Boundary value problems for the one-dimensional Willmore equation |
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Authors: | Klaus Deckelnick Hans-Christoph Grunau |
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Institution: | 1.Fakult?t für Mathematik,Otto-von-Guericke-Universit?t Magdeburg,Magdeburg,Germany |
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Abstract: | The one-dimensional Willmore equation is studied under Navier as well as under Dirichlet boundary conditions. We are interested
in smooth graph solutions, since for suitable boundary data, we expect the stable solutions to be among these. In the first
part, classical symmetric solutions for symmetric boundary data are studied and closed expressions are deduced. In the Navier
case, one has existence of precisely two solutions for boundary data below a suitable threshold, precisely one solution on
the threshold and no solution beyond the threshold. This effect reflects that we have a bending point in the corresponding
bifurcation diagram and is not due to that we restrict ourselves to graphs. Under Dirichlet boundary conditions we always have existence of precisely one
symmetric solution. In the second part, we consider boundary value problems with nonsymmetric data. Solutions are constructed
by rotating and rescaling suitable parts of the graph of an explicit symmetric solution. One basic observation for the symmetric
case can already be found in Euler’s work. It is one goal of the present paper to make Euler’s observation more accessible
and to develop it under the point of view of boundary value problems. Moreover, general existence results are proved. |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 53C21 34B15 35J65 |
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