Related papers: Median values, 1-harmonic functions, and functions…
Using, as main tool, the convergence theorem for discrete martingales and the mean value property of harmonic functions we solve, a particular case of, Dirichlet problem.
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain satisfying the uniform exterior cone condition. We establish existence and uniqueness of continuous solutions of the Dirichlet Problem associated to certain intrinsic nonlinear mean value…
We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties…
Results involving various mean value properties are reviewed for harmonic, biharmonic and metaharmonic functions. It is also considered how the standard mean value property can be weakened to imply harmonicity and belonging to other classes…
A recent result from [AtES24] allows one to define variational solutions of the Dirichlet problem for general continuous boundary data. We establish basic properties of this notion of solution and show that it coincides with the Perron…
We consider an energy functional combining the square of the local oscillation of a one--dimensional function with a double well potential. We establish the existence of minimal heteroclinic solutions connecting the two wells of the…
We prove that the gradient of any bounded subharmonic function is upper semi-continuous, provided that its super-level sets can be touched from the exterior by uniform $C^{1,\text{Dini}}$ domains at every point. This idea extends to a class…
We study the Dirichlet problem for least gradient functions for domains in metric spaces equipped with a doubling measure and supporting a (1,1)-Poincar\'e inequality when the boundary of the domain satisfies a positive mean curvature…
We collect several results concerning regularity of minimal laminations, and governing the various modes of convergence for sequences of minimal laminations. We then apply this theory to prove that a function has locally least gradient (is…
In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere outside their jump sets. As a tool,…
We generalize the well-known mean value inequality of subharmonic functions for a slightly more general function class. We also apply this generalized mean value inequality to weighted boundary behavior and nonintegrability questions of…
Asymptotic mean value properties, their converse and some other related results are considered for solutions to the $m$-dimensional Helmholtz equation (metaharmonic functions) and solutions to its modified counterpart (panharmonic…
We investigate the boundary behavior of variational solutions of Dirichlet problems for prescribed mean curvature equations at smooth boundary points where certain boundary curvature conditions are satisfied (which preclude the existence of…
We study the mean-value harmonic functions on open subsets of $\mathbb{R}^n$ equipped with weighted Lebesgue measures and norm induced metrics. Our main result is a necessary condition saying that all such functions solve a certain…
As well known, harmonic functions satisfy the mean value property, namely the average of the function over a ball is equal to its value at the center. This fact naturally raises the question on whether this is a characterizing feature of…
We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a $(1,1)$-Poincar\'e inequality.…
We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1,1)- Poincar\'e inequality. The notion of harmonicity is based on the Dirichlet…
We begin by shortly recalling a generalized mean value inequality for subharmonic functions, and two applications of it: first a weighted boundary behavior result (with some new references and remarks), and then a borderline case result to…
We give an analytic proof of the solution of Dirichlet Problem for continous functions satisfying a nonlinear mean value problem related to the p-laplace operator and certain stochastic games.
Mean value properties of solutions to the $m$-dimensional Helmholtz and modified Helmholtz equations are considered. An elementary derivation of these properties is given; it involves the Euler--Poisson--Darboux equation. Despite the…