Related papers: Harmonic functions on balls for x-dependent rectil…

Martin boundaries and integral representations of positive functions which are harmonic in a bounded domain $D$ with respect to Brownian motion are well understood. Unlike the Brownian case, there are two different kinds of harmonicity with…

We devise fast and provably accurate algorithms to transform between an $N\times N \times N$ Cartesian voxel representation of a three-dimensional function and its expansion into the {ball harmonics}, that is, the eigenbasis of the…

We establish two new estimates which control a function (after subtracting its average) in $L^1$ by only the $L^1$ norm of its radial derivative. While the interior estimate holds for all superharmonic functions, the boundary version is…

Our concern in this paper is to study the qualitative properties for harmonic functions related to the fractional Laplacian. Firstly we classify the polynomials in the whole space and in the half space for the fractional Laplacian defined…

Motivated by the mean value property of harmonic functions, we introduce the local and global median value properties for continuous functions of two variables. We show that the Dirichlet problem associated with the local median value…

We consider a class of eigenvalue problems for poly-harmonic operators, including Dirichlet and buckling-type eigenvalue problems. We prove an analyticity result for the dependence of the symmetric functions of the eigenvalues upon domain…

Finite-time coherent sets inhibit mixing over finite times. The most expensive part of the transfer operator approach to detecting coherent sets is the construction of the operator itself. We present a numerical method based on radial basis…

In this work we extend the theory of the classical Hardy space $H^1$ to the rational Dunkl setting. Specifically, let $\Delta$ be the Dunkl Laplacian on a Euclidean space $\mathbb{R}^N$. On the half-space $\mathbb{R}_+\times\mathbb{R}^N$,…

For functions defined on the $n$-dimensional hypercube $I_n (r) = \{{\bm{x}} \in \mathbb{R}^n ~\vert~ \vert x_i \vert \le r,~ i = 1, 2, \ldots , n\}$ and harmonic therein, we establish certain analogues of Gauss surface and volume…

The aim of this article is the explicit construction of some barrier functions ("fundamental solutions") for the Pucci-Heisenberg operators. Using these functions we obtain the continuity property, up to the boundary, for the viscosity…

We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value…

We introduce a technique to obtain pointwise upper and lower bounds for the Green's function of elliptic operators whose principal part is the Laplacian and that include a drift term diverging near the boundary like a power of the inverse…

A monotonicity property of Harnack inequality is proved for positive invariant harmonic functions in the unit ball.

Radial basis function methods are powerful tools in numerical analysis and have demonstrated good properties in many different simulations. However, for time-dependent partial differential equations, only a few stability results are known.…

We study boundary regularity for the inhomogeneous Dirichlet problem for $2s$-stable operators in generalized H\"older spaces. Moreover, we provide explicit counterexamples that showcase the sharpness of our results. Our approach directly…

We give a characterization of harmonic and subharmonic functions in terms of their mean values in balls and on spheres. This includes the converse of an inequality of Beardon's for subharmonic functions. We also obtain integral inequalities…

Kusuoka and Zhou have defined the Laplacian on the Sierpinski carpet using average values of functions on small cells and the graph structure of cell adjacency. We have implemented an algorithm that uses their method to approximate…

We extend a classical approximation result of harmonic functions in planar domains due to Bernstein and Walsch to the setting of harmonic functions in Riemann surfaces. This result gives an exact characterization of the rate at which a…

Positive solutions of homogeneous Dirichlet boundary value problems or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of their level surfaces is…

We study harmonic functions which admit a certain majorant in the unit ball in $\R^m $. We prove that when the majorant fulfills a doubling condition, the extremal growth or decay may occur only along small sets of radii, and we give…