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Open Access Research

Lp-estimates for a transmission problem of mixed elliptic-parabolic type

Robert Denk* and Tim Seger

Author Affiliations

Department of Mathematics and Statistics, University of Konstanz, Konstanz, 78457, Germany

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Boundary Value Problems 2014, 2014:22  doi:10.1186/1687-2770-2014-22

Published: 22 January 2014

Abstract

We consider the situation when an elliptic problem in a subdomain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M2">View MathML</a> of an n-dimensional bounded domain Ω is coupled via inhomogeneous canonical transmission conditions to a parabolic problem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M3">View MathML</a>. In particular, we can treat elliptic-parabolic equations in bounded domains with discontinuous coefficients. Using Fourier multiplier techniques, we prove an a priori estimate for strong solutions to the equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1">View MathML</a>-Sobolev spaces.

MSC: 35B45, 35M12.

Keywords:
transmission problem; elliptic-parabolic equation; a priori estimates