Parabolic Biased Infinity Laplacian Equation Related to the Biased Tug-of-War

Fang Liu; Feida Jiang

doi:10.1515/ans-2018-2019

Article Open Access

Parabolic Biased Infinity Laplacian Equation Related to the Biased Tug-of-War

Fang Liu and Feida Jiang

Published/Copyright: May 29, 2018

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From the journal Advanced Nonlinear Studies Volume 19 Issue 1

Abstract

In this paper, we study the parabolic inhomogeneous β-biased infinity Laplacian equation arising from the β-biased tug-of-war

u t - Δ ∞ β ⁢ u = f ⁢ ( x , t ) ,

where β is a fixed constant and Δ∞β is the β-biased infinity Laplacian operator

Δ ∞ β ⁢ u = Δ ∞ N ⁢ u + β ⁢ | D ⁢ u |

related to the game theory named β-biased tug-of-war. We first establish a comparison principle of viscosity solutions when the inhomogeneous term f does not change its sign. Based on the comparison principle, the uniqueness of viscosity solutions of the Cauchy–Dirichlet boundary problem and some stability results are obtained. Then the existence of viscosity solutions of the corresponding Cauchy–Dirichlet problem is established by a regularized approximation method when the inhomogeneous term is constant. We also obtain an interior gradient estimate of the viscosity solutions by Bernstein’s method. This means that when f is Lipschitz continuous, a viscosity solution u is also Lipschitz in both the time variable t and the space variable x. Finally, when f=0, we show some explicit solutions.

Keywords: Biased Infinity Laplacian; Existence; Comparison Principle; Lipschitz Estimate; Viscosity Solutions

MSC 2010: 35D40; 35K20; 35K67; 35Q91; 49K20; 35J57; 35J70; 49N60

1 Introduction

In this paper, we study the nonlinear degenerate and singular parabolic β-biased infinity Laplacian equation with an inhomogeneous term arising from the biased tug-of-war,

(1.1) u t - Δ ∞ β ⁢ u = f ⁢ ( x , t ) ,

where β is a fixed constant, the β-biased infinity Laplacian is denoted by

Δ ∞ β u = : Δ ∞ N u + β | D u |

and Δ∞N is the normalized infinity Laplacian operator

Δ ∞ N ⁢ u = | D ⁢ u | - 2 ⁢ Δ ∞ ⁢ u = | D ⁢ u | - 2 ⁢ 〈 D 2 ⁢ u ⁢ D ⁢ u , D ⁢ u 〉 .

When β=0, it is the well-known infinity Laplacian Δ∞N⁢u related to the unbiased tug-of-war. Here we are interested in the case β≠0 and we mainly establish uniqueness, existence and some regularity properties of viscosity solutions. Among them, we find that the biased operator shows some interesting facts different from the unbiased operator.

The β-biased infinity Laplacian was introduced in [34] to describe the β-biased ε-tug-of-war, and the existence and uniqueness of the viscosity solutions were analyzed by game theory under the Dirichlet boundary condition for the homogeneous elliptic version

Δ ∞ β ⁢ u = 0

with β≠0. For the inhomogeneous elliptic equation

(1.2) Δ ∞ β ⁢ u = f ,

existence and uniqueness of Dirichlet or Dirichlet–Neumann mixed boundary problems were proved by a probabilistic and Perron’s argument [4]. In the special case β=0 in (1.2), the unbiased equation was well studied in [35] by game theory, and in [27] by PDE methods. Especially when β=0 and f=0, that is the famous Aronsson equation related to the L∞ norm; see [5, 8] and the references therein. For more results about more general infinity Laplacian equations, one can see, e.g., [22, 29, 28, 26]. In these papers mentioned above, the uniqueness holds when the inhomogeneous term f is positive or negative. And some counterexamples were shown when f (the running payoff function in tug-of-war [35]) changes its sign.

Here we are interested in the corresponding parabolic version associated with the β-biased infinity Laplacian (1.1), which is not only meaningful in mathematical theory, but also necessary in numerical analysis and has many applications in optimal transportation and image reconstruction. It is interesting that the comparison principle is valid when the inhomogeneous term does not change its sign. This is one point that is different from the elliptic case.

For the parabolic homogeneous unbiased infinity Laplacian, i.e. β=0 and f=0 in (1.1), the existence and comparison principles were proved in viscosity sense by PDE arguments in [18]. Does [11] generalized the results to the normalized p-Laplacian equation

u t - 1 p ⁢ | D ⁢ u | 2 - p ⁢ Δ p ⁢ u = 0

for 1≤p≤∞. The normalized p-Laplacian is related to a random-turn two-person game named tug-of-war with noise [13, 16, 20, 31, 30, 36], and it is something quite different from the divergent p-Laplacian. For more results about the generalized infinity Laplacian, we refer to [1, 2, 3, 10, 24, 38, 37]. These parabolic equations involving the infinity Laplacian operator have received a lot of attention in the last decade, notably due to its applications to image processing [7, 12, 25], optimal mass transportation [14] etc. Obviously it is interesting to investigate the evolution equation corresponding to the β-biased infinity Laplacian operator.

By noticing that (1.1) is degenerate and singular when the gradient vanishes, the reasonable solutions to it are in the viscosity sense. In fact, if we let ν=D⁢u|D⁢u| to be the unit gradient vector, then the β-biased infinity Laplacian can be written as

Δ ∞ β ⁢ u = D ν 2 ⁢ u + β ⁢ | D ⁢ u | .

In the scope of viscosity solutions, Δ∞β is 1-homogeneous and bounded when the gradient vanishes. This property of scaling invariance is very useful not only in our proofs such as in proving the comparison principle, but also in applications such as in image processing.

In this paper, we first prove the uniqueness of the viscosity solutions to the Cauchy–Dirichlet problem

(1.3) { u t - Δ ∞ β ⁢ u = f in ⁢ Q T , u = g on ⁢ ∂ p ⁡ Q T ,

where g∈C⁢(∂p⁡QT) and a continuous function f does not change its sign. Based on the comparison principle and the 1-homogeneity of the parabolic β-biased operator, we also give some stability results. For f≡0 and β=0, the comparison result was obtained by Juutinen and Kawohl [18]. The method we use here is the classical perturbation argument for viscosity solutions. But in order to deal with the difficulty caused by the biased and inhomogeneous terms we should perturb the viscosity sub-solutions or super-solutions in addition. Thanks to the parabolic term and the homogeneity of degree one of the β-biased operator, we can establish a comparison principle when f≥0 or f≤0. It is interesting that the sign of the inhomogeneous term is important to the comparison principle, and this is mainly due to the high degeneracy of the biased infinity Laplacian operator. It should be pointed out that the comparison principle is valid when f is nonnegative or nonpositive for the parabolic equation. But for the elliptic case, the comparison principle is valid when f is strictly positive or negative [4]; for the unbiased case, see, e.g., [6, 22, 29, 28, 26, 32]. And it is an open problem whether the comparison principle is valid when f≥0 (f≤0) for the elliptic case.

Secondly, we obtain existence of the viscosity solutions to (1.3) when f is constant. Due to the high degeneracy and singularity of (1.1), we adopt the regularized equation

u t - ε ⁢ △ u - 1 | D ⁢ u | 2 + δ 2 ⁢ 〈 D 2 ⁢ u ⁢ D ⁢ u , D ⁢ u 〉 - β ⁢ | D ⁢ u | = f ,

and then establish the uniform estimates by the standard barrier method to obtain existence of the initial-boundary problem. But due to the biased term and the inhomogeneous term, we must compute it carefully to use suitable barrier function arguments. During this process, we also establish the Lipschitz continuity in t and Hölder continuity in x of the viscosity solutions. For existence results about the parabolic infinity Laplacian, one can see [3, 18, 38, 37] and the references therein.

Furthermore, we obtain the regularity estimates of the viscosity solutions to (1.1) when f is Lipschitz continuous. In order to obtain the local Lipschitz estimate in x, we utilize Bernstein’s method. And in this case we must overcome the difficulty caused by the singularity of the β-biased term. Hence we use another approximation to (1.1):

u t - ε ⁢ △ u - 1 | D ⁢ u | 2 + δ 2 ⁢ 〈 D 2 ⁢ u ⁢ D ⁢ u , D ⁢ u 〉 - β ⁢ | D ⁢ u | 2 + δ 2 = f ,

which is quite different from the unbiased case [18].

Finally, when f=0 we also give some explicit solutions such as solutions of separation of variables to (1.1) to give some insight to the features in the biased equation. We find that the solutions look like parabolic exponential cones, an analogue of comparisons with cones of infinity harmonic functions [8]. And it is interesting that the self-similarity solutions are not so easy to be shown due to the additional biased term. All of these show that it is meaningful to study the biased and unbiased cases from PDE perspective. And the results in this aspect may be helpful to connect the tug-of-war in differential game theory and the nonlinear PDEs.

The organization of this paper is in the following order: In Section 2, we give the definition of viscosity solutions to (1.1) by semi-continuous extension. In Section 3, we give a comparison principle of the viscosity solutions by the perturbation method when the inhomogeneous term does not change its sign. Some stability results are also established. In Section 4, we establish the existence of viscosity solutions of the initial-boundary problem with a constant source term by the compactness argument and the stability theory of viscosity solutions. In Section 5, we prove the interior Lipschitz estimate by Bernstein’s method and a compactness argument. In Section 6, we show some explicit solutions to (1.1) with the homogeneous term.

2 Viscosity Solutions

We always assume that T>0 and Ω⊂Rn is a bounded domain. Let QT=Ω×(0,T) be a space-time cylinder with the parabolic boundary

∂ p ⁡ Q T = { ∂ ⁡ Ω × [ 0 , T ] } ∪ { Ω × { 0 } } .

For a smooth function φ, the terms λmax⁢(D2⁢φ⁢(x,t)) and λmin⁢(D2⁢φ⁢(x,t)) denote the largest and smallest, respectively, eigenvalues to the Hessian matrix (D2⁢φ⁢(x,t)).

Notice that (1.1) is singular when the gradient vanishes. We can not use the distributional weak solutions. The reasonable solutions are in the viscosity sense based on the upper (lower) semi-continuous envelope discussed in [17]. We refer to [27] for details for the normalized infinity Laplacian equation; see also [15, 18, 38, 37]. Now we first give the definition of viscosity solutions based on semi-continuous extension of the operator.

Definition 2.1.

Suppose that f∈C⁢(QT) and u:QT→R is upper semi-continuous. If for every (x0,t0)∈QT and any C2⁢(QT) test function φ such that u-φ has a strict local maximum at the point (x0,t0), that is, u⁢(x0,t0)=φ⁢(x0,t0) and u⁢(x,t)<φ⁢(x,t) in a neighborhood of (x0,t0), there holds

φ t ⁢ ( x 0 , t 0 ) - Δ ∞ N ⁢ φ ⁢ ( x 0 , t 0 ) - β ⁢ | D ⁢ φ | ⁢ ( x 0 , t 0 ) ≤ f ⁢ ( x 0 , t 0 ) ⁢ if ⁢ D ⁢ φ ⁢ ( x 0 , t 0 ) ≠ 0 ,

φ t ⁢ ( x 0 , t 0 ) - λ max ⁢ ( D 2 ⁢ φ ⁢ ( x 0 , t 0 ) ) ≤ f ⁢ ( x 0 , t 0 ) ⁢ if ⁢ D ⁢ φ ⁢ ( x 0 , t 0 ) = 0 ,

then we say that u is a viscosity sub-solution of (1.1).

Similarly, suppose u:QT→R is lower semi-continuous. If for every (x0,t0)∈QT and any C2⁢(QT) test function φ such that u-φ has a strict local minimum at the point (x0,t0), that is, u⁢(x0,t0)=φ⁢(x0,t0) and u⁢(x,t)>φ⁢(x,t) in a neighborhood of (x0,t0), there holds

φ t ⁢ ( x 0 , t 0 ) - Δ ∞ N ⁢ φ ⁢ ( x 0 , t 0 ) - β ⁢ | D ⁢ φ | ⁢ ( x 0 , t 0 ) ≥ f ⁢ ( x 0 , t 0 ) ⁢ if ⁢ D ⁢ φ ⁢ ( x 0 , t 0 ) ≠ 0 ,

φ t ⁢ ( x 0 , t 0 ) - λ min ⁢ ( D 2 ⁢ φ ⁢ ( x 0 , t 0 ) ) ≥ f ⁢ ( x 0 , t 0 ) ⁢ if ⁢ D ⁢ φ ⁢ ( x 0 , t 0 ) = 0 ,

then we say that u is a viscosity super-solution of (1.1).

A continuous function u∈C⁢(QT) is called a viscosity solution of (1.1) if it is both a viscosity sub-solution and a viscosity super-solution.

Remark 2.2.

If u is twice differentiable with respect to x at the point (x,t), then the β-biased infinity Laplacian of u at (x,t) is defined to be the closed interval

Δ ∞ β ⁢ u ⁢ ( x , t ) = [ λ min ⁢ ( D 2 ⁢ u ⁢ ( x , t ) ) + β ⁢ | D ⁢ u | ⁢ ( x , t ) , λ max ⁢ ( D 2 ⁢ u ⁢ ( x , t ) ) + β ⁢ | D ⁢ u | ⁢ ( x , t ) ] .

When D⁢u⁢(x,t)≠0, then Δ∞β⁢u⁢(x,t) is the usual β-biased infinity Laplacian and

Δ ∞ β ⁢ u ⁢ ( x , t ) = 1 | D ⁢ u | 2 ⁢ ( x , t ) ⁢ 〈 D 2 ⁢ u ⁢ ( x , t ) ⁢ D ⁢ u ⁢ ( x , t ) , D ⁢ u ⁢ ( x , t ) 〉 + β ⁢ | D ⁢ u | ⁢ ( x , t ) .

Remark 2.2 means that at the points that the gradient D⁢φ of a test function vanishes, the viscosity solution of (1.1) should be viewed as φt, which belongs to the closed interval [λmin⁢(D2⁢φ)+f,λmax⁢(D2⁢φ)+f].

In the following, we will prove that if the gradient of a test function vanishes, one may assume that D2⁢φ=0, and thus Δ∞β⁢φ=0. This fact means that nothing needs to be tested if D⁢φ=0 and D2⁢φ≠0. The proof of this fact is based on the well-known perturbation argument in, e.g., [9, 18, 22, 27, 37]. The fact that we can reduce the number of test functions in the definition of viscosity solutions will be very useful. In fact, one can view the following lemma as a definition equivalent to Definition 2.1. We will prove the following Lemma 2.3. The idea comes from [18]; however, some of the details are quite different.

Lemma 2.3.

Suppose that f∈C⁢(QT), f≥0, and u:QT→R is an upper semi-continuous function with the property that for every (x0,t0)∈QT and any C2⁢(QT) test function φ such that u-φ has a strict local maximum at the point (x0,t0), the following holds:

φ t ⁢ ( x 0 , t 0 ) - Δ ∞ N ⁢ φ ⁢ ( x 0 , t 0 ) - β ⁢ | D ⁢ φ | ⁢ ( x 0 , t 0 ) ≤ f ⁢ ( x 0 , t 0 ) ⁢ ⁢ if ⁢ D ⁢ φ ⁢ ( x 0 , t 0 ) ≠ 0 ,

φ t ⁢ ( x 0 , t 0 ) ≤ f ⁢ ( x 0 , t 0 ) ⁢ ⁢ if ⁢ D ⁢ φ ⁢ ( x 0 , t 0 ) = 0 ⁢ and ⁢ D 2 ⁢ φ ⁢ ( x 0 , t 0 ) = 0 .

Then u is a viscosity sub-solution of (1.1).

Proof.

We argue by contradiction. Suppose that u is not a viscosity sub-solution, but satisfies the assumption of the lemma. Then there exist (x0,t0)∈QT and φ∈C2⁢(QT) such that u-φ has a strict local maximum at the point (x0,t0), D⁢φ⁢(x0,t0)=0, D2⁢φ⁢(x0,t0)≠0 and

(2.1) φ t ⁢ ( x 0 , t 0 ) - λ max ⁢ ( D 2 ⁢ φ ⁢ ( x 0 , t 0 ) ) - β ⁢ | D ⁢ φ | ⁢ ( x 0 , t 0 ) > f ⁢ ( x 0 , t 0 ) .

Let uδ⁢(x,t)=(1-δ)⁢u⁢(x,t) with δ>0 small enough. Set

(2.2) w j ⁢ ( x , t , y , s ) = u δ ⁢ ( x , t ) - φ ⁢ ( y , s ) - j 4 ⁢ | x - y | 4 - j 2 ⁢ ( t - s ) 2

and let (xj,tj,yj,sj) be a maximum point of wj in Q¯T×Q¯T. Since (x0,t0) is a strict local maximum for u-φ, there exists a strict local maximum (x0δ,t0δ) for uδ-φ and δ>0 small enough such that (x0δ,t0δ)→(x0,t0) as δ→0. By first choosing a δ>0 small enough and then j large enough, we have (xj,tj),(yj,sj)∈QT and

(2.3) ( x j , t j ) , ( y j , s j ) → ( x 0 δ , t 0 δ ) as ⁢ j → ∞ .

Set

ϕ ⁢ ( y , s ) = - j 4 ⁢ | x j - y | 4 - j 2 ⁢ ( t j - s ) 2 .

Then φ-ϕ has a local minimum at (yj,sj). By (2.1) and the continuity of

( x , t ) ↦ φ t ⁢ ( x , t ) - λ max ⁢ ( D 2 ⁢ φ ⁢ ( x , t ) ) - β ⁢ | D ⁢ φ | ⁢ ( x , t )

and f, we obtain

(2.4) φ t ⁢ ( y j , s j ) - λ max ⁢ ( D 2 ⁢ φ ⁢ ( y j , s j ) ) - β ⁢ | D ⁢ φ | ⁢ ( y j , s j ) > f ⁢ ( y j , s j ) + δ .

Noting

D 2 ⁢ φ ⁢ ( y j , s j ) ≥ D 2 ⁢ ϕ ⁢ ( y j , s j ) , φ t ⁢ ( y j , s j ) = ϕ t ⁢ ( y j , s j ) , D ⁢ φ ⁢ ( y j , s j ) = D ⁢ ϕ ⁢ ( y j , s j ) ,

we have

(2.5) ϕ t ⁢ ( y j , s j ) - λ max ⁢ ( D 2 ⁢ ϕ ⁢ ( y j , s j ) ) - β ⁢ | D ⁢ ϕ | ⁢ ( y j , s j ) > f ⁢ ( y j , s j ) + δ .

Similarly, uδ-ψ has a local maximum at (xj,tj), where

ψ ⁢ ( x , t ) = j 4 ⁢ | x - y j | 4 + j 2 ⁢ | t - s j | 2 .

We consider two cases: either xj≠yj or xj=yj for all j large enough.

Case 1: If xj=yj, then D⁢ϕ⁢(yj,sj)=0 and D2⁢ϕ⁢(yj,sj)=0. By (2.5), we get

(2.6) j ⁢ ( t j - s j ) > f ⁢ ( y j , s j ) + δ .

Since uδ-ψ=(1-δ)⁢(u-11-δ⁢ψ) has a local maximum at (xj,tj) and D⁢ψ⁢(xj,tj)=0 and D2⁢ψ⁢(xj,tj)=0, we have

(2.7) j ⁢ ( t j - s j ) ≤ ( 1 - δ ) ⁢ f ⁢ ( x j , t j ) ,

where we have used the assumption of the lemma and the fact that ψ is automatically a test function of uδ if 11-δ⁢ψ is a test function of u. Due to (2.3), the continuity of f and due to f≥0 in QT, we obtain that (2.7) contradicts (2.6).

Case 2: If xj≠yj, then D⁢ϕ⁢(yj,sj)≠0. we use parabolic jets and the parabolic maximum principle for semi-continuous functions; we refer the reader to [9, 33] for more details. The parabolic theorem of sums for wj implies that there exist n×n symmetric matrices Xj,Yj such that Yj-Xj is positive semi-definite and

( j ⁢ ( t j - s j ) , j ⁢ | x j - y j | 2 ⁢ ( x j - y j ) , X j ) ∈ P ¯ 2 , + ⁢ u δ ⁢ ( x j , t j ) ,

( j ⁢ ( t j - s j ) , j ⁢ | x j - y j | 2 ⁢ ( x j - y j ) , Y j ) ∈ P ¯ 2 , - ⁢ φ ⁢ ( y j , s j ) .

Using (2.4) and the fact that φ-ϕ has a local minimum at (yj,sj), we have

Since uδ-ψ has a local maximum at (xj,tj), xj≠yj and the assumption of the lemma, we have

Combining (2.8) and (2.9), for j large enough and δ small enough we have

0 ≤ f ⁢ ( y j , s j ) + δ - ( 1 - δ ) ⁢ f ⁢ ( x j , t j ) < 〈 ( X j - Y j ) ⁢ x j - y j | x j - y j | , x j - y j | x j - y j | 〉 ≤ 0 ,

where we have used the continuity of the nonnegative function f and the fact that Yj-Xj is positive semi-definite. This provides the desired contradiction. ∎

Remark 2.4.

For the homogeneous case f≡0, we can take δ=0 in the function of double variables (2.2), and Lemma 2.3 still holds.

Remark 2.5.

Lemma 2.3 is also valid for f≤0 if we replace 1-δ by 1+δ throughout the argument.

Remark 2.6.

It is easy to verify that a similar result still holds for viscosity super-solutions.

3 Comparison Principles and Stability Results

The main purpose of this section is to prove the uniqueness of the viscosity solution of (1.3). We also use the perturbation argument as above. It should be pointed out that we should consider the double perturbation so as to deal with the inhomogeneous term. And the 1-homogeneity of the parabolic β-biased infinity Laplacian operator makes this possible. For more comparison results about degenerate partial differential equations, one can see [3, 11, 17, 18]. Based on the comparison result and the homogeneity of the parabolic operator, we also establish some stability properties of the viscosity solutions.

Theorem 3.1.

Let f∈C⁢(QT) and f≥0. Assume that u and v are a viscosity sub-solution and a viscosity super-solution to (1.1), respectively. Moreover, u≤v on ∂p⁡QT in the sense that

(3.1) lim sup ( x , t ) → ( x 0 , t 0 ) ⁡ u ⁢ ( x , t ) ≤ lim inf ( x , t ) → ( x 0 , t 0 ) ⁡ v ⁢ ( x , t )

for all (x0,t0)∈∂p⁡QT, where both sides are not simultaneously -∞ or ∞. Then

u ≤ v in ⁢ Q T .

Proof.

By the compactness of the parabolic boundary ∂p⁡QT and by (3.1), we can assume that u is bounded from above and v from below [19]. In fact, we can suppose that v is a strict viscosity super-solution. Otherwise, we consider v¯=v+λT-t with λ>0 sufficiently small. Then by calculation it is easy to show that v¯>u on ∂p⁡QT-ϵ⁢(λ) and v¯ is a strict viscosity super-solution.

We argue by contradiction. Suppose that u-v has a strict positive interior maximum, that is,

u ⁢ ( x 0 , t 0 ) - v ⁢ ( x 0 , t 0 ) = sup Q T ⁡ ( u - v ) > 0 .

Denote uδ⁢(x,t)=(1-δ)⁢u⁢(x,t) with δ>0 small enough. As in the proof of Lemma 2.3, we take

w j ⁢ ( x , t , y , s ) = u δ ⁢ ( x , t ) - v ⁢ ( y , s ) - j 4 ⁢ | x - y | 4 - j 2 ⁢ ( t - s ) 2

and denote by (xj,tj,yj,sj) the maximum point of wj in Q¯T×Q¯T. Since (x0,t0) is a strict positive maximum for u-v, there exists a strict local maximum (x0δ,t0δ) for uδ-v and δ>0 small enough such that (x0δ,t0δ)→(x0,t0) as δ→0. By first choosing a δ>0 small enough and then j large enough, we have (xj,tj),(yj,sj)∈QT and

(3.2) ( x j , t j , y j , s j ) → ( x 0 δ , t 0 δ , x 0 δ , t 0 δ ) as ⁢ j → ∞ .

Let

φ ⁢ ( y , s ) = - j 4 ⁢ | x j - y | 4 - j 2 ⁢ ( t j - s ) 2 .

Then v-φ has a local minimum at (yj,sj).

Similarly, uδ-ϕ has a local maximum at (xj,tj), where

ϕ ⁢ ( x , t ) = j 4 ⁢ | x - y j | 4 + j 2 ⁢ | t - s j | 2 .

Once again, we consider two cases: either xj≠yj or xj=yj for j large enough.

Case 1: If xj=yj, by the definition of strict viscosity super-solutions we have

(3.3) j ⁢ ( t j - s j ) > f ⁢ ( y j , s j ) + δ .

On the other hand, since u is a viscosity sub-solution, we obtain

(3.4) j ⁢ ( t j - s j ) ≤ ( 1 - δ ) ⁢ f ⁢ ( x j , t j ) .

Due to (3.2), the continuity of f and due to f≥0 in QT, inequality (3.4) gives a contradiction to (3.3).

Case 2: If xj≠yj, we also use the parabolic theorem of sums for wj, which implies that there exist n×n symmetric matrices Xj,Yj such that Yj-Xj is positive semi-definite and

( j ⁢ ( t j - s j ) , j ⁢ | x j - y j | 2 ⁢ ( x j - y j ) , X j ) ∈ P ¯ 2 , + ⁢ u δ ⁢ ( x j , t j ) ,

( j ⁢ ( t j - s j ) , j ⁢ | x j - y j | 2 ⁢ ( x j - y j ) , Y j ) ∈ P ¯ 2 , - ⁢ v ⁢ ( y j , s j ) .

By the definition of strict viscosity super-solutions, we get

Since uδ-ϕ has a local maximum at (xj,tj), xj≠yj, by the definition of viscosity sub-solutions we have

Combining (3.5) and (3.6), we have

≤ ( 1 - δ ) ⁢ f ⁢ ( x j , t j ) .

Letting j→∞, we have

f ⁢ ( x 0 δ , t 0 δ ) + δ ≤ ( 1 - δ ) ⁢ f ⁢ ( x 0 δ , t 0 δ ) .

This is impossible because f≥0 in QT. ∎

Remark 3.2.

Theorem 3.1 is also valid for f≤0 in QT.

In the following theorem, we show that for a viscosity sub-solution inequality (3.7) holds for a set of test functions that is strictly larger than the one in Definition 2.1; see also Juutinen and Kawohl [18] for the homogeneous parabolic infinity Laplacian case.

Theorem 3.3.

Assume u is a viscosity sub-solution of (1.1) in QT. Then if (x0,t0)∈QT and φ∈C2⁢(QT) such that u⁢(x0,t0)=φ⁢(x0,t0) and u⁢(x,t)<φ⁢(x,t) for all (x,t)∈QT⋂{t≤t0} with (x,t)≠(x0,t0), there holds

(3.7) { φ t ⁢ ( x 0 , t 0 ) - Δ ∞ N ⁢ φ ⁢ ( x 0 , t 0 ) - β ⁢ | D ⁢ φ | ⁢ ( x 0 , t 0 ) ≤ f ⁢ ( x 0 , t 0 ) if ⁢ D ⁢ φ ⁢ ( x 0 , t 0 ) ≠ 0 , φ t ⁢ ( x 0 , t 0 ) - λ max ⁢ ( D 2 ⁢ φ ⁢ ( x 0 , t 0 ) ) ≤ f ⁢ ( x 0 , t 0 ) if ⁢ D ⁢ φ ⁢ ( x 0 , t 0 ) = 0 .

Proof.

We argue by contradiction once again. Suppose that there exist (x0,t0)∈QT and φ∈C2⁢(QT) satisfying the assumption of the theorem, but either

φ t ⁢ ( x 0 , t 0 ) - Δ ∞ N ⁢ φ ⁢ ( x 0 , t 0 ) - β ⁢ | D ⁢ φ | ⁢ ( x 0 , t 0 ) > f ⁢ ( x 0 , t 0 ) and D ⁢ φ ⁢ ( x 0 , t 0 ) ≠ 0 ,

φ t ⁢ ( x 0 , t 0 ) - λ max ⁢ ( D 2 ⁢ φ ⁢ ( x 0 , t 0 ) ) > f ⁢ ( x 0 , t 0 ) and D ⁢ φ ⁢ ( x 0 , t 0 ) = 0 .

By the continuity of f, these two inequalities mean that φ is a strict viscosity super-solution of (1.1) in Qε=Bε⁢(x0)×(t0-ε,t0) for some ε>0 small enough. Since sup∂p⁡Qε⁡(u-φ)<0, we have supQε⁡(u-φ)<0 by the comparison principle. This contradicts the fact that u⁢(x0,t0)=φ⁢(x0,t0). ∎

The comparison principle, Theorem 3.1, immediately implies the following uniqueness result.

Theorem 3.4.

Let f∈C⁢(QT) and either f≥0 or f≤0 in QT. Suppose u,v∈C⁢(Q¯T) are both viscosity solutions of (1.1) in QT and u=v on ∂p⁡QT. Then u=v in QT.

With the help of the comparison principle and the 1-homogeneity of the parabolic operator, we can easily get some stability results.

Corollary 3.5.

Suppose f∈C⁢(QT) and either f≥0 or f≤0 in QT. For j=1,2, suppose cj≠0, gj∈C⁢(∂p⁡QT) and uj∈C⁢(Q¯T) satisfy the problem

{ ( u j ) t - Δ ∞ N ⁢ u j - β ⁢ | D ⁢ u j | = c j ⁢ f ⁢ ( x , t ) in ⁢ Q T , u j = g j on ⁢ ∂ p ⁡ Q T

in the viscosity sense. Then there holds

∥ u 1 c 1 - u 2 c 2 ∥ L ∞ ⁢ ( Q T ) ≤ ∥ g 1 c 1 - g 2 c 2 ∥ L ∞ ⁢ ( ∂ p ⁡ Q T ) .

If, in particular, g1=g2=g∈C⁢(∂p⁡QT), we have

∥ u 1 c 1 - u 2 c 2 ∥ L ∞ ⁢ ( Q T ) ≤ | 1 c 1 - 1 c 2 | ⁢ ∥ g ∥ L ∞ ⁢ ( ∂ p ⁡ Q T ) .

Proof.

Let

v j = 1 c j ⁢ u j .

Then vj satisfies the problem

{ ( v j ) t - Δ ∞ N ⁢ v j - β ⁢ | D ⁢ v j | = f ⁢ ( x , t ) in ⁢ Q T , v j = 1 c j ⁢ g j on ⁢ ∂ p ⁡ Q T

in the viscosity sense for j=1,2. Therefore, by Theorem 3.1 and Remark 3.2 we have

∥ v 1 - v 2 ∥ L ∞ ⁢ ( Q T ) ≤ ∥ g 1 c 1 - g 2 c 2 ∥ L ∞ ⁢ ( ∂ p ⁡ Q T ) . ∎

Using the above corollary and the 1-homogeneity of the parabolic operator, we can immediately get the following stability result. And the following perturbation result is important not only in theory but also in numerical approximation.

Corollary 3.6.

Suppose f∈C⁢(QT) and either f≥0 or f≤0 in QT. Suppose cj→0, gj,g∈C⁢(∂p⁡QT) such that ∥gj-g∥L∞⁢(∂p⁡QT)→0, and uj,u∈C⁢(Q¯T) satisfy

{ ( u j ) t - Δ ∞ N ⁢ u j - β ⁢ | D ⁢ u j | = ( 1 + c j ) ⁢ f ⁢ ( x , t ) in ⁢ Q T , u j = g j on ⁢ ∂ p ⁡ Q T

and

{ u t - Δ ∞ N ⁢ u - β ⁢ | D ⁢ u | = f ⁢ ( x , t ) in ⁢ Q T , u = g on ⁢ ∂ p ⁡ Q T ,

respectively, in the viscosity sense. Then

u j → u uniformly in ⁢ Q T ⁢ as ⁢ j → ∞ .

Proof.

Corollary 3.5 implies

∥ u j 1 + c j - u ∥ L ∞ ⁢ ( Q T ) ≤ ∥ g j 1 + c j - g ∥ L ∞ ⁢ ( ∂ p ⁡ Q T ) .

It is easy to show that

∥ u j - u ∥ L ∞ ⁢ ( Q T ) ≤ | c j | ⁢ ∥ u ∥ L ∞ ⁢ ( Q T ) + ∥ g j - g ∥ L ∞ ⁢ ( ∂ p ⁡ Q T ) + | c j | ⁢ ∥ g ∥ L ∞ ⁢ ( ∂ p ⁡ Q T ) .

This means the desired result. ∎

4 Existence

In this section, we will prove the existence of viscosity solutions to (1.3) when f is constant. We adopt the compactness method introduced in, e.g., [2, 18, 38, 37]. The main existence result we will prove is the following theorem.

Theorem 4.1.

Let QT=Ω×(0,T), where Ω⊂Rn is a bounded domain. Suppose g∈C⁢(Rn+1) and f is constant. Then there exists a function u∈C⁢(Q¯T) such that u=g on ∂p⁡QT and

u t - △ ∞ β u = f

in QT in the viscosity sense.

In order to prove Theorem 4.1, we use the approximate procedure; cf. [2, 18, 37]. Because equation (1.1) is not only degenerate but also singular when the gradient vanishes, we consider the following regularized equation:

(4.1) u t - L ε , δ ⁢ u - β ⁢ | D ⁢ u | = f ,

where

(4.2) L ε , δ ⁢ u = ε ⁢ △ u + 1 | D ⁢ u | 2 + δ 2 ⁢ 〈 D 2 ⁢ u ⁢ D ⁢ u , D ⁢ u 〉 = ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ u i ⁢ j

with

(4.3) a i ⁢ j ε , δ ⁢ ( p ) = ε ⁢ δ i ⁢ j + p i ⁢ p j | p | 2 + δ 2 , 0 < ε < 1 , 0 < δ < 1 .

For the regularized equation (4.1) with smooth initial-boundary data, the existence of a smooth solution uε,δ is guaranteed by classical results in [23]. Next, we will establish the prior gradient estimates for smooth solutions uε,δ of (4.1), which are independent of 0<ε<1 and 0<δ<1. Thanks to the Arzela–Ascoli theorem, once we have uniform gradient estimates of uε,δ, we will obtain the convergence of uε,δ as ε,δ→0. Then the limit function is a solution to (1.1) by the stability of viscosity solutions [9]. In order to obtain the uniform estimates we will use the standard barrier method. Due to the biased term we must construct suitable barrier functions. During this procedure, we can establish the Lipschitz continuity in t and Hölder continuity in x. However, the Lipschitz continuity with respect to x can not be obtained by the same method. We will study it in the next section. In the following, we first establish a Lipschitz estimate for uε,δ⁢(x,⋅) at t=0.

Theorem 4.2 (Boundary Lipschitz Regularity at t=0).

Let f be continuous in QT and g∈C2⁢(Q¯T). Suppose that uε,δ is a smooth solution satisfying

{ u t - L ε , δ ⁢ u - β ⁢ | D ⁢ u | = f in ⁢ Q T , u = g on ⁢ ∂ p ⁡ Q T .

Then there exists a constant C1≥0 depending on ∥gt∥∞, ∥D⁢g∥∞, ∥D2⁢g∥∞, |β| and ∥f∥∞, but independent of 0<ε<1 and 0<δ<1, such that

| u ε , δ ⁢ ( x , t ) - g ⁢ ( x , 0 ) | ≤ C 1 ⁢ t in ⁢ Q T .

Moreover, if g is only continuous in x (possibly discontinuous in t) and bounded in t , then the modulus of continuity of uε,δ on Ω×[0,T¯] (for small T¯) can be estimated in terms of ∥g∥∞, |β|, diam⁡Ω, ∥f∥∞ and the modulus of continuity of g⁢(x,0) in x.

Proof.

We prove this theorem in two steps. Step 1. Suppose first that g∈C2⁢(Q¯T), and we consider the upper barrier function

v + ⁢ ( x , t ) = g ⁢ ( x , 0 ) + λ ⁢ t ,

where λ>0 is to be determined. By calculation, we have

v t + - L ε , δ ⁢ v + - β ⁢ | D ⁢ v + | = λ - [ ε ⁢ △ g ⁢ ( x , 0 ) + 1 | D ⁢ g ⁢ ( x , 0 ) | 2 + δ 2 ⁢ 〈 D 2 ⁢ g ⁢ ( x , 0 ) ⁢ D ⁢ g ⁢ ( x , 0 ) , D ⁢ g ⁢ ( x , 0 ) 〉 + β ⁢ | D ⁢ g | ⁢ ( x , 0 ) ]

≥ λ - ( 1 + ε ⁢ n ) ⁢ ∥ D 2 ⁢ g ∥ ∞ - | β | ⁢ ∥ D ⁢ g ∥ ∞

≥ f ⁢ ( x , t )

if λ≥∥f∥∞+(1+n)⁢∥D2⁢g∥∞+|β|⁢∥D⁢g∥∞. Therefore, v+ is a super-solution.

Clearly, v+⁢(x,0)=g⁢(x,0) for all x∈Ω. Moreover, for x∈∂⁡Ω and 0<t<T,

v + ⁢ ( x , t ) = g ⁢ ( x , 0 ) + λ ⁢ t ≥ g ⁢ ( x , t ) + ( λ - ∥ g t ∥ ∞ ) ⁢ t ≥ g ⁢ ( x , t )

if λ≥∥gt∥∞. This means v+≥u on ∂p⁡QT.

Thus, the classical comparison principle implies

u ε , δ ⁢ ( x , t ) ≤ v + ⁢ ( x , t ) = g ⁢ ( x , 0 ) + λ ⁢ t

for every (x,t)∈QT. Similarly, considering the lower barrier function

v - ⁢ ( x , t ) = g ⁢ ( x , 0 ) - λ ⁢ t ,

we obtain the symmetric inequality, and hence the Lipschitz estimate

| u ε , δ ⁢ ( x , t ) - g ⁢ ( x , 0 ) | ≤ C 1 ⁢ t

for 0<t<T and

C 1 = max ⁡ { ∥ g t ∥ ∞ , ∥ f ∥ ∞ + ( 1 + n ) ⁢ ∥ D 2 ⁢ g ∥ ∞ + | β | ⁢ ∥ D ⁢ g ∥ ∞ } .

Step 2. Now we assume that g is only continuous in x and let μ be its modulus of continuity. Let us fix a point x0∈Ω, by choosing 0<ρ<dist⁡(x0,∂⁡Ω), we have |g⁢(x,0)-g⁢(x0,0)|<μ whenever |x-x0|<ρ. Define

g ± ⁢ ( x , t ) = g ⁢ ( x 0 , 0 ) ± μ ± 2 ⁢ ∥ g ∥ ∞ ρ 2 ⁢ | x - x 0 | 2 .

Obviously, g-≤g≤g+ on ∂p⁡QT. Let u± be the unique classical solutions to (4.1) with boundary-initial data g±, respectively. Then

u - ≤ u ε , δ ≤ u +

in QT. Since g± are smooth, we have

| u ± ⁢ ( x 0 , t ) - g ± ⁢ ( x 0 , 0 ) | ≤ t ⁢ ( ∥ f ∥ ∞ + ( 1 + n ) ⁢ 4 ⁢ ∥ g ∥ ∞ ρ 2 + | β | ⁢ 4 ⁢ ∥ g ∥ ∞ ⁢ ( ρ + diam ⁡ Ω ) ρ 2 ) .

With the help of

| u ε , δ ⁢ ( x 0 , t ) - g ⁢ ( x 0 , 0 ) | ≤ u + ⁢ ( x 0 , t ) - u ε , δ ⁢ ( x 0 , t ) + t ⁢ ( ∥ f ∥ ∞ + ( 1 + n ) ⁢ 4 ⁢ ∥ g ∥ ∞ ρ 2 + | β | ⁢ 4 ⁢ ∥ g ∥ ∞ ⁢ ( ρ + diam ⁡ Ω ) ρ 2 ) + μ

and

| u ε , δ ⁢ ( x 0 , t ) - g ⁢ ( x 0 , 0 ) | ≤ u ε , δ ⁢ ( x 0 , t ) - u - ⁢ ( x 0 , t ) + t ⁢ ( ∥ f ∥ ∞ + ( 1 + n ) ⁢ 4 ⁢ ∥ g ∥ ∞ ρ 2 + | β | ⁢ 4 ⁢ ∥ g ∥ ∞ ⁢ ( ρ + diam ⁡ Ω ) ρ 2 ) + μ ,

we have

| u ε , δ ⁢ ( x 0 , t ) - g ⁢ ( x 0 , 0 ) | ≤ 1 2 ⁢ ( u + ⁢ ( x 0 , t ) - u - ⁢ ( x 0 , t ) ) + t ⁢ ( ∥ f ∥ ∞ + ( 1 + n ) ⁢ 4 ⁢ ∥ g ∥ ∞ ρ 2 + | β | ⁢ 4 ⁢ ∥ g ∥ ∞ ⁢ ( ρ + diam ⁡ Ω ) ρ 2 ) + μ

≤ 2 ⁢ t ⁢ ( ∥ f ∥ ∞ + ( 1 + n ) ⁢ 4 ⁢ ∥ g ∥ ∞ ρ 2 + | β | ⁢ 4 ⁢ ∥ g ∥ ∞ ⁢ ( ρ + diam ⁡ Ω ) ρ 2 ) + 2 ⁢ μ .

This finishes the proof. ∎

With the aid of the comparison principle and the fact that equation (4.1) is translation invariant with the constant inhomogeneous term, we can immediately obtain the full Lipschitz estimate in time as in [11, 18, 37].

Corollary 4.3 (Lipschitz Regularity in Time).

If f is constant, g∈C2⁢(Q¯T) and uε,δ is as in Theorem 4.2, then there exists a constant C1≥0 depending on ∥gt∥∞, ∥D⁢g∥∞, |β| and ∥D2⁢g∥∞, but independent of 0<ε≤1 and 0<δ<1, such that

| u ε , δ ⁢ ( x , t ) - u ε , δ ⁢ ( x , s ) | ≤ C 1 ⁢ | t - s |

for every x∈Ω and t,s∈(0,T). Moreover, if g is only continuous, then the modulus of continuity of u on QT can be estimated in terms of ∥g∥∞, |β|, diam⁡Ω and the modulus of continuity of g.

Proof.

Set

u ~ ε , δ ⁢ ( x , t ) = u ε , δ ⁢ ( x , t + σ ) , σ > 0 .

Then both u and u~ are smooth solutions to (4.1) in Qσ=Ω×(0,T-σ). Therefore, if g∈C2⁢(Q¯T), we have

| u ε , δ ⁢ ( x , t ) - u ε , δ ⁢ ( x , t + σ ) | ≤ sup Q σ ⁡ | u ε , δ - u ~ ε , δ | = sup ∂ p ⁡ Q σ ⁡ | u ε , δ - u ~ ε , δ | ≤ max ⁡ { C 1 ⁢ σ , ∥ g t ∥ ∞ ⁢ σ }

by Theorem 4.2. When g is only continuous, the proof is similar and we omit it. ∎

We next proceed to derive a Hölder estimate for uε,δ⁢(⋅,t) on the lateral boundary.

Theorem 4.4 (Hölder Regularity at the Lateral Boundary).

Let f be continuous in QT and g∈C2⁢(Q¯T). Suppose that uε,δ is a smooth solution satisfying

{ u t - L ε , δ ⁢ u - β ⁢ | D ⁢ u | = f in ⁢ Q T , u = g on ⁢ ∂ p ⁡ Q T .

Then for each 0<α<1, there exists a constant C∗≥1 depending on α, ∥g∥∞, ∥gt∥∞, ∥f∥∞, |β| and ∥D⁢g∥∞, but independent of ε and δ, such that

| u ε , δ ⁢ ( x , t 0 ) - g ⁢ ( x 0 , t 0 ) | ≤ C ∗ ⁢ | x - x 0 | α

for all (x0,t0)∈∂⁡Ω×(0,T), x∈Ω∩Br⁢(x0) and r>0 sufficiently small (depending on α and β). Moreover, if g is only continuous, then the modulus of continuity of uε,δ in x can be estimated in terms of ∥g∥∞, ∥f∥∞, |β| and the modulus of continuity of g in x.

Notice that Theorem 4.4 is also true if we assume the initial datum g only to be continuous.

Proof.

We prove this theorem in three steps. Step 1. For every (x0,t0)∈∂⁡Ω×(0,T) and 0<α<1, set

v + ⁢ ( x , t ) = g ⁢ ( x 0 , t 0 ) + C ∗ ⁢ | x - x 0 | α + λ ⁢ ( t 0 - t ) ,

where C∗>0 and λ>0 are to be determined. If |x-x0|≤1 and C*≥1, we have

v t + - L ε , δ ⁢ v + - β ⁢ | D ⁢ v + | = - λ + C * ⁢ α ⁢ | x - x 0 | α - 2 ⁢ ( - ε ⁢ ( n + α - 2 ) + 1 - α 1 + ( δ C * ⁢ α ⁢ | x - x 0 | α - 1 ) 2 - β ⁢ | x - x 0 | )

≥ - λ + C * ⁢ α ⁢ | x - x 0 | α - 2 ⁢ ( - ε ⁢ ( n + α - 2 ) + 1 - α 1 + ( δ α ) 2 - β ⁢ | x - x 0 | ) .

Now if δ≤α<1, we have

ε ⁢ ( n + α - 2 ) ≤ 1 - α 4 for ⁢ 0 < ε ≤ 1 - α 4 ⁢ ( n + α - 2 ) .

Choosing r=min⁡{1-α8⁢|β|,1}, we have

| β | ⁢ | x - x 0 | ≤ 1 - α 8 for ⁢ | x - x 0 | ≤ r .

Therefore, with this choice for r, δ and ε in the specified range, we get

v t + - L ε , δ ⁢ v + - β ⁢ | D ⁢ v + | ≥ - λ + C * ⁢ α ⁢ | x - x 0 | α - 2 ⁢ 1 - α 8 ≥ - λ + C * ⁢ α ⁢ r α - 2 ⁢ 1 - α 8 ≥ f ⁢ ( x , t ) ,

provided

C * ≥ max ⁡ { 8 α ⁢ ( 1 - α ) ⁢ r α - 2 ⁢ ( λ + ∥ f ∥ ∞ ) , 1 } .

Step 2. Let Q∗=(Ω∩Br⁢(x0))×(t0-t∗,t0), where t∗=min⁡{1,t0}. We want to prove first v+≥uε,δ on the lateral boundary ∂l⁡Q∗.

Case 1. If x∈∂⁡Ω∩Br⁢(x0), then

v + ⁢ ( x , t ) ≥ - ∥ g t ∥ ∞ ⁢ ( t 0 - t ) - ∥ D ⁢ g ∥ ∞ ⁢ | x - x 0 | + g ⁢ ( x , t ) + C * ⁢ | x - x 0 | + λ ⁢ ( t 0 - t ) ≥ g ⁢ ( x , t ) ,

provided C*≥∥D⁢g∥∞ and λ≥∥gt∥∞.

Case 2. If x∈Ω∩∂⁡Br⁢(x0), by the comparison principle we have

v + ⁢ ( x , t ) ≥ g ⁢ ( x 0 , t 0 ) + C * ⁢ r α ≥ ∥ g ∥ ∞ + ∥ f ∥ ∞ ⁢ T ≥ ∥ g ∥ ∞ + ∥ f ∥ ∞ ⁢ t ≥ u ε , δ ⁢ ( x , t ) ,

provided C*≥r-α⁢(2⁢∥g∥∞+∥f∥∞⁢T).

Step 3. We prove v+≥uε,δ on the bottom boundary ∂b⁡Q∗.

Case 1. If t∗=t0, since uε,δ=g on the bottom of Q∗=(Ω∩Br⁢(x0))×(0,t0), we have

v + ⁢ ( x , 0 ) ≥ g ⁢ ( x 0 , t 0 ) + C * ⁢ | x - x 0 | + λ ⁢ t 0 ≥ g ⁢ ( x 0 , t 0 ) + ∥ D ⁢ g ∥ ∞ ⁢ | x - x 0 | + ∥ g t ∥ ⁢ t 0 ≥ g ⁢ ( x , 0 ) ,

provided C*≥∥D⁢g∥∞ and λ≥∥gt∥∞.

Case 2. If t∗=t0-1, then Q∗=(Ω∩Br⁢(x0))×(t0-1,t0). By the comparison principle, we have

v + ⁢ ( x , t 0 - 1 ) ≥ g ⁢ ( x 0 , t 0 ) + λ ≥ ∥ g ∥ ∞ + ∥ f ∥ ∞ ⁢ ( T - 1 ) ≥ ∥ g ∥ ∞ + ∥ f ∥ ∞ ⁢ ( t 0 - 1 ) ≥ u ε , δ ⁢ ( x , t 0 - 1 )

if λ≥2⁢∥g∥∞+∥f∥∞⁢(T-1).

In summary, we have shown that v+≥uε,δ on ∂p⁡Q∗ if we choose

λ ≥ max ⁡ { ∥ g t ∥ ∞ , 2 ⁢ ∥ g ∥ ∞ + ∥ f ∥ ∞ ⁢ ( T - 1 ) } ,

C * ≥ max ⁡ { ∥ D ⁢ g ∥ ∞ , 2 ⁢ ∥ g ∥ ∞ + ∥ f ∥ ∞ ⁢ T , 8 α ⁢ ( 1 - α ) ⁢ r α - 2 ⁢ ( λ + ∥ f ∥ ∞ ) + 1 } ,

r = min ⁡ { 1 - α 8 ⁢ | β | , 1 } .

Hence we have v+≥uε,δ in Q∗ by the comparison principle. There also holds that

u ε , δ ⁢ ( x , t 0 ) ≤ v + ⁢ ( x , t 0 ) = g ⁢ ( x 0 , t 0 ) + C * ⁢ | x - x 0 | α

for x∈Ω∩Br⁢(x0). Considering the lower barrier

v - ⁢ ( x , t ) = g ⁢ ( x 0 , t 0 ) - C ∗ ⁢ | x - x 0 | α - λ ⁢ ( t 0 - t ) ,

we get the symmetric inequality. If g is only continuous, the argument is similar. ∎

Due to the translation invariant of the equation with the condition that f is constant and due to the comparison principle, we can extend the Hölder estimate to the interior of the domain; see, e.g., [18, 21, 37]. For w∈Rn, one has Ωw={x+w:x∈Ω} and Ωr={x∈Ω:dist⁡(x,∂⁡Ω)≥r}. By [x,y] we denote the closed segment of joining with the two endpoints x and y.

Corollary 4.5 (Hölder Regularity in Space).

Let f be constant and let g∈C2⁢(Q¯T). Suppose that u=uε,δ is a smooth solution satisfying

{ u t - L ε , δ ⁢ u - β ⁢ | D ⁢ u | = f in ⁢ Q T , u = g on ⁢ ∂ p ⁡ Q T .

Then for each 0<α<1, there exists a constant C∗≥1 depending on ∥g∥∞, ∥gt∥∞, |β| and ∥D⁢g∥∞, but independent of ε and δ small enough, such that

| u ⁢ ( x , t ) - u ⁢ ( y , t ) | ≤ C ∗ ⁢ | x - y | α

for all x,y∈Ω. Moreover, if g is only continuous, then the modulus of continuity of uε,δ in x can be estimated in terms of ∥g∥∞, |β| and the modulus of continuity of g in x.

Proof.

We prove this corollary in three steps. Step 1. For fixed 0<t<T, take a point z∈Br⁢(0) and let V=Ω∩Ωz. Define uz⁢(x,t)=u⁢(x-z,t). Noting that ∂⁡V⊆(∂⁡Ω∪∂⁡Ωz), we have that |u⁢(x,t)-uz⁢(x,t)|≤C∗⁢|z|α on ∂l⁡(V×(0,T)) by Theorem 4.4. Hence,

| u ⁢ ( x , t ) - u z ⁢ ( x , t ) | ≤ C ∗ ⁢ | z | α

for every x∈V by the comparison principle. This means that whenever x,y∈Ω∩Ωx-y or x,y∈Ω∩Ωy-x with |x-y|≤r, we have |u⁢(x,t)-u⁢(y,t)|≤C∗⁢|x-y|α. Especially, we have

| u ⁢ ( x , t ) - u ⁢ ( y , t ) | ≤ C ∗ ⁢ | x - y | α

for x,y∈Ω|x-y|.

Step 2. When |x-y|>r, using the comparison principle, we obtain the conclusion of the theorem with C∗=2⁢∥g∥∞rα. Therefore, Let us assume that |x-y|≤r and x-y∉Ω|x-y|. Suppose further [x,y]⊂Ω. In this case, we can take the two segments [x,w] and [w,y], where w=(x+y)/2 is the midpoint of [x,y]. Let z=y-w and note that w,y∈Ωz∩Ω and x,w∈Ω-z∩Ω. Hence from the first step of this proof we have

| u ⁢ ( x , t ) - u ⁢ ( y , t ) | ≤ | u ⁢ ( x , t ) - u ⁢ ( w , t ) | + | u ⁢ ( w , t ) - u ⁢ ( y , t ) | ≤ C ∗ ⁢ ( | x - w | α + | w - y | α ) ≤ 2 1 - α ⁢ C ∗ ⁢ | x - y | α .

Step 3. If the segment [x,y] is not completely in Ω, then we can certainly find w1,w2∈∂⁡Ω∩[x,y] such that [x,w1)∈Ω and (w2,y]∈Ω. By Theorem 4.4, we get

| u ⁢ ( x , t ) - u ⁢ ( w 1 , t ) | ≤ C ∗ ⁢ | x - w 1 | α

and

| u ⁢ ( w 2 , t ) - u ⁢ ( y , t ) | ≤ C ∗ ⁢ | w 2 - y | α .

| u ⁢ ( w 1 , t ) - u ⁢ ( w 2 , t ) | ≤ ∥ D ⁢ g ∥ ∞ ⁢ | w 1 - w 2 | ≤ C ∗ ⁢ | w 1 - w 2 | α ,

we easily get the conclusion. ∎

By calculation, it is clear that when β≥0 and f≥0, the function v+⁢(x,t)=C∗⁢|x-x0|+λ⁢(t0-t), where x0∈∂⁡Ω, is not a viscosity super-solution (in fact, not a classical super-solution) for any λ>0 and C∗>0. Similarly, when β≤0 and f≤0, the function v-⁢(x,t)=-C∗⁢|x-x0|-λ⁢(t0-t), where x0∈∂⁡Ω, is not a viscosity sub-solution for any λ>0 and C∗>0. This means that the Lipschitz estimate can not be obtained by the barrier method for the regularized equation (4.1). But it is interesting that we can get the Lipschitz estimate when we remove the viscous term; cf. [11, 18, 37]. We first show a Lipschitz estimate on the boundary.

Theorem 4.6 (Lipschitz Regularity in Space).

Let f be continuous in QT, and let g∈C2⁢(Q¯T). Suppose that uδ satisfies

(4.4) { u t - L 0 , δ ⁢ u - β ⁢ | D ⁢ u | = f in ⁢ Q T , u = g on ⁢ ∂ p ⁡ Q T

in the viscosity sense. Then there exist constants C∗≥0 and r>0 depending on ∥g∥∞, ∥gt∥∞, ∥f∥∞, |β| and ∥D⁢g∥∞, but independent of 0<δ<1 small enough, such that

| u ⁢ ( x , t 0 ) - g ⁢ ( x 0 , t 0 ) | ≤ C ∗ ⁢ | x - x 0 |

for all (x0,t0)∈∂⁡Ω×(0,T) and x∈Ω∩Br⁢(x0). Moreover, if g is only continuous, then the modulus of continuity of u on ∂⁡Ω×(0,T) can be estimated in terms of ∥g∥∞, ∥f∥∞, |β| and the modulus of continuity of g.

Proof.

We prove this theorem in five steps. Step 1. Let g∈C2⁢(Q¯T). For every (x0,t0)∈∂⁡Ω×(0,T), set

w + ⁢ ( x , t ) = g ⁢ ( x 0 , t 0 ) + C ∗ ⁢ | x - x 0 | + λ ⁢ ( t 0 - t ) - A ⁢ | x - x 0 | 2 ,

where C∗≥1 and λ,A>0 are to be determined. If we choose |x-x0|≤min⁡{1,12⁢A}, 0<δ≤1, C*≥2 and A≥λ+∥f∥∞+|β|⁢C∗, we have

w t + - L 0 , δ ⁢ w + - β ⁢ | D ⁢ w + | = - λ + 2 ⁢ A 1 + ( δ C ∗ - 2 ⁢ A ⁢ | x - x 0 | ) 2 - β ⁢ | C ∗ - 2 ⁢ A ⁢ | x - x 0 | |

≥ - λ + 2 ⁢ A 1 + ( 1 C ∗ - 2 ⁢ A ⁢ | x - x 0 | ) 2 - | β | ⁢ | C ∗ - 2 ⁢ A ⁢ | x - x 0 | |

≥ - λ + A - | β | ⁢ C ∗

≥ f ⁢ ( x , t ) .

This means that w+ is a super-solution.

Step 2. Let r=min⁡{1,12⁢A} and Q∗=(Ω∩Br⁢(x0))×(t0-t∗,t0), where t∗=min⁡{1,t0}. Now we first show w+≥uδ on the lateral boundary ∂l⁡Q∗.

Case 1. If x∈∂⁡Ω∩Br⁢(x0), then

w + ⁢ ( x , t ) ≥ g ⁢ ( x 0 , t 0 ) + ( C * - 1 2 ) ⁢ | x - x 0 | + λ ⁢ ( t 0 - t )

≥ - ∥ g t ∥ ∞ ⁢ ( t 0 - t ) - ∥ D ⁢ g ∥ ∞ ⁢ | x - x 0 | + g ⁢ ( x , t ) + ( C * - 1 2 ) ⁢ | x - x 0 | + λ ⁢ ( t 0 - t )

≥ g ⁢ ( x , t )

= u δ ⁢ ( x , t ) ,

with C*≥12+∥D⁢g∥∞ and λ≥∥gt∥∞.

Case 2. If x∈Ω∩∂⁡Br⁢(x0) and t∈(t0-t∗,t0), we have

w + ⁢ ( x , t ) = g ⁢ ( x 0 , t 0 ) + C * ⁢ | x - x 0 | + λ ⁢ ( t 0 - t ) - A ⁢ | x - x 0 | 2

≥ g ⁢ ( x 0 , t 0 ) + ( C * - 1 2 ) ⁢ | x - x 0 |

≥ ∥ g ∥ ∞ + ∥ f ∥ ∞ ⁢ T

≥ ∥ g ∥ ∞ + ∥ f ∥ ∞ ⁢ t

≥ u δ ⁢ ( x , t ) ,

provided C*≥12+2⁢∥g∥∞+∥f∥∞⁢T.

Step 3. We prove w+≥uδ on ∂b⁡Q∗.

Case 1. If t∗=t0, since uδ=g on the bottom of Q∗=(Ω∩Br⁢(x0))×(0,t0), we have

w + ⁢ ( x , 0 ) ≥ g ⁢ ( x 0 , t 0 ) + ( C * - 1 2 ) ⁢ | x - x 0 | + λ ⁢ t 0

≥ g ⁢ ( x 0 , t 0 ) + ∥ D ⁢ g ∥ ∞ ⁢ | x - x 0 | + ∥ g t ∥ ⁢ t 0

≥ g ⁢ ( x , 0 )

if C*≥12+∥D⁢g∥∞ and λ≥∥gt∥∞.

Case 2. If t∗=1, then Q∗=(Ω∩Br⁢(x0))×(t0-1,t0). By the comparison principle, we obtain

w + ⁢ ( x , t 0 - 1 ) ≥ g ⁢ ( x 0 , t 0 ) + λ + ( C * - 1 2 ) ⁢ | x - x 0 |

≥ ∥ g ∥ ∞ + ∥ f ∥ ∞ ⁢ ( T - 1 )

≥ ∥ g ∥ ∞ + ∥ f ∥ ∞ ⁢ ( t 0 - 1 )

≥ u δ ⁢ ( x , t 0 - 1 ) ,

provided λ≥2⁢∥g∥∞+∥f∥∞⁢(T-1).

Step 4. To sum up, we have shown that w+ is a super-solution and w+≥uδ on ∂p⁡Q∗ if we choose

λ ≥ max ⁡ { ∥ g t ∥ ∞ , 2 ⁢ ∥ g ∥ ∞ + ∥ f ∥ ∞ ⁢ ( T - 1 ) } ,

C * ≥ max ⁡ { 2 , 1 2 + ∥ D ⁢ g ∥ ∞ , 1 2 + 2 ⁢ ∥ g ∥ ∞ + ∥ f ∥ ∞ ⁢ T } ,

A ≥ λ + ∥ f ∥ ∞ + | β | ⁢ C * ,

r = min ⁡ { 1 , 1 2 ⁢ A } .

Therefore, there holds w+≥uδ in Q∗ by the comparison principle. Especially, we obtain

u δ ⁢ ( x , t 0 ) ≤ g ⁢ ( x 0 , t 0 ) + C * ⁢ | x - x 0 | - A ⁢ | x - x 0 | 2 ≤ g ⁢ ( x 0 , t 0 ) + C * ⁢ | x - x 0 |

for x∈Ω∩Br⁢(x0). The symmetric inequality can be obtained by a similar argument with the lower barrier

w - ⁢ ( x , t ) = g ⁢ ( x 0 , t 0 ) - C ∗ ⁢ | x - x 0 | - λ ⁢ ( t 0 - t ) + A ⁢ | x - x 0 | 2 .

Step 5. Finally, assume only that g is continuous. Let us fix a point (x0,t0)∈∂⁡Ω×(0,T), and for a given μ>0 choose 0<ρ<t0 such that |g⁢(x,t)-g⁢(x0,t0)|<μ whenever |x-x0|+|t-t0|<ρ. Set

g ± ⁢ ( x , t ) = g ⁢ ( x 0 , t 0 ) ± μ ± 4 ⁢ ∥ g ∥ ∞ ρ 2 ⁢ | x - x 0 | 2 ± 4 ⁢ ∥ g ∥ ∞ ρ ⁢ | t - t 0 | .

Thus if u± are the unique solutions to (4.4) with boundary-initial data g±, respectively, we have u-≤uδ≤u+ in QT due to the comparison principle again. Since g± are smooth, we have

| u ± ⁢ ( x , t 0 ) - g ± ⁢ ( x 0 , t 0 ) | ≤ C ∗ ⁢ | x - x 0 | .

It is easy to show

| u ⁢ ( x , t 0 ) - g ⁢ ( x 0 , t 0 ) | ≤ | u ⁢ ( x , t 0 ) - u ± ⁢ ( x , t 0 ) | + | u ± ⁢ ( x , t 0 ) - g ± ⁢ ( x 0 , t 0 ) | + | g ± ⁢ ( x 0 , t 0 ) - g ⁢ ( x 0 , t 0 ) |

≤ 1 2 ⁢ | u + ⁢ ( x , t 0 ) - u - ⁢ ( x , t 0 ) | + C ∗ ⁢ | x - x 0 | + μ

≤ 1 2 ⁢ | u + ⁢ ( x , t 0 ) - g + ⁢ ( x 0 , t 0 ) | + 1 2 ⁢ | g + ⁢ ( x 0 , t 0 ) - g - ⁢ ( x 0 , t 0 ) |

+ 1 2 ⁢ | u - ⁢ ( x , t 0 ) - g - ⁢ ( x 0 , t 0 ) | + C ∗ ⁢ | x - x 0 | + μ

≤ 2 ⁢ C ∗ ⁢ | x - x 0 | + 2 ⁢ μ .

The desired result is proved. ∎

By this Lipschitz boundary regularity, we can immediately get the interior Lipschitz estimate by the translation and comparison principle. The proof is similar to Corollary 4.5 and we omit it.

Corollary 4.7.

Let f be constant and let g∈C2⁢(Q¯T) . Suppose that uδ is a smooth solution satisfying (4.1). Then there exists a constant C∗≥1 depending on ∥g∥∞, ∥gt∥∞, |β| and ∥D⁢g∥∞, but independent of δ small enough, such that

| u δ ⁢ ( x , t ) - u δ ⁢ ( y , t ) | ≤ C ∗ ⁢ | x - y |

for all (x,y)∈Ω) and t∈(0,T) . Moreover, if g is only continuous, then the modulus of continuity of g in x on Ω×(0,T) can be estimated in terms of ∥g∥∞, |β| and the modulus of continuity of g in x.

By Corollaries 4.3 and 4.5, we can obtain the following global Hölder estimate of uε,δ.

Corollary 4.8 (Global Hölder Regularity in QT).

Let f be constant and let g∈C2⁢(Q¯T). Suppose that uε,δ is a smooth solution satisfying

{ u t - L ε , δ ⁢ u - β ⁢ | D ⁢ u | = f in ⁢ Q T , u = g on ⁢ ∂ p ⁡ Q T .

Then for each 0<α<1 there exists a constant C≥1 depending on ∥g∥∞, ∥gt∥∞, |β| and ∥D⁢g∥∞, but independent of ε and δ small enough, such that

| u ε , δ ⁢ ( x , t ) - u ε , δ ⁢ ( y , s ) | ≤ C ⁢ ( | x - y | α + | t - s | )

for all (x,t),(y,s)∈QT, where C=C1+C∗.

Similarly, by Corollaries 4.3 and 4.7 we can obtain the following global Lipschitz estimate of u0,δ.

Corollary 4.9 (Global Lipschitz Regularity in QT).

Let f be constant and let g∈C2⁢(Q¯T). Suppose that uδ is a smooth solution satisfying

{ u t - L 0 , δ ⁢ u - β ⁢ | D ⁢ u | = f in ⁢ Q T , u = g on ⁢ ∂ p ⁡ Q T .

Then there exists a constant C≥1 depending on ∥g∥∞, ∥gt∥∞, |β| and ∥D⁢g∥∞, but independent of δ small enough, such that

| u ⁢ ( x , t ) - u ⁢ ( y , s ) | ≤ C ⁢ ( | x - y | + | t - s | )

for all (x,t),(y,s)∈QT, where C=C1+C∗.

With these global uniform estimates at hand, we can immediately deduce Theorem 4.1 by the compactness method and the stability theory of viscosity solutions.

Proof of Theorem 4.1.

If g∈C2⁢(Q¯T) and uε,δ is the unique smooth solution to

{ u t - L ε , δ ⁢ u - β ⁢ | D ⁢ u | = f in ⁢ Q T , u = g on ⁢ ∂ p ⁡ Q T ,

Corollary 4.8 and the comparison principle imply that the family of functions {uε,δ} is uniformly bounded and equicontinuous. Therefore, by the Arzela–Ascoli compactness theorem, for some sequence εk→0, we have uεk,δ→uδ uniformly as εk→0, and uδ is the unique viscosity solution to (4.4) by the stability properties of viscosity solutions. Similarly, by Corollary 4.9 we conclude that uδ→u uniformly as δ→0, and u is a viscosity solution to (1.3) by the stability properties again.

The existence for a continuous data g follows by Corollaries 4.3, 4.5 and 4.7 and the stability principle of viscosity solutions again. In addition, we have

| u ⁢ ( x , t ) - u ⁢ ( x , s ) | ≤ 4 ⁢ μ + C 1 ⁢ | t - s |

for all t,s∈(0,T) and for every x∈Ω, where μ→0 as ε→0, δ→0 and C1 as in Corollary 4.3. In addition, we obtain that the viscosity solution u of (1.3) is Lipschitz continuous with respect to the time variable t and Hölder continuous in the space variable x. ∎

5 Lipschitz Estimate

As mentioned above, we can not obtain the interior Lipschitz continuity of the viscosity solution u of (1.1) with respect to the space variable x by the barrier method. In order to overcome this difficulty we follow the argument in [11, 18]. However, we must deal with the biased term and the leading term simultaneously in (1.1), and a precise calculation is required for the biased case. In the following, we consider another regularized equation

(5.1) u t - L ε , δ ⁢ u - β ⁢ | D ⁢ u | 2 + δ 2 = f ,

where Lε,δ is as in (4.2) with 0<ε≤1 and 0<δ≤1/3. We first employ Bernstein’s method to derive the interior gradient estimate for smooth solutions of (5.1). Then we use the smooth approximations to prove the interior Lipschitz estimate for viscosity solutions of (1.1).

Theorem 5.1.

Assume f∈C1⁢(Q¯T) and u=uε,δ∈C1⁢(Q¯T)∩C2⁢(QT) is a smooth solution of (5.1) in QT. Then there exists a positive constant C depending on β and n, which is independent of ε∈(0,1] and δ∈(0,1/3], such that the estimate

(5.2) | D ⁢ u | ⁢ ( x , t ) ≤ C ⁢ ( n , β ) ⁢ ( ∥ u ∥ ∞ + ∥ u ∥ ∞ ( dist ⁡ ( ( x , t ) , ∂ p ⁡ Q T ) ) 2 + ∥ f ∥ ∞ + ∥ D ⁢ f ∥ ∞ )

holds for all (x,t)∈QT.

Proof.

We consider the auxiliary function

w = ζ ⁢ v + κ ⁢ u 2 ,

where v=|D⁢u|2+δ2, κ>0 to be determined, and ζ is a smooth positive cut-off function that vanishes on ∂p⁡QT. Let (x0,t0) be a point where w attains its maximum in Q¯T. We first assume (x0,t0)∉∂p⁡QT. At the maximum point (x0,t0), we have

(5.3) D ⁢ w = 0

and

(5.4) ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ w i ⁢ j ≤ 0 ,

where ai⁢jε,δ is the operator defined in (4.3).

By (5.3), we have

(5.5) D ⁢ v = - 1 ζ ⁢ [ v ⁢ D ⁢ ζ + 2 ⁢ κ ⁢ u ⁢ D ⁢ u ]

at (x0,t0). Since (x0,t0)∉∂p⁡QT, equality (5.5) makes sense at (x0,t0). By (5.4), there holds

0 ≤ w t - ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ w i ⁢ j

= ζ ⁢ ( v t - ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ v i ⁢ j ) + v ⁢ ( ζ t - ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ ζ i ⁢ j )

+ 2 ⁢ κ ⁢ u ⁢ ( u t - ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ u i ⁢ j ) - 2 ⁢ ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ ζ j ⁢ v i - 2 ⁢ κ ⁢ ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ u i ⁢ u j

= ζ ⁢ ( v t - ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ v i ⁢ j ) + v ⁢ ( ζ t - ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ ζ i ⁢ j )

(5.6) + 2 ⁢ κ ⁢ u ⁢ ( β ⁢ v + f ) - 2 ⁢ ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ ζ j ⁢ v i - 2 ⁢ κ ⁢ | D ⁢ u | 2 ⁢ ( ε + | D ⁢ u | 2 v 2 )

at (x0,t0), where we have used equation (5.1). Now we study all the terms on the right-hand side of (5.6). By (5.5), we have

- 2 ⁢ ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ ζ j ⁢ v i = - 2 ⁢ ε ⁢ D ⁢ ζ ⋅ D ⁢ v - 2 v 2 ⁢ ( D ⁢ u ⋅ D ⁢ ζ ) ⁢ ( D ⁢ u ⋅ D ⁢ v )

= 2 ⁢ v ζ ⁢ ( ε ⁢ | D ⁢ ζ | 2 + ( D ⁢ u ⋅ D ⁢ ζ ) 2 v 2 ) + 4 ⁢ κ ⁢ u ζ ⁢ ( ε + | D ⁢ u | 2 v 2 ) ⁢ ( D ⁢ u ⋅ D ⁢ ζ )

≤ 2 ⁢ v ζ ⁢ ( 1 + ε ) ⁢ | D ⁢ ζ | 2 + 4 ⁢ κ ⁢ u ζ ⁢ ( ε + 1 ) ⁢ v ⁢ | D ⁢ ζ |

(5.7) ≤ 4 ⁢ v ⁢ ( 1 + ε ) ζ ⁢ ( | D ⁢ ζ | 2 + κ 2 ⁢ u 2 ) ,

where we have used Young’s inequality.

From the definition of ai⁢jε,δ, we have

(5.8) v ⁢ ( ζ t - ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ ζ i ⁢ j ) ≤ v ⁢ ( | ζ t | + ( n ⁢ ε + 1 ) ⁢ | D 2 ⁢ ζ | ) ≤ κ 4 ⁢ v 2 + 1 κ ⁢ ( | ζ t | + ( n + 1 ) ⁢ | D 2 ⁢ ζ | ) 2 .

Now we turn to the first term on the right-hand side of (5.6). By a direct calculation, we have

ζ ⁢ ( v t - ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ v i ⁢ j )

= ζ ⁢ ( 1 v 3 ⁢ ∑ j = 1 n ∑ i = 1 n ( u i ⁢ u i ⁢ j ) 2 - 1 v 5 ⁢ ( ∑ i , k = 1 n u i ⁢ u k ⁢ u i ⁢ k ) 2 - ε v ⁢ ( ∑ j = 1 n u i ⁢ j 2 ) + ε v 3 ⁢ ∑ k = 1 n ( ∑ i = 1 n u i ⁢ u i ⁢ k ) 2 ) .

Differentiating equation (5.1) with respect to xk, we have

(5.9) ( u k ) t - D k ⁢ ( a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ) ⁢ u i ⁢ j - a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ ( u k ) i ⁢ j = β ⁢ v k + f k

for k=1,…,n, and it is easy to show that

D k ⁢ ( a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ) = 2 v 2 ⁢ u i ⁢ u k ⁢ j - 2 v 3 ⁢ u i ⁢ u j ⁢ v k .

Multiplying ukv in (5.9) and adding with respect to k, we have

v t - ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ v i ⁢ j = ε v 3 ⁢ ∑ i = 1 n ( ∑ k = 1 n u k ⁢ u k ⁢ i ) 2 - ε v ⁢ ∑ k , i = 1 n ( u k ⁢ i ) 2 + 1 v 3 ⁢ ∑ k = 1 n ( ∑ i = 1 n u i ⁢ u k ⁢ i ) 2

- 1 v 5 ⁢ ( ∑ i , k = 1 n u i ⁢ u k ⁢ u k ⁢ i ) 2 + β v ⁢ ∑ k = 1 n u k ⁢ v k + 1 v ⁢ ∑ k = 1 n u k ⁢ f k

≤ 1 + ε v 3 ⁢ ∑ i = 1 n ( ∑ k = 1 n u k ⁢ u k ⁢ i ) 2 + β ⁢ D ⁢ u ⋅ D ⁢ v v + D ⁢ u ⋅ D ⁢ f v

(5.10) ≤ ( 1 + ε ) ⁢ | D ⁢ v | 2 v + | β | ⁢ | D ⁢ v | + | D ⁢ f | .

By (5.5), we have

( 1 + ε ) ⁢ | D ⁢ v | 2 v + | β | ⁢ | D ⁢ v | = ( 1 + ε ) v ⁢ ζ 2 ⁢ | v ⁢ D ⁢ ζ + 2 ⁢ κ ⁢ u ⁢ D ⁢ u | 2 + | β | ζ ⁢ | v ⁢ D ⁢ ζ + 2 ⁢ κ ⁢ u ⁢ D ⁢ u |

(5.11) ≤ 2 ⁢ ( 1 + ε ) ⁢ v ζ 2 ⁢ ( | D ⁢ ζ | 2 + 4 ⁢ κ 2 ⁢ u 2 ) + | β | ⁢ v ζ ⁢ ( | D ⁢ ζ | + 2 ⁢ κ ⁢ | u | ) .

Substituting (5.11) into (5.10), we obtain

(5.12) ζ ⁢ ( v t - ∑ i , j = 1 n a i ⁢ j ε , δ ⁢ ( D ⁢ u ) ⁢ v i ⁢ j ) ≤ 2 ⁢ ( 1 + ε ) ⁢ v ζ ⁢ ( | D ⁢ ζ | 2 + 4 ⁢ κ 2 ⁢ u 2 ) + | β | ⁢ v ⁢ ( | D ⁢ ζ | + 2 ⁢ κ ⁢ | u | ) + ζ ⁢ | D ⁢ f | .

Substituting (5.7), (5.8) and (5.12) into (5.6), we obtain

2 ⁢ κ ⁢ | D ⁢ u | 2 ⁢ ( ε + | D ⁢ u | 2 v 2 ) ≤ 4 ⁢ v ⁢ ( 1 + ε ) ζ ⁢ ( | D ⁢ ζ | 2 + κ 2 ⁢ u 2 ) + κ 4 ⁢ v 2 + 1 κ ⁢ ( | ζ t | + ( n + 1 ) ⁢ | D 2 ⁢ ζ | ) 2

+ 2 ⁢ κ ⁢ | u | ⁢ ( | β | ⁢ v + ∥ f ∥ ∞ ) + 2 ⁢ ( 1 + ε ) ⁢ v ζ ⁢ ( | D ⁢ ζ | 2 + 4 ⁢ κ 2 ⁢ u 2 ) + | β | ⁢ v ⁢ ( | D ⁢ ζ | + 2 ⁢ κ ⁢ | u | ) + ζ ⁢ | D ⁢ f |

≤ 12 ⁢ v ⁢ ( 1 + ε ) ζ ⁢ ( | D ⁢ ζ | 2 + κ 2 ⁢ u 2 ) + κ 4 ⁢ v 2 + 1 κ ⁢ ( | ζ t | + ( n + 1 ) ⁢ | D 2 ⁢ ζ | ) 2

+ 2 ⁢ | β | ⁢ v ⁢ ( | D ⁢ ζ | + κ ⁢ | u | ) + ζ ⁢ | D ⁢ f | + 2 ⁢ κ ⁢ | u | ⁢ ( | β | ⁢ v + ∥ f ∥ ∞ )

≤ 5 8 ⁢ κ ⁢ v 2 + 1152 κ ⁢ ζ 2 ⁢ ( | D ⁢ ζ | 2 + κ 2 ⁢ u 2 ) 2 + 1 κ ⁢ ( | ζ t | + ( n + 1 ) ⁢ | D 2 ⁢ ζ | ) 2

(5.13) + 8 ⁢ β 2 κ ⁢ ( | D ⁢ ζ | + κ ⁢ | u | ) 2 + ζ ⁢ | D ⁢ f | + 8 ⁢ κ ⁢ u 2 ⁢ β 2 + 2 ⁢ κ ⁢ | u | ⁢ ∥ f ∥ ∞ .

We consider the first case if |D⁢u⁢(x0,t0)|≥1. Then for δ∈(0,1/3], we obtain

(5.14) 2 ⁢ κ ⁢ | D ⁢ u | 2 ⁢ ( ε + | D ⁢ u | 2 v 2 ) = 2 ⁢ κ ⁢ v 2 ⁢ | D ⁢ u | 2 v 2 ⁢ ( ε + | D ⁢ u | 2 v 2 ) ≥ 2 ⁢ κ ⁢ v 2 ⁢ ( | D ⁢ u | 2 v 2 ) 2 ≥ 9 8 ⁢ κ ⁢ v 2 .

Combining (5.13) and (5.14), we get

(5.15) ( v ⁢ ζ ) 2 ≤ C ⁢ ( n ) κ 2 ⁢ ( ( | D ⁢ ζ | 2 + κ 2 ⁢ u 2 ) 2 + ( | ζ t | + | D 2 ⁢ ζ | ) 2 + β 2 ⁢ ( | D ⁢ ζ | + κ ⁢ | u | ) 2 ) + ζ 3 κ ⁢ | D ⁢ f | + C ⁢ ( n ) ⁢ ζ 2 ⁢ ( u 2 ⁢ β 2 + | u | ⁢ ∥ f ∥ ∞ )

at the point (x0,t0). For a fixed point (x,t)∈QT, we choose the cut-off function ζ such that

ζ ⁢ ( x , t ) = 1 , max ⁡ { ∥ D ⁢ ζ ∥ ∞ , ∥ ζ t ∥ ∞ } ≤ 1 dist ⁡ ( ( x , t ) , ∂ p ⁡ Q T ) , ∥ D 2 ⁢ ζ ∥ ∞ ≤ 1 ( dist ⁡ ( ( x , t ) , ∂ p ⁡ Q T ) ) 2 .

Since w attains its maximum at (x0,t0)∈QT, from (5.15) we have

| D ⁢ u | ⁢ ( x , t ) ≤ w ⁢ ( x , t ) ≤ w ⁢ ( x 0 , t 0 )

≤ C ⁢ ( n ) κ ⁢ ( ∥ D ⁢ ζ ∥ ∞ 2 + κ 2 ⁢ ∥ u ∥ ∞ 2 + ∥ ζ t ∥ ∞ + ∥ D 2 ⁢ ζ ∥ ∞ + | β | ⁢ ( ∥ D ⁢ ζ ∥ ∞ + κ ⁢ ∥ u ∥ ∞ ) )

+ ∥ D ⁢ f ∥ ∞ κ + C ⁢ ( n ) ⁢ ζ ⁢ ( | | u | | ∞ ⁢ | β | + | | u | | ∞ ⁢ ∥ f ∥ ∞ ) + κ ⁢ ∥ u ∥ ∞ 2

≤ C ⁢ ( n , β ) ⁢ ( ∥ u ∥ ∞ + ∥ u ∥ ∞ ( dist ⁡ ( ( x , t ) , ∂ p ⁡ Q T ) ) 2 + ∥ u ∥ ∞ dist ⁡ ( ( x , t ) , ∂ p ⁡ Q T ) + ∥ f ∥ ∞ + ∥ D ⁢ f ∥ ∞ )

≤ C ⁢ ( n , β ) ⁢ ( ∥ u ∥ ∞ + ∥ u ∥ ∞ ( dist ⁡ ( ( x , t ) , ∂ p ⁡ Q T ) ) 2 + ∥ f ∥ ∞ + ∥ D ⁢ f ∥ ∞ ) ,

where we have chosen κ=1/∥u∥∞ and the constant C⁢(n,β) depends only on β and the dimension n.

On the other hand, if |D⁢u⁢(x0,t0)|<1, since ∥ζ∥∞=1 and δ∈(0,1/3], we have

| D ⁢ u | ⁢ ( x , t ) ≤ w ⁢ ( x , t ) ≤ w ⁢ ( x 0 , t 0 ) ≤ 2 3 + ∥ u ∥ ∞ .

Finally, for the case (x0,t0)∈∂p⁡QT, since ζ=0 on ∂p⁡QT, it is easy to obtain

| D ⁢ u | ⁢ ( x , t ) ≤ w ⁢ ( x , t ) ≤ w ⁢ ( x 0 , t 0 ) ≤ ∥ u ∥ ∞ .

The desired interior gradient estimate (5.2) is proved. ∎

One should notice that if the inhomogeneous term f is Lipschitz continuous in a bounded domain, estimate (5.2) also holds for almost every (x,t)∈QT.

With the uniform estimate (5.2) at hand, we can immediately obtain the following Lipschitz estimate based on the compactness argument.

Theorem 5.2.

Let f be Lipschitz continuous in Q¯T and let f≥0 (f≤0). If u∈C⁢(Q¯T) is a viscosity solution of equation (1.1) in QT, then there exists a positive constant C depending on β and n such that

| D ⁢ u | ⁢ ( x , t ) ≤ C ⁢ ( n , β ) ⁢ ( ∥ u ∥ ∞ + ∥ u ∥ ∞ ( dist ⁡ ( ( x , t ) , ∂ p ⁡ Q T ) ) 2 + ∥ f ∥ ∞ + ∥ D ⁢ f ∥ ∞ )

for almost every (x,t)∈QT.

Proof.

We choose U⋐V⋐Ω open, σ′>σ>0, and consider the cylinders

Q 1 = V × ( σ , T - σ ) and Q 2 = U × ( σ ′ , T - σ ′ ) .

For ε∈(0,1] and δ∈(0,1/3], let uε,δ satisfy the following problem:

{ ( u ε , δ ) t - L ε , δ ⁢ u ε , δ - β ⁢ | D ⁢ u ε , δ | 2 + δ 2 = f in ⁢ Q 1 , u ε , δ ⁢ ( x , t ) = u ⁢ ( x , t ) on ⁢ ∂ p ⁡ Q 1 .

Then, for any (x,t)∈Q2, Theorem 5.1 and the maximum principle imply

| D ⁢ u ε , δ | ⁢ ( x , t ) ≤ C ⁢ ( n , β ) ⁢ ( ∥ u ∥ ∞ + ∥ u ∥ ∞ ( dist ⁡ ( ( x , t ) , ∂ p ⁡ Q T ) ) 2 + ∥ f ∥ ∞ + ∥ D ⁢ f ∥ ∞ ) .

By the Arzela–Ascoli compactness theorem, we obtain that the functions uε,δ converge uniformly as ε→0 and δ→0 to a locally Lipschitz continuous function u~. Then D⁢u~⁢(x,t) exists almost everywhere and

| D ⁢ u ~ | ⁢ ( x , t ) ≤ C ⁢ ( n , β ) ⁢ ( ∥ u ∥ ∞ + ∥ u ∥ ∞ ( dist ⁡ ( ( x , t ) , ∂ p ⁡ Q T ) ) 2 + ∥ f ∥ ∞ + ∥ D ⁢ f ∥ ∞ )

for almost every (x,t)∈Q2. By the stability principle of the viscosity solutions, we have that u~ satisfies

{ u ~ t - △ ∞ N u ~ - β ⁢ | D ⁢ u ~ | = f in ⁢ Q 1 , u ~ ⁢ ( x , t ) = u ⁢ ( x , t ) on ⁢ ∂ p ⁡ Q 1

in the viscosity sense. Then the comparison principle theorem implies u~=u in Q1. Hence we have

| D ⁢ u | ⁢ ( x , t ) ≤ C ⁢ ( n , β ) ⁢ ( ∥ u ∥ ∞ + ∥ u ∥ ∞ ( dist ⁡ ( ( x , t ) , ∂ p ⁡ Q T ) ) 2 + ∥ f ∥ ∞ + ∥ D ⁢ f ∥ ∞ )

for a.e. (x,t)∈Q2. Because the constant C⁢(n,β) does not depend on the choice of the subdomains in the above argument, the desired Lipschitz estimate is proved. ∎

6 Special Solutions

In this section, we will give some explicit solutions to the following homogeneous equation:

(6.1) u t - Δ ∞ N ⁢ u - β ⁢ | D ⁢ u | = 0 .

6.1 Separation of Variables

We are mainly interested in radial solutions of the form u⁢(x,t)=v⁢(r,t), where r=|x-x0|, x≠x0. By calculation, it is easy to see that v solves

v t - v r ⁢ r - β ⁢ | v r | = 0 ,

which is a one-dimensional evolution equation for the β-biased infinity Laplacian.

(i) Let us now look for solutions of the form

u ⁢ ( x , t ) = f ⁢ ( t ) ⁢ h ⁢ ( r ) ,

where r=|x-x0|,x≠x0. By direct calculation, we obtain Δ∞N⁢u=f⁢(t)⁢h′′⁢(r). Then u satisfies

u t - Δ ∞ N ⁢ u - β ⁢ | D ⁢ u | = 0

if and only if

f ′ ⁢ ( t ) ⁢ h ⁢ ( r ) = f ⁢ ( t ) ⁢ h ′′ ⁢ ( r ) + β ⁢ | f ⁢ ( t ) ⁢ h ′ ⁢ ( r ) | .

Then we get the following two ordinary differential: equations

f ′ ⁢ ( t ) + λ ⁢ f ⁢ ( t ) = 0 and h ′′ ⁢ ( r ) + β ⁢ h ′ ⁢ ( r ) + λ ⁢ h ⁢ ( r ) = 0 ,

with an arbitrary constant λ∈R. One can verify that

f ⁢ ( t ) = C ⁢ e - λ ⁢ t

and

h ⁢ ( | x - x 0 | ) = { C 1 ⁢ e - β + β 2 - 4 ⁢ λ 2 ⁢ | x - x 0 | + C 2 ⁢ e - β - β 2 - 4 ⁢ λ 2 ⁢ | x - x 0 | if ⁢ λ < β 2 4 , C 1 ⁢ e - β 2 ⁢ | x - x 0 | + C 2 ⁢ | x - x 0 | ⁢ e - β 2 ⁢ | x - x 0 | if ⁢ λ = β 2 4 , C 1 ⁢ e - β 2 ⁢ | x - x 0 | ⁢ cos ⁡ ( 4 ⁢ λ - β 2 ⁢ | x - x 0 | ) + C 2 ⁢ | x - x 0 | ⁢ e - β 2 ⁢ | x - x 0 | ⁢ sin ⁡ ( 4 ⁢ λ - β 2 ⁢ | x - x 0 | ) if ⁢ λ > β 2 4 .

Therefore, the functions

u ⁢ ( x , t ) = C ⁢ e - λ ⁢ t ⁢ e - β 2 ⁢ | x - x 0 | ⁢ cos ⁡ ( 4 ⁢ λ - β 2 ⁢ | x - x 0 | ) ⁢ if ⁢ λ > β 2 4 ,

u ⁢ ( x , t ) = C ⁢ e - λ ⁢ t ⁢ e - β 2 ⁢ | x - x 0 | ⁢ sin ⁡ ( 4 ⁢ λ - β 2 ⁢ | x - x 0 | ) ⁢ if ⁢ λ > β 2 4 ,

u ⁢ ( x , t ) = C ⁢ e - λ ⁢ t ⁢ e - β + β 2 - 4 ⁢ λ 2 ⁢ | x - x 0 | ⁢ if ⁢ λ < β 2 4 ,

u ⁢ ( x , t ) = C ⁢ | x - x 0 | ⁢ e - β 2 4 ⁢ t ⁢ e - β 2 ⁢ | x - x 0 |

are all classical solutions except the point x=x0. These functions seem like parabolic exponential cones with the vertex at the point x0, and the conical shape prevents testing from one side at the vertex. Therefore, these functions are only viscosity sub-solutions or viscosity super-solutions in the whole space, which depends on the sign of the constant coefficient C.

Furthermore, if we let

r = ∑ i = 1 k ( x i - x 0 , i ) 2 , i ∈ { 1 , 2 , … , k } ,

then we can similarly obtain the same equation

f ′ ⁢ ( t ) ⁢ h ⁢ ( r ) = f ⁢ ( t ) ⁢ h ′′ ⁢ ( r ) + β ⁢ | f ⁢ ( t ) ⁢ h ′ ⁢ ( r ) | .

Therefore, we can easily get the same exponential-cone type of solutions as above which only depend on the first k spacial variables. And in this case the singular set {x∈Rn:(x0,1,x0,2,…,x0,k,xk+1,…,xn)} is an (n-k)-dimensional subset. Note that the unbiased parabolic infinity Laplacian operator has a good invariance property, that is, if u is a solution of the equation

u t - Δ ∞ N ⁢ u = 0 ,

so is the function

u ϵ = u ⁢ ( x , t ) + ϵ ⁢ x n + 1

for any constant ϵ. But for the biased case, one can easily verify that this property does not hold any longer.

(ii) Set u⁢(x,t)=f⁢(t)+h⁢(r) with r=|x-x0|. By calculation, we obtain

f ′ ⁢ ( t ) = h ′′ ⁢ ( r ) + β ⁢ h ′ ⁢ ( r ) = λ .

Hence one can easily show that

u ⁢ ( x , t ) = λ ⁢ t + C 1 + C 2 ⁢ e - β ⁢ | x - x 0 | + λ β ⁢ | x - x 0 |

is a classical solution of (6.1) except the vertex x0. Especially,

u ⁢ ( x , t ) = β ⁢ t + | x - x 0 | + e - β ⁢ | x - x 0 |

is a solution like a parabolic exponential cone.

6.2 Traveling Waves

Finally, let us look for solutions of the form

u ⁢ ( x , t ) = h ⁢ ( x 1 - c ⁢ t ) .

Obviously, u satisfies

u t - Δ ∞ N ⁢ u - β ⁢ | D ⁢ u | = 0

if and only if the function h⁢(ω) satisfies

h ′′ = - ( c + β ) ⁢ h ′ .

We integrate twice to obtain

h ⁢ ( ω ) = - 1 c + β ⁢ e - ( c + β ) ⁢ ω ,

where we take the constants of integration equal to zero and ignore the stationary solutions (c=-β). Therefore, we obtain

u ⁢ ( x , t ) = - 1 c + β ⁢ e - ( c + β ) ⁢ ( x 1 - c ⁢ t ) .

Proposition 6.1.

For any unitary vector ν∈Rn and c∈R with c≠-β, the function

T ⁢ ( x , t ; ν , c ) = - 1 c + β ⁢ e - ( c + β ) ⁢ ( x ⋅ ν - c ⁢ t )

is a classical solution of (6.1) in Rn×R.

Notice that whenever u⁢(x,t) is a classical solution of (6.1), one can check that for any orthogonal n×n matrix O, for any (x0,t0)∈Rn×R, c∈R and any λ>0,

v ⁢ ( x , t ) = u ⁢ ( O ⁢ ( x - x 0 ) , t - t 0 ) + c

and

w ⁢ ( x , t ) = λ ⁢ u ⁢ ( x , t )

are also solutions in an appropriate domain. As usual, the argument can be transposed to viscosity solutions. This means that equation (6.1) has some symmetries. But we should note that the parabolic β-biased infinity Laplacian operator is not odd due to the β-biased term.

It should be pointed out that these two explicit solutions obtained by separation of variables are one-dimensional radial solutions. And it is not so easy to give more examples for the β-biased infinity Laplacian, even though for the unbiased case there are few explicit infinity harmonic functions known such as u⁢(x,y)=|x|4/3-|y|4/3. Although the biased parabolic normalized infinity Laplacian operator is 1-homogeneous, it seems difficult to find the similarity solutions. However, the similarity solutions can be obtained analogous to the heat equation for the unbiased case [18]. All of these facts show again that the equation discussed here is something quite different from the unbiased case.

Communicated by Guozhen Lu

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11501292

Award Identifier / Grant number: 11771214

Funding statement: This work was supported by National Natural Science Foundation of China (nos. 11501292, 11771214).

Acknowledgements

The authors would like to thank Professor Xiao-Ping Yang for his interest and valuable comments in this topic. They also wish to thank an anonymous referee for his/her careful checking of this paper.

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Received: 2017-09-17

Revised: 2018-04-26

Accepted: 2018-04-27

Published Online: 2018-05-29

Published in Print: 2019-02-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2018-2019

Keywords for this article

Biased Infinity Laplacian; Existence; Comparison Principle; Lipschitz Estimate; Viscosity Solutions

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