Regularity of optimal shapes for the Dirichlet's energy with volume constraint
ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 1, pp. 99-122.

In this paper, we prove some regularity results for the boundary of an open subset of ℝd which minimizes the Dirichlet’s energy among all open subsets with prescribed volume. In particular we show that, when the volume constraint is “saturated”, the reduced boundary of the optimal shape (and even the whole boundary in dimension 2) is regular if the state function is nonnegative.

DOI : 10.1051/cocv:2003038
Classification : 35R35, 49N60, 49Q10
Mots-clés : shape optimization, calculus of variations, free boundary, geometrical measure theory 
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     title = {Regularity of optimal shapes for the {Dirichlet's} energy with volume constraint},
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     url = {https://numdam.org/articles/10.1051/cocv:2003038/}
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Briancon, Tanguy. Regularity of optimal shapes for the Dirichlet's energy with volume constraint. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 1, pp. 99-122. doi : 10.1051/cocv:2003038. https://numdam.org/articles/10.1051/cocv:2003038/

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