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...ped and u is convex and nondifferentiable on parts of its ridge. □ 3 The normalized or game-theoretic ∞-Laplacian Recently the following operator has received considerable attention (see for instance =-=[13, 8, 11]-=-) in the PDE community ∆ N ∞u = 〈D 2 uDu, Du〉|Du| −2 .5 Let us therefore study the differential equation −uνν = 1 in Ω (3.1) under boundary conditions (1.2) and (1.3). A simple integration shows that...

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...ches u from above at x satisfies F∗p(∇ϕ(x), D2ϕ(x)) ≥ g(x). The above definitions are equivalent. The proof of this fact is based on the well-known fourth order perturbation argument, cf. [4], [8] or =-=[11]-=-. Proposition 4. Definitions 1, 2, and 3 are equivalent for 1 ≤ p ≤ ∞ and g ∈ C(Ω), g > 0 (or g < 0). Proof. We restrict ourselves to the case of finite p, since the case p = ∞ follows by a simple mod...

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...ty solutions for fully nonlinear problems. Many important results have been achieved and published in, e.g., [ACJ], [BB], [Ba], [BLW1], [BLW2], [BEJ], [Bh], [CE], [CEG], [CIL], [CY], [EG], [EY], [J], =-=[JK]-=-, [JLM1], [JLM2], [LM1], [LM2], [Ob]. On the other hand, the ∞-Laplace equation has been found to have some very interesting applications in areas such as image processing (see e.g. [CMS], [Sa]), mass...

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...ves. We are talking of the strongly degenerate equation ut − ∆∞u = 0 (1) where ( ∆∞u := D 2 u Du ) · |Du| Du (2) |Du| is the 1−homogeneous infinity Laplacian. The seminal paper of Juutinen and Kawohl =-=[11]-=-, where basic results on the existence, uniqueness and regularity of solutions are collected, is the first attempt to systematically study (1). One of the issues touched in that paper concerns the ass...

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...sidered in [MPR10], in connection with normalized p-parabolic equations and tug-of-war games, see also [BG] and [Doe11]. The parabolic equation involving a normalized infinity Laplacian is studied in =-=[JK06]-=-. 2. Preliminaries Recall that (Ω,F, {Fs},P) denotes a complete filtered probability space with a right-continuous filtration supporting a (n+1)-dimensional and {Fs}-adapted Brownian motion W = (W1, ....

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...omogeneous infinity Laplace equation 0u∞∆ =sis the Euler-Lagrange equation associated with L ∞ -svariational problem. See for details [1]-[5] and the references therein. Recently, Juutinen and Kawohl =-=[6]-=- con-sF. Lius489ssidered the degenerate and singular parabolic equations2 1 tu u Du ∞= ∆ .s(2)sThey proved the existence and uniqueness for both Dirichlet and Cauchy problems, established interior and...

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...on of an associated stochastic control problem (see e.g. [25, 26]). (d) For the infinity-Laplacian, the Dirichlet problem can be solved using a rather simple two-person stochastic game [39] (see also =-=[1, 2, 8, 22, 34, 40, 46, 47]-=- for related work including extensions to evolution problems and the p-Laplacian). Until recently, the only control-based interpretations of second-order elliptic or parabolic equations involved stoch...