Nonvariational basic parabolic systems of second order

Campanato, Sergio

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 2 (1991) no. 2, pp. 129-136

Résumé

\( \Omega \) is a bounded open set of \( \mathbb{R}^{n} \), of class \( C^{2} \) and \( T>0 \). In the cylinder \( Q = \Omega \times (0, T) \) we consider non variational basic operator \( a(H(u)) - \partial u / \partial t \) where \( a(\xi) \) is a vector in \( \mathbb{R}^{N} \), \( N \ge 1 \), which is continuous in \( \xi \) and satisfies the condition (A). It is shown that \( \forall f \in L^{2} (Q) \) the Cauchy-Dirichlet problem \( u \in W_{0}^{2,1} (Q) \), \( a(H(u)) - \partial u / \partial t = f \) in \( Q \), has a unique solution. It is further shown that if \( u \in W_{0}^{2,1} (Q) \) is a solution of the basic system \( a(H(u)) - \partial u / \partial t = 0 \) in \( Q \), then \( H(u) \) and \( \partial u / \partial t \) belong to \( H^{1}_{loc} (Q) \). From this the Hölder continuity in \( Q \) of the vectors \( u \) and \( D u \) are deduced respectively when \( n \le 4 \) and \( n = 2 \).