Preprint
Inserted: 1 mar 2024
Last Updated: 1 mar 2024
Year: 2024
Abstract:
Given a bounded domain $\Omega \subset \mathbb{R}^N$, we study the sharp constant $\lambda(A)$ of the classical Poincaré inequality for functions in $W^{1,\infty}(\Omega)$ that vanish in a hole $A \subset \Omega.$ Then, we prove existence of an optimal hole $A^\star$ that minimizes the Poincaré constant $\lambda(A)$, under some penalization on the volume of the hole $A.$ Moreover, we give a geometrical characterization of this optimal hole $A^\star$. In addition, we will consider the same shape optimization problem but in the case where the penalization is given by the Hausdorff measure of a rectifiable curve $A$.
On the other hand, we will also study the best constant $\lambda(A)$ of the Poincaré-Wirtinger inequality for functions $u \in W^{1,\infty}(\Omega)$ such that $\int_A u=0$. And, we will also consider a more general version of this Poincaré inequality where characteristic function of $A$ in the constraint $\int_A u=0$ is replaced by a probability measure $\nu$ (i.e., $\int_\Omega u\,\mathrm{d}\nu=0$). Moreover, we will show existence of an optimal hole $A^\star$ that optimizes $\lambda(A)$, among all subsets $A \subset \Omega$ of prescribed volume, and we will also characterize it. If the penalization on $A$ is part of the functional, then we discuss the cases where an optimal hole $A^\star$ exists and others where it does not.
Finally, we will prove existence, uniqueness and regularity of the optimal density $\nu^\star$ that minimizes the Poincaré constant $\overline{\lambda}(\nu)$ plus some penalization $F(\nu)$, among all probability measures $\nu$ over $\Omega$.
Keywords: Optimal transport, shape optimization, Poincare inequality
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