Recent zbMATH articles in MSC 31C15https://zbmath.org/atom/cc/31C152021-04-16T16:22:00+00:00WerkzeugOn simple eigenvalues of the fractional Laplacian under removal of small fractional capacity sets.https://zbmath.org/1456.310072021-04-16T16:22:00+00:00"Abatangelo, Laura"https://zbmath.org/authors/?q=ai:abatangelo.laura"Felli, Veronica"https://zbmath.org/authors/?q=ai:felli.veronica"Noris, Benedetta"https://zbmath.org/authors/?q=ai:noris.benedettaAuthors' abstract: We consider the eigenvalue problem for the restricted fractional Laplacian in a bounded domain with homogeneous Dirichlet boundary conditions. We introduce the notion of fractional capacity for compact subsets, with the property that the eigenvalues are not affected by the removal of zero fractional capacity sets. Given a simple eigenvalue, we remove from the domain a family of compact sets which are concentrating to a set of zero fractional capacity and we detect the asymptotic expansion of the eigenvalue variation; this expansion depends on the eigenfunction associated to the limit eigenvalue. Finally, we study the case in which the family of compact sets is concentrating to a point.
Reviewer: Dian K. Palagachev (Bari)Various concepts of Riesz energy of measures and application to condensers with touching plates.https://zbmath.org/1456.310082021-04-16T16:22:00+00:00"Fuglede, Bent"https://zbmath.org/authors/?q=ai:fuglede.bent"Zorii, Natalia"https://zbmath.org/authors/?q=ai:zorii.nataliaLet \(n\geq 3\) and \(0<\alpha \leq 2\). The \(\alpha \)-Riesz potential of a signed Radon measure \(\mu \) on \(\mathbb{R}^{n}\) is defined by \(\kappa_{\alpha }\mu (x)=\int \left\vert x-y\right\vert ^{\alpha -n}d\mu (y)\), and the usual \(\alpha \)-Riesz energy is given by \(E_{\alpha }(\mu )=\int \kappa_{\alpha }\mu ~d\mu \). However, this notion of energy is unsuitable for studying minimum energy problems for condensers with touching plates. This motivates the authors to develop further the theory of weak \(\alpha \)-Riesz energy, which they recently introduced in [Potential Anal. 51, No. 2, 197--217 (2019; Zbl 1432.31006)]. This is defined by \(\int (\kappa _{\alpha /2}\mu )^{2}dm\), where \(m\) denotes the Lebesgue measure on \(\mathbb{R}^{n}\), and coincides with \(E_{\alpha}(\mu )\) when \(\mu \) is a positive measure. The authors investigate minimum weak \(\alpha \)-Riesz energy problems with external fields (in both the constrained and unconstrained settings) for condensers with plates that may touch. They also describe the relationship between minimum weak \(\alpha \)-Riesz energy problems over signed measures associated with generalized condensers \((A_{1},A_{2})\) and minimum \(\alpha \)-Green energy problems over positive measures carried by \(A_{1}\). They succeed in recovering and improving results that were announced in [\textit{P. D. Dragnev} et al., Potential Anal. 44, No. 3, 543--577 (2016; Zbl 1338.31008)], but which relied on a lemma that turned out to be erroneous.
Reviewer: Stephen J. Gardiner (Dublin)A concept of weak Riesz energy with application to condensers with touching plates.https://zbmath.org/1456.310092021-04-16T16:22:00+00:00"Zorii, Natalia"https://zbmath.org/authors/?q=ai:zorii.nataliaSummary: We proceed further with the study of minimum weak Riesz energy problems for condensers with touching plates, initiated jointly with \textit{B. Fuglede} [Potential Anal. 51, No. 2, 197--217 (2019; Zbl 1432.31006)]. Having now added to the analysis constraint and external source of energy, we obtain a Gauss type problem, but with weak energy involved. We establish sufficient and/or necessary conditions for the existence of solutions to the problem and describe their potentials. Treating the solution as a function of the condenser and the constraint, we prove its continuity relative to the vague topology and the topologies determined by the weak and standard energy norms. We show that the criteria for the solvability thus obtained fail in general once the problem is reformulated in the setting of standard energy, thereby justifying an advantage of weak energy when dealing with condensers with touching plates.