Sur les zéros de solutions d’inégalités différentielles elliptiques. (On zeros of solutions of elliptic differential inequalities). (French) Zbl 0654.35036

Robbiano, Luc

Commun. Partial Differ. Equations 12, 903-919 (1987).

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

Keywords:

Lipschitz continuous; Hausdorff measure; harmonic polynomial

Citations:

Zbl 0593.35047

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