On an isoperimetric problem with power-law potentials and external attraction
The authors consider the following energy (or shape functional:) \[ E(\Omega):=P(\Omega)+\int_\Omega\!\!\int_\Omega K(x-y) u(x) u(y) dxdy -\int_\Omega V(x) dx \] where \(\Omega\subset \mathbb{R}^N\), \(N\geq 3\) is a set of finite perimeter, \(P(\Omega)\) is the classical perimeter (in the sense of De Giorgi), \(K(x)\) is a power law potential: \[ K(x)=\frac{1}{p} |x|^{-p} - \frac{1}{q} |x|^{-q}\,, \] and \(V\) is a function in \(L^1_{\mathrm{loc}}(\mathbb{R}^N)\), \(V\geq 0\) and \(V(x)\) tends to 0 when \(|x|\to \infty\). They are interested in the minimization of the functional \(E\), in the class of sets of finite perimeter, with the volume constraint: \(|\Omega|=m\). The authors prove that, when \(0<q<p<N\), there exists a minimizing domain for any volume constraint (for any \(m\)). The proof relies on the use of the concentration-compactness Lemma for sets of finite perimeter due to \textit{R. L. Frank} and \textit{E. H. Lieb} [SIAM J. Math. Anal. 47, No. 6, 4436--4450 (2015; Zbl 1332.49042)]. Then, the authors look at the minimization problem in the case \(q=-2, 0<p<N\) and \(V\equiv 0\). They prove that, for \(m\) sufficiently large, the unique minimizer is the ball. They use a relaxation technique together with the optimality condition to derive this property.
scientific article; zbMATH DE number 7138379
scientific article