⊂ E such that E\S is a �2+� hypersurface,with � ��S� = 0 for every � > n− 8, and satisfies the geometric equation

� = g on E\S� (26)

Proof. Let �1� �2 be two minima of the functional Fc and E1� E2 be the respectivesupports. By Proposition 3.7, if Ei1 and E

j2 are connected components respectively

of E1 and E2 then either Ei1 ∩ Ej2 = ∅ or Ei1 = Ej2. In this case there exists a constant

k such that �1 = �2 + k on Ei1 = Ej2. We then define E as the union of all theconnected components of the supports of the minima of the functional Fc.

We claim that the connected components of E are finite. Fix Ei connectedcomponent of E and �i solution to (9) with support Ei. From (25) we obtain thatPer�Ei��

n� ≤ maxQ g�Ei�. This, combined with the isoperimetric inequality (4), givesthat �Ei� ≥ �Cn/maxQ g�n, which implies our claim.

If E is connected, the uniqueness up to addition of constants of the minimizersis a consequence of Proposition 3.7.

We now show the regularity of E. Let � ≥ 0 be a minimizer of Fc and assumewithout loss of generality that E = E�. Since � = �+ is also a minimizer forall ∈ �, from the proof of Proposition 3.7 we know that the subgraphs � =

�y� z� ∈ Q×� � z < ��y�� (locally) minimize the functional �c defined in (21), forall ∈ �. In particular, since � → E ×� locally in the L1-topology, as → +�,by compactness of quasi minimizers of the area functional [1] we have that E ×�is also a minimizer of �c under compact perturbations. The thesis then follows byclassical regularity theory for minimal surfaces with prescribed curvature [1, 21]. �

Remark 3.11. When n = 1, (26) reduces to

g = 0 on E�

In particular, E �= Q necessarily implies minQ g ≤ 0.

Remark 3.12. Let � � E→ � be a minimizer of Fc with maximal support, as in theproof of Proposition 3.10, and let � = ec�

cbe the corresponding minimizer of Gc.

Since Gc is a convex functional on L2�Q�, by the general theory of subdifferentials

in [3, 8] there exist a vector field �� = � � Q→ �n, with �� ≤ 1 and div�� ∈ L2�Q�,and a function h� = h � Q→ �, with 0 ≤ h ≤ 1, such that∫

Q�−div ��y�+ ch�y�− g�y�� w −��dy ≥ 0 in Q� (27)

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for all w ∈ BVper�Q� such that w ≥ 0. Moreover, for all y ∈ E�,

h�y� = c��y�√c2� 2�y�+ �D��y��2

��y� = D��y�√c2� 2�y�+ �D��y��2

�

If we apply inequality (27) to w = � + �F , where F ⊆ Q is a set of finite perimeter,we obtain

Per�F�Q�+∫F�ch�y�− g�y�� dy ≥ 0� (28)

In particular, (25) and (28) imply that E is a minimum for the functional

��F� = Per�F��n�+∫V�ch�y�− g�y�� dy F ⊆ Q�

Remark 3.13. We observe that, if � a solution to (8) such that ec� ∈ BVper�Q� forsome c > 0, which by regularity amounts to say that � is bounded from above,then necessarily Fc�� = 0 so that c ≤ c (see Corollary 3.2). Moreover, if c < c, theninfE� � = −�. This means that our variational method selects the fastest travelingwave solutions to (1) which are bounded from above [23]. However, there mightexist other traveling wave solutions with c > c, which are not in BVper�Q� (see forinstance [22]).

3.1. Existence of Classical Traveling Waves

In this subsection we state some condition on the forcing term g under whichequation (9) admits a bounded solution � in Q. This problem can be restatedas following: find sufficient conditions on g, under which the maximal support Edefined in Proposition 3.10 coincides with Q.

Remark 3.14. Observe that a first necessary condition on g, under which equation(9), with c > 0, admits a bounded solution � in Q is that

∫Qg > 0. In fact, if

∫Qg = 0

and � is a bounded solution to (8), then c = 0. In [5] we show that condition (7) issufficient to get the existence of a bounded smooth solution to (8) on Q with c = 0.Proposition 3.7 shows that this condition is essentially optimal for the existence ofstationary wave solutions.

We consider a solution � to (9) with boundary conditions (10) and maximalsupport E. Let � = ec�

c. We recall that by (25)

Per�E��n� =∫E�g�y�− ch�y�� dy ≤ max

Qg�E�� (29)

where h = h� is the function defined in (27), In Remark 3.12. Since by (28)∫Qch�y�− g�y�dy ≥ 0�

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we also have

Per�E��n� ≤∫Q\Ech�y�− g�y�dy� (30)

From inequality (30), recalling 0 ≤ h ≤ 1 and that ∫Qg ≤ c ≤ maxQ g, it follows

Per�E��n� ≤(maxQg −min

Qg

)�Q\E�� (31)

Assume now �Q\E� > 0. Recalling the isoperimetric inequality (4), from (29)and (31) we get(

maxQg −min

Qg

)1

21n

≥(maxQg −min

Qg

)�Q\E� 1n ≥ Cn or �Q\E� >

12�

In particular, if

maxQg −min

Qg < Cn 2

1n (32)

we necessarily have �E� ≤ 1/2 and, from (29),

maxQg ≥ Per�E��

n�

�E� ≥ Cn�E�− 1n ≥ Cn2 1n � (33)

If minQ g ≤ 0, then (32) implies that , in contradiction with (33).If minQ g > 0, from (33) we get

12≥ �E� ≥

(Cn

maxQ g

)n�

From (31) it then follows(maxQg −min

Qg

)(1−

(Cn

maxQ g

)n)≥(maxQg −min

Qg

)�1− �E��

≥ Cn�E� n−1n

≥ Cn(Cn

maxQ g

)n−1�

So if minQ g, we necessarily have E = Q if either maxQ g < Cn2 1n or maxQ g ≥ Cn2 1nand maxQ g −minQ g < Cn

(Cn

maxQ g

)n−1(1− ( CnmaxQ g )n)−1.

Collecting the previous results above and recalling Remark 3.11 we get thefollowing proposition.

Proposition 3.15. Assume that∫Qg > 0. Then equation (9) admits a bounded solution

� in Q if one of the following conditions is verified.

– minQ g ≤ 0 and maxQ g −minQ g < Cn21/n;

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– g > 0 on Q and maxQ g < Cn21/n;

– g > 0 on Q, maxQ g ≥ Cn21/n and maxQ g −minQ g < maxQ g((maxQ g

Cn

)n − 1)−1;– n = 1 and g > 0 on Qwhere Cn is the isoperimetric constant appearing in (4) (and C1 = 2).

Remark 3.16. Observe that the assumptions in the previous Proposition assurethe existence of classical traveling wave solutions to (1), i.e. solutions of the formct + ��x�, where � is a smooth, �n-periodic solution to (9).

Remark 3.17. In [19] Lions and Souganidis showed that (9) admits a (periodic)solution over all Q if g does not change sign and satisfies the condition

∃� ∈ �0� 1� s.t. minx∈Q

(�g2�x�− �n− 1�2�Dg�x��) > 0�

In [9] Cardaliaguet et al. proved that, when n = 1 and ∫ 10 g�y�dy > 0, thefollowing condition implies the solvability of the cell problem:

0 ≤∫ 10g�y�dy − min

z∈�0�1�g�z� < 2� (34)

4. Stability and Long-Time Behavior

If u is a solution to (1), then w�t� y� = u�t� y�− ct is a solution to

wt = tr[(

I − Dw⊗Dw1+ �Dw�2

)D2w

]+ g√1+ �Dw�2 − c in �0�+��×Q (35)

with periodic boundary conditions and initial datum w�0� y� = u0�y�. Note that wis the unique solution to (35), and it is also a classical solution, see Theorem 2.3.Standard comparison gives that �min g − c�t − �u0�� ≤ w�t� x� ≤ �max g − c�t +�u0�� for every t ≥ 0, x ∈ �n. Moreover, under the assumption (6), w is bounded(from below) uniformly in t.

Lemma 4.1. Let w be the solution to (35) and � be any solution to (9), then

w�t� y�− ��y� ≥ minQ�u0 − �� ∀ t ≥ 0� y ∈ Q� (36)

Moreover, if there exists a solution � to (9) in Q, then there exists a constant M ,depending only on �u0�� such that �w�t� y�� ≤ M for every t ≥ 0 and y ∈ Q.

Proof. We fix a � solution to (9), and let E = E� (see Proposition 3.7). We recallthat by Corollary 3.9, � satisfies the boundary conditions (11) on E�.

We shall prove that

m�t� �= minx∈Q�w�t� x�− ��x��

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is nondecreasing in t. Obviously this is sufficient to prove that minx∈E�w�t� x�− ��x��is nondecreasing in t. We fix s ≥ 0 and observe that w�t + s� x� is the solution to

vt�t� x� = tr[(

I − Dv⊗Dv1+ �Dv�2

)D2v

]+ g �x�√1+ �Dv�2 − c in �0�+��× E

with initial datum v�0� x� = w�s� x�, and with boundary conditions v�t� x� = w�t +s� x� on E for all t ≥ 0. Notice that ��y�+minŷ∈Q�w�s� ŷ�− ��ŷ�� is a regular(stationary) subsolution to the same problem. Moreover by Corollary 3.9 we havethat w�t + s� x�− ��x�+miny∈Q�w�s� y�− ��y�� can attain its minima only in theinterior of E. So we can apply comparison principle arguments (see [4]) to concludethat w�t + s� x�− ��x� ≥ miny∈Q�w�s� y�− ��y�� for every t ≥ 0 and x ∈ Q.

Finally, if there exists a solution � to (9) in the whole Q, then ��x�+ �u0�� +�� and ��x�− �u0�� − �� are, respectively, a supersolution and a subsolutionto (1) and we conclude by the standard comparison principle. �

Remark 4.2. Note that if there is a solution to (9) in the whole Q, a similarargument gives that the quantity

maxx∈Q�w�t� x�− ��x��

is nonincreasing in t.

Lemma 4.3. Let w be a solution to (35). Then for all � > 0 there exists a constantC > 0, depending on u0, g and �, such that �wt� ≤ C for all t ≥ �.Proof. Recalling Theorem 2.3, we define

C �=∥∥∥∥tr[(I − Dw�� ·�⊗Dw�� ·�1+ �Dw�� ·��2

)D2w�� ·�

]+ g�x�√1+ �Dw�� ·��2 − c∥∥∥∥

L��Q�< +�

Then S�t� x� = Ct + w�t� ·� is a supersolution to (35) and s�t� x� = −Ct + w�t� ·� isa subsolution for all t > �. Then by comparison [4] we obtain −Ct ≤ w�t� x�−w�� x� ≤ Ct. Moreover, for every fixed s > �, we get that w�t� x�+ supx �w�s� x�−w�� x�� and w�t� x�− supx �w�s� x�− w�� x�� are respectively a supersolution and asubsolution to (35) with initial data w�s� x�. So, again by comparison, and recallingthe previous estimate, for every � ≤ s ≤ t we obtain

−Cs ≤ w�t + s� x�− w�t� x� ≤ Cs��

The estimate in Lemma 4.3 imlies that, for all t > 0 the function w�t� ·� satisfiesin the viscosity sense

−C − g�x� ≤ div(

Dw�t� x�√1+ �Dw�t� x��2

)≤ C + c − g�x� in �n ∀t ≥ �� (37)

So, this gives in particular that the curvature of the graph of w�t� ·� is uniformlybounded with respect to t ∈ ��+��.Proposition 4.4. Let �w�t� ⊂ Q×� be the graph of w�t� ·�. Then, for all � > 0, �w�t�are hypersurfaces of class �1+�, for all � ∈ �0� 1�, uniformly in t ∈ ��+��.

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Proof. Assume by contradiction the statement to be false. Then we can find�xn� tn� ∈ Q× �0�+�� such that, for all � > 0, the hypersurfaces �w�tn� ∩ B��xn� tn�are not uniformly �1+�. Letting w̃n�x� �= w�x� tn�− w�xn� tn�, from (37) we have that

−div(

Dw̃n�x�√1+ �Dw̃n�x��2

)= hn�x�� (38)

with �hn�� ≤ C̃ for some C̃ independent of n. As a consequence w̃n is a minimizerof the prescribed curvature functional∫

Q

(√1+ �Du�2 − hnu

)dy�

By the compactness theorem for quasi minimizers of the perimeter [1] the graphs�w̃n of w̃n converge locally in the L

1-topology, up to a subsequence, to a limithypersurface �� of class �1+�. We can also assume that xn → x for some x ∈ Q,and let �� be the normal vector to �� at �x� 0�. However, by Theorem 2.5 thereexists � > 0 such that �w̃n ∩ B��x� 0� and �� ∩ B��x� 0� can all be written as graphsin the direction given by ��. Therefore, by elliptic regularity for minimizers of theprescribed curvature functional [21], the sets �w̃n ∩ B��x� 0� are uniformly of class�1+� for all � ∈ �0� 1�, thus leading to a contradiction. �

The following lemma that will be useful in the following.

Lemma 4.5. Let Fc�v� =∫Qecv�y�

(√1+ �Dv�y��2 − g�y�

c

)dy the functional defined

in (12). Then for every (smooth) solution w to the equation in (35),

0 ≤ Fc�w�t� ·�� ≤ Fc�u0� for all t > 0� (39)

Proof. For every solution w to (35), using the definition of the functional Fc, we get

dFc�w�t� ·��dt

=∫Qecwwt

[−div

(Dw√

1+ �Dw�2)− g + c√

1+ �Dw�2]

= −∫Q

ecww2t√1+ �Dw�2 ≤ 0� (40)�

The first result on the asymptotic behavior of the solutions u to (1) is about theconvergence of u�t�x�

tas t→ +�.

Proposition 4.6. Let u be the solution to (1) and E be the maximal support defined inProposition 3.10. Then

limt→+�

maxx∈Q u�t� x�t

= c� and limt→+�

u�t� x�

t= c locally uniformly in E�

Moreover if there exists a bounded solution to (9),

limt→+�

u�t� x�

t= c uniformly in Q�

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In particular there exists a constant C ∈ � such that

minQu0�x� ≤ M�t� �= max

Q�u�t� x�− ct� ≤ C + log�1+ t�

c� (41)

Proof. Recall that if the stationary problem (9) has a bounded solution, Lemma 4.1gives an uniform bound on w�t� x� = u�t� x�− ct, and then we obtain the result.

We observe, recalling Lemma 4.1, that to prove the general statement it issufficient to prove (41). The lower bound on M is an immediate consequence ofLemma 4.1, just by choosing � as the maximal nonpositive solution to (9).

We define f�t� x� �= 2cecw�t�x�

2 , so that f 2t �t� x� = w2t �t� x�ecw�t�x�. Integrating (40)between 0 and T , we obtain

C ≥ Fc�u0� ≥ Fc�u0�− Fc�w�T� ·� =∫ T0

∫Q

f 2t �t� x�√1+ �Dw�t� x��2 dx dt (42)

for some constant C > 0 depending only on u0 and g.Given � > 0 we let

M��t� �= maxx

1�B��

∫B��x�f�t� y�dy�

where we identify f with its periodic extension on �n. Given a point x̄ where M��t�attains its maximum, thanks to Proposition 4.4 we can choose � < 2/c, independentof t and x̄, such that �Dw�x� t�� ≤ 1 for every x ∈ B��x̄�. Notice that

2cec�M�t�−��

2 ≤ M��t� ≤2cecM�t�2 for all t ≥ 0�

so that, in order to prove the second inequality in (41), it is enough to show that

M��t� ≤ K(1+√t

)� (43)

for some constant K > 0 (possibly depending on �). Given t ≥ 0, we let

��t� �={x ∈ Q � M��t� =

1�B��

∫B��x�f�t� y�dy

}�

Using the fact that �Dw�x� t�� ≤ 1 on B��x̄�t��, from (42) and Jensen’s inequalitywe get

C̃ = 2C�B��≥∫ T0

maxx̄�t�∈��t�

1�B��

∫B��x̄�t��

f 2t �t� x�dx dt

≥∫ T0

(maxx̄�t�∈��t�

1�B��

∫B��x̄�t��

ft�t� x�dx

)2dt (44)

=∫ T0M ′��t�

2dt ≥ 1T

(∫ T0

�M ′��t��dt)2�

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From (44) we then have

M��T� ≤ M��0�+∫ T0

�M ′��t��dt ≤ M��0�+√C̃T

which gives (43). �

We now prove our main convergence result.

Theorem 4.7. Let u�t� x� be the unique solution to (1) with periodic boundaryconditions, let M�t� �= maxQ w�t� y�, and let

w̃�t� x� �= w�t� x�−M�t� = u�t� x�−maxx∈Q�u�t� x�� ≤ 0�

Then, for any sequence tn → +� there exist a subsequence tnk and a function � � E� →� (possibly depending on the subsequence tnk) such that

w�tnk� x� −→{��x� locally in �1+��E��−� locally uniformly in Q\E�

(45)

as k→ +�, for all � ∈ �0� 1�. Moreover � is a generalized traveling wave solutionto (9).

Proof. We let

W�t� y� �= ecw�t�y�

c� W̃ �t� y� �= e

cw̃�t�y�

c= e−cM�t�W�t� y� ≤ 1

c�

Notice that from (35) it follows that W satisfies the equation

Wt =√c2W 2 + �DW �2

div DW√c2W 2 + �DW �2

+ g− c2W in �0�+��×Q�

(46)

By (39) and (41), for all t ≥ 0 we haveGc�W̃ �t� ·�� = Fc�w̃�t� ·�� = e−cM�t�Fc�w�t� ·�� ≤ e−c�minQ u0�Fc�u0��

In particular,∫Q

√c2W̃ 2�t� y�+ �DW̃�t� y��2dy = Gc�W̃ �t� ·��+

∫g�y�W̃ �t� y�dy ≤ C

for all t ≥ 0, where C depends only on u0 and g. Hence, up to extracting asubsequence tnk , W̃ �tnk � ·� ⇀ W� weakly* in BVper�Q�, as k→ +�. Notice that, asin the previous section, the epigraph of w̃�t� ·� is, for every t > 0, a minimizer of theprescribed curvature functional

� �→ Perc��n ×��−∫�eczg̃t�y�dy dz

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where g̃t is an appropriate bounded function, depending on t. It therefore satisfiesthe lower density bound (24), which implies W� �≡ 0. We claim that

Gc�W�� = 0� (47)

We introduce the modified functional, for t > 0,

G̃c�t�W� �=∫Q

(√c2W 2 + �DW �2 − g̃tW

)dy

where

g̃t�y� �= g�y�−Wt�t� y�√

c2W 2�t� y�+ �DW�t� y��2∈ L��Q�� g̃t�� ≤ C�

with C independent of t. From (46) it follows by direct computation thatG̃c�t�W�t� ·�� = 0, hence also G̃c�t�W̃ �t� ·�� = 0. Recalling (40), up to extracting afurther subsequence, we can assume that

tGc�W�tnk� ·�� = tFc�w�tnk � ·��

= −∫Q

ecw�tnk �y�w2t �tnk � y�√1+ �Dw�tnk� y��2

dy

= −∫Q

W 2t �tnk � y�√c2W 2�tnk � y�+ �DW�tnk� y��2

dy

→ 0 (48)

as k→ +�.Since Gc�v� ≥ 0 for every v, to prove the claim (47) it is sufficient to show that

Gc�W�� ≤ 0. We get, using the convexity of Gc and the definition of the modifiedfunctional G̃c�t,

Gc�W�� ≤ lim infk→+�

Gc�W̃ �tnk � y��

= lim infk→+�

G̃c�tnk �W̃ �tnk � y��− ∫Q W̃�tnk � y�Wt�tnk � y�√c2W 2�tnk � y�+ �DW�tnk� y��2dy

= lim infk→+�

−∫Q

W̃�tnk � y�Wt�tnk � y�√c2W 2�tnk � y�+ �DW�tnk� y��2

dy

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since G̃c�W̃ �tnk � y�� = 0. Using the Hölder inequality, (48) and the definition of W̃ ,we obtain

lim infk→+�

∫Q

−W̃ �tnk � y�Wt�tnk � y�√c2W 2�tnk � y�+ �DW�tnk� y��2

dy

≤ lim infk→+�

∫Q

W 2t �tnk � y�√c2W 2�tnk � y�+ �DW�tnk� y��2

dy

12 (∫

Q

e−cM�t�

c2dy

) 12

= 0

which proves our claim. In particular, � �= log�cW��/c � E� → �−��+�� is atraveling wave solution of (9) with c = c.

Let us now prove (45). Given y ∈ E�, by Theorem 2.5 there exists r > 0 suchthat Br�y� ⊂ E� and �Dw̃�tnk � y��L��Br �y�� is uniformly bounded in k. By standardelliptic regularity [16] it then follows that the functions w̃�tnk � ·� are uniformlybounded in �1+��Br�y�� for all � ∈ �0� 1�, so that they converge to � locally in�1+��E��.

Fix now y ∈ Q\E� and take r > 0 such that Br�y� ⊂ Q\E�. Assume bycontradiction that there exist c ∈ � and yk ∈ Br�y�, k ∈ , such that w̃�tnk � yk� ≥ cfor all k. By the density estimate (24) this would imply

∫QW̃�tnk � y�dy ≥ c′ for some

c′ ∈ �, contradicting the fact that W̃ �tnk � y�→ W� in L1�Q�, with W� ≡ 0 in Br�y�.We thus proved (45). �

Remark 4.8. If the functional Fc admits a unique minimizer �̄ � E� → � up toan additive constant (for instance if the maximal support E is connected, seeProposition 3.10), then instead of (45) we have

limt→+�w�t� x� =

{��x�−maxE� �̄ locally in �1+��E��−� locally uniformly in Q\E�

(49)

for all � ∈ �0� 1�.Corollary 4.9. Let u�t� x� be the unique solution to (1) with periodic boundaryconditions, and assume that there exist bounded solutions to (9) in Q (seeProposition 3.15). Then

u�t� x�− ct −→ ��x� in �1+��Q�� as t→ +��where � is a bounded solution to (9).

Proof. By Lemma 4.1 and Remark 4.2, it is enough to prove that w�tn� x�→ ��x�uniformly along a subsequence tn → +�. This result can be obtained by repeatingthe same argument as in the proof of Theorem 4.7. �

Remark 4.10. A straightforward adaptation of the argument in Corollary 4.9 givesthat, under assumption (7),

u�t� x�→ ��x� in �1+��Q�� as t→ +��

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where � is a stationary solution of the parabolic equation (1) (whose existence hasbeen shown in [5]).

Remark 4.11. The results of this paper can be easily extended to equation (1)considered on a bounded open set ! ⊂ �n with Lipschitz boundary, and withNeumann boundary conditions on !.

Acknowledgments

The authors thank Guy Barles and Cyrill Muratov for inspiring discussions on thisproblem.

References

[1] Ambrosio. L. (1997). Corso Introduttivo Alla Teoria Geometrica Della Misura EdAlle Superfici Minime [Introductory Course on Geometric Measure Theory andMinimal Surfaces]. Pisa, Italy: Edizioni della Scuola Normale Superiore.

[2] Ambrosio, L., Fusco, N., Pallara, D. (2000). Functions of Bounded Variation andFree Discontinuity Problems. Oxford: Oxford Mathematical Monographs.

[3] Andreu-Vaillo, F., Caselles, V., Mazón, J. (2004). Parabolic QuasilinearEquations Minimizing Linear Growth Functionals. Basel: Birkhäuser Verlag.

[4] Barles, G., Biton, S., Bourgoing, M., Ley, O. (2003). Uniqueness results forquasilinear parabolic equations through viscosity solutions methods. Calc. Var.Partial Differential Equations 18:159–179.

[5] Barles, G., Cesaroni, A., Novaga, M. (2011). Homogenization of fronts inhighly heterogeneous media. SIAM J. Math. Anal. 43:212–227.

[6] Barles, G., Porretta, A., Tabet Tchamba, T. (2010). On the large time behaviorof solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobiequations. J. Math. Pures Appl. 94:497–519.

[7] Barles, G., Souganidis, P.E. (2001). Space-time periodic solutions and long-timebehavior of solutions to quasi-linear parabolic equations. SIAM J. Math. Anal.32:1311–1323.

[8] Brezis, H. (1973). Operateurs Maximaux Monotones [Maximal MonotoneOperators]. Amsterdam: North-Holland.

[9] Cardaliaguet, P., Lions, P.-L., Souganidis, P.E. (2009). A discussion about thehomogenization of moving interfaces. J. Math. Pures Appl. 91:339–363.

[10] Chambolle, A., Thouroude, G. (2009). Homogenization of interfacial energiesand construction of plane-like minimizers in periodic media through a cellproblem. Netw. Heterog. Media 4:127–152.

[11] Chen, X., Lou, B. (2009). Traveling waves of a curvature flow in almostperiodic media. J. Differential Equations 247:2189–2208.

[12] Chou, K.-S., Kwong, Y.-C. (2001). On quasilinear parabolic equations whichadmit global solutions for initial data with unrestricted growth. Calc. Var.Partial Differential Equations 12:281–315.

[13] Da Lio, F. (2008). Large time behavior of solutions to parabolic equations withNeumann boundary conditions. J. Math. Anal. Appl. 339:384–398.

[14] Dirr, N., Karali, G., Yip, N.-K. (2008). Pulsating wave for mean curvature flowin inhomogeneous medium. European J. Appl. Math. 19:661–699.

Dow

nloa

ded

by [

Uni

vers

ita d

i Pad

ova]

at 0

6:13

12

Apr

il 20

13


[15] Ecker, K., Huisken, G. (1989). Mean curvature evolution of entire graphs.Annals of Math. 130:453–471.

[16] Gilbarg, D., Trudinger, N.S. (1983). Elliptic Partial Differential Equations ofSecond Order. Berlin: Springer.

[17] Giusti, E. (1978). On the equation of surfaces of prescribed mean curvature.Invent. Math. 46:111–137.

[18] Giusti, E. (1984). Minimal Surfaces and Functions of Bounded Variation.Monographs in Mathematics, Vol. 80. Basel: Birkhäuser Verlag.

[19] Lions, P.-L., Souganidis, P.E. (2005). Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications.Annales IHP – Analyse Nonlinéaire 22:667–677.

[20] Lou, B. (2009). Periodic traveling waves of a mean curvature flow inheterogeneous media. Discrete Contin. Dyn. Syst. 25:231–249.

[21] Massari, U. (1974). Esistenza e regolarità delle ipersuperfice di curvatura mediaassegnata in �n. Arch. Rational Mech. Anal. 55:357–382.

[22] Monneau, R., Roquejoffre, J.M., Roussier-Michon, V. (2013). Travellinggraphs for the forced mean curvature motion in an arbitrary space dimension.Ann. Sci. Ec. Norm. Supér 46:215–246.

[23] Muratov, C. (2004). A global variational structure and propagation ofdisturbances in reaction-diffusion systems of gradient type. Discrete Cont. Dyn.Syst. B 4:867–892.

[24] Muratov, C., Novaga, M. (2008). Front propagation in infinite cylinders. II.The sharp reaction zone limit. Calc. Var. Part. Diff. Eqs. 31:521–547.

[25] Namah, G., Roquejoffre, J.M. (1997). Convergence to periodic fronts in aclass of semilinear parabolic equations. Nonlinear Differential Equations Appl.4:521–536.

[26] Ninomiya, H., Taniguchi, M. (2001). Stability of traveling curved fronts in acurvature flow with driving force. Methods Appl. Anal. 8:429–450.

[27] Tamanini, I. (1982). Boundaries of Caccioppoli sets with Hölder-continuousnormal vector. J. Reine Angew. Math. 334:27–39.

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i Pad

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Long-Time Behavior of the Mean Curvature Flow with ...acesar/docs/... · Long-Time Behavior 781 We are particularly interested in the asymptotic behavior as t→+ of solutions to

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