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Smooth regularity for solutions of the Levi Monge-Ampère equation

Francesca Lascialfari, Annamaria Montanari (2001)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We present a smooth regularity result for strictly Levi convex solutions to the Levi Monge-Ampère equation. It is a fully nonlinear PDE which is degenerate elliptic. Hence elliptic techniques fail in this situation and we build a new theory in order to treat this new topic. Our technique is inspired to those introduced in [3] and [8] for the study of degenerate elliptic quasilinear PDE’s related to the Levi mean curvature equation. When the right hand side has the meaning of total curvature of a...

Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Jacques Giacomoni, Ian Schindler, Peter Takáč (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We investigate the following quasilinear and singular problem, t o 2 . 7 c m - Δ p u = λ u δ + u q in Ω ; u | Ω = 0 , u > 0 in Ω , t o 2 . 7 c m (P) where Ω is an open bounded domain with smooth boundary, 1 < p < , p - 1 < q p * - 1 , λ > 0 , and 0 < δ < 1 . As usual, p * = N p N - p if 1 < p < N , p * ( p , ) is arbitrarily large if p = N , and p * = if p > N . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 0 1 , p ( Ω ) . While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions...

Soluzioni di viscosità

Italo Capuzzo Dolcetta (2001)

Bollettino dell'Unione Matematica Italiana

This is the expanded text of a lecture about viscosity solutions of degenerate elliptic equations delivered at the XVI Congresso UMI. The aim of the paper is to review some fundamental results of the theory as developed in the last twenty years and to point out some of its recent developments and applications.

Some possibly degenerate elliptic problems with measure data and non linearity on the boundary

Thierry Gallouët, Yannick Sire (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

The goal of this paper is to study some possibly degenerate elliptic equation in a bounded domain with a nonlinear boundary condition involving measure data. We investigate two types of problems: the first one deals with the laplacian in a bounded domain with measure supported on the domain and on the boundary. A second one deals with the same type of data but involves a degenerate weight in the equation. In both cases, the nonlinearity under consideration lies on the boundary. For the first problem,...

Some results on strongly nonlinear anisotropic differential equations

L. Bougoffa, A. El Khalil, S. El Manouni (2010)

Applicationes Mathematicae

The paper concerns the existence of weak solutions to nonlinear elliptic equations of the form A(u) + g(x,u,∇u) = f, where A is an operator from an appropriate anisotropic function space to its dual and the right hand side term is in L 1 + m with 0 < m < 1. We assume a sign condition on the nonlinear term g, but no growth restrictions on u.

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