Related papers: Boundary value problems and Hardy spaces for singu…
Consider operators $L_{V}:=\Delta + V$ in a bounded Lipschitz domain $\Omega\subset \mathbb{R}^N$. Assume that $V\in C^\alpha(\Omega)$ satisfies $|V(x)| \leq \bar a\,\mathrm{dist}(x,\partial\Omega)^{-2}$ in $\Omega$ and that $L_V$ has a…
In this paper, we investigate the Dirichlet boundary value problem on Cartan-Hadamard manifolds, focusing on the non-existence of bounded (viscosity) solutions to semi-linear elliptic equations of the form $\Delta u + f(u) = 0$ in domains…
This paper is concerned with an inverse boundary value problem for the Helmholtz equation over a bounded domain. The aim is to reconstruct two constant coefficients together with the location and shape of a Dirichlet polygonal obstacle from…
We study the boundedness of some sublinear operators on weighted Morrey spaces under certain size conditions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator,…
In this work, we develop the classical theory of monotone and pseudomonotone operators in the class of convex constrained Dirichlet-type problems involving fractional Riesz gradients in bounded and in unbounded domains…
We establish the Hopf boundary point lemma for the Schr\"odinger operator $-\Delta + V$ involving potentials $V$ that merely belong to the space $L^{1}_{loc}(\Omega)$. More precisely, we prove that among all supersolutions $u$ of $-\Delta +…
For an $SU(2)\times U(1)$-invariant $S^3$ boundary the classical Dirichlet problem of Riemannian quantum gravity is studied for positive-definite regular solutions of the Einstein equations with a negative cosmological constant within…
Based on the need of studying the fractional boundary value problems by using variational methods, in this paper, we introduce a fundamental theory framework of fractional Sobolev space in one dimension, study the regularity of weak…
We generalize the Strichartz estimates for Schr\"odinger operators on compact manifolds of Burq, G\'erard and Tzvetkov [10] by allowing critically singular potentials $V$. Specifically, we show that their $1/p$--loss $L^p_tL^q_x(I\times…
In this paper I consider the inverse boundary value problem for a quasilinear, anisotropic, elliptic equation of the form $\nabla\cdot(\gamma\nabla u+|\nabla u|^{p-2}\nabla u)=0$, where $\gamma$ is a smooth, matrix valued, function with a…
Let $(\Sigma^2,ds^2)$ be a compact Riemannian surface, possibly with boundary, and consider Schr\"odinger-type operators of the form $L=\Delta+V-aK$ together with natural Robin and Steklov-type boundary conditions incorporating a boundary…
We establish existence and regularity results for boundary value problems arising from the first variation of the Willmore energy in the graphical setting. Our focus lies on two-dimensional surfaces with fixed clamped boundary conditions,…
This paper concerns positive solutions to the boundary value problems of the scalar field equation in the half space with a Sobolev supercritical nonlinearity and an inhomogeneous Dirichlet boundary condition, admitting a nontrivial…
In this article, by applying the well known method for dealing with $p$-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and…
We extend representation formulas that generalize the similarity principle of solutions to the Vekua equation to certain classes of meta-analytic functions. Also, we solve a generalization of the higher-order Schwarz boundary value problem…
The paper is concerned with the following $n\times n$ Dirac type equation$$Ly=-iB(x)^{-1}(y'+Q(x)y)=\lambda y, \quad B(x)=B(x)^*,\quad y={\rm col}(y_1,\ldots,y_n),\quad x\in[0,\ell],$$ on a finite interval $[0,\ell]$. Here $Q$ is a summable…
We study global regularity for solutions of quasilinear elliptic equations of the form $\div \A(x,u,\nabla u) = \div \F $ in rough domains $\Omega$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed…
We present uniqueness and existence in weighted Sobolev spaces of the equation $$ u_t=(au_{xx}+bu_x+cu)+ \xi |u|^{1+\lambda} {\dot{B}}, \quad\,\, t>0, \, x\in (0,1) $$ with initial data $u(0,\cdot)=u_0$ and zero boundary data. Here…
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…
End-point maximal $L^1$-regularity for parabolic initial-boundary value problems is considered. For the inhomogeneous Dirichlet and Neumann data, maximal $L^1$-regularity for initial-boundary value problems is established in time end-point…