Related papers: A second order approach to the Kato square root pr…
We consider nonlinear second order elliptic problems of the type \[ -\Delta u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] where $\Omega$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general…
In general, the system of $2$nd-order partial differential equations made of the Euler-Lagrange equations of classical field theories are not compatible for singular Lagrangians. This is the so-called second-order problem. The first aim of…
For a second-order strongly elliptic differential operator on an exterior domain in R^n it is known from works of Birman and Solomiak that a change of the boundary condition from the Dirichlet condition to an elliptic Neumann or Robin…
We demonstrate that solving the classical problems mentioned in the title on quadrature domains when the given boundary data is rational is as simple as the method of partial fractions. A by-product of our considerations will be a simple…
We prove regularity for a class of boundary value problems for first order elliptic systems, with boundary conditions determined by spectral decompositions, under coefficient differentiability conditions weaker than previously known. We…
We study a semilinear elliptic problem with a singular nonlinear term of the type $g(u)=-u^{-1}$, using a variational approach. Note that the minus sign is important since the corresponding term in the Euler-Lagrange functional is concave.…
This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This theory finds applications in multi-wave imaging, greedy methods to…
This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value…
We consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. The extension of the result…
We study the Kato problem for degenerate divergence form operators. This was begun by Cruz-Uribe and Rios who proved that given an operator $L_w=-w^{-1}{\rm div}(A\nabla)$, where $w\in A_2$ and $A$ is a $w$-degenerate elliptic measure (i.e,…
We prove that a Kato class potential $V$ defined on an open, bounded set in $\mathbb{R}^3$ with Lipschitz boundary is uniquely determined by the Dirichlet-to-Neumann operator associated to the equation $-\Delta u + Vu = 0\,.$
By using some recent results for divergence form equations, we study the $L_p$-solvability of second-order elliptic and parabolic equations in nondivergence form for any $p\in (1,\infty)$. The leading coefficients are assumed to be in…
An arbitrary order finite difference method for curved boundary domains with Cartesian grid is proposed. The technique handles in a universal manner Dirichlet, Neumann or Robin condition. We introduce the Reconstruction Off-site Data (ROD)…
The first-order approach to boundary value problems for second-order elliptic equations in divergence form with transversally independent complex coefficients in the upper half-space rewrites the equation algebraically as a first-order…
We consider the linear second order PDO's $$ \mathscr{L} = \mathscr{L}_0 - \partial_t : = \sum_{i,j =1}^N \partial_{x_i}(a_{i,j} \partial_{x_j} ) - \sum_{j=i}^N b_j \partial_{x_j} - \partial _t,$$and assume that $\mathscr{L}_0$ has…
In this paper, we revisit the nonoverlapping domain decomposition methods for solving elliptic problems with high contrast coefficients. Some interesting results are discovered. We find that the Dirichlet-Neumann algorithm and Robin-Robin…
In this article, we present a simpler and alternative proof of the solvability of the regularity problem - that is, the Dirichlet problem with boundary data in $\dot W^{1,p}$ - for uniformly elliptic operators on $\mathbb{R}^n_+$ under a…
This paper concerns about the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumption…
Wiener's criterion for the regularity of a boundary point with respect to the Dirichlet problem for the Laplace equation has been extended to various classes of elliptic and parabolic partial differential equations. They include linear…
We consider a slightly subcritical elliptic system with Dirichlet boundary conditions and a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of…