DOI: 10.1007/s00245-006-0883-0 Appl Math Optim 55:145–161 (2007) © 2007 Springer Science+Business Media, Inc. Generalized Harmonic Functions and the Dewetting of Thin Films ∗ Giles Auchmuty 1 and Petr Klouˇ cek 2 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 gauchmut@nsf.gov 2 TLC 2 , University of Houston, 4800 Calhoun, TX 77204, USA and Institut de Math` ematiques, Universit` e de Neuchˆ atel, Rue Emile Argand 11, CH-2007 Neuchˆ atel, Switzerland kloucek@mac.com 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 1 (). 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 functions, Dewetting. 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.