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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}
}
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  • Atomic force microscopy is used to characterize the evolution of film morphology produced by heavy-ion bombardment. Pt films, 3 and 5 nm thick, are deposited on SiO{sub 2} substrates and subsequently bombarded by 800 keV Kr{sup +}. Ion doses of >2x10{sup 14} initiate pattern formation and the dewetting of Pt films from the substrate. The film morphology becomes increasingly disconnected with increasing dose; at the highest doses, ({approx}2x10{sup 16} cm{sup -2}), isolated nanoparticles are formed with a uniform spacing. The results are explained by the nucleation of bare substrate patches and subsequent coarsening of the morphology by the molten zonesmore » created by individual Kr{sup +} impacts. (c) 2000 American Institute of Physics.« less
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