Higher regularity for solutions to elliptic systems in divergence form subject to mixed boundary conditions

  • Robert Haller-Dintelmann
  • Hannes MeinlschmidtEmail author
  • Winnifried Wollner


This work combines results from operator and interpolation theory to show that elliptic systems in divergence form admit maximal elliptic regularity on the Bessel potential scale \( H ^{s-1}_D(\varOmega )\) for \(s>0\) sufficiently small, if the coefficient in the main part satisfies a certain multiplier property on the spaces \( H ^{s}(\varOmega )\). Ellipticity is enforced by assuming a Gårding inequality, and the result is established for spaces incorporating mixed boundary conditions with very low regularity requirements for the underlying spatial set. To illustrate the applicability of our results, two examples are provided. Firstly, a phase-field damage model is given as a practical application where higher differentiability results are obtained as a consequence to our findings. These are necessary to show an improved numerical approximation rate. Secondly, it is shown how the maximal elliptic regularity result can be used in the context of quasilinear parabolic equations incorporating quadratic gradient terms.


Maximal elliptic regularity Non-Lipschitz coefficients Second-order divergence operators Elliptic system Mixed boundary conditions Phase-field damage 

Mathematics Subject Classification

Primary 35B65 Secondary 35J57 35J25 



The authors express their gratitude to the anonymous referee for useful comments and Joachim Rehberg (WIAS Berlin) for valuable discussions. Hannes Meinlschmidt is grateful for support of his former institution TU Darmstadt; Winnifried Wollner acknowledges funding by the DFG priority program 1962.


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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Johann Radon Institute for Computational and Applied Mathematics (RICAM)LinzAustria
  3. 3.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany

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