Removable singular sets for equations of the form $\sum\dfrac{\partial}{\partial x_i}a_{ij}(x)\dfrac{\partial u}{\partial x_j}=f(x,u,\nabla u)$

M. V. Tuvaev

Matematičeskie zametki, Tome 52 (1992) no. 3, pp. 146-153.

Résumé

The following uniformly elliptic equation is considered: $$ \sum\frac{\partial}{\partial x_i}a_{ij}(x)\frac{\partial u}{\partial x_j}=f(x,u,\nabla u), \qquad x\in\Omega\subset\mathbf{R}^n, $$ with measurable coefficients. The function $f$ satisfies the condition $$ f(x,u,\nabla u)u\geqslant C|u|^{\beta_1+1}|\nabla u|^{\beta_2}, \qquad \beta_1>0, \quad 0\leqslant\beta_2\leqslant2, \quad \beta_1+\beta_2>1. $$ It is proved that if $u(x)$ is a generalized (in the sense of integral identity) solution in the domain $\Omega\setminus K$, where the compactum $K$ has Hausdorff dimension $\alpha$, and if $\dfrac{2\beta_1+\beta_2}{\beta_1+\beta_2-1}$, $u(x)$ will be a generalized solution in the domain $\Omega$. Moreover, the sufficient removability conditions for the singular set are, in some sense, close to the necessary conditions.