We prove local Lipschitz continuity of the solution to the state equation in two kinds of shape optimization problems with constraint on the volume: the minimal shaping for the Dirichlet energy, with no sign condition on the state function, and the minimal shaping for the first eigenvalue of the Laplacian. This is a main first step for proving regularity of the optimal shapes themselves. (2000): 49Q10, 35R35, 49K20, 35J20
Mathematics Subject Classification
This paper deals with the classical Bernoulli free boundary problem. We are interested in solving some shape optimization problems related to this free boundary problem. We prove the continuous dependence of the solution with respect to the data K , working with Hausdorff convergence. We can deduce an existence result for a large class of shape optimization problems. Finally, we give some ideas for a numerical method, based on the use of conformal mappings, to solve such problems in two dimensions.
Abstract.In this paper we study the existence of critical points of a functional depending on O C I2 through its perimeter and the solution of the Diriclilet problem in 0, under the constraint that the measure of £1 is given. We give a sufficient condition for the existence of critical points using the implicit function theorem.
In this paper, we deal with the continuity with respect to the domain of the solutions of a first boundary value problem of fourth order in dimension 2 and 3. These dimensions are those involved in applications and are critical for this question of continuity. Indeed, continuity holds in dimension 1 thanks to Sobolev embeddings while homogenization may occur in dimensions higher than or equal to 4. Specific phenomena appear in dimensions 2 and 3. Here, we provide a necessary and sufficient condition for continuity with respect to the complementary Hausdorff metric. A main point is that the condition involves only the regularity of the limit domain and not the sequence of approximating domains. We then study various sufficient conditions for continuity in terms of the H2-capacity and we analyze a discontinuous case on an explicit example.
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