Regularity of boundary points for degenerate quasilinear second-order elliptic equations (Q1072001)

The paper deals with regularity of boundary points relative to the Dirichlet problem for degenerate quasilinear elliptic equations \[ (1)\quad \phi (| x-x^ 0|)\sum^{n}_{i,k=1}a_{ik}(x)u_{x_ ix_ k}+b(x,u,\nabla u)=0. \] The coefficients are determined in the boundary region \({\mathcal D}\subset R^ n\), \(n\geq 3\), and satisfy the following conditions \[ (2)\quad \alpha | \xi |^ 2\leq \sum^{n}_{i,k=1}a_{ik}(x)z_ iz_ k\leq \alpha^{-1}| \xi |^ 2;\quad \alpha >0,\quad x\in {\mathcal D},\quad \xi \in R^ n, \] \[ (3)\quad | a_{ik}(x)-a_{ik}(y)| \leq \omega_ r(| x- y|);\quad x,y\in \bar {\mathcal D}\cap \Omega^{x^ 0}_{r/2,r},\quad r>0, \] \[ (4)\quad | b(x,u,\nabla u)| \leq d| \nabla u|^ 2,\quad d=const. \] Here \(x^ 0\in \partial {\mathcal D}\), \(\omega_ r(t)\) for any \(r>0\) is a non decreasing function of t and \(\omega_ r(0)=0\), \(\phi (r)>0\) for \(r>0\), \(\phi (0)=0\), \(\Omega^{x_ 0}_{r_ 1,r_ 2}=\{x:\quad r_ 1\leq | x-x_ 0| \leq r_ 2\}.\) Let \(\nu_ r(E)\) be a capacity of the set E generated by the kernel \(| x-y|^{2-n} \exp \{j\int^{diam}_{| x- y|}(\omega_ r(t)/t)dt\},\) where \(j>0\) is a constant depending on \(\alpha\), d and n. It is proved, that if conditions (2)-(4) are satisfied for the coefficients of the equation (1), then for regularity of the point \(x_ 0\in \partial {\mathcal D}\) is sufficient that \[ \int_{0}(\nu_ R(\Omega^{x_ 0}_{R/\sigma,R}\setminus {\mathcal D})\phi (R)\exp \{j\int^{diam {\mathcal D}}_{R}(\omega_ r(t)/t)dt\}/R^{n- 2})(1/R)dR=\infty. \] The estimate of the continuity modulo of solution to the first boundary problem in the regular boundary point is also obtained.