We study the potential which minimizes the fundamental gap of the Schrödinger operator under the total mass constraint. We consider the relaxed potential and prove a regularity result for the optimal one, we also give a description of it. A consequence of this result is the existence of an optimal potential under constraints.
We show existence of nonconstant stable equilibria for the Neumann reaction-diffusion problem on domains with fractures inside. We also show that the stability properties of all hyperbolic equilibria remain unchanged under domain perturbation in a quite general sense, covered by the theory of Mosco convergence.
We consider the problem of placing a Dirichlet region made by small balls of given radius in a given domain subject to a force in order to minimize the compliance of the configuration. Then we let tend to infinity and look for the Γ-limit of suitably scaled functionals, in order to get informations on the asymptotical distribution of the centres of the balls. This problem is both linked to optimal location and shape optimization problems.
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