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Existence and regularity of optimal shapes for elliptic operators with drift

  • Emmanuel Russ
  • Baptiste Trey
  • Bozhidar VelichkovEmail author
Article
  • 65 Downloads

Abstract

This paper is dedicated to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift \(L = -\Delta + V(x) \cdot \nabla \) with Dirichlet boundary conditions, where V is a bounded vector field. In the first instance, we prove the existence of a principal eigenvalue \(\lambda _1(\Omega ,V)\) for a bounded quasi-open set \(\Omega \) which enjoys similar properties to the case of open sets. Then, given \(m>0\) and \(\tau \ge 0\), we show that the minimum of the following non-variational problem
$$\begin{aligned} \min \Big \{\lambda _1(\Omega ,V)\ :\ \Omega \subset D\ \text {quasi-open},\ |\Omega |\le m,\ \Vert V\Vert _{L^\infty }\le \tau \Big \}. \end{aligned}$$
is achieved, where the box \(D\subset {\mathbb {R}}^d\) is a bounded open set. The existence when V is fixed, as well as when V varies among all the vector fields which are the gradient of a Lipschitz function, are also proved. The second interest and main result of this paper is the regularity of the optimal shape \(\Omega ^*\) solving the minimization problem
$$\begin{aligned} \min \Big \{\lambda _1(\Omega ,\nabla \Phi )\ :\ \Omega \subset D\ \text {quasi-open},\ |\Omega |\le m\Big \}, \end{aligned}$$
where \(\Phi \) is a given Lipschitz function on D. We prove that the optimal set \(\Omega ^*\) is open and that its topological boundary \(\partial \Omega ^*\) is composed of a regular part, which is locally the graph of a \(C^{1,\alpha }\) function, and a singular part, which is empty if \(d<d^*\), discrete if \(d=d^*\) and of locally finite \({\mathcal {H}}^{d-d^*}\) Hausdorff measure if \(d>d^*\), where \(d^*\in \{5,6,7\}\) is the smallest dimension at which there exists a global solution to the one-phase free boundary problem with singularities. Moreover, if D is smooth, we prove that, for each \(x\in \partial \Omega ^{*}\cap \partial D\), \(\partial \Omega ^*\) is \(C^{1,1/2}\) in a neighborhood of x.

Mathematics Subject Classification

49Q10 35R35 47A75 

Notes

Acknowledgements

The authors have been partially supported by Agence Nationale de la Recherche (ANR) with the projects GeoSpec (LabEx PERSYVAL-Lab, ANR-11-LABX-0025-01). The third author was also partially supported by the project CoMeDiC (ANR-15-CE40-0006).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Emmanuel Russ
    • 1
  • Baptiste Trey
    • 1
  • Bozhidar Velichkov
    • 2
    Email author
  1. 1.Université Grenoble Alpes, CNRS UMR 5582, Institut FourierGièresFrance
  2. 2.Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”Università degli Studi di Napoli Federico IINapoliItaly

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