The transmission problem in domains with a corner point for the Laplace operator in weighted Hölder spaces

We prove the existence and uniqueness of a solution to the elliptic transmission problem in nonsmooth domains in the weighted Hölder space. The coercive estimates of the solution are given.     

Quasilinear elliptic system in exterior domains with dependence on the gradient

In this paper, we deal with a quasilinear elliptic system in exterior domains with dependence on the gradient and coupling of the equations not only inside of the domain, but also on the boundary. We prove the existence of positive, negative or sign changing weak solutions. Our approach relies on an aproximation argument and an adequate elliptic “a priori” estimate.     

The Green function for the mixed problem for the linear Stokes system in domains in the plane

We construct the Green function for the mixed boundary value problem for the linear Stokes system in a two‐dimensional Lipschitz domain.     

On a generalization of the Neumann problem for the Laplace equation

We investigate solvability of a fractional analogue of the Neumann problem for the Laplace equation. As a boundary operator we consider operators of fractional differentiation in the Hadamard sense. The problem is solved by reduction to an integral Fredholm equation. A theorem on existence and uniqueness of the problem solution is proved.     

On the existence of three solutions for the Dirichlet problem on the Sierpinski gasket

We apply a recently obtained three-critical-point theorem of B. Ricceri to prove the existence of at least three solutions of certain two-parameter Dirichlet problems defined on the Sierpinski gasket. We also show the existence of at least three nonzero solutions of certain perturbed two-parameter Dirichlet problems on the Sierpinski gasket, using both the mountain pass theorem of Ambrosetti and Rabinowitz and that of Pucci and Serrin.     

On the Besov regularity of conformal maps and layer potentials on nonsmooth domains

We study the Besov regularity of conformal mappings for domains with rough boundary based on the well-posedness for the Dirichlet problem with Besov data. Also, sharp invertibility results for the classical layer potential operators on Sobolev-Besov spaces on the boundary of curvilinear polygons are obtained.     

The mixed problem for the Lamé system in a class of Lipschitz domains

We consider the mixed problem for the Lamé system

     

Traces of functions in Hardy and Besov spaces on Lipschitz domains with applications to compensated compactness and the theory of Hardy and Bergman type spaces

In this paper we study conditions guaranteeing that functions defined on a Lipschitz domain Ω have boundary traces in Hardy and Besov spaces on ∂Ω. In turn these results are used to develop a new approach to the theory of compensated compactness and the theory of non-locally convex Hardy and Bergman type spaces.