Regularity of solutions of the parabolic normalized p-Laplace equation

Research Article

Fredrik Arbo Høeg* and Peter Lindqvist

Regularity of solutions of the parabolic normalized p-Laplace equation

https://doi.org/10.1515/anona-2018-0091 Received April 17, 2018; accepted May 31, 2018

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 toL².

Keywords:Non-linear equation, regularity theory, time derivative MSC 2010:35K92, 35K10

1 Introduction

We consider viscosity solutions of thenormalized p-Laplaceequation

∂u

∂t = |∇u|^2−pdiv(|∇u|^p−2∇u), 1<p< ∞, (1.1) in Ω_T=Ω× (0,T), Ω being a domain inℝⁿ. Formally, the equation reads

∂u

∂t =∆u+ (p−2)|∇u|⁻²

n

∑

i,j=1

∂u

∂x_i

∂u

∂x_j

∂²u

∂x_i∂x_j.

In the linear casep=2, we have the heat equationut=∆u, and also forn=1, the equation reduces to the heat equationu_t = (p−1)u_xx. At the limitp=1, we obtain the equation for motion by mean curvature. We aim at showing that the time derivative^∂u_∂t exists in the Sobolev sense and belongs toL²

loc(Ω_T). We also study the second derivatives ^∂

2u

∂xi∂xj.

There has been some recent interest in connection with stochastic game theory, where the equation ap- pears, cf. [7]. From our point of view, the work [3] is of actual interest, because there it is shown that the time derivativeutof 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 likeut∈L^∞

loc(Ω_T)is in doubt. Our main result is the following, where we unfortunately have to restrictp.

Theorem 1.1. Suppose that u=u(x,t)is a viscosity solution of the normalized p-Laplace equation inΩ_T. If

6

5 <p< ¹⁴5, then the Sobolev derivatives^∂u_∂t and _∂x^∂2u

i∂xj exist and belong to L²

loc(Ω_T).

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

T

∫

0

∫

Ω

uϕ_tdx dt= −

T

∫

0

∫

Ω

Uϕ dx dt, ϕ∈C^∞

0 (Ω_T),

*Corresponding author: Fredrik Arbo Høeg,Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway, e-mail: [email protected]

Peter Lindqvist,Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway, e-mail: [email protected]

where we have to prove that the functionU, which is the right-hand side of equation (1.1), belongs toL²

loc(Ω_T). Thus, the second spatial derivativesD²uare crucial (local boundedness of∇uwas proven in [2, 3] 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 Ω_T. If 1<p<2, then the Sobolev derivative^∂u_∂t exists and belongs to L²

loc(Ω_T).

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

∂u^ϵ

∂t = (|∇u^ϵ|²+ϵ²)^2−p2 div((|∇u^ϵ|²+ϵ²)^p−22 ∇u^ϵ). (1.2) Here the classical parabolic regularity theory is applicable. The equation was studied by Does in [3], where an estimate of the gradient∇u^ϵwas 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 foru^ϵ, which are passed over to the solutionuof (1.1).

Analogous results seem to be possible to reach through theCordes condition. This also restricts the range of valid exponentsp. We have refrained from this approach, mainly since the absence of zero (lateral) bound- ary values produces many undesired terms to estimate. Finally, we mention that the limits ⁶

5and¹⁴

5 in Theo- rem 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 functionf: Ω_T → ℝis

∇f = ( ∂f

∂x1

, . . . ,

∂f

∂xn) and its Hessian matrix is

(D²f)^ij= ∂²f

∂x_i∂x_j, |D²f|²=

n

∑

i,j=1

( ∂²f

∂x_i∂x_j)

2

. We shall, occasionally, use the abbreviation

u_j= ∂u

∂x_j, u_jk = ∂²u

∂x_j∂x_k

for partial derivatives. Young’s inequality

|ab| ≤δ|a|^p p + (1

δ)^q−

1|b|^q q ,

1 p+1

q =1 is often referred to. Finally, the summation convention is used when convenient.

Viscosity solutions. The normalizedp-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 vis- cosity solutions works well. The existence and uniqueness of viscosity solutions of the normalizedp-Laplace equation was established in [2]. We recall the definition.

Definition 2.1. We say that an upper semi-continuous functionuis aviscosity subsolutionof equation (1.1) if for allϕ∈C²(Ω_T), we have

ϕ_t≤ (δ_ij+ (p−2)ϕ_xiϕ_xj

|∇ϕ|² )ϕ_xixj

at any interior point(x,t)whereu−ϕattains a local maximum, provided∇ϕ(x,t) ̸=0. Further, at any interior

point(x,t)whereu−ϕattains a local maximum and∇ϕ(x,t) =0, we require ϕ_t≤ (δ_ij+ (p−2)η_iη_j)ϕ_xixj

for someη∈ ℝⁿ, with|η| ≤1.

Definition 2.2. We say that a lower semi-continuous functionuis aviscosity supersolutionof equation (1.1) if for allϕ∈C²(Ω_T), we have

ϕ_t≥ (δ_ij+ (p−2)ϕ_xiϕ_xj

|∇ϕ|² )ϕ_xixj

at any interior point(x,t)whereu−ϕattains a local minimum, provided∇ϕ(x,t) ̸=0. Further, at any interior point(x,t)whereu−ϕattains a local minimum and∇ϕ(x,t) =0, we require

ϕ_t≥ (δ_ij+ (p−2)η_iη_j)ϕ_xixj for someη∈ ℝⁿ, with|η| ≤1.

Definition 2.3. A continuous functionuis aviscosity solutionif it is both a viscosity subsolution and a vis- cosity supersolution.

For a detailed discussion on the definition at critical points, we refer to [5]. The reason behind the choice of η∈ ℝⁿis 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 ofu^ϵ(and also on its gradient). If we first assume thatu^ϵisC¹on the parabolic boundary∂parΩ_T, we get bounds on the gradient in all of Ω_T. This follows from the following maximum principle.

Proposition 2.4(Maximum principle). Let u^ϵbe a solution of equation(1.2). If∇u^ϵ∈C¹(Ω_T), then max

Ω_T

{|∇u^ϵ|} = max

∂parΩT{|∇u^ϵ|}.

Proof. With some modifications, a proof can be extracted from [3]. We give a direct proof. To this end, consider V^ϵ(x,t) = |∇u^ϵ|²+ϵ².

To find the partial differential equation satisfied byV^ϵ, we calculate¹ V^ϵ_i =2u^ϵ_νu^ϵ_iν, V_ij^ϵ=2u^ϵ_νju^ϵ_iν+2u^ϵ_νu^ϵ_ijν, u^ϵ_iu^ϵ_jV_ij^ϵ = 1

2|∇V^ϵ|²+2u^ϵ_iu^ϵ_ju^ϵ_νu^ϵ_ijν. Writing equation (1.1) in the form

u^ϵ_t = (δ_ij+ (p−2) u^ϵ_iu^ϵ_j

|∇u^ϵ|²+ϵ²)u^ϵ_ij, we find

1 2

V_t^ϵ=u^ϵ_ν ∂

∂x_νu^ϵ_t =u^ϵ_ν∆u^ϵ_ν− p−2

2(V^ϵ)²|⟨∇u^ϵ,∇V^ϵ⟩|²+ p−2 V^ϵ (1

4|∇V^ϵ|²+1 2

u^ϵ_νu^ϵ_μV_νμ^ϵ ). Rearranging and using

∆V^ϵ=2|D²u^ϵ|²+2⟨∇u^ϵ,∇∆u^ϵ⟩, we arrive at the following differential equation forV^ϵ:

V_t^ϵ=∆V^ϵ−2|D²u^ϵ|²− p−2

(V^ϵ)²|⟨∇u^ϵ,∇V^ϵ⟩|²+p−2 V^ϵ {1

2|∇V^ϵ|²+u^ϵ_νu^ϵ_μV_νμ^ϵ }. (2.1) Let

w(x,t) = |∇u^ϵ(x,t)|²+ϵ²−αt=V^ϵ(x,t) −αt forα>0.

1Sum over repeated indices.

Suppose thatw^ϵhas aninteriormaximum point at(x0,t0). At this point,V^ϵ(x0,t0) >0, otherwise we would haveV^ϵ(x,t) ≡0 in Ω_T, in which case there is nothing to prove. By the infinitesimal calculus,

∇w(x₀,t₀) =0 and w_t(x₀,t₀) ≥0,

where we have included the caset0=T. Further, the matrixD²w(x0,t0)is negative semidefinite. Using equa- tion (2.1) and noting that∇w= ∇V^ϵandD²w=D²V^ϵ, we get, at(x0,t0),

0≤w_t =V_t^ϵ−α

=∆V^ϵ−2|D²u^ϵ|²− p−2

(V^ϵ)²|⟨∇u^ϵ,∇V^ϵ⟩|²+p−2 V^ϵ {1

2|∇V^ϵ|²+u^ϵ_νu^ϵ_μV_νμ^ϵ } −α

= (δij+ (p−2) u^ϵ_iu^ϵ_j

V^ϵ )w^ϵ_ij−2|D²u^ϵ|²−α≤ −α, since the matrixA, with elementsA_ij=δ_ij+ (p−2)^u

ϵ iu^ϵ_j

V^ϵ , is positive semidefinite. To avoid the contradiction α≤0,wmust attain its maximum on the parabolic boundary.

Hence, for any(x,t) ∈Ω_T, we have

V^ϵ(x,t) −αt≤ max

∂parΩT{V^ϵ(x,t) −αt} ≤ max

∂parΩT

V^ϵ(x,t). We finish the proof by sendingα→0⁺.

With no assumptions foru^ϵon the parabolic boundary, we need a stronger result, taken from [3, p. 381].

Theorem 2.5. Let u^ϵbe a solution of equation(1.2), with u^ϵ(x, 0) =u₀(x). Then

|∇u^ϵ(x,t)| ≤Cn,p‖u0‖^L∞(ΩT){1+ ( 1

dist((x,t),∂_parΩ_T))

2

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

|∇u^ϵ(x,t)| ≤Cn,p‖u^ϵ( ⋅,t0)‖∞{1+ ( 1

dist((x,t),∂_parΩ_T))

2

}

forx∈D⊂⊂Ω and 0<t0≤t≤T−t0. The estimate

‖∇u^ϵ‖L^∞(D×[t0,T−t0])≤C‖u^ϵ‖L^∞(ΩT){1+ ( 1

dist(D,∂parΩ_T))

2

} (2.2)

follows. (Here one can pass to the limit asϵ→0.)

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]. Iff is not smooth, we perform astrictlyinterior approximation, so that no boundary integrals appear (which is possible sinceξ ∈C^∞

0 ).

Lemma 2.6(Miranda–Talenti). Let ξ ∈C^∞₀(Ω_T)and f ∈L²(0,T,W^2,2(Ω)). Then

T

∫

0

∫

Ω

|∆(ξf)|²dx dt=

T

∫

0

∫

Ω

|D²(ξf)|²dx dt.

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 u^ϵbe the classical solution of the regularized equation(1.2)with boundary values

u=u^ϵ on ∂_parΩ_T. Then u^ϵ→u uniformly on compact subsets ofΩ_T.

Proof. By Theorem 2.5, we can use Ascoli’s theorem to extract a convergent subsequenceu^ϵj converging locally uniformly to some continuous function, namely,u^ϵj →v. We claim thatvis a viscosity solution of equation (1.1). The lemma then follows by uniqueness.

We demonstrate thatvis a viscosity subsolution. (A symmetric proof shows thatvis a viscosity super- solution.) Assume thatv−ϕattains a strict local maximum atz0= (x0,t0). Sinceu^ϵ→vlocally uniformly, there are points

z_ϵ→z0

such thatu^ϵ−ϕattains a local maximum atz_ϵ. If∇ϕ(z0) ̸=0, then∇ϕ(z_ϵ) ̸=0 for allϵ>0 small enough, and atz_ϵ, we have

ϕ_t ≤ (δ_ij+ (p−2) ϕ_xiϕ_xj

|∇ϕ|²+ϵ²)ϕ_xixj. (3.1)

Lettingϵ→0, we see thatvsatisfies Definition 2.3 when∇ϕ(z0) ̸=0. If∇ϕ(z0) =0, let η_ϵ= ∇ϕ(z_ϵ)

√|∇ϕ(z_ϵ)|²+ϵ² .

Since|η_ϵ| ≤1, there is a subsequence such thatη_ϵk →ηwhenk→ ∞for someη∈ ℝⁿ, with|η| ≤1. Passing to the limitϵ_k→0 in equation (3.1), we see thatvis a viscosity subsolution.

Our proof of Theorem 1.1 consists in showing that the second derivativesD²u^ϵbelong locally toL²with a bound independent ofϵ. Once this is established, we see that

(|∇u^ϵ|²+ϵ²)^2−p2 div((|∇u^ϵ|²+ϵ²)^p−22 ∇u^ϵ) =∆u^ϵ+ p−2

|∇u^ϵ|²+ϵ²⟨∇u^ϵ,D²u^ϵ∇u^ϵ⟩ ≤Cp,n|D²u^ϵ|. Hence, for any bounded subdomainD⊂⊂Ω_T,

󵄩󵄩󵄩󵄩(|∇u^ϵ|²+ϵ²)^2−p2 div((|∇u^ϵ|²+ϵ²)^p−22 ∇u^ϵ)󵄩󵄩󵄩󵄩^L2(D)≤C,

withCindependent ofϵ. By this uniform bound, there exists a subsequence such that, asj→ ∞, (|∇u^ϵj|²+ϵ²_j)

2−p

2 div((|∇u^ϵj|²+ϵ²_j)

p−2

2 ∇u^ϵj) →U weakly inL²(D). In particular, this means thatU∈L²(D)and for anyϕ∈C^∞₀ (D), we have

jlim→∞

T

∫

0

∫

D

ϕ(|∇u^ϵj|²+ϵ²_j)^2−p2 div((|∇u^ϵj|²+ϵ²_j)^p−22 ∇u^ϵj)dx dt=

T

∫

0

∫

D

ϕU dx dt.

Ifuis the unique viscosity solution of (1.1), we invoke Lemma 3.1 and the calculations above to find, for any test functionϕ∈C^∞₀ (D),

T

∫

0

∫

D

u∂ϕ

∂t dx dt= lim

j→∞

T

∫

0

∫

D

u^ϵj∂ϕ

∂t dx dt

= −lim

j→∞

T

∫

0

∫

D

ϕ(|∇u^ϵj|²+ϵ²_j)

2−p

2 div((|∇u^ϵj|²+ϵ²_j)

p−2

2 ∇u^ϵj)dx dt

= −

T

∫

0

∫

D

ϕU dx dt.

This shows that the Sobolev derivativeu_texists and, since the previous equation holds for any subdomain D⊂⊂Ω_T, we conclude that ^∂u_∂t =U∈L²

loc(Ω_T). To complete the proof of Theorem 1.1, it remains to establish the missing local bound of‖D²u^ϵ‖L²uniformly inϵ.

4 The differentiated equation

We shall derive a fundamental identity. Let

v^ϵ= |∇u^ϵ|², V^ϵ= |∇u^ϵ|²+ϵ². Differentiating equation (1.2) with respect to the variablexj, we obtain

∂

∂tu^ϵ_j = 2−p

2 (V^ϵ)^−p2v^ϵ_jdiv((V^ϵ)^p−22 ∇u^ϵ) + (V^ϵ)^2−p2 div[((V^ϵ)^p−22 ∇u^ϵ)_j]. Takeξ∈C^∞

0(Ω_T), withξ ≥0. Multiply both sides of the equation byξ²V^ϵu^ϵ_j and sumjfrom 1 ton. Integrate over Ω_T, using integration by parts and keeping in mind thatξis compactly supported in Ω_T, to obtain

−1 2

T

∫

0

∫

Ω

ξξ_tV^ϵdx dt= 2−p 2

T

∫

0

∫

Ω

ξ²(V^ϵ)^−p2⟨∇u^ϵ,∇v^ϵ⟩div((V^ϵ)^p−22 ∇u^ϵ)dx dt

−

T

∫

0

∫

Ω

∂

∂x_j{(V^ϵ)

p−2 2 u^ϵ_k} ∂

∂x_k{ξ²(V^ϵ)

2−p

2 u^ϵ_j}dx dt.

Writing out the derivatives gives the fundamental formula I+II :=

T

∫

0

∫

Ω

ξ²|D²u^ϵ|²dx dt+ p−2 2

T

∫

0

∫

Ω

1

V^ϵξ²⟨∇u^ϵ,∇v^ϵ⟩∆u^ϵdx dt

= 1 2

T

∫

0

∫

Ω

ξξ_tV^ϵdx dt+ (2−p)

T

∫

0

∫

Ω

1

V^ϵξ⟨∇u^ϵ,∇v^ϵ⟩⟨∇u^ϵ,∇ξ⟩dx dt−

T

∫

0

∫

Ω

ξ⟨∇v^ϵ,∇ξ dx dt

=: III+IV−V.

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

5 Estimate of the second derivatives

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

|∇v^ϵ|²= |2D²u^ϵ∇u^ϵ|²≤4|D²u^ϵ|²v^ϵ. (5.1) One dimension. As an exercise, we show that in this case, the second derivatives are locally bounded inL² for any 1<p< ∞. In one dimension, equation (1.1) reads

ut= |ux|^2−p ∂

∂x{|ux|^p−2ux} = (p−1)uxx.

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

T

∫

0

∫

Ω

ξ²(∂²u^ϵ

∂x² )

2

(1+ (p−2) (^∂u_∂x^ϵ)²

(^∂u_∂x^ϵ)²+ϵ² −δ(|p−2| +1))dx dt

≤ 1 2

T

∫

0

∫

Ω

ξξ_tV^ϵdx dt+ |p−2| +1 δ

T

∫

0

∫

Ω

V^ϵ|∇ξ|²dx dt.

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

1+ (p−2) (^∂u_∂x^ϵ)²

(^∂u_∂x^ϵ)²+ϵ² ≥min{1,p−1} >0.

It follows that ^∂

2u^ϵ

∂x² ∈L²locally for anyp∈ (1,∞).

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

2 1

V^ϵξ²⟨∇u^ϵ,∇v^ϵ⟩∆u^ϵ= 2−p 2

1

V^ϵξ⟨∇u^ϵ,∇v^ϵ⟩{∆(ξu^ϵ) −2⟨∇u^ϵ,∇ξ⟩ −u^ϵ∆ξ}.

Upon this rewriting, the term IV disappears from the equation. We focus our attention on the term involving

∆(ξu^ϵ). By Lemma 2.6,

T

∫

0

∫

Ω

|D²(ξu^ϵ)|²dx dt=

T

∫

0

∫

Ω

|∆(ξu^ϵ)|²dx dt.

Differentiating, we see that

(ξu^ϵ)ⁱ=ξiu^ϵ+ξu^ϵ_i, (ξu^ϵ)^ij=ξiju^ϵ+u^ϵ_iξj+ξiu^ϵ_j +ξu^ϵ_ij. It follows that

|D²(ξu^ϵ)|²=ξ²|D²u^ϵ|²+f(u^ϵ,∇u^ϵ,D²u^ϵ), wheref(u^ϵ,∇u^ϵ_i,D²u^ϵ)depends only linearly on the second derivativesu^ϵ_ij:

f(u^ϵ,∇u^ϵ,D²u^ϵ) = (u^ϵ)²|D²ξ|²+4u^ϵ⟨∇ξ,D²ξ∇u^ϵ⟩ +4ξ⟨∇ξ,D²u^ϵ∇u^ϵ⟩

+2|∇ξ|²|∇u^ϵ|²+2|⟨∇u^ϵ,∇ξ⟩|²+2u^ϵξtrace{(D²ξ)(D²u^ϵ)}. By Young’s inequality, we obtain

2−p 2

T

∫

0

∫

Ω

1

V^ϵξ⟨∇u^ϵ,∇v^ϵ⟩∆(ξu^ϵ)dx dt≤ 5 4(2−p)

T

∫

0

∫

Ω

ξ²|D²u^ϵ|²dx dt+2−p 4

T

∫

0

∫

Ω

f(u^ϵ,∇u^ϵ,D²u^ϵ)dx dt.

Inserting this into the main equation gives I^∗ := (1−5

4(2−p))

T

∫

0

∫

Ω

ξ²|D²u^ϵ|²dx dt≤ 1 2

T

∫

0

∫

Ω

ξξ_tV^ϵdx dt−

T

∫

0

∫

Ω

ξ⟨∇v^ϵ,∇ξ⟩dx dt

+2−p 2

T

∫

0

∫

Ω

f(u^ϵ,u^ϵ_i,u^ϵ_ij)dx dt

+2−p 2

T

∫

0

∫

Ω

1

V^ϵξ⟨∇u^ϵ,∇v^ϵ⟩u^ϵ∆ξ dx dt.

=: III−V+VI+VII.

All terms containingD²u^ϵcan be absorbed by the new main term I^∗. To this end, we use Young’s inequal- ity with a small parameterδ>0 to balance the terms.² For term V, we have

T

∫

0

∫

Ω

ξ⟨∇v^ϵ,∇ξ⟩dx dt≤δ

T

∫

0

∫

Ω

ξ²|D²u^ϵ|²dx dt+1 δ

T

∫

0

∫

Ω

V^ϵ|∇ξ|²dx dt.

Similarly, for term VII,

T

∫

0

∫

Ω

1

V^ϵξ⟨∇u^ϵ,∇v^ϵ⟩u^ϵ∆ξ dx dt≤2δ1 T

∫

0

∫

Ω

ξ²|D²u^ϵ|²+ 1 δ₁

T

∫

0

∫

Ω

|u^ϵ|²|∆ξ|²dx dt.

2The parameterδis to be made so small that terms likeδ∫₀^T∫_Ωξ²|D²u^ϵ|²dx dtcan be moved over to the left-hand side.

Using similar inequalities for the term involvingf(u^ϵ,∇u^ϵ,D²u^ϵ)and choosing the parameters small enough in Young’s inequality, we find,

T

∫

0

∫

Ω

ξ²|D²u^ϵ|²dx dt≤C ∬

{ξ≠0}

((u^ϵ)²+ |∇u^ϵ|²)dx dt, (5.2)

whereCis independent ofϵbut depends on‖ξ‖C², 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 indepen- dent ofϵ.

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

6 The case 1 < p < 2

In this section, we give a proof of Theorem 1.2. To this end, letξ∈C^∞₀ (Ω_T), with 0≤ξ≤1. We claim that

T

∫

0

∫

Ω

ξ²(∂u^ϵ

∂t )

2

dx dt≤4‖V^ϵ‖²_∞{

T

∫

0

∫

Ω

|∇ξ|²dx dt+ 1 p

T

∫

0

∫

Ω

ξ|ξ_t|dx dt}, (6.1)

where the supremum norm ofV^ϵ= |∇u^ϵ|²+ϵ²is taken locally, over the support ofξ. Here,u^ϵis 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(|∇u^ϵ|²+ϵ²)^p−22 ξ²u^ϵ_t yields ξ²(|∇u^ϵ|²+ϵ²)^p−22 (u^ϵ_t)²=ξ²u^ϵ_tdiv((|∇u^ϵ|²+ϵ²)^p−22 ∇u^ϵ)

=div(ξ²u^ϵ_t(|∇u^ϵ|²+ϵ²)^p−22 ∇u^ϵ) − (|∇u^ϵ|²+ϵ²)^p−22 ⟨∇u^ϵ,∇(ξ²u^ϵ_t)⟩. The integral of the divergence term vanishes by Gauss’s theorem and, upon integration, we have

T

∫

0

∫

Ω

ξ²(V^ϵ)^p−22 (u^ϵ_t)²dx dt= −

T

∫

0

∫

Ω

(V^ϵ)^p−22 ⟨∇u^ϵ,∇(ξ²u^ϵ_t)⟩dx dt

= −2

T

∫

0

∫

Ω

ξ(V^ϵ)

p−2

2 ⟨∇u^ϵ,∇ξ⟩u^ϵ_t dx dt−

T

∫

0

∫

Ω

ξ²(V^ϵ)

p−2

2 ⟨∇u^ϵ,∇u^ϵ_t⟩dx dt.

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

󵄨󵄨󵄨󵄨2ξ(V^ϵ)^p−22 ⟨∇u^ϵ,∇ξ⟩u^ϵ_t󵄨󵄨󵄨󵄨 ≤σξ²(V^ϵ)^p−22 (u^ϵ_t)²+1

σ(V^ϵ)^p−22 |∇u^ϵ|²|∇ξ|², and integrating.

For the last term, the decisive observation is that 1

p

∂

∂t(|∇u^ϵ|²+ϵ²)^p2 = (|∇u^ϵ|²+ϵ²)^p−22 ⟨∇u^ϵ,∇u^ϵ_t⟩ = (V^ϵ)^p−22 ⟨∇u^ϵ,∇u^ϵ_t⟩. We use this in the last integral on the right-hand side to obtain

−

T

∫

0

∫

Ω

ξ²(V^ϵ)^p−22 ⟨∇u^ϵ,∇u^ϵ_t⟩dx dt= −

T

∫

0

∫

Ω

∂

∂t{ξ²

p(V^ϵ)^p2}dx dt+2 p

T

∫

0

∫

Ω

ξξ_t(V^ϵ)^p2 dx dt

= − ∫

Ω

[ξ² p(V^ϵ)

p 2]

t=T t=0

dx+2 p

T

∫

0

∫

Ω

ξξ_t(V^ϵ)

p 2 dx dt

= 2 p

T

∫

0

∫

Ω

ξξ_t(V^ϵ)^p2 dx dt.

To sum up, we have now the final estimate 1

2

T

∫

0

∫

Ω

ξ²(V^ϵ)^p−22 (u^ϵ_t)²dx dt≤2

T

∫

0

∫

Ω

(V^ϵ)^p−22 |∇u^ϵ|²|∇ξ|²dx dt+ 2 p

T

∫

0

∫

Ω

ξξ_t(V^ϵ)^p2dx dt

≤2

T

∫

0

∫

Ω

(V^ϵ)^p2|∇ξ|²dx dt+ 2 p

T

∫

0

∫

Ω

ξξt(V^ϵ)^p2dx dt.

So far, our calculations are valid in the full range 1<p< ∞. For 1<p<2, we have (V^ϵ)^p−22 ≥ ‖V^ϵ‖

p−2 2

∞ ,

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

Acknowledgment: We thank Amal Attouchi for valuable help with a proof.

Funding: Supported by the Norwegian Research Council (grant 250070).

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