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...context of certain random tug-of-war games [2, 16], mass transportation problems [10] and several other applications, such as image reconstruction and enhancement [6]. See also the recent approach of =-=[14]-=- to a two-phase problem of mixed type. In the next section, we introduce the infinity obstacle problem and obtain a solution u∞, passing to the limit, as p → ∞, in a sequence of solutions up to the ob...

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... 1t ) 1/(n−1). If n = 2, this implies the log-Hölder continuity. In higher dimensions the assumption ∇p ∈ Ln log L2(n−1)(Ω) would suffice for this. Remark 1.7. In the last two years, several authors =-=[26, 31, 32, 35]-=- have considered pi(·)-Laplacian type equations when pi → ∞. This leads to equations somewhat similar to our, namely −p(·)∆∞u − |∇u|2 log |∇u| ∇u · ∇p = 0. Also in this context, the Harnack inequality...

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...ons to image processing have kept the subject as the focus of an intensive research activity. Although a natural extension of the theory, the problem addressed here is a continuation of recent papers =-=[19]-=- (where the case of a variable exponent that equals infinity in a subdomain of Ω is considered) and [20] (where the Dirichlet case was treated). Closely related to this work is [18], where the authors...

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...s (see, e.g., [6], [10], [20], [21] and the references therein). Also limits of the eigenvalue problem related to the p-Laplacian have been exhaustively examined, see [14], [15], [22]. See also [17], =-=[18]-=-, [19], [22], [23] and [24] for limits as p(x)→∞. On the other hand, anisotropic problems like (1.1) have been analyzed for many years, specially concerning regularity of the solutions, see for exampl...

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...ents, viscosity solutions. 2000 Mathematics Subject Classification. 35J20, 35J60, 35J70. 1 2 J.J. MANFREDI, J.D. ROSSI AND J.M. URBANO addressed here is a follow-up from a recent paper of the authors =-=[12]-=-, where the case of a variable exponent that equals infinity in a subdomain of Ω is treated. Closely related to this work is [11], where the authors prove existence and uniqueness (via a comparison pr...

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... is a viscosity solution of (3.1). Let us now start with the proof of Theorem 4.1. Proving the fact that a p(x)-superharmonic function is a viscosity supersolution is straightforward, cf. for example =-=[22]-=-: Suppose first that v is p(x)-superharmonic. To see that v is a viscosity supersolution, assuming the opposite we find a function ϕ ∈ C2 (Ω) such that v(x0) = ϕ(x0), v(x) > ϕ(x) for all x ̸= x0, Dϕ(x...

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...uch problems. See [9, 20] and also the recent survey [14] and references therein. Although a natural extension of the theory, the problem addressed here is a natural continuation of recent papers. In =-=[22]-=-, the authors treat the case of a variable exponent that equals infinity in a subdomain of Ω and in [21], [23], the limit of p(x)-harmonic functions is studied, that is, the limit as p(x)→∞ of solutio...

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...[7] for more details about general theory of viscosity solutions, and to [14] for viscosity solutions related to the ∞−Laplacian and the p−Laplacian operators. The following result can be shown as in =-=[15]-=-[Proposition 2.4]. Lemma 2.3. A continuous weak solution to the equation (2.6) { −∆pu = λα|u|α−2uvβ Ω, u = 0, ∂Ω, 8 D. BONHEURE, J. D. ROSSI AND N. SAINTIER is a viscosity solution to (2.5). Now, we h...

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...arrive at mappings of finite distortion, we must allow Kp(·) to be unbounded, which happens if either p− = 1 or p+ =∞. Interest in such situation has arisen recently for the equation −∆p(·)u = 0, see =-=[31, 32, 33, 44, 48, 49]-=-. In this paper we pursue the lower limit, p− = 1. Theorem 1.3. Let Ω ⊂ R2 be a bounded C2 domain and let g ∈ C1,γ(∂Ω). Suppose that p ∈ C(Ω) such that • the set Y := {p = 1} has vanishing Hausdorff 1...