We review some recent results on the equilibrium shapes of charged liquid drops. We show that the natural variational model is ill-posed and how this can be overcome by either restricting the class of competitors or by adding penalizations in the functional. The original contribution of this note is twofold. First, we prove existence of an optimal distribution of charge for a conducting drop subject to an external electric field. Second, we prove that there exists no optimal conducting drop in this...
): ∈ 𝒜, ℋ() = }, where ℋ
,,
} ⊂ R . The cost functional ℰ() is the Dirichlet energy of defined through the Sobolev functions on vanishing on the points
. We analyze the existence of a solution in both the families of connected sets and of metric graphs. At the end, several explicit examples are discussed.
We consider the Schrödinger operator on , where is a given domain of . Our goal is to study some optimization problems where an optimal potential has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.
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