Regularity of the solutions of degenerate elliptic equations in divergent form

A priori estimates of the solution to the Dirichlet problem and of its first derivatives in terms of weighted Lebesgue norms are obtained for linear and quasilinear equations with degeneracy from A _p Muckenhoupt classes.     

On the continuity of solutions to degenerate elliptic equations

The local behavior of solutions to a degenerate elliptic equation

     

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.     

Local behaviour of singular solutions for nonlinear elliptic equations in divergence form

We consider the following class of nonlinear elliptic equations $$\begin{array}{ll}{-}{\rm div}(\mathcal{A}(|x|)\nabla u) +u^q=0\quad {\rm in}\; B_1(0)\setminus\{0\}, \end{array}$$ where q > 1 and ${\mathcal{A}}$ is a positive C ¹(0,1] function which is regularly varying at zero with index ${\vartheta}$ in (2?N,2). We prove that all isolated singularities at zero for the positive solutions are removable if and only if ${\Phi\not\in L^q(B_1(0))}$ , where ${\Phi}$ denotes the fundamental solution of ${-{\rm div}(\mathcal{A}(|x|)\nabla u)=\delta_0}$ in ${\mathcal D'(B_1(0))}$ and δ₀ is the Dirac mass at 0. Moreover, we give a complete classification of the behaviour near zero of all positive solutions in the more delicate case that ${\Phi\in L^q(B_1(0))}$ . We also establish the existence of positive solutions in all the categories of such a classification. Our results apply in particular to the model case ${\mathcal{A}(|x|)=|x|^\vartheta}$ with ${\vartheta\in (2-N,2)}$ .     

Discontinuous superlinear elliptic equations of divergence form

We prove existence and global Hölder regularity of the weak solution to the Dirichlet problem $\left\{ {\begin{array}{lc} {{\rm div} \left( a^{ij} (x,u)D_{j} u \right) = b(x,u,Du) \quad {\rm in}\, \Omega \subset {\mathbb R}^{n}, \, n \ge 2,} \\ {u = 0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,{\rm on}\, \partial\Omega \in C^{1}. } \\ \end{array}} \right.$ The coefficients a ^ij (x, u) are supposed to be VMO functions with respect to x while the term b(x, u, Du) allows controlled growth with respect to the gradient Du and satisfies a sort of sign-condition with respect to u. Our results correct and generalize the announcements in Ragusa (Nonlinear Differ Equ Appl 13:605–617, 2007, Erratum in Nonlinear Differ Equ Appl 15:277–277, 2008).     

Multiplicity of positive solutions for a class of elliptic equations in divergence form

We prove results concerning the existence and multiplicity of positive solutions for the quasilinear equation

     

Regularity of solutions of linear and quasilinear equations of elliptic type in divergence form

Translated from Matematicheskie Zametki, Vol. 53, No. 1, pp. 68–82, January, 1993.     

Boundary values of the solutions of degenerate elliptic equations

One determines the natural boundary values of the solutions of degenerate elliptic equations. It is shown that the natural boundary values can be expressed in terms of the Dirichlet data with the aid of the classical pseudodifferential operator.Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 123–139, 1986.In conclusion, the author expresses his deep gratitude to M. Z. Solomyak for useful discussions on the formulation of the problem and on the results of the paper.     

On the theory of degenerate elliptic equations

Siberian Mathematical Journal -     

Remarks on the Hölder continuity of solutions to elliptic equations in divergence form

The classical proofs of the De Giorgi–Nash–Moser Theorem are based on the iteration of some inequality through countably many concentric balls. In this note, we present a new approach to the Hölder continuity of solutions to elliptic equations in divergence form, which avoids any form of discrete iteration. In particular, we prove that a suitable energy function satisfies a differential inequality, whose integration yields a new proof of the crucial step in the regularity result.     

Existence of solutions for degenerate quasilinear elliptic equations

This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L¹(Ω) and f∈(L^p′(Ω))^N.     

On differentiability of solutions to homogeneous elliptic equations of divergence type

Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 6, pp. 1279–1286, November–December, 1994.     

Fundamental solutions of degenerate or singular elliptic equations

Exitence of fundamental solutions is established for a class of degenerate or singular, second-order, linear elliptic partial differential equations. This class contains, for example, Tricomi's equation in the upper half-plane which arises in the study of aerodynamics; the equation of Weinstein's generalized axially symmetric potential theory which arises in the study of fluid dynamics and elasticity; and Schrödinger's equation with a singular potential which arises in quantum mechanics.     

Behaviors of solutions to a class of nonlinear degenerate parabolic equations not in divergence form

In this note we study the nonexistence and long time behavior of solutions for a class of nonlinear degenerate parabolic equations of the non-divergence type.