Asymptotic symmetry of solutions of nonlinear partial differential equations

For a large class of partial differential equations on exterior domains or on ?^N we show that any solution tending to a limit from one side as x goes to infinity satisfies the property of “asymptotic spherical symmetry”. The main examples are semilinear elliptic equations, quasilinear degenerate elliptic equations, and first-order Hamilton-Jacobi equations.     

On the existence of solutions for quasilinear elliptic problems with radial potentials on exterior ball

In this paper, we are concerned with a class of quasilinear elliptic problems with radial potentials and a mixed nonlinear boundary condition on exterior ball domain. Based on a compact embedding from a weighted Sobolev space to a weighted L^s space, the existence of nontrivial solutions is obtained via variational methods.     

Finding Gateaux-Saddles by a Local Minimax Method

Motivated by quasilinear elliptic PDEs in physical applications, Gateaux-saddles of a class of functionals J:H→{±∞}∪?, which are only Gateaux-differentiable at regular points, are considered. Since mathematical results and numerical methods for saddles of 𝒞¹ or locally Lipschitz continuous functionals in the literature are not applicable, the main objective of this article is to introduce a new mixed norm strong-weak topology approach such that a mathematical framework of a local minimax method is established to handle the singularity issue and to use the Gateaux-derivative of J for finding multiple Gateaux-saddles. Algorithm implementations on weak form and error control are presented. Numerical examples solving quasilinear elliptic problems from physical applications are successfully carried out to illustrate the method. Some interesting solution properties are to be numerically observed and open for analytical verification for the first time.     

Unloading near a shear band: a free boundary problem for the wave equation

ABSTRACT

For solutions on unbounded domains of boundary value problems for a class of quasilinear elliptic equations which are not uniformly elliptic, we prove that the solutions have the same bounds as those of the boundary data.     

Diseguaglianza di Harnack per equazioni quasilineari ellittiche sopra superficie di curvatura media assegnata

Summary In this paper we prove a Harnack type inequality for non-negative solutions and supersolutions of second order quasilinear elliptic equations on hypersurfaces (inR ⁿ) of L^p prescribed mean curvature, with p>n. In the last section an application to non-parametric surfaces of Lipschitz mean curvature is given.

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Entrata in Redazione il 13 giugno 1977.     

Generalized solutions for a class of non-uniformaly elliptic equations in divergence form

We study a general class of quasilinear non-uniformly elliptic pdes in divergence from with linear growth in the gradient. We examine the notions of BV and viscosity solutions and derive for such generalized solutions various a priori pointwise and integral estimates, including a Harnack inequality. In particular we prove that viscosity solutions are unique (on strictly convex domains), are contained in the space BV ^loc and are C ^1,α almost everywhere.     

Graphs with prescribed mean curvature on hyperbolic spaces

We study a quasilinear elliptic equation in the unit ball ofℝ ^m . Using this result we get the existence of graphs with prescribed curvature on hyperbolic spacesℍ ^m inℍ ^m ×ℝ.     

On nonnegative solutions of elliptic and parabolic equations

A theorem on the nonexistence of a nonnegative nontrivial generalized solution inR ⁿ is proved for general quasilinear second-order degenerate elliptic equations. Analogous results are obtained for a large class of systems of partial differential equations, second-order parabolic and inverse parabolic equations, which are nonlinear and may be degenerate.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 114–136, 1992.     

On the continuity of minimizers for quasilinear functionals

In this paper we establish a continuity result for local minimizers of some quasilinear functionals that satisfy degenerate elliptic bounds. The non-negative function which measures the degree of degeneracy is assumed to be exponentially integrable. The minimizers are shown to have a modulus of continuity controlled by log log(1/|x|)⁻¹. Our proof adapts ideas developed for solutions of degenerate elliptic equations by J. Onninen, X. Zhong: Continuity of solutions of linear, degenerate elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 103–116.     

Two Positive Solutions of a Quasilinear Elliptic Dirichlet Problem

For a class of second order quasilinear elliptic equations we establish the existence of two non–negative weak solutions of the Dirichlet problem on a bounded domain, Ω. Solutions of the boundary value problem are critical points of C ¹–functional on H₀¹(W){H_0^1(\Omega)}. One solution is a local minimum and the other is of mountain pass type.