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

Poisson problems for semilinear Brinkman systems on Lipschitz domains in Rn

The purpose of this paper is to combine a layer potential analysis with the Schauder fixed point theorem to show the existence of solutions of the Poisson problem for a semilinear Brinkman system on bounded Lipschitz domains in Rn (n 65 2) with Dirichlet or Robin boundary conditions and data in L2-based Sobolev spaces. We also obtain an existence and uniqueness result for the Dirichlet problem for a special semilinear elliptic system, called the Darcy\u2013Forchheimer\u2013 Brinkman system

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 $b$-calculus considered by Melrose and Schulze and to the $c$-calculus considered by Melrose and Mazzeo-Melrose. Our calculi, however, are different and, while some of their properties follow from those of the $b$- or $c$-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 $b$-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 $|x−y|^{α−n}$, where n ≥ 2 and α ∈ (−1, 1), considered for compact (n−1)-dimensional $C^{\infty}$-manifolds Γ immersed into $\mathbb{R}^n$. 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 $H^{β/2}(Γ)$, 0 0 the set |x−y|≤δ of Γ×Γ is cut out

Riesz minimal energy problems on Ck−1,1C^{k−1,1}-manifolds

In $\mathbb{R}^n, n\ge 2$, we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel $|x − y|^{α−n}$, where 1 (α−1)/2, each $Γ_l$ being charged with Borel measures with the sign $α_l := ±1$ 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 $H^{−ε/2}(Γ)$, where ε := α−1 and $Γ := \bigcup_{l∈L} Γ_l$. 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

Rapid solution of minimal Riesz energy problems

In $\mathbb{R}^n$, $n\ge 2$ we obtain the numerical solution to both the unconstrained and constrained Gauss variational problems, considered for the Riesz kernel $\|x-y\|^{\alpha - n}$ of order $1 (\alpha-1)/2$ each $\Gamma_i$ being charged with Borel measures with the sign $\alpha_i := \pm 1$ 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 $H^{-\varepsilon/2}(\Gamma)$, where $\varepsilon := \alpha - 1$ and $\Gamma = \Gamma_1 \cup \Gamma_2$ (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 $\mathcal{H}$-matrix approximation. This particularly enables the application of a preconditioner for the iterative solution of the first order optimality system. Numerical results in $\mathbb{R}^3$ are given to demonstrate our approach