The length of level lines of solutions of elliptic equations in the plane (Q1114851)

The author gives estimates for the length of \(x\in G:\quad u(x)=t\}\) where t is real number, G a subregion of a two-dimensional domain \(\Omega\) and u: \(\Omega\) \(\to {\mathbb{R}}\) denotes a nontrivial solution of the uniformly elliptic equation \[ 0=a_{11} u_{x_ 1x_ 1}+2a_{12} u_{x_ 1x_ 2}+a_{22} u_{x_ 2x_ 2}+b_ 1 u_{x_ 1}+b_ 2 u_{x_ 2} \] with sufficiently smooth coefficients. The estimate for the length of \(\{x\in G:\quad u(x)=t\}\) involves gradient bounds on subdomains \(G_ 1\) such that \(G\subset \subset G_ 1\subset \subset \Omega\) and is based on the fact that the critical points of u form a discrete subset of \(\Omega\) which in turn implies an integration by parts formula for the divergence of the field \(Du/| Du|\) on domains \(\{x\in G:\quad u(x)<t\}.\)