Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions

By M. Meyries, Roland Schnaubelt. 2012

46 citations recorded by the source corpus.

We develop a maximal regularity approach in temporally weighted Lp‐spaces for vector‐valued parabolic initial‐boundary value problems with inhomogeneous boundary conditions, both of static and of relaxation type. Normal ellipticity and conditions of Lopatinskii‐Shapiro type are the basic structural assumptions. The weighted framework allows to reduce the initial regularity and to avoid compatibility conditions at the boundary, and it provides an inherent smoothing effect of the solutions. Our main tools are interpolation and trace theory for anisotropic Slobodetskii spaces with temporal weights, operator‐valued functional calculus, as well as localization and perturbation arguments.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ccd469d43e1b2922cf01beb687d8c56841a30c05

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

A structurally damped plate equation with Dirichlet-Neumann boundary conditions (2015, background): “4 of [26] for a more detailed exposition in a somewhat different situation).”
Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights (2012, background): “In the companion paper [25] we establish the necessary theory for linear inhomogeneous initial-boundary value problems, based on the present study of the underlying function spaces.”
On quasilinear parabolic evolution equations in weighted Lp-spaces II (2009, methodology): “The idea of employing time-weights is by no means new, at has been employed before in papers by [6, 22, 23] and others.”
Maximal Regularity for Non-autonomous Equations with Measurable Dependence on Time (2014, background): “For more details and applications to evolution equations we refer to [39, 59, 67, 72, 73, 83].”
Traces and embeddings of anisotropic function spaces (2012, unknown): “If more than one boundary condition is involved, then mixed derivative (or Newton polygon) embeddings are important to determine the optimal regularities of all boundary inhomogeneities, see [11, 12, 13, 15, 22, 27, 28] and Section 3.”