Appl Math Optim 55:145–161 (2007)
2007 Springer Science+Business Media, Inc.
Generalized Harmonic Functions and the
Dewetting of Thin Films
and Petr Klouˇcek
Division of Mathematical Sciences, National Science Foundation,
Arlington, VA 22230, USA
Department of Mathematics, University of Houston,
4800 Calhoun, TX 77204-3008, USA
, University of Houston,
4800 Calhoun, TX 77204, USA
Institut de Math`ematiques, Universit`e de Neuchˆatel,
Rue Emile Argand 11, CH-2007 Neuchˆatel, Switzerland
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
(). Scales of spaces of harmonic functions and of boundary traces
are deﬁned and the solutions are characterized as limits of classical harmonic func-
tions 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.
Key Words. Steklov eigenvalue problem, Trace spaces, Generalized harmonic
AMS Classiﬁcation. 35P05, 35J05, 35J55, 46F99, 65N25, 76A20.
The second author was supported in part by Grants NSF ACI-0325081 and NSF CCR-0306503 and by
the European Commission via MEXC-CT-2005-023843.