Regularity for free interface variational problems in a general class of gradients
| Autor: | Adolfo Arroyo-Rabasa |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Applied Mathematics
010102 general mathematics Scalar (mathematics) 01 natural sciences Omega 010101 applied mathematics Combinatorics Mathematics - Analysis of PDEs 49J20 49J35 49N60 49Q20 (primary) 49J45 (secondary) Lipschitz domain Optimization and Control (math.OC) Bounded function Free interface Saddle point FOS: Mathematics 0101 mathematics Borel set Mathematics - Optimization and Control Analysis Mathematics Analysis of PDEs (math.AP) |
| DOI: | 10.48550/arxiv.1603.00940 |
| Popis: | 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 $$ (u,A) \quad \mapsto \quad \int_\Omega 2fu \; \text{d}x \; - \int_{\Omega \cap A} \sigma_1 \mathscr A u \cdot \mathscr A u \; \text{d}x \; - \int_{\Omega \setminus A} \sigma_2\mathscr A u\cdot \mathscr A u \; \text{d}x \; + \; \text{Per}(A;\overline \Omega),$$ where $\Omega$ is a bounded Lipschitz domain, $A\subset \mathbb R^N$ is a Borel set, $u:\Omega \subset \mathbb R^N \to \mathbb R^d$, $\mathscr A$ 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 $(w,A)$, that the topological boundary of $A \cap \Omega$ is locally a $\rm{C}^1$-hypersurface up to a closed set of zero $\mathscr H^{N-1}$-measure. Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s00526-016-1085-5 |
| Databáze: | OpenAIRE |
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