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We extend to objects of general convex form an earlier criterion given by Finn for ensuring a local energy minimum corresponding to floating of a heavy horizontal cylinder on a liquid surface. We obtain a formal analogue of Finn´s criterion, and additionally a condition on interfacial surface energies under which an object of given shape with density exceeding that of the liquid must float if it is scaled to be sufficiently small. In analogy with Finn´s earlier work, we show also in the more physical case presently considered that if a body is lowered rigidly from a position above the undisturbed liquid surface to a position below the surface, the liquid configuration must change during the motion in a discontinuous way. Please note: This paper contains printing mistakes. The correct version you find here: http://www.oldenbourg-link.com/doi/pdf/10.1524/anly.2009.0931

We derive an energy bound, estimating a weighted Dirichlet integral of two solutions for the nonparametric equation with prescribed mean curvature in n dimensions in terms of the L 1 -norm for the difference of their values on the boundary. Furthermore, a similar estimate is established for solutions of the equation div F p (·,∇ u )= n H (·, u ), where F(x,p) denotes an elliptic Lagrangian with linear growth in p . These results are used to remove singularities of solutions to these equations.

Plateau’s problem is solved for a certain class of properly embedded unbounded curves in R 3 , i. e. the existence of simply connected properly immersed minimal surfaces is shown which have a given curve from this class as their boundary. The class of admissible boundary contours includes all properly embedded piecewise C 1, α curves with polynomial ends. The constructed surfaces have quadratic area growth.

Given a Riemannian metric g in R 2 with negative Gauss curvature, we study the problem of local isometric embedding of g into R 3 . We focus on two different geometric initial conditions.

We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemannian geometry of the target. These maps are called affine harmonic. We show an existence result for affine harmonic maps in a given homotopy class when the target has nonpositive sectional curvature and some global nontriviality condition is met. An example shows that such a condition is necessary. The analytical part is made difficult by the absence of a variational structure underlying affine harmonic maps. We therefore need to combine estimation techniques from geometric analysis and PDE theory with global geometric considerations.

We prove that weak solutions of systems with skew-symmetric structure, which possess a continuous boundary trace, have to be continuous up to the boundary. This applies, e.g., to the H -surface system Δ u = 2 H(u) ∂ x 1 u ∧∂ x 2 u with bounded H and thus extends an earlier result by P. Strzelecki and proves the natural counterpart of a conjecture by E. Heinz. Methodically, we use estimates below natural exponents of integrability and a recent decomposition result by T. Rivière.

Variational derivation of the expression of the solution of Monge’s problem posed on compact manifolds (possibly with boundary), assuming all data are smooth, the solution is a diffeomorphism and the cost function satisfies a generating type condition.