Related papers: Infinite dimension of solutions for Dirichlet prob…
It is developed the theory of the Dirichlet problem for harmonic functions. On this basis, for the nondegenerate Beltrami equations in the quasidisks and, in particular, in the smooth domains, it is proved the existence of regular solutions…
We consider the Dirichlet problem for the biharmonic equation on an arbitrary convex domain and prove that the second derivatives of the variational solution are bounded in all dimensions.
In this paper, we study the infinity harmonic functions with linear growth rate at infinity defined on exterior domains. We show that such functions must be asymptotic to planes or cones at infinity. We also establish the solvability of…
We characterize the set of positive harmonic functions with Dirichlet boundary conditions in unbounded domains which are union of several different chambers. We analyze the asymptotic behavior of the solutions in connection with the changes…
We prove the existence of infinitely many solutions to a class of non-symmetric Dirichlet problems with exponential nonlinearities. Here the domain $\Omega \subset\subset \mathbb{R}^{2l}$ where $2l$ is the order of the equation. Considered…
It is proved the existence of single-valued analytic solutions in the unit disk and multivalent analytic solutions in domains bounded by a finite collection of circles for the Riemann-Hilbert problem with coefficients of sigma-finite…
An existence result is shown for the asymptotic Dirichlet problem for harmonic maps from the product of the hyperbolic planes to the hyperbolic space, where the Dirichlet data is given on the distinguished boundary (the product of the…
We show existence and uniqueness for the solutions of the regularity and the Neumann problems for harmonic functions on Lipschitz domains with data in the Hardy spaces H^p, p>2/3, where This in turn implies that solutions to the Dirichlet…
With no criteria of the index type, it is proved the existence of a solution for the Riemann-Hilbert problem in the fairly general setting of arbitrary Jordan domains, measurable coefficients and measurable boundary data. The theorem is…
Dirichlet problem in an $n$-dimensional billiard space is investigated. In particular, the system of ODEs $\ddot x(t) = f(t,x(t))$ together with Dirichlet boundary conditions $x(0) = A$, $x(T) = B$ in an $n$-dimensional interval $K$ with…
It is proved the existence of multivalent solutions for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The…
We provide infinitely many solutions of a Dirichlet problem on balls.
For a linear difference equation with the coefficients being computable sequences, we establish algorithmic undecidability of the problem of determining the dimension of the solution space including the case when some additional prior…
In the context of the correspondence between real functions on the unit circle and inner analytic functions within the open unit disk, that was presented in previous papers, we show that the constructions used to establish that…
We study the obstacle problem for unbounded sets in a proper metric measure space supporting a (p,p)-Poincare inequality. We prove that there exists a unique solution. We also prove that if the measure is doubling and the obstacle is…
It is shown that the Dirichlet problem for the slab $(a,b) \times \mathbb{R}^{d}$ with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given…
For a class of oscillatory resonant problems, involving Dirichlet problems for semilinear PDE's on balls and rectangles in $R^n$, we show the existence of infinitely many solutions, and study the global solution set. The first harmonic of…
This article is devoted to the study of the so-called incompressible limit for solutions of the compressible liquid crystals system. We consider the problem in the whole space $\mathbb{R}^{\mathbb{N}}$ and a bounded domain of…
In this paper we prove that in a three-manifold with finitely many expansive ends, such that each end has a neighborhood where the curvature is bounded above by a negative constant, the Dirichlet problem at infinity is solvable, and hence…
It was recently established that a function which is harmonic on an infinite cylinder and vanishes on the boundary necessarily extends to an entire harmonic function. This paper considers harmonic functions on an annular cylinder which…