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Zap. Nauchn. Sem. LOMI, 1985, Volume 146, Pages 92–101 (Mi znsl5328)  

On estimates for the solutions to the Dirichlet problem for the Laplacian in exterior domains

P. Maremonti, V. A. Solonnikov


Abstract: Let $u(x)$ be the solution of the exterior Dirichlet problem for the equation $\Delta u=f$ vanishing at the infinity. It is shown that the coercive estimate $\| D^2u\|_{Lp)}\leq c\| f\|_{L_p}$ holds for $p<n/2$ In the case $p\geq n/2$ this estimate is established for solutions of the exterior Dirichlet problem that do not vanish at the infinity but may tend to a certain constant or even blow up as a linear function (for $p>n$). Bibl. – 2.
Пусть $u$ – решение уравнения $\Delta u=f$ с финитной функцией $f$ по внешней области $\Omega\subset\mathbf{R}^u$ и с условиями $u|_{\partial\Omega}=0$, $u\to0$ при $|x|\to\infty$. Показано, что коэрцитивная оценка $\|D^2u\|_{L_p(\Omega)}\leq c\|f\|$ справедлива лишь при $p<n/2$. При $p\geq n/2$ она имеет место для решения внешней задачи Дирихле, которая не исчезает на бесконечности, а может стремиться к постоянной или даже к линейной (при $p>n$) функции. Библ. – 2 назв.

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Bibliographic databases:
UDC: 517.956.225

Citation: P. Maremonti, V. A. Solonnikov, “On estimates for the solutions to the Dirichlet problem for the Laplacian in exterior domains”, Differential geometry, Lie groups and mechanics. Part VII, Zap. Nauchn. Sem. LOMI, 146, "Nauka", Leningrad. Otdel., Leningrad, 1985, 92–101

Citation in format AMSBIB
\Bibitem{MarSol85}
\by P.~Maremonti, V.~A.~Solonnikov
\paper On estimates for the solutions to the Dirichlet problem for the Laplacian in exterior domains
\inbook Differential geometry, Lie groups and mechanics. Part~VII
\serial Zap. Nauchn. Sem. LOMI
\yr 1985
\vol 146
\pages 92--101
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl5328}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=836550}
\zmath{https://zbmath.org/?q=an:0605.35021}


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