Generalized Harmonic Functions and the Dewetting of Thin Films |
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Authors: | Giles Auchmuty Petr Kloucek |
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Institution: | (1) Division of Mathematical Sciences, National Science Foundation, Arlington, VA 22230, USA and Department of Mathematics, University of Houston, 4800, Calhoun, TX 77204-3008, USA;(2) University of Houston, 4800 Calhoun, TX 77204, USA and Institut de Mathematiques, Universite de Neuchatel, Rue Emile Argand 11, CH-2007, Neuchatel, Switzerland |
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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 H1Ω. 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. |
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