A varifold solution of the nonlinear wave equation of a membrane (Q1058069)

This note is concerned with the nonlinear wave equation \[ D^ 2_ tu(t,x)-\sum^{n}_{j=1}D_ j\{D_ ju(t,x)(1+| Du(t,x)|^ 2)^{-1/2}\}=0 \] subject to the following initial and boundary conditions: \(u(0,x)=u_ 0(x)\), \(D_ tu(0,x)=u_ 1(x)\), \(u(t,x)=g(x)\) for \(x\in \partial U\). Here u is a bounded domain in \({\mathbb{R}}^ n\) with the boundary \(\partial U\) and \(D_ j=\partial /\partial x\) \((j=1,...,n)\) while \(D_ t=\partial /\partial t\). The objective of the authors consists in treating this problem by virtue of the theory of varifolds. After briefly defining the notion of a varifold solution, they phrase the main results in two theorems. One of them states the necessary conditions for the existence of such a solution. The second theorem shows an extremal property of the solution. The note does not contain the proofs.