Equations of mean curvature type in 2 independent variables

On Two Mean Curvature Equations

This paper is an exposition of some results concerning two distinct mean curvature equations which have been obtained recently by the author and others. We also propose some unsolved problems and new questions arising from ours developments.

Constant mean curvature surfaces and integrable equations

Mean Curvature Blowup in Mean Curvature Flow

In this note we establish that finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M t →֒ (N, h) are characterised by the blow up of the mean curvature.

The classical solvability of the contact angle problem for generalized equations of mean curvature type

In this paper, mean curvature type equations with general potentials and contact angle boundary conditions are considered. We extend the ideas of Ural’tseva, formulating sharper hypotheses for the existence of a classical solution. Corner stone for these results is a method to estimate quantities on the boundary of the free surface. We moreover provide alternative proofs for the higher-order es...

Fattening and Comparison Principle for Level-Set Equations of Mean Curvature Type

In this paper, we give several applications of the discrete game approach to partial differential equations (PDEs). We first present a rigorous game-theoretic proof of fattening phenomenon for motion by curvature with figure-eight–shaped initial curves without using parabolic PDE theory. The proof is based on a comparison between the game value and its inverse. Accompanied by the example of fig...