This article is cited in 2 papers

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

M. V. Lomonosov Moscow State University

Abstract: 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 $\dfrac{2\beta_1+\beta_2}{\beta_1+\beta_2-1}<n-\alpha$, $u(x)$ 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.

UDC: 517.9

Received: 30.03.1989

 English version: Mathematical Notes, 1992, 52:3, 983–989

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