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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, $t o 2.7 c m \{\begin{matrix} - Δ_{p} u = \frac{λ}{u^{δ}} + u^{q} & in Ω; \\ {u |}_{\partial Ω} = 0, u > 0 & in Ω, \end{matrix} t o 2.7 c m (P)$ where $Ω$ is an open bounded domain with smooth boundary, $1 < p < \infty$ , $p - 1 < q \leq p^{*} - 1$ , $λ > 0$ , and $0 < δ < 1$ . As usual, $p^{*} = \frac{N p}{N - p}$ if $1 < p < N$ , $p^{*} \in (p, \infty)$ is arbitrarily large if $p = N$ , and $p^{*} = \infty$ 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...

Solution to a semilinear problem on type II regions determined by the Fučik spectrum.

Tomiczek, Petr (2003)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Solutions faibles des équations elliptiques du deuxième ordre

Franchi (1985)

Publications mathématiques et informatique de Rennes

Solutions fortes pour des problèmes aux limites du second ordre dégénérés

M. Langlais (1977/1978)

Séminaire Équations aux dérivées partielles (Polytechnique)

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.

Solvability of degenerate elliptic boundary value problems: another approach

Alois Kufner, Salvatore Leonardi (1994)

Mathematica Bohemica

Using a Hardy-type inequality, the authors weaken certain assumptions from the paper [1] and derive existence results for equations with a stronger degeneration.

Currently displaying 321 – 340 of 453

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Displaying 321 – 340 of 453

Remarks on the gradient of an infinity-harmonic function.

Removable singularities for Hölder continuous quasiregular mappings in the plane.

Renormalized entropy solutions for degenerate nonlinear evolution problems.

Renormalized solutions for nonlinear degenerate elliptic problems with L1 data.

Represenation of a p-harmonic function near a critical point in the plane.

Rough isometries and p-harmonic functions with finite Dirichlet integral.

Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in $ℝ^{n}$

Semidifferentials, quadratic forms and fully nonlinear elliptic equations of second order

Separation theorem with respect to sub-topical functions and abstract convexity.

Smooth regularity for solutions of the Levi Monge-Ampère equation

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

Solution to a semilinear problem on type II regions determined by the Fučik spectrum.

Solutions faibles des équations elliptiques du deuxième ordre

Solutions fortes pour des problèmes aux limites du second ordre dégénérés

Soluzioni di viscosità

Solvability of degenerate elliptic boundary value problems: another approach

Solvability of nonlinear Dirichlet problem for a class of degenerate elliptic equations.

Some boundary problems for the equations with strong degeneration

Some Hardy type inequalities in the Heisenberg group.

Some logarithmic function spaces, entropy numbers, applications to spectral theory [Book]

Currently displaying 321 – 340 of 453