726 research outputs found
Minimal energy problems for strongly singular Riesz kernels
We study minimal energy problems for strongly singular Riesz kernels on a
manifold. Based on the spatial energy of harmonic double layer potentials, we
are motivated to formulate the natural regularization of such problems by
switching to Hadamard's partie finie integral operator which defines a strongly
elliptic pseudodifferential operator on the manifold. The measures with finite
energy are shown to be elements from the corresponding Sobolev space, and the
associated minimal energy problem admits a unique solution. We relate our
continuous approach also to the discrete one, which has been worked out earlier
by D.P. Hardin and E.B. Saff.Comment: 31 pages, 2 figure
Essentially translation invariant pseudodifferential operators on manifolds with cylindrical ends
We study two classes (or calculi) of pseudodifferential operators defined on
manifolds with cylindrical ends: the class of pseudodifferential operators that
are ``translation invariant at infinity'' and the class of ``essentially
translation invariant operators'' that have appeared in the study of layer
potential operators on manifolds with straight cylindrical ends. Both classes
are close to the -calculus considered by Melrose and Schulze and to the
-calculus considered by Melrose and Mazzeo-Melrose. Our calculi, however,
are different and, while some of their properties follow from those of the -
or -calculi, many of their properties do not. In particular, the
``essentially translation invariant calculus'' is spectrally invariant, a
property not enjoyed by the ``translation invariant at infinity'' calculus or
the -calculus. For our calculi, we provide easy, intuitive proofs of the
usual properties: stability for products and adjoints, mapping and boundedness
properties for operators acting between Sobolev spaces, regularity properties,
existence of a quantization map and topological properties of our algebras, the
Fredholm property. Since our applications will be to the Stokes operator, we
systematically work in the setting of Agmon-Douglis-Nirenberg-elliptic
operators.Comment: 39 page
IN MEMORY OF GAETANO FICHERA
Dear Dr. Matelda Fichera, Prof. Dr. Maria Pia Colautti, Dr. Anna MariaFichera, Dr. Massimo Fichera,Gaetano Fichera passed away at the age of 74 June 1st just 10 years ago. He left behind his dear wife Dr. Mathelda Fichera after 44 years of happy togetherness - and the world of mathematics.Dear Dr. Matelda Fichera, Prof. Dr. Maria Pia Colautti, Dr. Anna MariaFichera, Dr. Massimo Fichera,Gaetano Fichera passed away at the age of 74 June 1st just 10 years ago. He left behind his dear wife Dr. Mathelda Fichera after 44 years of happy to- getherness - and the world of mathematics
On the Dirichlet problem in elasticity for a domain exterior to an arc
AbstractWe consider here a Dirichlet problem for the two-dimensional linear elasticity equations in the domain exterior to an open arc in the plane. It is shown that the problem can be reduced to a system of boundary integral equations with the unknown density function being the jump of stresses across the arc. Existence, uniqueness as well as regularity results for the solution to the boundary integral equations are established in appropriate Sobolev spaces. In particular, asymptotic expansions concerning the singular behavior for the solution near the tips of the arc are obtained. By adding special singular elements to the regular splines as test and trial functions, an augmented Galerkin procedure is used for the corresponding boundary integral equations to obtain a quasi-optimal rate of convergence for the approximate solutions
Minimal energy problems for strongly singular Riesz kernels
We study minimal energy problems for strongly singular Riesz kernels , where n ≥ 2 and α ∈ (−1, 1), considered for compact (n−1)-dimensional -manifolds Γ immersed into . Based on the spatial energy of harmonic double layer potentials, we are motivated to formulate the natural regularization of such a minimization problem by switching to Hadamard's partie finie integral operator which defines a strongly elliptic pseudodifferential operator of order β = 1 − α on Γ. The measures with finite energy are thus elements from the Sobolev space , 0 0 the set |x−y|≤δ of Γ×Γ is cut out
Rapid solution of minimal Riesz energy problems
In , we obtain the numerical solution to both the unconstrained and constrained Gauss variational problems, considered for the Riesz kernel of order each being charged with Borel measures with the sign prescribed. Using the fact that such problems over a cone of Borel measures can alternatively be formulated as minimum problems over the corresponding cone of surface distributions belonging to the Sobolev–Slobodetski space , where and (see [17]), we approximate the sought density by piecewise constant boundary elements and apply the primal-dual active set strategy to impose the desired inequality constraints. The boundary integral operator which is defined by the Riesz kernel under consideration is e ciently approximated by means of an -matrix approximation. This particularly enables the application of a preconditioner for the iterative solution of the first order optimality system. Numerical results in are given to demonstrate our approach
Riesz minimal energy problems on -manifolds
In , we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel , where 1 (α−1)/2, each being charged with Borel measures with the sign prescribed. We show that the Gauss variational problem over a cone of Borel measures can alternatively be formulated as a minimum problem over the corresponding cone of surface distributions belonging to the Sobolev–Slobodetski space , where ε := α−1 and . An equivalent formulation leads in the case of two manifolds to a nonlinear system of boundary integral equations involving simple layer potential operators on Γ. A corresponding numerical method is based on the Galerkin–Bubnov discretization with piecewise constant boundary elements. Wavelet matrix compression is applied to sparsify the system matrix. Numerical results are presented to illustrate the approach
Riesz minimal energy problems on -manifolds
In , we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel , where 1 (α−1)/2, each being charged with Borel measures with the sign prescribed. We show that the Gauss variational problem over a cone of Borel measures can alternatively be formulated as a minimum problem over the corresponding cone of surface distributions belonging to the Sobolev–Slobodetski space , where ε := α−1 and . An equivalent formulation leads in the case of two manifolds to a nonlinear system of boundary integral equations involving simple layer potential operators on Γ. A corresponding numerical method is based on the Galerkin–Bubnov discretization with piecewise constant boundary elements. Wavelet matrix compression is applied to sparsify the system matrix. Numerical results are presented to illustrate the approach
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