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Research talks;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 58A10 | enregistrements trouvés : 60

$L^{r}$-Helmholtz-Weyl decomposition in 3D exterior domains Kozono, Hideo | CIRM H Nouveau

Differential geometric methods in mathematical physics :proceedings of the 14th international conference held in salamanca#June 24-29 Garcia, P. L. ; Perez-Rendon, A. | Springer-Verlag 1987 Nouveau

Differential geometry. Vol. Ipartial differential equations on manifoldsproceedings of a summer research institute on ...held at the university of CaliforniaJuly 9-27 Greene, Robert ; Yau, S. T. | American Mathematical Society 1993 Nouveau

Méthodes de géométrie différentielle en topologie algébrique Karoubi, M. ; Leruste, C. | Universite Paris VII Nouveau

Méthodes de géométrie différentielle en topologie algébrique Karoubi, M. ; Leruste, C. | Universite Paris VII Nouveau

Méthodes de géométrie différentielle en topologie algébrique Partie II Karoubi, M. ; Leruste, C. | Universite Paris VII 1979 Nouveau

Séminaire Bourbaki. Vol. 1983/84 :exposés 615-632 | Société Mathématique de France 1985 Nouveau

Séminaire Bourbaki. Vol. 1994/95exposés 790-804 | Société Mathématique de France 1996 Nouveau

A geometric approach to differential forms Bachman, David | Birkhäuser 2006 Nouveau

An introduction to manifolds Tu, Loring W. | Springer 2011 Nouveau

An Introduction to Manifolds Tu, Loring W. | Springer 2008 Nouveau

Analysis II Lang, Serge | Addison-Wesley Publishing Company 1969 Nouveau

Analysis on real and complex manifolds Narasimhan, R. | Masson et Cie, Editeurs;North-Holland 1968 Nouveau

Cohomology and differential forms Vaisman, Izu | Marcel Dekker, Inc. 1972 Nouveau

Complex Tori and Abelian varieties Debarre, Olivier | American Mathematical Society 2005 Nouveau

Cours de mathématiques spéciales. Vol. 5 :applications de l'analyse à la géométrie Ramis, E. ; Deschamps, C. ; Odoux , J. | Masson 1992 Nouveau

Curvature and characteristic classes Dupont, Johan L. | Springer-Verlag 1978 Nouveau

Diffeology Iglesias-Zemmour, Patrick | American Mathematical Society 2013 Nouveau

Differentiable manifolds Chern, Shiing-Shen | Department Of Mathematics 1959 Nouveau

Differentiable manifolds Matsushima, Yozo ; Kobayashi, E. T. | Marcel Dekker, Inc. 1972 Nouveau

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Differential geometric methods in mathematical physics :
proceedings of the 14th international conference held in salamanca#June 24-29 Garcia, P. L. ; Perez-Rendon, A. | Springer-Verlag 1987 Nouveau

Differential geometry. Vol. I
partial differential equations on manifolds
proceedings of a summer research institute on ...
held at the university of California
July 9-27 Greene, Robert ; Yau, S. T. | American Mathematical Society 1993 Nouveau

Séminaire Bourbaki. Vol. 1983/84 :
exposés 615-632 | Société Mathématique de France 1985 Nouveau

Séminaire Bourbaki. Vol. 1994/95
exposés 790-804 | Société Mathématique de France 1996 Nouveau

Cours de mathématiques spéciales. Vol. 5 :
applications de l'analyse à la géométrie Ramis, E. ; Deschamps, C. ; Odoux , J. | Masson 1992 Nouveau