The Functional Equation
The Generalized Ahlfors-Heins Theorem in Certain d-Dimensional Cones.
The inverse problem for elliptic equations from Dirichlet to Neumann map in multiply connected domains.
The Martin compactification of a plane domain
We prove that the Martin compactification of a plane domain is homeomorphic to a subset of the two-dimensional sphere.
The Method of Lines for Poisson's Equation with Nonlinear or Free Boundary Conditions.
The minimum modulus of a subharmonic function
The Neumann problem for the 2-D Helmholtz equation in a domain, bounded by closed and open curves.
The Poisson Space of the Koranyi Ball.
The problem and singular integrals on Orlicz spaces
The regularity of the trace for minimal surfaces
The Riesz kernels do not give rise to higher dimensional analogues of the Menger-Melnikov curvature.
Ever since the discovery of the connection between the Menger-Melnikov curvature and the Cauchy kernel in the L2 norm, and its impressive utility in the analytic capacity problem, higher dimensional analogues have been coveted. The lesson from 1-sets was that any such (nontrivial, nonnegative) expression, using the Riesz kernels for m-sets in Rn, even in any Lk norm (k ∈ N), would probably carry nontrivial information on whether the boundedness of these kernels in the appropriate norm implies rectifiability...
The Sobolev spaces of harmonic functions
The solution of a singular integral equation with some applications in potential theory.
The solution of an open problem given by H. Haruki and T. M. Rassias.
The successive approximation method for the Dirichlet problem in a planar domain
The Dirichlet problem for the Laplace equation for a planar domain with piecewise-smooth boundary is studied using the indirect integral equation method. The domain is bounded or unbounded. It is not supposed that the boundary is connected. The boundary conditions are continuous or p-integrable functions. It is proved that a solution of the corresponding integral equation can be obtained using the successive approximation method.
The transmission problem with boundary conditions given by real measures
The unique solvability of the problem Δu = 0 in G⁺ ∪ G¯, u₊ - au_ = f on ∂G⁺, n⁺·∇u₊ - bn⁺·∇u_ = g on ∂G⁺ is proved. Here a, b are positive constants and g is a real measure. The solution is constructed using the boundary integral equation method.
Theoretical and numerical study of a free boundary problem by boundary integral methods
In this paper we study a free boundary problem appearing in electromagnetism and its numerical approximation by means of boundary integral methods. Once the problem is written in a equivalent integro-differential form, with the arc parametrization of the boundary as unknown, we analyse it in this new setting. Then we consider Galerkin and collocation methods with trigonometric polynomial and spline curves as approximate solutions.
Theoretical and numerical study of a free boundary problem by boundary integral methods
In this paper we study a free boundary problem appearing in electromagnetism and its numerical approximation by means of boundary integral methods. Once the problem is written in a equivalent integro-differential form, with the arc parametrization of the boundary as unknown, we analyse it in this new setting. Then we consider Galerkin and collocation methods with trigonometric polynomial and spline curves as approximate solutions.
Theoretical aspects and numerical computation of the time-harmonic Green's function for an isotropic elastic half-plane with an impedance boundary condition
This work presents an effective and accurate method for determining, from a theoretical and computational point of view, the time-harmonic Green's function of an isotropic elastic half-plane where an impedance boundary condition is considered. This method, based on the previous work done by Durán et al. (cf. [Numer. Math.107 (2007) 295–314; IMA J. Appl. Math.71 (2006) 853–876]) for the Helmholtz equation in a half-plane, combines appropriately analytical and numerical techniques, which has an important...
Thin vortex tubes in the stationary Euler equation
In this paper we outline some recent results concerning the existence of steady solutions to the Euler equation in with a prescribed set of (possibly knotted and linked) thin vortex tubes.