Recent zbMATH articles in MSC 31Chttps://zbmath.org/atom/cc/31C2021-04-16T16:22:00+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)The quasisuperminimizing constant for the minimum of two quasisuperminimizers in \(R^n\).https://zbmath.org/1456.310102021-04-16T16:22:00+00:00"Björn, Anders"https://zbmath.org/authors/?q=ai:bjorn.anders"Björn, Jana"https://zbmath.org/authors/?q=ai:bjorn.jana"Mirumbe, Ismail"https://zbmath.org/authors/?q=ai:mirumbe.ismailQuasiminimizers were introduced in [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 79--107 (1984; Zbl 0541.49008)] by \textit{M. Giaquinta} and \textit{E. Giusti}, who noticed that they belong to the De Giorgi class. A function \(u\) in the Sobolev space \(W^{1,p}_{\text{loc}}(\Omega)\) is a \(Q\)-quasiminimizer in the domain \(\Omega\) in \(\mathbb{R}^n\) if
\[
\int_{\operatorname{supp}\{\phi\}}|\nabla u|^pdx\le Q \int_{\operatorname{supp}\{\phi\}}|\nabla u+\nabla\phi|^pdx
\]
for all \(\phi\in W^{1,p}_0(\Omega)\). Here \(1<p<\infty\) and \(Q\) is a constant. \(Q\)-quasisuperminimizers are defined analogously, but under the restriction \(\phi\ge 0\) for the test functions.
If \(u_1\) is a \(Q_1\)-quasisuperminimizers and \(u_2\) a \(Q_2\)-quasisuperminimizer, then \(\max\{u_1,u_2\}\) is a \(Q\)-quasisuperminimizer for some \(Q\). The sharpness of a formula for a nearly optimal \(Q\) is the topic of the present work. Explicit functions like \(|x|^\alpha(\log^+(|x|))^\beta\) are utilized. In the important borderline case \(p=n\ge 2\) the following example is constructed in the ring domain \(1/e<|x|<1\). Let \(Q_2\ge Q_1\ge 1\) be given. Two functions \(u_1\) and \(u_2\) are exhibited: \(u_j\) is a \(Q_j\)-quasisuperminimizer \((j= 1,2)\), but \(\max\{u_1,u_2\}\) is not a \(Q_2\)-quasisuperminimizer.
Some numerical tables for the constants \(Q\) are constructed for radial functions.
Reviewer: Peter Lindqvist (Trondheim)Hyperbolic harmonic functions and hyperbolic Brownian motion.https://zbmath.org/1456.601942021-04-16T16:22:00+00:00"Eriksson, Sirkka-Liisa"https://zbmath.org/authors/?q=ai:eriksson.sirkka-liisa"Kaarakka, Terhi"https://zbmath.org/authors/?q=ai:kaarakka.terhiSummary: We study harmonic functions with respect to the Riemannian metric
\[ds^2=\frac{dx_1^2+\cdots +dx_n^2}{x_n^{\frac{2\alpha}{n-2}}}\] in the upper half space \(\mathbb{R}_+^n=\{(x_1,\dots,x_n) \in \mathbb{R}^n :x_n>0\}\). They are called \(\alpha\)-hyperbolic harmonic. An important result is that a function \(f\) is \(\alpha\)-hyperbolic harmonic íf and only if the function \(g(x) =x_n^{-\frac{2-n+\alpha}{2}}f(x)\) is the eigenfunction of the hyperbolic Laplace operator \(\triangle_h=x_n^2\triangle -(n-2) x_n\frac{\partial}{\partial x_n}\) corresponding to the eigenvalue \(\frac{1}{4} ((\alpha+1)^2-(n-1)^2)=0\). This means that in case \(\alpha =n-2\), the \(n-2\)-hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincaré upper half-space. We are presenting some connections of \(\alpha\)-hyperbolic functions to the generalized hyperbolic Brownian motion. These results are similar as in case of harmonic functions with respect to usual Laplace and Brownian motion.The Liouville theorem for \(p\)-harmonic functions and quasiminimizers with finite energy.https://zbmath.org/1456.350552021-04-16T16:22:00+00:00"Björn, Anders"https://zbmath.org/authors/?q=ai:bjorn.anders"Björn, Jana"https://zbmath.org/authors/?q=ai:bjorn.jana"Shanmugalingam, Nageswari"https://zbmath.org/authors/?q=ai:shanmugalingam.nageswariSummary: We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite \(p\)th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global \(p\)-Poincaré inequality. The geometric conditions are that either (a) the measure has a sufficiently strong volume growth at infinity, or (b) the metric space is annularly quasiconvex (or its discrete version, annularly chainable) around some point in the space. Moreover, on the weighted real line \(\mathbf{R}\), we characterize all locally doubling measures, supporting a local \(p\)-Poincaré inequality, for which there exist nonconstant quasiminimizers of finite \(p\)-energy, and show that a quasiminimizer is of finite \(p\)-energy if and only if it is bounded. As \(p\)-harmonic functions are quasiminimizers they are covered by these results.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.