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Partial Differential Equations

It is known that in 3D exterior domains Ω with the compact smooth boundary $\partial \Omega$, two spaces $X^{r}_{har}\left ( \Omega \right )$ and $V^{r}_{har}\left ( \Omega \right )$ of $L^{r}$-harmonic vector fields $h$ with $h\cdot v\mid _{\partial \Omega }= 0$ and $h\times v\mid _{\partial \Omega }= 0$ are both of finite dimensions, where $v$ denotes the unit outward normal to $\partial \Omega$. We prove that for every $L^{r}$-vector field $u$, there exist $h\in X^{r}_{har}\left ( \Omega \right )$, $w\in H^{1,r}\left ( \Omega \right )^{3}$ with div $w= 0$ and $p\in H^{1,r}\left ( \Omega \right )$ such that $u$ is uniquely decomposed as $u= h$ + rot $w$ + $\bigtriangledown p$.
On the other hand, if for the given $L^{r}$-vector field $u$ we choose its harmonic part $h$ from $V^{r}_{har}\left ( \Omega \right )$, then we have a similar decomposition to above, while the unique expression of $u$ holds only for $1< r< 3$. Furthermore, the choice of $p$ in $H^{1,r}\left ( \Omega \right )$ is determined in accordance with the threshold $r= 3/2$.
Our result is based on the joint work with Matthias Hieber, Anton Seyferd (TU Darmstadt), Senjo Shimizu (Kyoto Univ.) and Taku Yanagisawa (Nara Women Univ.). It is known that in 3D exterior domains Ω with the compact smooth boundary $\partial \Omega$, two spaces $X^{r}_{har}\left ( \Omega \right )$ and $V^{r}_{har}\left ( \Omega \right )$ of $L^{r}$-harmonic vector fields $h$ with $h\cdot v\mid _{\partial \Omega }= 0$ and $h\times v\mid _{\partial \Omega }= 0$ are both of finite dimensions, where $v$ denotes the unit outward normal to $\partial \Omega$. We prove that for every $L^{r}$-vector field $u...

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Documents 35B45 | enregistrements trouvés : 35

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Boundary value problems of mathematical physics XIII Ladyzhenskaya, O. A. | American Mathematical Society 1988 Nouveau

Boundary value problems of mathematical physics XIV :collection of papers Ladyzhenskaya, O. A. | American Mathematical Society 1991 Nouveau

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Short wave radiation problems in inhomogeneous mediaasymptotic solutions Bloom, Clifford O. ; Kazarinoff, Nicholas D. | Springer-Verlag 1976 Nouveau

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Surveys in differential geometry 2014. Vol. XIX : regularity and evolution of nonlinear equations.Essays dedicated to Richard Hamilton, Leon Simon, and Karen Uhlenbeck Cao, Huai-Dong ; Schoen, Richard ; Yau, Shing-Tung | International Press 2015 Nouveau

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Nonlinear operators and the calculus of variations
summer school held at Sept. 8-19 Gossez, J. P. ; Lami Dozo, E.J. ; Mawhin, J. L. ; Waelbroeck, L. | Springer-Verlag 1976 Nouveau

The AMSI-ANU workshop on spectral theory and harmonic analysis.
Proceedings of the workshop
Canberra # july 13-17, 2009 Hassell, Andrew ; McIntosh, Alan ; Taggart, Robert | Australian National University;Centre for Mathematics and its Applications 2010 Nouveau

Boundary value problems of mathematical physics XIV :
collection of papers Ladyzhenskaya, O. A. | American Mathematical Society 1991 Nouveau

Short wave radiation problems in inhomogeneous media
asymptotic solutions Bloom, Clifford O. ; Kazarinoff, Nicholas D. | Springer-Verlag 1976 Nouveau

Surveys in differential geometry 2014. Vol. XIX : regularity and evolution of nonlinear equations.
Essays dedicated to Richard Hamilton, Leon Simon, and Karen Uhlenbeck Cao, Huai-Dong ; Schoen, Richard ; Yau, Shing-Tung | International Press 2015 Nouveau