Planar biharmonic vector fields; potentials and traces

This paper describes eigenfunction approximation, and representations, of planar biharmonic vector fields with prescribed normal and tangential boundary data. These enable the characterization of the dependence of the solutions on the data including convergence in various norms. These problems that have been extensively studied using finite element algorithms; here various analytical results about the solutions are obtained. Spectral representations for the solutions of some scalar harmonic and biharmonic boundary value problems are first described and their dependence on boundary data is summarized. For biharmonic problems, the solutions are described using spaces with bases of DBS eigenfunctions in a manner similar to the use of harmonic Steklov eigenfunctions for solving harmonic boundary value problems. These methods are then used to obtain eigenfunction expansions of the scalar potential and stream function of a biharmonic field with given normal and tangential boundary traces. The potentials are $C^\infty$ and bounded and criteria for the solutions to be in specific spaces of fields are found. Some convergence results are stated and bounds of some norms are found. With further conditions on the traces, orthogonal expansions for the vorticity and the boundary trace of the vorticity of the flow are found.

Keywords

biharmonic solenoidal planar vector fields, Dirichlet biharmonic Steklov eigenfunctions, normal and tangential boundary data, Steklov representations

2010 Mathematics Subject Classification

Primary 46E22. Secondary 33E20, 35J40, 46E35.

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For Roland with fond memories of over 45 years of collegiality and friendship

Received 29 June 2023

Accepted 20 July 2023

Published 14 November 2023