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 Mat. Zametki, 1992, Volume 52, Issue 3, Pages 146–153 (Mi mz4711)

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.

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

Bibliographic databases:

UDC: 517.9

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

Citation in format AMSBIB
\Bibitem{Tuv92} \by M.~V.~Tuvaev \paper Removable singular sets for equations of the form $\sum\dfrac{\partial}{\partial x_i}a_{ij}(x)\dfrac{\partial u}{\partial x_j}=f(x,u,\nabla u)$ \jour Mat. Zametki \yr 1992 \vol 52 \issue 3 \pages 146--153 \mathnet{http://mi.mathnet.ru/mz4711} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1194139} \zmath{https://zbmath.org/?q=an:0789.35032} \transl \jour Math. Notes \yr 1992 \vol 52 \issue 3 \pages 983--989 \crossref{https://doi.org/10.1007/BF01209621} 

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This publication is cited in the following articles:
1. S. V. Pikulin, “Convergence of a family of solutions to a Fujita-type equation in domains with cavities”, Comput. Math. Math. Phys., 56:11 (2016), 1872–1900
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