Related papers: Solution operator of inhomogenuous Dirichlet probl…
We consider the Dirichlet problem on general, possibly nonsmooth bounded domain, for elliptic linear equation with uniformly elliptic divergence form operator. We investigate carefully the relationship between weak, soft and the…
In this short note, we solve a Dirichlet problem for a fully nonlinear elliptic equation. The operator is introduced by S. Donaldson and it is relevant to the geometry of the space of volume forms.
We develop a formula for the diagonal values of the Hadamard coefficients associated to a normally hyperbolic operator on a globally hyperbolic spacetime in terms of the advanced and retarded Green's operators. We develop a local formula as…
We study a class of local, first-order, stationary mean-field games (MFGs) on bounded domains with nonstandard mixed boundary conditions: prescribed inflow on $\Gamma_N$ and a relaxed Signorini-type exit condition on $\Gamma_D$…
The Dirac-Dolbeault operator for a compact K\"ahler manifold is a special case of a Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows to express the values of the sections of the…
A well-known result says that the Euclidean unit ball is the unique fixed point of the polarity operator. This result implies that if, in $\mathbb{R}^n$, the unit ball of some norm is equal to the unit ball of the dual norm, then the norm…
We study the boundary behavior of solutions to the Dirichlet problems for integro-differential operators with order of differentiability $s \in (0, 1)$ and summability $p>1$. We establish a nonlocal counterpart of the Wiener criterion,…
We investigate composition-differentiation operators acting on the Dirichlet space of the unit disk. Specifically, we determine characterizations for bounded, compact, and Hilbert-Schmidt composition-differentiation operators. In addition,…
The purpose of this paper is to investigate some spectral properties of Sturm-Liouville type problems with interior singularities. Some of the mathematical aspects necessary for developing own technique presented. By applying this technique…
In this study, we address the challenge of obtaining a Green's function operator for linear partial differential equations (PDEs). The Green's function is well-sought after due to its ability to directly map inputs to solutions, bypassing…
The main result of the paper is an extension of the Dirichlet problem from (closures of) bounded open domains U to arbitrary compact subsets X of the complex plane, i.e. the closure of the corresponding space of functions which are harmonic…
It is shown that if a complete set of mutually commuting operators is formed by constants of motion, then, up to a factor that only depends on the time, each common eigenfunction of such operators is a solution of the Schr\"odinger…
Problem for the first order differential equation with an unbounded operator coefficient in Banach space and integral nonlocal condition is considered. An exponentially convergent algorithm is proposed and justified for the numerical…
In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $\sum 1/\lambda_{i}^{\rm rad}=\int V(s)/S(s)ds$. We also obtain upper and lower…
In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex…
Integral operators play an important role modeling various physical and biological processes. In this article we consider such a nonlinear integro-differential equation. We study several properties of equilibrium solutions of the operator…
We solve the Dirichlet problem for Monge-Amp\`ere equation for $(n-1)$-PSH functions possibly with degenerate right-hand side, through deriving a quantitative version of boundary estimate under the assumption of $(n-1)$-PSH subsolutions. In…
A generalized variant of the Calder\'on problem from electrical impedance tomography with partial data for anisotropic Lipschitz conductivities is considered in an arbitrary space dimension $n \geq 2$. The following two results are shown:…
We present an explicit construction of the solution to the Dirichlet boundary value problem for the radial Schr\"odinger equation in the unit ball, with a complex-valued potential $V$ satisfying the condition $\int_0^1r|V(r)|dr<\infty$. The…
Necessary and sufficient conditions for existence of bounded on the entire real axis solutions of Schrodinger equation are obtained under assumption that the homogeneous equation admits an exponential dichotomy on the semi-axes. Bounded…