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This article is cited in 2 scientific papers (total 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.
Received: 30.03.1989
Citation:
M. V. Tuvaev, “Removable singular sets for equations of the form $\sum\dfrac{\partial}{\partial x_i}a_{ij}(x)\dfrac{\partial u}{\partial x_j}=f(x,u,\nabla u)$”, Mat. Zametki, 52:3 (1992), 146–153; Math. Notes, 52:3 (1992), 983–989
Linking options:
https://www.mathnet.ru/eng/mzm4711 https://www.mathnet.ru/eng/mzm/v52/i3/p146
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