Related papers: The mixed problem for the Laplacian in Lipschitz d…
We consider singular perturbed eigenvalue problem for Laplace operator in a two-dimensional domain. In the boundary we select a set depending on a character small parameter and consisting of a great number of small disjoint parts. On this…
We deal with homogeneous Dirichlet and Neumann boundary-value problems for anisotropic elliptic operators of p-Laplace type. They emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly…
We consider nonhomogeneous fractional $p$-Laplace equations defined on a bounded nonsmooth domain which goes beyond the Lipschitz category. Under a sufficient flatness assumption on the domain in the sense of Reifenberg, we establish…
This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions: \begin{align*} \begin{cases} div(A(x,\nabla u)) &=…
We use novel integral representations developed by the second author to prove certain rigorous results concerning elliptic boundary value problems in convex polygons. Central to this approach is the so-called global relation, which is a…
In this paper, we consider Dirichlet boundary value problem involving the anisotropic $p(x)$-Laplacian, where $p(x)= (p_1(x), ..., p_n(x))$, with $p_i(x)> 1$ in $\overline{\Omega}$. Using the topological degree constructed by Berkovits, we…
In a cylindrical space-time domain with a convex, spatial base, we establish a local Lipschitz estimate for weak solutions to parabolic systems with Uhlenbeck structure up to the lateral boundary, provided homogeneous Dirichlet data are…
A new transform pair which can be used to solve mixed boundary value problems for Laplace's equation and the complex Helmholtz equation in bounded convex planar domains is presented. This work is an extension of Crowdy (2015, CMFT, 15,…
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:…
In the present article we present a particular combination of boundary problems for the inhomogeneous tri-analytic equation: the Neumann-(Dirichlet-Neuman) problem and the (Dirichlet-Neumann)-Dirichlet problem. In order to obtain the…
The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the…
We solve the Neumann problem in the half space $\mathbb{R}^{n+1}_+$, for higher order elliptic differential equations with variable self-adjoint $t$-independent coefficients, and with boundary data in $L^p$, where…
A new method is introduced for studying boundary value problems for a class of linear PDEs with {\it variable} coefficients. This method is based on ideas recently introduced by the author for the study of boundary value problems for PDEs…
The probabilistic representation of weak solutions to a parabolic boundary value problem is established in the following framework. The boundary value problem consists of a second order parabolic equation defined on a time-varying Lipschitz…
We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map at selected frequencies as the data. A conditional Lipschitz stability estimate for the inverse problem holds in the case of…
We consider Dirichlet problems for fully nonlinear mixed local-nonlocal non-translation invariant operators. For a bounded $C^2$ domain $\Omega \subset \mathbb{R}^d,$ let $u\in C(\mathbb{R}^d)$ be a viscosity solution of such Dirichlet…
We present a simple discretization by radial basis functions for the Poisson equation with Dirichlet boundary condition. A Lagrangian multiplier using piecewise polynomials is used to accommodate the boundary condition. This simplifies…
In this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary,…
The thesis studies linear and semilinear Dirichlet problems driven by different fractional Laplacians. The boundary data can be smooth functions or also Radon measures. The goal is to classify the solutions which have a singularity on the…
The existence of Dirichlet minimizing multiple-valued functions for given boundary data has been known since pioneering work of F. Almgren. Here we prove a multiple-valued analogue of the classical Plateau problem of the existence of…