Geometric Control of the Robin Laplacian Eigenvalues: The Case of Negative Boundary Parameter
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Geometric Control of the Robin Laplacian Eigenvalues: The Case of Negative Boundary Parameter Dorin Bucur1
· Simone Cito2
Received: 15 October 2018 © Mathematica Josephina, Inc. 2019
Abstract This paper is motivated by the study of the existence of optimal domains maximizing the kth Robin Laplacian eigenvalue among sets of prescribed measure, in the case of a negative boundary parameter. We answer positively to this question and prove an existence result in the class of measurable sets and for quite general spectral functionals. The key tools of our analysis rely on tight isodiametric and isoperimetric geometric controls of the eigenvalues. In two dimensions of the space, under simply connectedness assumptions, further qualitative properties are obtained on the optimal sets. Keywords Robin Laplacian · Spectrum · Geometric control · Maximal eigenvalues Mathematics Subject Classification 49Q10 · 49R05 · 35P15
1 Introduction Let ⊂ Rd be a bounded Lipschitz domain and β > 0 be a fixed positive real number. A number λ ∈ R is an eigenvalue of the Robin problem for the Laplace operator with boundary parameter −β if there exists a nonzero function u ∈ H 1 () solving, in the
Dorin Bucur is member of the Institut Universitaire de France.
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Dorin Bucur [email protected] Simone Cito [email protected]
1
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
2
Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, 73100 Lecce, Italy
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D. Bucur, S. Cito
weak sense, the problem
⎧ ⎨−u = λu in ∂u ⎩ = βu on ∂ ∂n (here n is the outer normal on ∂), i.e. u ∈ H 1 (), u = 0, ∀ v ∈ H 1 () =λ uv dx.
(1)
∇u · ∇v dx − β
∂
uv dHd−1
The eigenvalues, counted with their multiplicity are ordered and denoted λ1,β () ≤ λ2,β () ≤ · · · → +∞. In general, λ1,β () < 0 and, if is connected, λ1,β () is simple. Starting with some index, all eigenvalues become positive. Precisely, for every k ∈ N, the kth eigenvalue, is given by the Rayleigh min-max formula |∇u|2 dx − β u 2 dHd−1 ∂ λk,β () = min max , (2) S∈Sk u∈S\{0} u 2 dx
where Sk denotes the family of all k-dimensional subspaces of H 1 (). We are motivated by the following shape optimization problem sup{λk,β () : ⊂ Rd , || = m}.
(3)
It has been conjectured by Bareket in 1977 that for k = 1 the solution to the problem above is the ball. In 2015, Freitas and Krejcirik proved in [14] that even in R2 the solution is, in general, not the disc. Precisely, if the boundary parameter β is larger than a fixed threshold β1 (depending on the area m), then the first eigenvalue of a suitably chosen annulus is strictly greater than the same eigenvalue computed for the disc of the same measure. On the opposite sense, if β is smaller than another threshold β2 , then the ball is the only maximizer. A natural conjecture, still unsolved, is that the solution is the disc in the class of simply connected two-dimensional sets. For higher order eigenvalues, several aspects of the problem
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