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- Title
Existence and regularity of optimal shapes for elliptic operators with drift.
- Authors
Russ, Emmanuel; Trey, Baptiste; Velichkov, Bozhidar
- Abstract
This paper is dedicated to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift L = - Δ + V (x) · ∇ with Dirichlet boundary conditions, where V is a bounded vector field. In the first instance, we prove the existence of a principal eigenvalue λ 1 (Ω , V) for a bounded quasi-open set Ω which enjoys similar properties to the case of open sets. Then, given m > 0 and τ ≥ 0 , we show that the minimum of the following non-variational problem min { λ 1 (Ω , V) : Ω ⊂ D quasi-open , | Ω | ≤ m , ‖ V ‖ L ∞ ≤ τ }. is achieved, where the box D ⊂ 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 Ω ∗ solving the minimization problem min { λ 1 (Ω , ∇ Φ) : Ω ⊂ D quasi-open , | Ω | ≤ m } , where Φ is a given Lipschitz function on D. We prove that the optimal set Ω ∗ is open and that its topological boundary ∂ Ω ∗ is composed of a regular part, which is locally the graph of a C 1 , α function, and a singular part, which is empty if d < d ∗ , discrete if d = d ∗ and of locally finite H d - d ∗ Hausdorff measure if d > d ∗ , where d ∗ ∈ { 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 ∈ ∂ Ω ∗ ∩ ∂ D , ∂ Ω ∗ is C 1 , 1 / 2 in a neighborhood of x.
- Subjects
VECTOR fields; ELLIPTIC operators; HAUSDORFF measures; STRUCTURAL optimization; GEOMETRIC shapes
- Publication
Calculus of Variations & Partial Differential Equations, 2019, Vol 58, Issue 6, pN.PAG
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-019-1653-6