skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Generalized Harmonic Functions and the Dewetting of Thin Films

Abstract

This paper describes the solvability of Dirichlet problems for Laplace's equation when the boundary data is not smooth enough for the existence of a weak solution in H{sup 1}{omega}. Scales of spaces of harmonic functions and of boundary traces are defined and the solutions are characterized as limits of classical harmonic functions in special norms. The generalized harmonic functions, and their norms, are defined using series expansions involving harmonic Steklov eigenfunctions on the domain. It is shown that the usual trace operator has a continuous extension to an isometric isomorphism of specific spaces. This provides a characterization of the generalized solutions of harmonic Dirichlet problems. Numerical simulations of a model problem are described. This problem is related to the dewetting of thin films and the associated phenomenology is described.

Authors:
 [1];  [2]
  1. Division of Mathematical Sciences, National Science Foundation, Arlington, VA 22230 (United States) and Department of Mathematics, University of Houston, 4800 (United States), E-mail: gauchmut@nsf.gov
  2. University of Houston, 4800 Calhoun, TX 77204 (United States) and Institut de Mathematiques, Universite de Neuchatel, Rue Emile Argand 11, CH-2007 (Switzerland), E-mail: kloucek@mac.com
Publication Date:
OSTI Identifier:
21067413
Resource Type:
Journal Article
Resource Relation:
Journal Name: Applied Mathematics and Optimization; Journal Volume: 55; Journal Issue: 2; Other Information: DOI: 10.1007/s00245-006-0883-0; Copyright (c) 2007 Springer; www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DIRICHLET PROBLEM; EIGENFUNCTIONS; EQUATIONS; MATHEMATICAL SOLUTIONS; MATHEMATICAL SPACE; SERIES EXPANSION; SIMULATION; THIN FILMS

Citation Formats

Auchmuty, Giles, and Kloucek, Petr. Generalized Harmonic Functions and the Dewetting of Thin Films. United States: N. p., 2007. Web. doi:10.1007/S00245-006-0883-0.
Auchmuty, Giles, & Kloucek, Petr. Generalized Harmonic Functions and the Dewetting of Thin Films. United States. doi:10.1007/S00245-006-0883-0.
Auchmuty, Giles, and Kloucek, Petr. Thu . "Generalized Harmonic Functions and the Dewetting of Thin Films". United States. doi:10.1007/S00245-006-0883-0.
@article{osti_21067413,
title = {Generalized Harmonic Functions and the Dewetting of Thin Films},
author = {Auchmuty, Giles and Kloucek, Petr},
abstractNote = {This paper describes the solvability of Dirichlet problems for Laplace's equation when the boundary data is not smooth enough for the existence of a weak solution in H{sup 1}{omega}. Scales of spaces of harmonic functions and of boundary traces are defined and the solutions are characterized as limits of classical harmonic functions in special norms. The generalized harmonic functions, and their norms, are defined using series expansions involving harmonic Steklov eigenfunctions on the domain. It is shown that the usual trace operator has a continuous extension to an isometric isomorphism of specific spaces. This provides a characterization of the generalized solutions of harmonic Dirichlet problems. Numerical simulations of a model problem are described. This problem is related to the dewetting of thin films and the associated phenomenology is described.},
doi = {10.1007/S00245-006-0883-0},
journal = {Applied Mathematics and Optimization},
number = 2,
volume = 55,
place = {United States},
year = {Thu Mar 15 00:00:00 EDT 2007},
month = {Thu Mar 15 00:00:00 EDT 2007}
}