The affine Cauchy problem

Gröbner bases and their application to the Cauchy problem on finitely generated affine monoids

For finitely generated submonoids of the integer lattice and submodules over the associated monoid algebra, we investigate Gröbner bases with respect to generalised term orders. Up to now, this theory suffered two disadvantages: The algorithm for computing the Gröbner bases was slow and it was not known whether there existed generalised term orders for arbitrary finitely generated submonoids. T...

Cauchy Problems for Discrete Affine Minimal Surfaces

In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with ...

The Cauchy Problem for Membranes

We show existence and uniqueness for timelike minimal submanifolds (world volume of p-branes) in ambient Lorentz manifolds admitting a time function in a neighborhood of the initial submanifold. The initial value formulation introduced and the conditions imposed on the initial data are given in purely geometric terms. Only an initial direction must be prescribed in order to provide uniqueness f...

Inhomogeneous Two-Parameter Abstract Cauchy Problem

Lectures on Cauchy Problem

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