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The length of level lines of solutions of elliptic equations in the plane. (English) Zbl 0664.35016
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 $$\{x\in G:\quad u(x)<t\}.$$
Reviewer: M.Fuchs

MSC:
 35J15 Second-order elliptic equations 35B35 Stability in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs
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References:
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