Abstract
Minimizing the so-called “Dirichlet energy” with respect to the domain under a volume constraint is a standard problem in shape optimization which is now well understood. This article is devoted to a prototypal non-linear version of the problem, where one aims at minimizing a Dirichlet-type energy involving the solution to a semilinear elliptic PDE with respect to the domain, under a volume constraint. One of the main differences with the standard version of this problem rests upon the fact that the criterion to minimize does not write as the minimum of an energy, and thus most of the usual tools to analyze this problem cannot be used. By using a relaxed version of this problem, we first prove the existence of optimal shapes under several assumptions on the problem parameters. We then analyze the stability of the ball, expected to be a good candidate for solving the shape optimization problem, when the coefficients of the involved PDE are radially symmetric.
Highlights
We assume that D is a centered ball D = B(0, RD) strictly containing a centered ball B∗ = B(0, R) of volume m, in other words we assume that R < RD
We investigate the local stability of the ball B∗: we will prove that the ball is always a critical point, and show that we obtain different stability results, related to the non-negativity of the second shape derivative of the Lagrangian, depending on f and g
Jρ is monotonous on the set of stable quasi-open sets and to use the continuity of Jρ with respect to the γ-convergence to establish its monotonicity on Om
Summary
The issue of minimizing the Dirichlet energy (in the linear case) with respect to the domain is a basic and academical shape optimization problem under PDE constraint, which is well understood. This problem reads: Let d ∈ N∗ and D be a C 2 compact set of Rd. Given g ∈ L2(D) and m |D|, minimize the Dirichlet energy. In the perturbed version of the Dirichlet problem we will deal with, the linear PDE solved by uΩ is changed into a nonlinear one but the functional to minimize remains the same Since, in this case, the problem is not ”energetic” anymore (in the sense described above), the PDE constraint cannot be incorporated into the shape functional.
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