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On the Harnack principle for strongly elliptic systems with nonsmooth coefficients. (English) Zbl 0949.35046

“Let \(\Omega\subseteq \mathbb{R}^n\) be a domain, \(m\in\mathbb{N}\) and \(\lambda, q> 0\). Then, there exists \(r (=r(\lambda, q))> 1\) such that for every \(0< p< q\), whenever \(\vec u_1,\vec u_2,\dots,\vec u_m\) are weak solutions of a strongly elliptic system with \(m\) equations of ellipticity \(\lambda\) satisfying \((\vec u_1\vec u_2\cdots\vec u_m)\in{\mathcal P}_r\) a.e. and \(\Omega'\subseteq \Omega\) is a subdomain, the following inequalities hold: \[ \begin{aligned} \sup_{1\leq j\leq m}\||\vec u_j|\|_{L^q(\Omega')} &= C\inf_{1\leq j\leq m} \||\vec u_j|\|_{L^p(\Omega')},\\ \sup_{1\leq j\leq m} \||\vec u_j |^{-1}\|_{L^q(\Omega')} &\leq C\inf_{1\leq j\leq m} \||\vec u_j|^{-1}\|_{L^p(\Omega')},\end{aligned} \] where \(C (= C(n,m,\lambda, q,p,\Omega,\Omega'))\) is a positive constant.” The systems considered have merely bounded measurable coefficients. Here “\((\vec u_1\vec u_2\cdots\vec u_m)\in{\mathcal P}_r\)” means that the matrix whose \(j\)th column is \(\vec u_j\) satisfies a certain ‘uniform positiveness’ assumption. Examples concerning the sharpness of the result are also discussed.
Reviewer: C.Muscalu

MSC:

35J45 Systems of elliptic equations, general (MSC2000)
35B50 Maximum principles in context of PDEs
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