Elliptic and parabolic boundary value problems in weighted function spaces. (English) Zbl 1502.35059
Summary: In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted \(L_q\)-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt \(A_{\infty}\)-class. In the Besov space case we have the restriction that the microscopic parameter equals to \(q\). Going beyond the \(A_p\)-range, where \(p\) is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators.
MSC:
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46E40 | Spaces of vector- and operator-valued functions |
| 47G30 | Pseudodifferential operators |
Keywords:
Bessel potential; Lopatinskii-Shapiro; maximal regularity; mixed-norm; Poisson operator; smoothing; Sobolev spaces; Triebel-Lizorkin spaces; UMD spaces; inhomogeneous boundary conditionsReferences:
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