Related papers: Solution operator of inhomogenuous Dirichlet probl…
We present explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable…
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…
We present a nonvariational setting for the Neumann problem for the Poisson equation for solutions that are H\"{o}lder continuous and that may have infinite Dirichlet integral. We introduce a distributional normal derivative on the boundary…
On the basis of the Green function method, analytical solutions of the diffusion equation which describes nonstationary migration of nonequilibrium interstitial impurity atoms have been derived. It is supposed that the initial distribution…
We compute Poisson kernels for integer weight parameter standard weighted biharmonic operators in the unit disc with Dirichlet boundary conditions. The computations performed extend the supply of explicit examples of such kernels and…
This introduction to Green's functions is based on their role as kernels of differential equations. The procedures to construct solutions to a differential equation with an external source or with an inhomogeneity term are put together to…
We show an example of a sequence of elliptic operators in the unit ball with drifts that diverge as the inverse distance to the boundary, for which we do not get uniform upper estimates for the Green's function with the pole at the origin.…
We obtain simple formulas for the matrix elements of the resolvent operator (the Green's function) in any finite set of square integrable basis. These formulas are suitable for numerical computations whether the basis elements are…
In this work, we transform the equation in the upper half space first studied by Caffarelli and Silvestre to an equation in the Euclidean unit ball $\mathbb{B}^n$. We identify the Poisson kernel for the equation in the unit ball. Using the…
General formula for causal Green's function of linear differential operator of given degree in one variable is given according to coefficient functions of differential operator as a series of integrals. The solution also provides analytic…
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.
A Green's function based solver for the modified Bessel equation has been developed with the primary motivation of solving the Poisson equation in cylindrical geometries. The method is implemented using a Discrete Hankel Transform and a…
In this paper we derive quantitative boundary H\"older estimates, with explicit constants, for the inhomogeneous Poisson problem in a bounded open set $D\subset \mathbb{R}^d$. Our approach has two main steps: firstly, we consider an…
We show continuity in generalized weighted Morrey spaces of sub-linear integral operators generated by some classical integral operators and commutators. The obtained estimates are used to study global regularity of the solution of the…
This paper considers the Helmholtz problem in the exterior of a ball with Dirichlet boundary conditions and radiation conditions imposed at infinity. The differential Helmholtz operator depends on the complex wavenumber with non-negative…
We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in $R^n$ for the perturbed polyharmonic operator $(-\Delta)^m +q$ with $q\in L^{n/2m}$, $n>2m$, determines the potential $q$ in the set…
In this paper, it is investigated for an inhomogeneous Dirichlet problem with $L^p$ boundary data for polyharmonic equation in the upper half-plane. By using higher order Poisson kernels and Pompeiu operators, which are respectively due to…
We present a method of solving partial differential equations on the $n$-dimensional unit sphere using methods based on the continuous wavelet transform derived from approximate identities. We give an explicit analytical solution to the…
We give a uniform estimate and an inequality for solutions of an equation with Dirichlet boundary condition.
The Green's functions for the Laplace equation respectively satisfying the Dirichlet and Neumann boundary conditions on the upper side of an infinite plane with a circular hole are introduced and constructed. These functions enables…