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...⊂ Ω and each v ∈ C(V ) u∣∣ ∂V = v ∣∣ ∂V implies esssupV |∇u| ≤ esssupV |∇v|. The equation finds applications in image processing and many other fields, we refer the reader to two recent survey papers =-=[3, 28]-=- for detailed discussions on the latest developments on PDE analysis and applications of the infinity-Laplacian equation. Like the equation of prescribed Gauss curvature, we have some flexibility in d...

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...solves the p-Laplace equation �pup = div(|Dup| p−2 Dup) = 0 in �. (1.3) If up → u∞ uniformly in � as p ↑+∞,thenu∞is an infinity harmonic function in �. We refer to a recent survey article by Crandall =-=[3]-=- for more backgrounds and information of equation (1.1). For 1 ≤ p ≤+∞,ifup is a p-harmonic function in �\{x0}, thenx0is called an isolated singularity of up. x0 is called a removable singularity, if ...

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...vertex at a boundary point and positive opening, which lie above both the obstacle and the boundary data. For more on comparison with cones and the characterization of infinity harmonic functions see =-=[7]-=-. To be concrete, for y ∈ ∂Ω and b = (b1, b2), with b1 ≥ 0, we consider the cones Kby(x) = b1|x− y|+ b2 such that Kby(x) ≥ F (x), x ∈ ∂Ω and Kby(x) ≥ Ψ(x), x ∈ Ω. Note that, since the vertex of the co...

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... as a limit of p-harmonic functions for p → ∞, [3]. It can also be shown that this corresponds to define the ∞-Laplacian as ∆∞u := 〈D2u∇u,∇u〉 , (1.2) and to say that u is ∞-harmonic whenever ∆∞u = 0, =-=[1, 2, 8, 11, 12]-=-. See also [4] for a nonlocal ∞−Laplacian. This operator appear in applications (for example, in image processing, [7]) and is related with game theory, [14]. The optimal regularity of ∞-harmonic func...

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... considerable attention recently. In particular, existence, uniqueness and local regularity have become topics of great interest. For greater motivation and context, we direct the reader to the works =-=[1, 4, 8, 9, 18]-=-. Our current work is more along the lines of [5, 6, 16, 17]. From hereon, we will often refer to (1.1) as the eigenvalue problem. One of the main tasks is to be able to characterize the principal or ...

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...vertex at a boundary point and positive opening, which lie above both the obstacle and the boundary data. For more on comparison with cones and the characterization of infinity harmonic functions see =-=[7]-=-. To be concrete, for y ∈ ∂Ω and b = (b1, b2), with b1 ≥ 0, we consider the cones Kby(x) = b1|x− y|+ b2 such that Kby(x) ≥ F (x), x ∈ ∂Ω and Kby(x) ≥ Ψ(x), x ∈ Ω. Note that, since the vertex of the co...

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...ng considerable attention recently. In particular, existence, uniqueness and local regularity have been topics of great interest. For greater motivation and context, we direct the reader to the works =-=[3, 4, 8, 9, 14, 15, 17]-=-. Moreover, degenerate parabolic equations, in general, are of great interest, see for instance, [13]. An additional reason for studying (1.1) is the connection that it has with the following problem ...