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

• Published in: Annali di matematica pura ed applicata Vol. 198; no. 4; pp. 1227 - 1241
• Main Authors: Haller-Dintelmann, Robert, Meinlschmidt, Hannes, Wollner, Winnifried
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2019; Springer Nature B.V
• Subjects: Boundary conditions; Damage assessment; Divergence; Ellipticity; Interpolation; Mathematics; Mathematics and Statistics; Regularity; Maximal elliptic regularity; Mixed boundary conditions; Second-order divergence operators; Phase-field damage; Primary 35B65; Elliptic system; Secondary 35J57; 35J25; Non-Lipschitz coefficients
• ISSN: 0373-3114; 1618-1891; 1618-1891
• DOI: 10.1007/s10231-018-0818-9

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 D s - 1 ( Ω ) for s > 0 sufficiently small, if the coefficient in the main part satisfies a certain multiplier property on the spaces H s ( Ω ) . 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.