The maximum principle and local Lipschitz estimates near the lateral boundary for solutions of second-order parabolic equations

Abstract

ellipticity constants and the maximum moduli of the coefficients of the equation. In this paper we initiate the study of a similar problem for linear second-order parabolic equations. We shall show that a regular solution of a second-order parabolic equation satisfies a Lipschitz condition (in a parabolic metric) near the lateral part of the parabolic boundary of a region if the boundary surface has the property of strict (with a Dini condition) p-paraboloidness from without with respect to the region, and if natural conditions are satisfied for the behavior of the boundary values of the solution in the neighborhood under consideration. For n = 1, the class of surfaces satisfying a strict (with a Dini condition) Pz paraboloidness condition from without will include Gevrey curves and even curves of the type F ~ [7], where the modulus of continuity ~ satisfies a Dini condition; for n -> 2, it includes surfaces with the following two properties: 1) Their cuts by hyperplanes orthogonal to the Ot axis have a continuously varying outer normal with a modulus of continuity that satisfies a Dini condition, and they lie in a half-space specified by the outer normal; 2) the cuts by hyperplanes parallel to the Ot axis satisfy with respect to t a Dini-- H6lder condition with an Windex" l/2 + ~2, where the modulus of continuity ~2 likewise satisfies a Dini condition. The obtained conditions of local regularity (in the Lipschitz sense) are exact. We shall Show that there exist surfaces near which a regular solution (that reaches a strict extremum at a boundary point) of a parabolic equation cannot satisfy a Lipschitz condition whatever the behavior of the solution at the boundary. The surfaces have the following properties: 1) Their cuts by a hyperplane orthogonal to the Ot ~is (and passing through the boundary point of an extremum) have a continuously varying outer normal with a modulus of continuity that does not satisfy a Dini condition; 2) these surfaces are located in a half-space specified by the outer normal in such a way that the region near these surfaces has a very "narrow" completion. Let us note that Dini's condition occurs also in the requirement towards the behavior of the solution at the boundary, and we show that Dini's condition is also an exact (i. e., necessary) condition for local regularity of the solution in the Lipschitz sense near the lateral part of a parabolic boundary.