The integral equation methods for the perturbed Helmholtz eigenvalue problems.
The Martin compactification of the polydisc at the bottom of the positive spectrum.
The mean curvature measure
We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the...
The multiple layer potential for the biharmonic equation in variables
The definition of multiple layer potential for the biharmonic equation in is given. In order to represent the solution of Dirichlet problem by means of such a potential, a singular integral system, whose symbol determinant identically vanishes, is considered. The concept of bilateral reduction is introduced and employed for investigating such a system.
The Neumann problem for the Laplace equation on general domains
The solution of the weak Neumann problem for the Laplace equation with a distribution as a boundary condition is studied on a general open set in the Euclidean space. It is shown that the solution of the problem is the sum of a constant and the Newtonian potential corresponding to a distribution with finite energy supported on . If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on a necessary and sufficient condition for the solvability...
The obstacle problem for the biharmonic operator
The Pluripolar Hull and the Fine Topology
We show that the projections of the pluripolar hull of the graph of an analytic function in a subdomain of the complex plane are open in the fine topology.
The Poisson's problem for the Laplacian with Robin boundary condition in non-smooth domains.
The problem and singular integrals on Orlicz spaces
The p-Royden and p-Harmonic Boundaries for Metric Measure Spaces
Let p be a real number greater than one and let X be a locally compact, noncompact metric measure space that satisfies certain conditions. The p-Royden and p-harmonic boundaries of X are constructed by using the p-Royden algebra of functions on X and a Dirichlet type problem is solved for the p-Royden boundary. We also characterize the metric measure spaces whose p-harmonic boundary is empty.
The Robin problem in potential theory (Preliminary communication)
The simple layer potential for the biharmonic equation in variables
A theory of the «simple layer potential» for the classical biharmonic problem in is worked out. This hinges on the study of a new class of singular integral operators, each of them trasforming a vector with scalar components into a vector whose components are differential forms of degree one.
The Sobolev capacity on metric spaces.
The third boundary value problem in potential theory
The third boundary value problem in potential theory for domains with a piecewise smooth boundary
The paper investigates the third boundary value problem for the Laplace equation by the means of the potential theory. The solution is sought in the form of the Newtonian potential (1), (2), where is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure . Denote by the corresponding operator on the space of signed measures on the boundary of the investigated domain . If there is such that the essential spectral radius of is...
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.
Théorème de Keldych dans la théorie axiomatique de Bauer des fonctions harmoniques
Théorie du potentiel associée à certains systèmes différentiels.
Théorie du potentiel dans les domaines lipschitziens, d'après G. C. Verchota
Theory of Bessel potentials. II
Dans cette partie de la théorie des potentiels besseliens on considère les restrictions de potentiels de la classe aux domaines ouverts . On cherche à caractériser de manière intrinsèque la classe ainsi obtenue.On attaque ce problème en définissant de manière directe (§ 2) une classe qui, pour des domaines assez réguliers, est égale à .L’égalité est équivalente à l’existence d’un opérateur-extension , linéaire et continu, tel que soit une extension de . Si un tel opérateur transforme...