Representations of Solutions of Laplacian Boundary Value Problems on Exterior Regions

Representations of Solutions of Laplacian Boundary Value Problems on Exterior Regions This paper treats the well-posedness and representation of solutions of Poisson’s equation on exterior regions $U\subsetneq{\mathbb{R}}^{N}$ with N ≥3. Solutions are sought in a space E 1 ( U ) of finite energy functions that decay at infinity. This space contains H 1 ( U ) and existence-uniqueness theorems are proved for the Dirichlet, Robin and Neumann problems using variational methods with natural conditions on the data. A decomposition result is used to reduce the problem to the evaluation of a standard potential and the solution of a harmonic boundary value problem. The exterior Steklov eigenproblems for the Laplacian on U are described. The exterior Steklov eigenfunctions are proved to generate an orthogonal basis for the subspace of harmonic functions and also of certain boundary trace spaces. Representations of solutions of the harmonic boundary value problem in terms of these bases are found, and estimates for the solutions are derived. When U is the region exterior to a 3-d ball, these Steklov representations reduce to the classical multi-pole expansions familiar in physics and engineering analysis. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Representations of Solutions of Laplacian Boundary Value Problems on Exterior Regions

Loading next page...
 
/lp/springer_journal/representations-of-solutions-of-laplacian-boundary-value-problems-on-8IJE0O4Mr0
Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer Science+Business Media New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-013-9215-3
Publisher site
See Article on Publisher Site

Abstract

This paper treats the well-posedness and representation of solutions of Poisson’s equation on exterior regions $U\subsetneq{\mathbb{R}}^{N}$ with N ≥3. Solutions are sought in a space E 1 ( U ) of finite energy functions that decay at infinity. This space contains H 1 ( U ) and existence-uniqueness theorems are proved for the Dirichlet, Robin and Neumann problems using variational methods with natural conditions on the data. A decomposition result is used to reduce the problem to the evaluation of a standard potential and the solution of a harmonic boundary value problem. The exterior Steklov eigenproblems for the Laplacian on U are described. The exterior Steklov eigenfunctions are proved to generate an orthogonal basis for the subspace of harmonic functions and also of certain boundary trace spaces. Representations of solutions of the harmonic boundary value problem in terms of these bases are found, and estimates for the solutions are derived. When U is the region exterior to a 3-d ball, these Steklov representations reduce to the classical multi-pole expansions familiar in physics and engineering analysis.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Feb 1, 2014

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off