L2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients
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- 作者：Kaj Nyströ ; m kaj.nystrom@math.uu.se
- 关键词：35K20 ; 31B10
- 刊名：Journal of Differential Equations
- 出版年：2017
- 出版时间：5 February 2017
- 年：2017
- 卷：262
- 期：3
- 页码：2808-2939
- 全文大小：1197 K
- 卷排序：262
文摘
We consider parabolic operators of the form∂t+L,L=−divA(X,t)∇, in R+n+2:={(X,t)=(x,xn+1,t)∈Rn×R×R:xn+1>0}, n≥1n≥1. We assume that A is a (n+1)×(n+1)(n+1)×(n+1)-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate xn+1xn+1 as well as of the time coordinate t . For such operators we prove that the boundedness and invertibility of the corresponding layer potential operators are stable on L2(Rn+1,C)=L2(∂R+n+2,C) under complex, L∞L∞ perturbations of the coefficient matrix. Subsequently, using this general result, we establish solvability of the Dirichlet, Neumann and Regularity problems for ∂t+L∂t+L, by way of layer potentials and with data in L2L2, assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix.