On boundary values in $L_p$, $p>1$, of solutions of elliptic equations

A. K. Gushchin ; V. P. Mikhailov

Sbornik. Mathematics, Tome 36 (1980) no. 1, pp. 1-19

Résumé

The behavior near the boundary of generalized solutions of a second order elliptic equation $$ \sum_{i,j=1}^n\frac\partial{\partial x_i}\biggl(a_{ij}(x)\frac{\partial u}{\partial x_j}\biggr)=f,\qquad x\in Q=\{|x|1\}\subset\mathbf R_n. $$ in $W_p^1(Q)$, $p>1$, is studied. It is shown that under a certain condition on the right side of the equation, the boundedness of the function $\|x\|_{L_p(\|x\|=r)}$, $\frac12\leqslant r1$, is necessary and sufficient for the existence of a limit for the solution $u(rw)$, $\frac12\leqslant r1$, $|w|=1$, in $L_p(\|w\|=1)$ as $r\to1-0$. Moreover, the summability of the function $(1-|x|)|u(x)|^{p-2}|\nabla u(x)|^2$ is also a necessary and sufficient condition for the existence of a limit in $ L_p$ on the boundary. Bibliography: 10 titles.