tailieunhanh - The regularity in Lp sobolev spaces for cauchy dirichlet problem for second order parabolic equations in a domain with conical points

In this paper, we study the regularity of the solution for Cauchy-Dirichlet problem for the second order parabolic equations in domains with conical points. The regularity of the solution in Lp-weighted Sobolev spaces will be established. | JOURNAL OF SCIENCE OF HNUE Natural Sci. 2012 Vol. 57 No. 3 pp. 48-52 THE REGULARITY IN LP -SOBOLEV SPACES FOR CAUCHY-DIRICHLET PROBLEM FOR SECOND-ORDER PARABOLIC EQUATIONS IN A DOMAIN WITH CONICAL POINTS Nguyen Thanh Anh and Tran Manh Cuong Hanoi National University of Education Abstract. In this paper we study the regularity of the solution for Cauchy- Dirichlet problem for the second order parabolic equations in domains with conical points. The regularity of the solution in Lp -weighted Sobolev spaces will be established. Keywords Parabolic equation Cauchy-Dirichlet problem nonsmooth do- mains conical points regularity. 1. Introduction The Lp -theory of second order parabolic equations have been studied widely under various regularity assumptions on the coefficients and the domains. Let us mention some works related to this topic. For the case of continuous leading coeffi- cients and smooth domains the Wp2 1 -solvability has been known for a long time see for example 5 . Bramanti and Cerutti established the Wp2 1 -solvability the Cauchy- Dirichlet problem for second order parabolic equations with VMO coefficients in domains being of the C 1 1 class in 2 . Byun obtained Wp1 -solvability for second or- der divergence parabolic equations with small BMO coefficients in Lipschitz domains with small Lipschitz constants in 3 . In domains of a more general class analogous estimates were received by Alkhutov and Gordeev in 1 for parabolic equations with continuous coefficients. The present paper is concerned with Lp -estimates for of the Dirichlet-Cauchy problem for parabolic equations of second order in domains with conical points. The unique existence of weak solutions of such problems is reduced in 1 . Our goal is to study the regularity of the weak solutions in Lp -weighted Sobolev spaces. 2. Preliminaries and the statement of the main result Suppose that G is a bounded domain in Rn n gt 2 with the boundary G infinitely smooth everywhere except at the origin 0 of .