The maximum principle and boundary α-estimates of the solution of the first boundary value problem for a parabolic equation in a non-cylindrical region

Abstract

In the theory of boundary value problems for second-order parabolic equations a priori estimates of the smoothness of the solutions are of considerable value. We note that (2 + α)- a priori estimates of the Schauder type right up to the boundary have been obtained for solutions both of the first boundary value problem [1, 2] and for the second and third boundary value problems [3–5]. The Hölder constants of the solution in these estimates depend both on the maximum of the moduli and on the Holder constants of the coefficients of the equation, which belong to the class C α . Another group of a priori estimates, originating in the work of Giorgi [6], touches on α- a priori estimates of general solutions of the first boundary value problem for parabolic equations in selfconjugate form [7]. The Holder constants in these estimates depend only on the constants of parabolicity and the maxima of the moduli of the coefficients of the equations, which are of most value for applications to non-linear equations. A simple proof of α- a priori estimates, but only near the boundary, can be obtained by proceeding from the maximum principle, if we use the method of Pucci [9], who investigated for this purpose the first boundary value problem for a second-order elliptic equation. In [in] Pucci's method was used to derive α- a priori estimates near the boundary of the solution of the first boundary value problem for a parabolic equation in a cylindrical region. In our paper Pucci's method (see [9]), modified in [10], is used to derive α- a priori estimates near the boundary for the solution of the first boundary value problem for a second-order parabolic equation in non-cylindrical regions in a fairly general form. The paper consists of three sections. In Section 1 the basic results are formulated. In Section 2 supplementary lemmas are quoted, which make use of a number of results in the theory of thermal potentials [11, 12]. In Section 3 the proofs of Theorems 2 and 3 are given.