L2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:262

Article

Nystrom, Kaj¹ 

[1] Uppsala Univ, Dept Math, S-75106 Uppsala, Sweden

关键词: Second order parabolic operator;    Complex coefficients;    Boundary value problems;    Layer potentials;    Kato problem;   

DOI  :  10.1016/j.jde.2016.11.011

来源: Elsevier

【 摘 要 】

We consider parabolic operators of the form partial derivative(t) + L, L = -div A(X,t)del, in R-+(n+2) := {(X, = (x, x(n+1), t) is an element of R-n x R x R x(n+1) > 0), n >= 1. We assume that A is a (n + 1) x (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 x(n+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 L-2(Rn+1, C) = L-2(partial derivative R-+(n+2), C) under complex, L-infinity perturbations of the coefficient matrix. Subsequently, using this general result, we establish solvability of the Dirichlet, Neumann and Regularity problems for partial derivative(t) + L, by way of layer potentials and with data in L-2, assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix. (C) 2016 Elsevier Inc. All rights reserved.

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