On a partial regularity up to the boundary of weak solutions to quasilinear parabolic systems with quadratic growth
The paper under review is concerned with the following quasilinear parabolic systems in a cylinder \(Q=\Omega \times (0,T)\) \((\Omega \subset \mathbb R^{n},n \geq 2)\): (1) \(\frac{\partial u^{k}}{\partial t}- \frac{\partial}{\partial x_{\alpha}}( A^{\alpha\beta}_{kl}(x,t,u) \frac{\partial u^{l}}{\partial x_{\beta}}) + b^{k} (x,t,u,\nabla u)=0\) \((k= 1, \cdots , N).\) Here the coefficients \(A^{\alpha \beta}_{kl}\) are uniformly continuous and bounded on \(\overline{Q} \times \mathbb R^{N}\) and satisfy the elliptic condition \[ A^{\alpha \beta}_{kl}(x,t,u)\zeta^{k}_{\alpha}\zeta^{l}_{\beta}\geq \nu |\zeta |^{2} \qquad \forall \zeta \in \mathbb R^{nN} \qquad (\nu \text{ const } > 0) , \] and \(b^{k}\) are Carathéodory functions on \(Q \times \mathbb R^{N} \times \mathbb R^{nN}\) and satisfy the growth condition \[ |b (x,t,u,\zeta)|\leq a|\zeta|^{2}+L\quad \forall (x,t,u,\zeta)\in Q \times \mathbb R^{N} \times \mathbb R^{nN} \] (\(a\)= const \(\geq 0\), \(L=\) const \(> 0\)). The main result of the paper is as follows. Let \(\partial \Omega \in C^{2}\). Let \(u \in W^{1,1}_{2}(Q; \mathbb R^{N})\cap L^{\infty}(Q; \mathbb R^{N})\) be a weak solution to (1) under Dirichlet or Neumann boundary conditions on \(\partial \Omega \times (0,T)\). Assume that \(2a||u||_{L^{\infty}(Q; \mathbb R^{N})}< \nu\). Then there exists an open set \(Q' \subset\overline{Q} \times (0,T)\) such that \(u\) is Hölder continuous in \(Q'\) and the \((n-\varepsilon)\)-dimensional Hausdorff measure of \((\overline{\Omega} \times (0,T))\setminus Q' \) is zero.
Dirichlet or Neumann boundary conditions
partial Hölder continuity of weak solutions
scientific article; zbMATH DE number 1541246