Regularity for free interface variational problems in a general class of gradients

Article

Abstract

We present a way to study a wide class of optimal design problems with a perimeter penalization. More precisely, we address existence and regularity properties of saddle points of energies of the form
$$\begin{aligned} (u,A) \quad \mapsto \quad \int _\Omega 2fu \,\mathrm {d}x \; - \int _{\Omega \cap A} \sigma _1\mathscr {A}u\cdot \mathscr {A}u \, \,\mathrm {d}x \; - \int _{\Omega {\setminus } A} \sigma _2\mathscr {A}u\cdot \mathscr {A}u \, \,\mathrm {d}x \; + \; \text {Per }(A;\overline{\Omega }), \end{aligned}$$
where \(\Omega \) is a bounded Lipschitz domain, \(A\subset \mathbb {R}^N\) is a Borel set, \(u:\Omega \subset \mathbb {R}^N \rightarrow \mathbb {R}^d\), \(\mathscr {A}\) is an operator of gradient form, and \(\sigma _1, \sigma _2\) are two not necessarily well-ordered symmetric tensors. The class of operators of gradient form includes scalar- and vector-valued gradients, symmetrized gradients, and higher order gradients. Therefore, our results may be applied to a wide range of problems in elasticity, conductivity or plasticity models. In this context and under mild assumptions on f, we show for a solution (wA), that the topological boundary of \(A \cap \Omega \) is locally a \(\mathrm {C}^1\)-hypersurface up to a closed set of zero \(\mathscr {H}^{N-1}\)-measure.

Mathematics Subject Classification

Primary 49J20 49J35 49N60 49Q20 Secondary 49J45 

Notes

Acknowledgements

I wish to extend many thanks to Prof. Stefan Müller for his advice and fruitful discussions in this beautiful subject. I would also like to thank the reviewer and the editor for their patience and care which derived in the correct formulation of Theorem 1.5. The support of the University of Bonn and the Hausdorff Institute for Mathematics is gratefully acknowledged. The research conducted in this paper forms part of the author’s Ph.D. thesis at the University of Bonn.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikBonnGermany

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