Regularity of solutions of divergence form elliptic equations

The aim of this paper is to study local regularity in the Morrey spaces of the first derivatives of the solutions of an elliptic second order equation in divergence form

where is assumed to be in some spaces and the coefficients belong to the space

     

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

     

On the boundedness of solutions for degenerate nonlinear elliptic equations

We establish the boundedness of solutions of Dirichlet Problem for a class of degenerate nonlinear elliptic equations. To prove the result we follow a modification of Moser's method.     

Dahlberg's bilinear estimate for solutions of divergence form complex elliptic equations

We consider divergence form elliptic operators , defined in , where the coefficient matrix is , uniformly elliptic, complex and -independent. Using recently obtained results concerning the boundedness and invertibility of layer potentials associated to such operators, we show that if in , then for any vector-valued we have the bilinear estimate

where and where is the usual non-tangential maximal operator. The result is new even in the case of real symmetric coefficients and generalizes an analogous result of Dahlberg for harmonic functions on Lipschitz graph domains. We also identify the domain of the generator of the Poisson semigroup for the equation in

     

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.     

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.     

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.     

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.