Related papers: Boundary integral formula for harmonic functions o…
We propose to relax the classic Cauchy-Riemann equations for a mapping. We support the interest of such a proposal by looking at one specific situation in 3D, and proving the existence of pairs of harmonic conjugate functions with respect…
We explore for compact Riemannian surfaces whose boundary consists of a single closed geodesic the relationship between orthospectrum and boundary length. More precisely, we establish a uniform lower bound on the boundary length in terms of…
We study various properties of the gradients of solutions to harmonic functions on Lipschitz surfaces. We improve an exponential bound of Naber and Valtorta on the size of the superlevel sets for the frequency function to a sharp quadratic…
In this note, we find an equivalent boundary integral equation to the classical $\bar{\partial}$-Neumann problem. The new equation contains an equivalent regularity to the global regularity of the $\bar{\partial}$-Neumann problem. We also…
We study two-dimensional classically integrable field theory with independent boundary condition on each end, and obtain three possible generating functions for integrals of motion when this model is an ultralocal one. Classically…
We construct the Green function for the mixed boundary value problem for the linear Stokes system in a two-dimensional Lipschitz domain.
Boundary value problems on the unit sphere arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Robust numerical methods play an important role in solving these problems. In this article,…
Boundary conforming coordinates are commonly used in plasma physics to describe the geometry of toroidal domains, for example, in three-dimensional magnetohydrodynamic equilibrium solvers. The magnetohydrodynamic equilibrium configuration…
We prove partial and full boundary regularity for manifold constrained $p(x)$-harmonic maps.
The moduli spaces of compact and connected Riemann surfaces has been a central topic in modern mathematics in recent years. Thus their homological dimensions become important invariants. Motivated by the emergence mathematical counterparts…
Given a mean-convex domain $\Omega\subset \R^n$ with boundary of class $C^{2,1}$, we provide a representation formula for a boundary integral of the type \[ \int_{\partial \Omega} f(k(x)) \, d\mathcal{H}^{n-1} \] where $k\geq 0$ is the mean…
An intuitive probabilistic alternative for the construction of the Martin boundary is presented along with a construction of maximal representing measures for positive harmonic functions.
We construct harmonic functions in the quarter plane for discrete Laplace operators. In particular, the functions are conditioned to vanish on the boundary and the Laplacians admit coefficients associated with transition probabilities of…
Recently published formulas for the surface and regular solid spherical harmonics and for the expansion of the product of two normalized associated Legendre functions with different centers in ellipsoidal coordinates (Telhat Ozdogan, Metin…
We study boundary value problems for harmonic functions on certain domains in the level-$l$ Sierpinski gaskets $\mathcal{SG}_l$($l\geq 2$) whose boundaries are Cantor sets. We give explicit analogues of the Poisson integral formula to…
It is established that if a harmonic function $u$ on the unit disk $\mathbb D$ in $\mathbb C$ has angular limits on a measurable set $E$ of the unit circle $\partial\mathbb D$, then its conjugate harmonic function $v$ in $\mathbb D$ also…
We study boundary integral formulations for an interior/exterior initial boundary value problem arising from the thermo-elasto-dynamic equations in a homogeneous and isotropic domain. The time dependence is handled, based on Lubich's…
Given a continuous function on the boundary of a bounded open set in $\mathbb{R}^d$ there exists a unique bounded harmonic function, called the Perron solution, taking the prescribed boundary values at least at all regular points (in the…
We consider shape optimization problems for general integral functionals of the calculus of variations that may contain a boundary term. In particular, this class includes optimization problems governed by elliptic equations with a Robin…
We construct an expression for the Green function of a differential operator satisfying nonlocal, homogeneous boundary conditions starting from the fundamental solution of the differential operator. This also provides the solution to the…