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Zeitschrift für Analysis und ihre Anwendungen

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Volume 19, Issue 3, 2000, pp. 747–762
DOI: 10.4171/ZAA/978

Published online: 2000-09-30

Capillary Surfaces in Non-Cylindrical Domains

G. Schindlmayr[1]

(1) Dreieich, Germany

This paper is concerned with the capillary problem in a class of non-cylindrical domains in $K \subset \mathbb R^{n+1}$ obtained by scaling a bounded cross-section ­ $\Omega \subset \mathbb R^n$ along the vertical axis. The capillary surfaces are described in two different ways. In the first model, they are described as the boundary of a Caccioppoli set and in a second model, after transforming $K$ to a cylinder, they are described as graphs of functions on $\Omega$­. The volume of the fluid is prescribed. For both models, the energy functional is derived and declared on the appropriate function space consisting of $BV$-functions. Main results are existence and a priori bounds of minimizers, using the direct methods in the calculus of variations. For the special case of a cone over the domain $\Omega$­, a criterion is given to assure that the tip is not filled with liquid. Another point of examination concerns modelling the volume restriction by means of a Lagrange multiplier.

Keywords: Equilibrium capillary surfaces, BV-functions, existence, Lagrange multiplier

Schindlmayr G.: Capillary Surfaces in Non-Cylindrical Domains. Z. Anal. Anwend. 19 (2000), 747-762. doi: 10.4171/ZAA/978