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Sharp conditions for regularity of generalized solutions of fourth order quasilinear elliptic systems. (English. Russian original) Zbl 0602.35027

Sov. Math., Dokl. 30, 743-746 (1984); translation from Dokl. Akad. Nauk SSSR, 279, 820-824 (1984).
The following Dirichlet problem is considered for the fourth order quasilinear system for the vector valued function \(u=(u_ 1,...,u_ N):\) \[ Lu=\sum_{0\leq | \alpha |,| \beta | \leq 2}(- 1)^{| \alpha |} D^{\alpha}a_{\alpha}(x;u,...,D^{\beta}u)=0,\quad x\in \Omega, \] (\(\partial^ ku/\partial v^ k)|_{\partial \Omega}=(\partial^ kf(x)/\partial v^ k)|_{\partial \Omega}\), \(k=0,1;\)
\(\Omega\) being a smooth bounded domain contained in \({\mathbb{R}}^ m\) and v the interior normal to its boundary \(\partial \Omega\). The following assumptions are made if \(f\in W_ q^{(4)}(\Omega)\), \(q>m/2;\)
(i) \(\forall u\in W_ q^{(4)}(\Omega)\) and for some \(q>m/2\), \(a_{\alpha}(x;u,...,D^{\beta}u)\), \(Lu\in L_ q.\)
(ii) For a.e. \(x\in \Omega\), \(a_{\alpha}(x;p_ 0,...,p_{\beta})\in C^{(1)}({\mathbb{R}}^ N_ p)\) in any bounded domain of variation of \(p_{\beta}.\)
(iii) Let A be the matrix \(\partial a_{\alpha}^{(i)}/\partial p_{\beta}^{(k)}\), \(0\leq | \alpha |,| \beta | \leq 2\), \(i,k=1,...,N\). Then \(\exists \mu,\mu '\in {\mathbb{R}}_+\) such that \[ \mu | \xi |^ 2\leq <A\xi,\xi >\leq \mu '| \xi |^ 2. \] Under assumptions i)-iii) some sharp conditions are given in order that \(u\in C^{(1+\gamma)}(\Omega ')\), where \(\Omega\) ’ is contained in \(\Omega\), and some \(\gamma >0\).
Reviewer: A.Bove

MSC:

35J35 Variational methods for higher-order elliptic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
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