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Simple examples of bounded domains are considered for which the presence of peculiar corners and edges in the boundary causes that the double layer potential operator acting on the space of all continuous functions on can for no value of the parameter be approximated (in the sub-norm) by means of operators of the form (where is the identity operator and is a compact linear operator) with a deviation less then ; on the other hand, such approximability turns out to be possible for...
We study the appropriate versions of parabolicity stochastic completeness and related Liouville properties for a general class of operators which include the p-Laplace operator, and the non linear singular operators in non-diagonal form considered by J. Serrin and collaborators.
Two kinds of orthogonal decompositions of the Sobolev space W̊₂¹ and hence also of for bounded domains are given. They originate from a decomposition of W̊₂¹ into the orthogonal sum of the subspace of the -solenoidal functions, k ≥ 1, and its explicitly given orthogonal complement. This decomposition is developed in the real as well as in the complex case. For the solenoidal subspace (k = 0) the decomposition appears in a little different form.
In the second kind decomposition the -solenoidal...
The α-harmonic measure is the hitting distribution of symmetric α-stable processes upon exiting an open set in ℝⁿ (0 < α < 2, n ≥ 2). It can also be defined in the context of Riesz potential theory and the fractional Laplacian. We prove some geometric estimates for α-harmonic measure.
Noting that a resolvent is associated with a convolution kernel satisfying the domination principle if and only if has the dominated convergence property, we give some remarks on the existence of a resolvent.
When one is restricted to a Stolz domain in a half plane we prove that internal thinness of a set at the origin structly implies minimal thinness there. Furthermore this result extends to the half plane itself. We also work out some relations among the concepts of minimal thinness, semi-thinness and finite logarithmic length. Finally we show that a theorem of Ahlfors and Heins can be improved.
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