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.     

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.     

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.     

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.     

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.     

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.     

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 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.     

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.     

Nonexistence of Positive Supersolutions of Elliptic Equations via the Maximum Principle

We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of ? ⁿ . The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the p-Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as well as exterior domains and cone-like domains.     

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.     

A non-smooth three critical points theorem with applications in differential inclusions

We extend a recent result of Ricceri concerning the existence of three critical points of certain non-smooth functionals. Two applications are given, both in the theory of differential inclusions; the first one concerns a non-homogeneous Neumann boundary value problem, the second one treats a quasilinear elliptic inclusion problem in the whole \mathbb R^N{\mathbb R^N}.     

On a quasilinear system involving the operator curl

This paper concerns a quasilinear system involving the operator curl. This system is an approximation of the anisotropic Ginzburg–Landau system which describes the Meissner state of type II superconductors. The existence of the weak solutions of the quasilinear system is proved by applying a variational method to a modified functional, and the C ^2+α regularity of the weak solutions H is established without assuming the boundedness of curl H.     

Existence results for hemivariational inequalities with measures

We prove existence results for multivalued quasilinear elliptic problems of hemivariational inequality type with measure data right-hand sides. In case of L ¹-data, we study existence and enclosure behaviors of solutions by an appropriate sub-supersolution approach. The proofs of our results are based on general existence theory for multivalued pseudomonotone operators, and approximation-, truncation-, and special test function techniques.     

High-order difference schemes for two-dimensional elliptic equations

A high-order finite-difference approximation is proposed for numerical solution of linear or quasilinear elliptic differential equation. The approximation is defined on a square mesh stencil using nine node points and has a truncation error of order h⁴. Several test problems, including one modeling convection-dominated flows, are solved using this and existing methods. The results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency.     

Necessary and Sufficient Conditions of Existence for Positive Solution to a Class of Quasilinear Elliptic Systems in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

In this paper, we establish the necessary and sufficient conditions of existence for a positive solution to a class of non-variational quasilinear elliptic systems in R ^N . The sufficient condition of existence result bases on the Mountain Pass Lemma and the sub-super solution methods, and the necessary condition is a consequence of a Picone’s identity. The system models some phenomena in different physical and other natural sciences: non-Newtonian mechanics, nonlinear elasticity and glaciology, combustion theory, population biology and so on.     

Singular quasilinear elliptic problems with indefinite weights and critical potential

The Dirichlet problem for a quasilinear sub-critical inhomogeneous elliptic equation with critical potential and singular coefficients, which has indefinite weights in RN , is studied in this paper. We discuss the corresponding eigenvalue problems by the variational techniques and Picone’s identity, and obtain the existence of non-trivial solutions for the inhomogeneous Dirichlet problem by using Hardy inequality, Mountain Pass Lemma in conjunction with the property of eigenvalues.     

A counter-example to the boundary regularity of solutions to elliptic quasilinear systems

It is shown that solutions to the Dirichlet problem for quasilinear elliptic systems in a domain ofR ⁿ n3 with smooth boundary datum can be singular at the boundary.     

Boundary blow-up solutions of p-Laplacian elliptic equations with lower order terms

We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type −Δ _p u = a(x)u ^m −b(x)f(u) with p >  1 and 0 <  m <  p−1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p =  2 to the quasilinear case with p >  1.     

Error estimates for the numerical approximation of a quaslinear Neumann problem under minimal regularity of the data

The finite element based approximation of a quasilinear elliptic equation of non monotone type with Neumann boundary conditions is studied. Minimal regularity assumptions on the data are imposed. The consideration is restricted to polygonal domains of dimension two and polyhedral domains of dimension three. Finite elements of degree k ≥ 1 are used to approximate the equation. Error estimates are established in the L ²(Ω) and H ¹(Ω) norms for convex and non-convex domains. The issue of uniqueness of a solution to the approximate discrete equation is also addressed.