On the continuity of solutions to elliptic equations with variable order of nonlinearity

We study the p-Laplacian with variable exponent p(x) bounded away from unity and infinity. We obtain a sufficient condition on p(x) under which all solutions of the p-Laplace equation are continuous at a fixed point of a domain, and find an estimate for the modulus of continuity of solutions.     

Increased Integrability of the Gradient of the Solution to the Zaremba Problem for the Poisson Equation

Doklady Mathematics - An estimate is obtained for the increased integrability of the gradient of the solution to the Zaremba problem in a bounded plane domain with a Lipschitz boundary and rapidly...     

On the Regularity of a Boundary Point for the <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)-Laplacian

The Dirichlet problem for the p(x)-Laplacian with a continuous boundary function is considered, and a sufficient condition is found for the continuity of its solution at a boundary point, assuming that the exponent p(x) satisfies the log-Hölder continuity condition at this point.     

Harnack inequality for a class of second-order degenerate elliptic equations

A second-order degenerate elliptic equation in divergence form with a partially Muckenhoupt weight is studied. In a model case, the domain is divided by a hyperplane into two parts, and in each part the weight is a power function of |x| with the exponent less than the dimension of the space in absolute value. It is well known that solutions of such equations are H?lder continuous, whereas the classical Harnack inequality is missing. In this paper, we formulate and prove the Harnack inequality corresponding to the second-order degenerate elliptic equation under consideration.     

Hölder Continuity of Solutions of Nondivergent Degenerate Second-Order Elliptic Equations

One considers a degenerate nondivergent second-order linear elliptic equation. In the model case, the matrix of its higher-order coefficients is diagonal and its elements are powers of the moduli of the independent variables. A sufficient condition on the power exponents is obtained which ensures interior a priori Hölder estimates of the solutions.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript><Emphasis Type="Bold">-</Emphasis>solvability of the Dirichlet problem for the heat equation in noncylindrical domains with isolated characteristic points at the boundary

We study the solvabitlity of the Dirichlet problem for the heat operator in weighted Sobolev L _p -spaces in noncylindrical paraboloid type domains with isolated characteristic points at the boundary. For any p > 1 we find a necessary and sufficient L _p -solvability condition and establish an L _p -estimate. The results are formulated in terms of Muckenhoupt type conditions on the weight. Bibliography: 10 titles.