05-31
Boundary value problems for the one-dimensional Willmore equation
by Deckelnick, K.; Grunau, H.-Ch.
Preprint series: 05-31, Preprints
The paper is published: Calc. Var. Partial Differ. Equ. 30, 293-314 (2007).
- MSC:
- 34B15 Nonlinear boundary value problems
- 53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions, See also {58G30}
- 35J65 Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE
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 {\it 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.
Keywords: Willmore equation, one dimensional, boundary value problem
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