ON A CERTAIN DEGENERATE PARABOLIC EQUATION ASSOCIATED WITH THE INFINITY-LAPLACIAN

. The comparison, uniqueness and existence of viscosity solutions to the Cauchy-Dirichlet problem are proved for a degenerate parabolic equation of the form u t = ∆ ∞ u , where ∆ ∞ denotes the so-called inﬁnity-Laplacian given by ∆ ∞ u = P Ni,j =1 u x i u x j u x i x j . Our proof relies on a coercive regulariza- tion of the equation, barrier function arguments and the stability of viscosity solutions.

1. Introduction. Aronsson [2] introduces the so-called infinity-Laplacian ∆ ∞ given by to investigate the existence of absolutely minimizing Lipschitz extensions (AMLE's for short) of functions g defined only on the boundary ∂Ω of a domain Ω in R N into Ω. Here the AMLE of g into Ω means a function u ∈ W 1,∞ (Ω) satisfying that u = g on ∂Ω and that for every open subset U of Ω and φ ∈ W 1,∞ (U ), if u − φ ∈ W 1,∞ 0 (U ), then More precisely, the following elliptic problem is proposed in [2] as an Euler equation of the above variational problem for smooth AMLE's: Furthermore, various problems related to elliptic equations associated with the infinity Laplacian, e.g., the regularity of solutions, Harnack's inequality, limiting problems associated with p-Laplacian as p → +∞, eigenvalue problem, L ∞inequality of the Poincaré type, have been vigorously studied by many authors (see, e.g., [4], [6], [5], [7], [10], [11], [12], [14], [17], [22]). On the other hand, to the best of the authors' knowledge, parabolic problems associated with the infinity-Laplacian have not been studied yet except in [9], [21] and [16].
This paper is concerned with the following parabolic problem: where Ω is a bounded domain in R N with boundary ∂Ω, PQ denotes the parabolic boundary of Q = Ω×(0, T ) and u t denotes the time-derivative of u = u(x, t) (see the notation in the end of this section). The main purpose of this paper is to investigate the comparison, uniqueness and existence of viscosity solutions u = u(x, t) of the Cauchy-Dirichlet problem (3), (4). Another type of parabolic equation associated with the infinity-Laplacian is also studied by Juutinen and Kawohl in [16], where they treat the following: They investigate the existence and uniqueness of solutions of the Cauchy-Dirichlet problem for (5) with initial-boundary data ϕ, and moreover, they deal with the Cauchy problem for the case Ω = R N as well. To prove the existence, they introduce approximate problems of the form (u ε,δ ) t = ε∆u ε,δ + ∆ ∞ u ε,δ /(|Du ε,δ | 2 + δ) with ε, δ > 0, and establish boundary Hölder estimates of their solutions by constructing barrier functions.
To prove the existence for (3), (4), we introduce the following approximate problems with ε > 0: and prove the existence of classical solutions u ε for the Cauchy-Dirichlet problems for (6) with initial-boundary data ϕ. Moreover, as in [16], we employ barrier function arguments to establish a priori estimates for the solutions u ε . Our proof of establishing a priori estimates is inspired by [16].
In the next section, we state our main results on the comparison, uniqueness and existence of viscosity solutions of the Cauchy-Dirichlet problem (3), (4). Section 3 is devoted to our proof of the existence result.
Notation: Throughout this paper, we use the following notation: and D 2 denotes the N × N matrix whose (i, j)-th element is D 2 ij . Furthermore, we also use the Einstein summation convention, where we sum over repeated Greek indices. As for the definitions of function spaces such as C 2,1 , H α and H , /2 and (semi-)norms, we refer the reader to [19, pp. 7-8]. Moreover, we denote by Lip(Q) the class of Lipschitz continuous functions in Q, and we simply denote by | · | ∞ the sup-norm in the corresponding space if no confusion arises.
2. Main results. Before stating our main results, we give a couple of notation and definitions to be used. Set where S N denotes the set of all symmetric N × N matrices. We are then concerned with viscosity solutions of (3) given in the following.
Moreover, u ∈ C(Q) is said to be a viscosity solution in Q of (3) if it is both a viscosity subsolution and a viscosity supersolution in Q of (3).
Applying Theorem 8.2 and related remarks of [8], the comparison principle for (3), (4) is immediately derived, and moreover, it also implies the continuous dependence on initial-boundary data ϕ and the uniqueness of solutions.

Theorem 1 (Comparison and uniqueness).
Let Ω be a bounded domain in R N with boundary ∂Ω and let u ∈ U SC(Q) and v ∈ LSC(Q) be a viscosity subsolution and a viscosity supersolution in In particular, let ϕ 1 , ϕ 2 ∈ C(Q) and let u 1 and u 2 be viscosity solutions in Q of (3), (4) with the initial-boundary data ϕ 1 and ϕ 2 , respectively. Then it follows that which also implies the uniqueness of solutions.
Proof of Theorem 1. Due to Theorem 8.2 of [8], the comparison part follows immediately. Now, let u 1 and u 2 be viscosity solutions of (3), (4) with the initial-boundary data ϕ 1 and ϕ 2 , respectively, and put w ± (x, t) Then the functions w − and w + become a viscosity subsolution and a viscosity supersolution of (3), (4) with ϕ replaced by ϕ 1 respectively. Thus we have which implies (7). In particular, if ϕ 1 = ϕ 2 on PQ, then the uniqueness of solutions follows.
As for the existence of solution, we first introduce the following assumption.
We have several steps to establish a Hölder estimate for u ε in Q. The first step is concerned with a Lipschitz estimate for u ε (x, ·) at t = 0 (see Lemma 2), and the second step yields a Lipschitz estimate at any t ∈ (0, T ) (see Lemma 3). In the third step, we estimate a Hölder constant of u ε (·, t) on ∂Ω (see Lemma 4). Hence these three steps imply a boundary Hölder estimate on PQ (see Lemma 5). Finally, we derive a global Hölder estimate for u ε in Q from the boundary Hölder estimate (see Lemma 6). Our derivations of these estimates are due to the similar barrier function argument as in [16], and we also employ the translation invariance of the equation (10) to extend Lipschitz and Hölder estimates established only on the boundary, e.g., t = 0, ∂Ω, PQ, as in [18] (a similar argument using the translation invariance of an equation is also found in [13,Corollary 2.11]). Lemma 2 (Lipschitz estimate for u ε (x, ·) at t = 0). Let Ω be a bounded domain in R N with boundary ∂Ω and let u ∈ C(Q) ∩ C 2,1 (Q) be a classical solution in Q = Ω × (0, T ) of the Cauchy-Dirichlet problem (10), (11) with ϕ ∈ C 2,1 (Q). Then it follows that |u(x, t) − ϕ(x, 0)| ≤ M 1 t for all t ∈ (0, T ) and x ∈ Ω, (12) where M 1 := 2(|Dϕ| 2 ∞ + 1)|D 2 ϕ| ∞ + |ϕ t | ∞ . Proof of Lemma 2. Put w ± (x, t) = ϕ(x, 0) ± M 1 t and observe that We can also deduce that P (w − t (x, t), Dw − (x, t), D 2 w − (x, t)) ≥ 0 for all (x, t) ∈ Q and w − ≤ ϕ on PQ. Therefore the classical comparison principle ensures that w − ≤ u ≤ w + in Q. Hence we obtain (12).
By using the translation invariance of the equations (10) and the above lemma, we can obtain a Lipschitz estimate for u ε (x, ·) in (0, T ).
We next establish a Hölder estimate for u ε (·, t) on ∂Ω.
Proof of Lemma 6. Let h : . We then find that v still remains to be a classical solution in Q+h of (10), (11) with ϕ replaced by ϕ(· − h x , · − h t ). Then, by Lemma 5, we can assure that, for ( Furthermore, since v ± M 3 |h| α,1 also become classical solutions in Q ∩ (Q + h) of (10), the classical comparison theorem ensures that v − M 3 |h| α,1 ≤ u ≤ v + M 3 |h| α,1 in Q ∩ (Q + h). From the arbitrariness of h, we can verify (15).
By virtue of the global Hölder estimate for u ε in Lemma 6 and Ascoli-Arzela's compactness theorem, taking a sequence ε n → +0, we can deduce that u εn → u uniformly on Q as ε n → +0. We also note that P ε (s, p, X) → P (s, p, X) as ε → +0, for all (s, p, X) ∈ R × R N × S N .