Regularity for solutions of nonlinear elliptic equations with degenerate coercivity

We prove regularity results for solutions of some nonlinear Dirichlet problems for an equation in the form

On coercivity and irregularity for some nonlinear degenerate elliptic systems

We study the ‘universal’ strong coercivity problem for variational integrals of degenerate p-Laplacian type by mixing finitely many homogenous systems. We establish the equivalence between universal p-coercivity and a generalized notion of p-quasiconvex extreme points. We then give sufficient conditions and counterexamples for universal coercivity. In the case of noncoercive systems we give examples showing that the corresponding variational integral may have infinitely many non-trivial minimizers in W ₀^1,p which are nowhere C ¹ on their supports. We also give examples of universally p-coercive variational integrals in W ₀^1,p for p ⩾ with L ^∞ coefficients for which uniqueminimizers under affine boundary conditions are nowhere C ¹.      

Higher integrability for nonlinear elliptic equations with variable growth and discontinuous coefficients

In this paper, based on the theory of variable exponent spaces, we study the higher integrability for a class of nonlinear elliptic equations with variable growth and discontinuous coefficients. Under suitable assumptions, we obtain a local gradient estimate in Orlicz space for weak solution.     

Solutions for systems of nonlinear elliptic equations with nonlinear boundary conditions

We look for solutions of systems of nonlinear elliptic equations with nonlinear boundary conditions and values in some compact convex set M. If the nonlinear terms satisfy a sign condition on the boundary of M and the inhomogeneous terms assume their values in this set existence of solutions is proved. The proof is based on the homotopy invariance of the Leray-Schauder degree and Weinberger's strong maximum principle.     

Global bifurcation for a class of degenerate elliptic equations with variable exponents

We are concerned with the following nonlinear problem

     

Strongly nonlinear degenerate elliptic equations with discontinuous coefficients. II

We use energy methods to prove the existence and uniqueness of solutions of the Dirichlet problem for an elliptic nonlinear second-order equation of divergence form with a superlinear tem [i.e., g(x, u)=v(x) a(x)⋎u⋎ ^p−1u,p>1] in unbounded domains. Degeneracy in the ellipticity condition is allowed. Coefficients a _i,j(x,r) may be discontinuous with respect to the variable r. University of Catania, Italy. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1601–1609, December, 1997.     

Strongly nonlinear degenerate elliptic equations with discontinuous coefficients. I

This paper is concerned with the existence and uniqueness of variational solutions of the strongly nonlinear equation $$ - \sum\limits_1^m {_i \frac{\partial }{{\partial x_i }}\left( {\sum\limits_1^m {_j a_{i,j} (x, u(x))\frac{{\partial u(x)}}{{\partial x_j }}} } \right) + g(x, u(x)) = f(x)} $$ with the coefficients a _i,j (x, s) satisfying an eHipticity degenerate condition and hypotheses weaker than the continuity with respect to the variable s. Furthermore, we establish a condition on f under which the solution is bounded in a bounded open subset Ω of R^m.     

A maximum rank problem for degenerate elliptic fully nonlinear equations

The solutions to the Dirichlet problem for two degenerate elliptic fully nonlinear equations in n + 1 dimensions, namely the real Monge–Ampère equation and the Donaldson equation, are shown to have maximum rank in the space variables when n ≤ 2. A constant rank property is also established for the Donaldson equation when n = 3.     

Results for degenerate nonlinear elliptic equations involving a Hardy potential

Consider the problem

     

Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations

We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for operators of the form \(\nabla ^2 \psi + L(x,\psi ,\nabla \psi )\) which are non-decreasing in \(\psi \).     

Nonlinear elliptic equation with lower order term and degenerate coercivity

In this article, we study the regularizing effects of the lower order term in the case of a nonlinear elliptic problem with degenerate coercivity. We show that the presence of certain lower order terms has a regularizing effect on the solutions. The obtained results extend some existing ones.     

Boundedness of solutions for anisotropic degenerate elliptic equations with nonstandard growth

Under nonstandard growth conditions, following Moser's iteration technique, we prove boundedness of solutions for fourth order nonlinear elliptic equation in divergence form.     

A priori properties of solutions of nonlinear equations with degenerate coercivity and L 1-data

A Dirichlet problem for a second-order nonlinear elliptic equation in the general divergent form with a right-hand side from L ¹ is considered. The high-order coefficients in the equation are assumed to satisfy the degenerate coercivity condition. The main results concern a priori properties of summability and some estimates of entropy solutions of this problem. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 16, Differential and Functional Differential Equations. Part 2, 2006.     

The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source

The Cauchy problem for a degenerate parabolic equation with a source and variable coefficient of the form

     

Deterministic homogenization of nonlinear degenerate elliptic operators with nonstandard growth

We study the deterministic homogenization of nonlinear degenerate elliptic equations with nonstandard growth.One fundamental in this topic is to extend the classical compactness results of theΣ-convergence method to the Orlicz spaces.We also show that one can homogenize nonlinear Dirichlet problems in a general way by leaning on a simple abstract hypothesis contrary to what has been done in the determinstic homogenization theory.     

SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER

The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem.Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension,the existence of solutions of the above problem is proved.In this article,the complex analytic method is used,namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed,afterwards the above problem for the degenerate elliptic equations of second order is solved.     

Existence and regularity of multiple solutions for infinitely degenerate nonlinear elliptic equations with singular potential

In this paper, we study the Dirichlet problem for a class of infinitely degenerate nonlinear elliptic equations with singular potential term. By using the logarithmic Sobolev inequality and Hardy's inequality, the existence and regularity of multiple nontrivial solutions have been proved.