Estimates on the boundary of a domain for the first derivatives of functions satisfying an elliptic or parabolic inequality
The authors investigate boundary regularity of solutions of the Dirichlet problem for a quasilinear elliptic equation in nondivergence form: \[ a^{ij}(x,u,Du)D_{ij}u+a(x,u,Du)=0\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega, \] where \(\Omega\) is a domain in \({\mathbb{R}}^ n\), under fairly weak hypotheses. Specifically if there are positive constants \(\nu\leq \mu\) such that \[ \nu | \xi |^ 2\leq a^{ij}(x,u,Du)\xi_ i\xi_ j\leq \mu | \xi |^ 2\text{ for all } x\in \Omega,\quad \xi \in {\mathbb{R}}^ n, \] and a nonnegative constant \(\mu_ 1\) and functions b, \(\Phi\) in \(L^ q(\Omega)\), for some \(q>n\), such that \[ | a(x,u,Du)| \leq \mu,\quad | Du|^ 2+b(x)| Du| +\Phi (x), \] and if \(\partial \Omega \in W^{2,q}\), then the normal derivative exists on \(\partial \Omega\) and is bounded and Hölder continuous there. Moreover the norms can be estimated in terms of the data. Analogous results for parabolic problems are also proved.
scientific article; zbMATH DE number 4161189