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Sur les zéros de solutions d’inégalités différentielles elliptiques. (On zeros of solutions of elliptic differential inequalities). (French) Zbl 0654.35036
It is shown that a function $$u\in C^{1+\delta}(\Omega)$$, $$\delta >0$$, for which $$| Au| \leq C_ k(| u| +| \nabla u|)$$ on compact sets $$K\subset \Omega$$, $$\Omega$$ open in $${\mathbb{R}}^ m$$, $$m\geq 2$$, A being a second-order elliptic operator with Lipschitz continuous coefficients, satisfies: $$\forall \epsilon >0:$$ $$H^{m- 2+\epsilon}(S)=0$$, where $$S=\{x\in \Omega |$$ $$u(x)=0$$, $$\nabla u(x)=0\}$$, $$H^ k$$ denoting the k-dimensional Hausdorff measure, and $$H^{m-1+\epsilon}(\{x\in \Omega |$$ $$u(x)=0\})=0$$. Moreover, if $$u(0)=0$$, $$\nabla u(0)=0$$, $$u\neq 0$$, there is $$n\geq 2$$ with $$u=P_ n+\Gamma_ n$$, where $$P_ n$$ is a harmonic polynomial of degree n, $$P_ n\neq 0$$, and $$\Gamma_ n$$ satisfies: $$| \Gamma_ n(x)| \leq C| x|^{n+\delta /2}$$, $$| \nabla \Gamma_ n(x)| \leq C| x|^{n-1+\delta /2}$$ on $$\{$$ $$x|$$ $$| x| \leq 1\}$$, provided $$A=\Delta +\sum^{m}_{i,j=1}\partial_ ia_{ij}(x)\partial_ x$$ with $$a_{ij}(0)=0$$. - This paper generalizes results from L. A. Caffarelli and A. Friedman [J. Differ. Equations 60, 420-433 (1985; Zbl 0593.35047)], in which $$A=\Delta$$.
Reviewer: R.Racke

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35R45 Partial differential inequalities and systems of partial differential inequalities 35J15 Second-order elliptic equations 35B65 Smoothness and regularity of solutions to PDEs
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