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Advances in Nonlinear Analysis

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Regularity of solutions of the parabolic normalized p-Laplace equation

Fredrik Arbo Høeg
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  • Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
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/ Peter Lindqvist
Published Online: 2018-07-12 | DOI: https://doi.org/10.1515/anona-2018-0091

Abstract

The parabolic normalized p-Laplace equation is studied. We prove that a viscosity solution has a time derivative in the sense of Sobolev belonging locally to L2.

Keywords: Non-linear equation; regularity theory; time derivative

MSC 2010: 35K92; 35K10

1 Introduction

We consider viscosity solutions of the normalized p-Laplace equation

ut=|u|2-pdiv(|u|p-2u),1<p<,(1.1)

in ΩT=Ω×(0,T), Ω being a domain in n. Formally, the equation reads

ut=Δu+(p-2)|u|-2i,j=1nuxiuxj2uxixj.

In the linear case p=2, we have the heat equation ut=Δu, and also for n=1, the equation reduces to the heat equation ut=(p-1)uxx. At the limit p=1, we obtain the equation for motion by mean curvature. We aim at showing that the time derivative ut exists in the Sobolev sense and belongs to Lloc2(ΩT). We also study the second derivatives 2uxixj.

There has been some recent interest in connection with stochastic game theory, where the equation appears, cf. [7]. From our point of view, the work [3] is of actual interest, because there it is shown that the time derivative ut of the viscosity solutions exists and is locally bounded, provided that the lateral boundary values are smooth. Thus, the boundary values control the time regularity. If no such assumptions about the behaviour at the lateral boundary Ω×(0,T) are made, a conclusion like utLloc(ΩT) is in doubt. Our main result is the following, where we unfortunately have to restrict p.

Theorem 1.1.

Suppose that u=u(x,t) is a viscosity solution of the normalized p-Laplace equation in ΩT. If 65<p<145, then the Sobolev derivatives ut and 2uxixj exist and belong to Lloc2(ΩT).

We emphasize that no assumptions on the boundary values are made for this interior estimate. Our method of proof is based on a verification of the identity

0TΩuϕt𝑑x𝑑t=-0TΩUϕ𝑑x𝑑t,ϕC0(ΩT),

where we have to prove that the function U, which is the right-hand side of equation (1.1), belongs to Lloc2(ΩT). Thus, the second spatial derivatives D2u are crucial (local boundedness of u was proven in [3, 2] and interior Hölder estimates for the gradient in [6]). The elliptic case has been studied in [1].

In the range 1<p<2, one can bypass the question of second derivatives.

Theorem 1.2.

Suppose that u=u(x,t) is a viscosity solution of the normalized p-Laplace equation in ΩT. If 1<p<2, then the Sobolev derivative ut exists and belongs to Lloc2(ΩT).

To avoid the problem of vanishing gradient, we first study the regularized equation

uϵt=(|uϵ|2+ϵ2)2-p2div((|uϵ|2+ϵ2)p-22uϵ).(1.2)

Here the classical parabolic regularity theory is applicable. The equation was studied by Does in [3], where an estimate of the gradient uϵ was found with Bernstein’s method. We shall prove a maximum principle for the gradient. Further, we differentiate equation (1.2) with respect to the space variables and derive estimates for uϵ, which are passed over to the solution u of (1.1).

Analogous results seem to be possible to reach through the Cordes condition. This also restricts the range of valid exponents p. We have refrained from this approach, mainly since the absence of zero (lateral) boundary values produces many undesired terms to estimate. Finally, we mention that the limits 65 and 145 in Theorem 1.1 are evidently an artifact of the method. It would be interesting to know whether the theorem is valid in the whole range 1<p<. In any case, our method is not capable to reach all exponents.

2 Preliminaries

Notation.

The gradient of a function f:ΩT is

f=(fx1,,fxn)

and its Hessian matrix is

(D2f)ij=2fxixj,|D2f|2=i,j=1n(2fxixj)2.

We shall, occasionally, use the abbreviation

uj=uxj,ujk=2uxjxk

for partial derivatives. Young’s inequality

|ab|δ|a|pp+(1δ)q-1|b|qq,1p+1q=1

is often referred to. Finally, the summation convention is used when convenient.

Viscosity solutions.

The normalized p-Laplace equation is not in divergence form. Thus, the concept of weak solutions with test functions under the integral sign is problematic. Fortunately, the modern concept of viscosity solutions works well. The existence and uniqueness of viscosity solutions of the normalized p-Laplace equation was established in [2]. We recall the definition.

Definition 2.1.

We say that an upper semi-continuous function u is a viscosity subsolution of equation (1.1) if for all ϕC2(ΩT), we have

ϕt(δij+(p-2)ϕxiϕxj|ϕ|2)ϕxixj

at any interior point (x,t) where u-ϕ attains a local maximum, provided ϕ(x,t)0. Further, at any interior point (x,t) where u-ϕ attains a local maximum and ϕ(x,t)=0, we require

ϕt(δij+(p-2)ηiηj)ϕxixj

for some ηn, with |η|1.

Definition 2.2.

We say that a lower semi-continuous function u is a viscosity supersolution of equation (1.1) if for all ϕC2(ΩT), we have

ϕt(δij+(p-2)ϕxiϕxj|ϕ|2)ϕxixj

at any interior point (x,t) where u-ϕ attains a local minimum, provided ϕ(x,t)0. Further, at any interior point (x,t) where u-ϕ attains a local minimum and ϕ(x,t)=0, we require

ϕt(δij+(p-2)ηiηj)ϕxixj

for some ηn, with |η|1.

Definition 2.3.

A continuous function u is a viscosity solution if it is both a viscosity subsolution and a viscosity supersolution.

For a detailed discussion on the definition at critical points, we refer to [5]. The reason behind the choice of ηn is given in [5, Section 2]. The viscosity solutions of equation (1.2) are defined in a similar manner, except that now ϕ(x,t)=0 is not a problem.

Maximum principle for the gradient.

In order to estimate the time derivative, we need bounds on the second derivatives of uϵ (and also on its gradient). If we first assume that uϵ is C1 on the parabolic boundary parΩT, we get bounds on the gradient in all of ΩT. This follows from the following maximum principle.

Proposition 2.4 (Maximum principle).

Let uϵ be a solution of equation (1.2). If uϵC1(Ω¯T), then

maxΩ¯T{|uϵ|}=maxparΩT{|uϵ|}.

Proof.

With some modifications, a proof can be extracted from [3]. We give a direct proof. To this end, consider

Vϵ(x,t)=|uϵ|2+ϵ2.

To find the partial differential equation satisfied by Vϵ, we calculate1

Viϵ=2uνϵuiνϵ,Vijϵ=2uνjϵuiνϵ+2uνϵuijνϵ,uiϵujϵVijϵ=12|Vϵ|2+2uiϵujϵuνϵuijνϵ.

Writing equation (1.1) in the form

utϵ=(δij+(p-2)uiϵujϵ|uϵ|2+ϵ2)uijϵ,

we find

12Vtϵ=uνϵxνutϵ=uνϵΔuνϵ-p-22(Vϵ)2|uϵ,Vϵ|2+p-2Vϵ(14|Vϵ|2+12uνϵuμϵVνμϵ).

Rearranging and using

ΔVϵ=2|D2uϵ|2+2uϵ,Δuϵ,

we arrive at the following differential equation for Vϵ:

Vtϵ=ΔVϵ-2|D2uϵ|2-p-2(Vϵ)2|uϵ,Vϵ|2+p-2Vϵ{12|Vϵ|2+uνϵuμϵVνμϵ}.(2.1)

Let

w(x,t)=|uϵ(x,t)|2+ϵ2-αt=Vϵ(x,t)-αtfor α>0.

Suppose that wϵ has an interior maximum point at (x0,t0). At this point, Vϵ(x0,t0)>0, otherwise we would have Vϵ(x,t)0 in ΩT, in which case there is nothing to prove. By the infinitesimal calculus,

w(x0,t0)=0andwt(x0,t0)0,

where we have included the case t0=T. Further, the matrix D2w(x0,t0) is negative semidefinite. Using equation (2.1) and noting that w=Vϵ and D2w=D2Vϵ, we get, at (x0,t0),

0wt=Vtϵ-α=ΔVϵ-2|D2uϵ|2-p-2(Vϵ)2|uϵ,Vϵ|2+p-2Vϵ{12|Vϵ|2+uνϵuμϵVνμϵ}-α=(δij+(p-2)uiϵujϵVϵ)wijϵ-2|D2uϵ|2-α-α,

since the matrix A, with elements Aij=δij+(p-2)uiϵujϵVϵ, is positive semidefinite. To avoid the contradiction α0, w must attain its maximum on the parabolic boundary.

Hence, for any (x,t)ΩT, we have

Vϵ(x,t)-αtmaxparΩT{Vϵ(x,t)-αt}maxparΩTVϵ(x,t).

We finish the proof by sending α0+. ∎

With no assumptions for uϵ on the parabolic boundary, we need a stronger result, taken from [3, p. 381].

Theorem 2.5.

Let uϵ be a solution of equation (1.2), with uϵ(x,0)=u0(x). Then

|uϵ(x,t)|Cn,pu0L(ΩT){1+(1dist((x,t),parΩT))2}.

Note that no condition on the lateral boundary Ω×[0,T] was used. By continuity,

|uϵ(x,t)|Cn,puϵ(,t0){1+(1dist((x,t),parΩT))2}

for xDΩ and 0<t0tT-t0. The estimate

uϵL(D×[t0,T-t0])CuϵL(ΩT){1+(1dist(D,parΩT))2}(2.2)

follows. (Here one can pass to the limit as ϵ0.)

The proof of the lemma below, a simple special case of the Miranda–Talenti lemma, can be found for smooth functions in [4, p. 308]. If f is not smooth, we perform a strictly interior approximation, so that no boundary integrals appear (which is possible since ξC0).

Lemma 2.6 (Miranda–Talenti).

Let ξC0(ΩT) and fL2(0,T,W2,2(Ω)). Then

0TΩ|Δ(ξf)|2dxdt=0TΩ|D2(ξf)|2dxdt.

3 Regularization

The next lemma tells us that the solutions of (1.2) converge locally uniformly to the viscosity solution of (1.1).

Lemma 3.1.

Let u be a viscosity solution of equation (1.1) and let uϵ be the classical solution of the regularized equation (1.2) with boundary values

u=uϵon parΩT.

Then uϵu uniformly on compact subsets of ΩT.

Proof.

By Theorem 2.5, we can use Ascoli’s theorem to extract a convergent subsequence uϵj converging locally uniformly to some continuous function, namely, uϵjv. We claim that v is a viscosity solution of equation (1.1). The lemma then follows by uniqueness.

We demonstrate that v is a viscosity subsolution. (A symmetric proof shows that v is a viscosity supersolution.) Assume that v-ϕ attains a strict local maximum at z0=(x0,t0). Since uϵv locally uniformly, there are points

zϵz0

such that uϵ-ϕ attains a local maximum at zϵ. If ϕ(z0)0, then ϕ(zϵ)0 for all ϵ>0 small enough, and at zϵ, we have

ϕt(δij+(p-2)ϕxiϕxj|ϕ|2+ϵ2)ϕxixj.(3.1)

Letting ϵ0, we see that v satisfies Definition 2.3 when ϕ(z0)0. If ϕ(z0)=0, let

ηϵ=ϕ(zϵ)|ϕ(zϵ)|2+ϵ2.

Since |ηϵ|1, there is a subsequence such that ηϵkη when k for some ηn, with |η|1. Passing to the limit ϵk0 in equation (3.1), we see that v is a viscosity subsolution. ∎

Our proof of Theorem 1.1 consists in showing that the second derivatives D2uϵ belong locally to L2 with a bound independent of ϵ. Once this is established, we see that

(|uϵ|2+ϵ2)2-p2div((|uϵ|2+ϵ2)p-22uϵ)=Δuϵ+p-2|uϵ|2+ϵ2uϵ,D2uϵuϵCp,n|D2uϵ|.

Hence, for any bounded subdomain DΩT,

(|uϵ|2+ϵ2)2-p2div((|uϵ|2+ϵ2)p-22uϵ)L2(D)C,

with C independent of ϵ. By this uniform bound, there exists a subsequence such that, as j,

(|uϵj|2+ϵj2)2-p2div((|uϵj|2+ϵj2)p-22uϵj)Uweakly in L2(D).

In particular, this means that UL2(D) and for any ϕC0(D), we have

limj0TDϕ(|uϵj|2+ϵj2)2-p2div((|uϵj|2+ϵj2)p-22uϵj)𝑑x𝑑t=0TDϕU𝑑x𝑑t.

If u is the unique viscosity solution of (1.1), we invoke Lemma 3.1 and the calculations above to find, for any test function ϕC0(D),

0TDuϕt𝑑x𝑑t=limj0TDuϵjϕt𝑑x𝑑t=-limj0TDϕ(|uϵj|2+ϵj2)2-p2div((|uϵj|2+ϵj2)p-22uϵj)𝑑x𝑑t=-0TDϕU𝑑x𝑑t.

This shows that the Sobolev derivative ut exists and, since the previous equation holds for any subdomain DΩT, we conclude that ut=ULloc2(ΩT). To complete the proof of Theorem 1.1, it remains to establish the missing local bound of D2uϵL2 uniformly in ϵ.

4 The differentiated equation

We shall derive a fundamental identity. Let

vϵ=|uϵ|2,Vϵ=|uϵ|2+ϵ2.

Differentiating equation (1.2) with respect to the variable xj, we obtain

tujϵ=2-p2(Vϵ)-p2vjϵdiv((Vϵ)p-22uϵ)+(Vϵ)2-p2div[((Vϵ)p-22uϵ)j].

Take ξC0(ΩT), with ξ0. Multiply both sides of the equation by ξ2Vϵujϵ and sum j from 1 to n. Integrate over ΩT, using integration by parts and keeping in mind that ξ is compactly supported in ΩT, to obtain

-120TΩξξtVϵ𝑑x𝑑t=2-p20TΩξ2(Vϵ)-p2uϵ,vϵdiv((Vϵ)p-22uϵ)𝑑x𝑑t-0TΩxj{(Vϵ)p-22ukϵ}xk{ξ2(Vϵ)2-p2ujϵ}𝑑x𝑑t.

Writing out the derivatives gives the fundamental formula

I+II:=0TΩξ2|D2uϵ|2𝑑x𝑑t+p-220TΩ1Vϵξ2uϵ,vϵΔuϵ𝑑x𝑑t=120TΩξξtVϵdxdt+(2-p)0TΩ1Vϵξuϵ,vϵuϵ,ξdxdt-0TΩξvϵ,ξdxdt=:III+IV-V.

In the next section we shall bound the main term I uniformly with respect to ϵ.

5 Estimate of the second derivatives

We shall provide an estimate of the main term I. First, we record the elementary inequality

|vϵ|2=|2D2uϵuϵ|24|D2uϵ|2vϵ.(5.1)

One dimension. As an exercise, we show that in this case, the second derivatives are locally bounded in L2 for any 1<p<. In one dimension, equation (1.1) reads

ut=|ux|2-px{|ux|p-2ux}=(p-1)uxx.

We absorb the terms IV and V, using Young’s inequality and inequality (5.1). For any δ>0,

0TΩξ2(2uϵx2)2(1+(p-2)(uϵx)2(uϵx)2+ϵ2-δ(|p-2|+1))𝑑x𝑑t120TΩξξtVϵ𝑑x𝑑t+|p-2|+1δ0TΩVϵ|ξ|2𝑑x𝑑t.

Applying Theorem 2.5 we see that the right-hand side is bounded by a constant independent of ϵ>0. We have

1+(p-2)(uϵx)2(uϵx)2+ϵ2min{1,p-1}>0.

It follows that 2uϵx2L2 locally for any p(1,).

General n. We assume for the moment that 1<p<2. We rewrite the term II involving the Laplacian as

2-p21Vϵξ2uϵ,vϵΔuϵ=2-p21Vϵξuϵ,vϵ{Δ(ξuϵ)-2uϵ,ξ-uϵΔξ}.

Upon this rewriting, the term IV disappears from the equation. We focus our attention on the term involving Δ(ξuϵ). By Lemma 2.6,

0TΩ|D2(ξuϵ)|2dxdt=0TΩ|Δ(ξuϵ)|2dxdt.

Differentiating, we see that

(ξuϵ)i=ξiuϵ+ξuiϵ,(ξuϵ)ij=ξijuϵ+uiϵξj+ξiujϵ+ξuijϵ.

It follows that

|D2(ξuϵ)|2=ξ2|D2uϵ|2+f(uϵ,uϵ,D2uϵ),

where f(uϵ,uiϵ,D2uϵ) depends only linearly on the second derivatives uijϵ:

f(uϵ,uϵ,D2uϵ)=(uϵ)2|D2ξ|2+4uϵξ,D2ξuϵ+4ξξ,D2uϵuϵ+2|ξ|2|uϵ|2+2|uϵ,ξ|2+2uϵξtrace{(D2ξ)(D2uϵ)}.

By Young’s inequality, we obtain

2-p20TΩ1Vϵξuϵ,vϵΔ(ξuϵ)𝑑x𝑑t54(2-p)0TΩξ2|D2uϵ|2𝑑x𝑑t+2-p40TΩf(uϵ,uϵ,D2uϵ)𝑑x𝑑t.

Inserting this into the main equation gives

I*:=(1-54(2-p))0TΩξ2|D2uϵ|2𝑑x𝑑t120TΩξξtVϵ𝑑x𝑑t-0TΩξvϵ,ξ𝑑x𝑑t+2-p20TΩf(uϵ,uiϵ,uijϵ)𝑑x𝑑t+2-p20TΩ1Vϵξuϵ,vϵuϵΔξ𝑑x𝑑t.=:III-V+VI+VII.

All terms containing D2uϵ can be absorbed by the new main term I*. To this end, we use Young’s inequality with a small parameter δ>0 to balance the terms.2 For term V, we have

0TΩξvϵ,ξ𝑑x𝑑tδ0TΩξ2|D2uϵ|2𝑑x𝑑t+1δ0TΩVϵ|ξ|2𝑑x𝑑t.

Similarly, for term VII,

0TΩ1Vϵξuϵ,vϵuϵΔξdxdt2δ10TΩξ2|D2uϵ|2+1δ10TΩ|uϵ|2|Δξ|2dxdt.

Using similar inequalities for the term involving f(uϵ,uϵ,D2uϵ) and choosing the parameters small enough in Young’s inequality, we find,

0TΩξ2|D2uϵ|2𝑑x𝑑tC{ξ0}((uϵ)2+|uϵ|2)𝑑x𝑑t,(5.2)

where C is independent of ϵ but depends on ξC2, provided that 1-54(2-p)>0, i.e., p>65. This is now a decisive restriction. Invoking Lemma 3.1 and estimate (2.2), we deduce that the majorant in (5.2) is independent of ϵ.

A symmetric proof when p>2 shows that equation (5.2) holds when p<145.

6 The case 1<p<2

In this section, we give a proof of Theorem 1.2. To this end, let ξC0(ΩT), with 0ξ1. We claim that

0TΩξ2(uϵt)2dxdt4Vϵ2{0TΩ|ξ|2dxdt+1p0TΩξ|ξt|dxdt},(6.1)

where the supremum norm of Vϵ=|uϵ|2+ϵ2 is taken locally, over the support of ξ. Here, uϵ is the solution of the regularized equation (1.2). This is enough to complete the proof of Theorem 1.2, in virtue of Theorem 2.5.

Multiplying the regularized equation (1.2) by (|uϵ|2+ϵ2)p-22ξ2utϵ yields

ξ2(|uϵ|2+ϵ2)p-22(utϵ)2=ξ2utϵdiv((|uϵ|2+ϵ2)p-22uϵ)=div(ξ2utϵ(|uϵ|2+ϵ2)p-22uϵ)-(|uϵ|2+ϵ2)p-22uϵ,(ξ2utϵ).

The integral of the divergence term vanishes by Gauss’s theorem and, upon integration, we have

0TΩξ2(Vϵ)p-22(utϵ)2𝑑x𝑑t=-0TΩ(Vϵ)p-22uϵ,(ξ2utϵ)𝑑x𝑑t=-20TΩξ(Vϵ)p-22uϵ,ξutϵ𝑑x𝑑t-0TΩξ2(Vϵ)p-22uϵ,utϵ𝑑x𝑑t.

The first integral on the right-hand side can be absorbed by the left-hand side by choosing σ=12 in

|2ξ(Vϵ)p-22uϵ,ξutϵ|σξ2(Vϵ)p-22(utϵ)2+1σ(Vϵ)p-22|uϵ|2|ξ|2,

and integrating.

For the last term, the decisive observation is that

1pt(|uϵ|2+ϵ2)p2=(|uϵ|2+ϵ2)p-22uϵ,utϵ=(Vϵ)p-22uϵ,utϵ.

We use this in the last integral on the right-hand side to obtain

-0TΩξ2(Vϵ)p-22uϵ,utϵ𝑑x𝑑t=-0TΩt{ξ2p(Vϵ)p2}𝑑x𝑑t+2p0TΩξξt(Vϵ)p2𝑑x𝑑t=-Ω[ξ2p(Vϵ)p2]t=0t=T𝑑x+2p0TΩξξt(Vϵ)p2𝑑x𝑑t=2p0TΩξξt(Vϵ)p2𝑑x𝑑t.

To sum up, we have now the final estimate

120TΩξ2(Vϵ)p-22(utϵ)2𝑑x𝑑t20TΩ(Vϵ)p-22|uϵ|2|ξ|2𝑑x𝑑t+2p0TΩξξt(Vϵ)p2𝑑x𝑑t20TΩ(Vϵ)p2|ξ|2𝑑x𝑑t+2p0TΩξξt(Vϵ)p2𝑑x𝑑t.

So far, our calculations are valid in the full range 1<p<. For 1<p<2, we have

(Vϵ)p-22Vϵp-22,

where the supremum norm is taken over the support of ξ. Hence, equation (6.1) holds for 1<p<2 and the proof of Theorem 1.2 is complete.

Acknowledgements

We thank Amal Attouchi for valuable help with a proof.

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Footnotes

  • 1

    Sum over repeated indices. 

  • 2

    The parameter δ is to be made so small that terms like δ0TΩξ2|D2uϵ|2𝑑x𝑑t can be moved over to the left-hand side. 

About the article

Received: 2018-04-17

Accepted: 2018-05-31

Published Online: 2018-07-12

Published in Print: 2019-03-01


Funding Source: Norges Forskningsråd

Award identifier / Grant number: 250070

Supported by the Norwegian Research Council (grant 250070).


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 7–15, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0091.

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