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Regularity of optimal shapes for the Dirichlet’s energy with volume constraint

Tanguy Briancon — 2004

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove some regularity results for the boundary of an open subset of ℝ d 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.

Regularity of optimal shapes for the Dirichlet's energy with volume constraint

Tanguy Briancon — 2010

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove some regularity results for the boundary of an open subset of d 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.

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