We study the Dirichlet problem for p-harmonic functions on metric spaces with respect to arbitrary compactifications. A particular focus is on the Perron method, and as a new approach to the invariance problem we introduce Sobolev-Perron solutions. We obtain various resolutivity and invariance results, and also show that most functions that have earlier been proved to be resolutive are in fact Sobolev-resolutive. We also introduce (Sobolev)-Wiener solutions and harmonizability in this nonlinear context, and study their connections to (Sobolev)-Perron solutions, partly using Q-compactifications.
This paper establishes a conjecture of Gustafsson and Khavinson, which relates the analytic content of a smoothly bounded domain in R-N to the classical isoperimetric inequality. The proof is based on a novel combination of partial balayage with optimal transport theory. (C) 2018 Elsevier Inc. All rights reserved.
The Bergman p-analytic content (1 <= p < infinity) of a planar domain Omega measures the L-p (Omega)-distance between (z) over bar and the Bergman space A(p) (Omega) of holomorphic functions. It has a natural analogue in all dimensions which is formulated in terms of harmonic vector fields. This paper investigates isoperimetric inequalities for Bergman p-analytic content in terms of the St. Venant functional for torsional rigidity, and addresses the cases of equality with the upper and lower bounds.
This note verifies a conjecture of Armitage and Goldstein that annular domains may be characterized as quadrature domains for harmonic functions with respect to a uniformly distributed measure on a sphere.
This note addresses a problem of Dvoretzky concerning the harmonic measure of the set of boundary points of a domain in Euclidean space that are of angular type.
Gustafsson and Lin recently published a significant result concerning Laplacian growth problems that start from a simply connected planar domain. However, the validity of their result depends on the verification of a particular conjecture. This paper provides the missing proof.
This note verifies a conjecture of Kral, that a continuously differentiable function, which is subharmonic outside its critical set, is subharmonic everywhere.
It is known that corners of interior angle less than pi/2 in the boundary of a plane domain are initially stationary for Hele-Shaw flow arising from an arbitrary injection point inside the domain. This paper establishes the corresponding result for Laplacian growth of domains in higher dimensions. The problem is treated in terms of evolving families of quadrature domains for subharmonic functions.
Recent work on two-phase free boundary problems has led to the investigation of a new type of quadrature domain for harmonic functions. This paper develops a method of constructing such quadrature domains based on the technique of partial balayage, which has proved to be a useful tool in the study of one-phase quadrature domains and Hele-Shaw flows.
In this article, we introduce the concept of harmonic balls in sub-domains of ^{ n }, through a mean-value property for a subclass of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept fails to exist due to the fact that analytic functions cannot have prescribed data on the boundary. Nevertheless, a two-phase version of the problem does exist, and gives rise to the generalization of the well-known Schwarz function to the case of a two-phase Schwarz function. Our primary goal is to derive simple properties for these problems, and tease the appetites of experts working on Schwarz function and related topics. Hopefully these two concepts will provoke further study of the topic.
Let (X, d(X), mu) be a metric measure space where X is locally compact and separable and mu is a Borel regular measure such that 0 amp;lt; mu(B(x, r)) amp;lt; infinity for every ball B(x, r) with center x is an element of X and radius r amp;gt; 0. We define chi to be the set of all positive, finite non- zero regular Borel measures with compact support in X which are dominated by mu, and M = X boolean OR {0}. By introducing a kind of mass transport metric d(M) on this set we provide a new approach to first order Sobolev spaces on metric measure spaces, first by introducing such for functions F : X -amp;gt; R, and then for functions f : X -amp;gt; [-infinity, infinity] by identifying them with the unique element F-f : X -amp;gt; R defined by the mean- value integral: Ff(eta) - 1/vertical bar vertical bar eta vertical bar vertical bar integral f d eta. In the final section we prove that the approach gives us the classical Sobolev spaces when we are working in open subsets of Euclidean space R-n with Lebesgue measure. (C) 2016 Elsevier Ltd. All rights reserved.
This survey describes recent advances on quadrature domains that were made in the context of the ESF Network on Harmonic and Complex Analysis and its Applications (2007–2012). These results concern quadrature domains, and their two-phase counterparts, for harmonic, subharmonic and analytic functions.