Related papers: Infinite dimension of solutions for Dirichlet prob…

We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for incompressible MHD equations in a bounded smooth domain of three spatial dimensions with initial density being allowed to have…

In [18], fundamental solutions for the generalized bi-axially symmetric Helmholtz equation were constructed in $R_2^ + = \left\{ {\left( {x,y} \right):x > 0,y > 0} \right\}.$ They contain Kummer's confluent hypergeometric functions in three…

We study the Dirichlet problem $-\div(|\nabla u|^{p(x)-2} \nabla u) =0 $ in $\Omega$, with $u=f$ on $\partial \Omega$ and $p(x) = \infty$ in $D$, a subdomain of the reference domain $\Omega$. The main issue is to give a proper sense to what…

We consider the Dirichlet boundary value problem for graphical maximal submanifolds inside Lorentzian type ambient spaces, and obtain general existence and uniqueness results which apply to any codimension.

After calculating the Dushnik-Miller dimension of Minkowski spaces to be countable infinity, we define a novel notion of dimension for ordered spaces recovering the correct manifold dimension and obtain a corresponding obstruction for the…

In this article we discuss density of products of biharmonic functions vanishing on an arbitrarily small part of the boundary. We prove that one can use three or more such biharmonic functions to construct a dense subset of smooth symmetric…

We prove the solvability of a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary. We also prove a factorization result for flat hermitian metrics on doubly connected domains.

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…

A system of boundary-domain integral equations is derived from the bidimensional Dirichlet problem for the diffusion equation with variable coefficient using the novel parametrix from [22] different from the one in [5,18]. Mapping…

We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solutions which grow at most like the distance to the boundary to a power given by the natural scaling exponent of the equation; in other words,…

We study function spaces and extension results in relation with Dirichlet problems involving integrodifferential operators. For such problems, data are prescribed on the complement of a given domain in the Euclidean space. We introduce a…

In this paper we extend our findings in [3] and answer further questions regarding continuity and discontinuity of seminorms on infinite-dimensional vector spaces.

We are concerned with the half-space Dirichlet problem \[\begin{array}{ll} -\Delta v+v=|v|^{p-1}v & \textrm{in}\ \mathbb{R}^N_+, v=c\ \textrm{on}\ \partial\mathbb{R}^N_+, &\lim_{x_N\to \infty}v(x',x_N)=0\ \textrm{uniformly in}\…

In this paper, we consider a biharmonic equation with respect to the Dirichlet problem on a domain of a locally finite graph. Using the variation method, we prove that the equation has two distinct solutions under certain conditions.

We consider a unique continuation problem where the Dirichlet trace of the solution is known to have finite dimension. We prove Lipschitz stability of the unique continuation problem and design a finite element method that exploits the…

For an infinite penny graph, we study the finite-dimensional property for the space of harmonic functions, or ancient solutions of the heat equation, of polynomial growth. We prove the asymptotically sharp dimensional estimate for the above…

We consider the Dirichlet problem for two types of degenerate elliptic Hessian equations . New results about solvability of the equations in the $C^{1,1}$ space are provided.

We consider singular solutions to quasilinear elliptic equations under zero Dirichlet boundary condition. Under suitable assumptions on the nonlinearity we deduce symmetry and monotonicity properties of positive solutions via an improved…

It is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain, with Dirichlet boundary conditions, are unique up to multiplication by a positive constant.

Based on projective representations of smooth Deligne cohomology groups, we introduce an analogue of the space of conformal blocks to compact oriented (4k+2)-dimensional Riemannian manifolds with boundary. For the standard…