L p -Regularity for the Cauchy-Dirichlet Problem for Parabolic Equations in Convex Polyhedral Domains
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Lp -Regularity for the Cauchy-Dirichlet Problem for Parabolic Equations in Convex Polyhedral Domains Vu Trong Luong1 · Nguyen Thanh Anh2 · Do Van Loi3
Received: 6 November 2014 / Revised: 1 December 2015 / Accepted: 7 December 2015 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016
Abstract In this paper, we study the regularity of the solution of the initial-boundary value problem with Dirichlet boundary conditions for second-order divergence parabolic equations in a domain of polyhedral type. We establish several results on the regularity of the solution in weighted Lp -Sobolev spaces. Keywords Parabolic equation · Polyhedral domains · Regularity of solutions Mathematical Subject Classification (2010) 35B65 · 35D30 · 35K20
1 Introduction The Lp -theory of second-order parabolic equations has 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 coefficients and smooth domains, the Wp2,1 -solvability has been known for a long time, see, for example, [7]. In [2], Bramanti and Cerutti established the Wp2,1 -solvability of the Cauchy-Dirichlet problem for second-order
Vu Trong Luong
[email protected] Nguyen Thanh Anh [email protected] Do Van Loi [email protected] 1
Department of Mathematics, Tay Bac University, Sonla, Vietnam
2
Vietnam Education Publishing House, Hanoi, Vietnam
3
Department of Mathematics, Hong Duc University, Thanhhoa, Vietnam
V. T. Luong et al.
parabolic equations with VMO coefficients in domains being of the C 1,1 class. For the case of nonsmooth domains, let us mention the works [3] and [1] in which the unique existence of weak solutions in Wp1 -Sobolev spaces was established. In [8], coercive estimates for strong solutions to the Dirichlet problem and to the Neumann problem for the heat operator in a dihedral angle were obtained. In this paper, we are concerned with the regularity of the weak solution obtained in [1] in the case the domain is of polyhedral type. For elliptic boundary value problems in domains of this type, many results on the solvability and the regularity in weighted Lp Sobolev spaces are known (see the monograph [6] and the references therein). Basing on the regularity results for elliptic boundary value problems in [6] together with the solvability result for the Cauchy-Dirichlet problem for parabolic equations in [1], we will establish several results on the regularity of the weak solution in weighted Lp -Sobolev spaces. Our method has similarities with [4] in which the authors considered only the case of weighted Sobolev spaces with the L2 -norms. Our paper is organized as follows. In Section 2, we introduce some notations and preliminaries. Section 3 is devoted to studying the regularity in time of the weak solution. This is an intermediate step to investigate the global regularity of the solution in Section 4.
2 Notations and Preliminaries Let G be a bo
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