Chapitre V. Juifs, Grecs et Égyptiens

Barbu, Daniel

doi:10.4000/books.pulg.8867

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Elliptic Problem1

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2020

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“…If h ∈ L ∞ (Ω), we obtain (12) with q = ∞ by taking the limit as q → ∞ (see Ch.1 §3 Theorem 1 of [58]).…”

Section: Elliptic Problemmentioning

confidence: 99%

Solvability of Doubly Nonlinear Parabolic Equation with $p$-Laplacian

Uchida¹

2020

Preprint

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In this paper, we consider a doubly nonlinear parabolic equation ∂tβ(u) − ∇ • α(x, ∇u) ∋ f with the homogeneous Dirichlet boundary condition in a bounded domain, where β : R → 2 R is a maximal monotone graph satisfying 0 ∈ β(0) and ∇•α(x, ∇u) stands for a generalized p-Laplacian. Existence of solution to the initial boundary value problem of this equation has been investigated in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on β. However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for 1 < p < 2. Main purpose of this paper is to show the solvability of the initial boundary value problem for any p ∈ (1, ∞) without any conditions for β except 0 ∈ β(0). We also discuss the uniqueness of solution by using properties of entropy solution.

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“…If h ∈ L ∞ (Ω), we obtain (12) with q = ∞ by taking the limit as q → ∞ (see Ch.1 §3 Theorem 1 of [58]).…”

Section: Elliptic Problemmentioning

confidence: 99%