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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 58A10 | enregistrements trouvés : 60

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

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

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

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

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

Variétés différentielles et analytiquesfascicule de résultats, paragraphes 8 à 15 Bourbaki, N. | Hermann 1971 Nouveau

Differential forms in algebraic topology Bott, Raoul ; Tu, Loring W. | Springer-Verlag 1982 Nouveau

The Hodge-Laplacian :boundary value problems on Riemannian manifolds Mitrea, Dorina ; Mitrea, Irina ; Mitrea, Marius ; Taylor, Michael | Walter De Gruyter 2016 Nouveau

From Frenet to Cartan : the method of moving frames Clelland, Jeanne N. | American Mathematical Society 2017 Nouveau

Differentiable manifolds:forms, currents, harmonic forms De Rham, Georges ; Smith, F. R. ; Chern, S. S. | Springer-Verlag 1984 Nouveau

Winding around:the winding number in topology, geometry, and analysis Roe, John | American Mathematical Society;Mathematics Advanced Study Semesters 2015 Nouveau

Synthetic differential geometry Kock, Anders | Cambridge University Press 2006 Nouveau

Differential forms:theory and practice Weintraub, Steven H. | Academic Press 2014 Nouveau

Rational homotopy theory and differential forms Griffiths, Phillip ; Morgan, John | Birkhäuser 2013 Nouveau

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

The pullback equation for differential forms Csato, Gyula ; Dacorogna, Bernard ; Kneuss, Olivier | Birkhäuser 2012 Nouveau

Mathematical analysis:foundations and advanced techniques for functions of several variables Giaquinta, Mariano ; Modica, Giuseppe | Birkhäuser 2012 Nouveau

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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

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

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

Variétés différentielles et analytiques
fascicule de résultats, paragraphes 8 à 15 Bourbaki, N. | Hermann 1971 Nouveau

The Hodge-Laplacian :
boundary value problems on Riemannian manifolds Mitrea, Dorina ; Mitrea, Irina ; Mitrea, Marius ; Taylor, Michael | Walter De Gruyter 2016 Nouveau

Differentiable manifolds:
forms, currents, harmonic forms De Rham, Georges ; Smith, F. R. ; Chern, S. S. | Springer-Verlag 1984 Nouveau

Winding around:
the winding number in topology, geometry, and analysis Roe, John | American Mathematical Society;Mathematics Advanced Study Semesters 2015 Nouveau

Differential forms:
theory and practice Weintraub, Steven H. | Academic Press 2014 Nouveau

Mathematical analysis:
foundations and advanced techniques for functions of several variables Giaquinta, Mariano ; Modica, Giuseppe | Birkhäuser 2012 Nouveau