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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 35J25 | enregistrements trouvés : 97

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

Numerical analysis lancaster 1984 :proceedings of the S.E.R.C. summer school held in lancaster#July 15 - Aug. 3 Turner, P. R. | Springer-Verlag 1985 Nouveau

Equations elliptiques du second ordre a coefficients discontinus Stampacchia, Guido | Les Presse de l'Université de Montréal 1966 Nouveau

Calculus of variations and partial differential equationsproceedings of a conference held in trento,italy,june 16-21,1986 Hildebrandt, S. ; Kinderlehrer, D. ; Miranda, M. | Springer-Verlag 1988 Nouveau

Calculus of variations and partial differential equationsproceedings of a conference held in trento,italy,june 16-21,1986 Hildebrandt, S. ; Kinderlehrer, D. ; Miranda, M. | Springer-Verlag 1988 Nouveau

Nonlinear analysis and its applications to differential equations :papers from the Autumn School on ... held at the University of Lisbon#Sept.14 - Oct. 23 Grossinho, M. R. ; Ramos, M. ; Rebelo, C. | Birkhäuser 2001 Nouveau

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

Actes du colloque de théorie de Hodge :Luminy#Juin 1 - 6 | Société Mathématique de France 1989 Nouveau

Actes des journées mathématiques à la mémoire de Jean Leray Guillopé, Laurent ; Robert, Didier | Société Mathématique de France 2004 Nouveau

Nonlinear elliptic partial differential equations.Workshop in celebration of Jean-Pierre Gossez's 65th birthdayBruxelles # september 2-4, 2009 Bonheure, Denis ; Cuesta, Mabel ; Lami Dozo, Enrique J. ; Takac, Peter ; Van Schaftingen, Jean ; Willem, Michel | American Mathematical Society 2011 Nouveau

Control and optimization with PDE constraints.Based on the international workshop on control and optimization of PDEsMariatrost # october 10-14, 2011 Bredies, Kristian ; Clason, Christian ; Kunisch, Karl ; von Winckel, Gregory | Birkhäuser 2013 Nouveau

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

New trends in shape optimization.Based on a workshop held at the Friedrich-Alexander University Erlangen-NürnbergErlangen # September 2013 Pratelli, Aldo ; Leugering, Günter | Birkhäuser 2015 Nouveau

Imaging, multi-scale and high contrast partial differential equations.Seoul ICM 2014 satellite conferenceDaejeon # August 7-9, 2014 Ammari, Habib ; Capdeboscq, Yves ; Kang, Hyeonbae ; Sim, Imbo | American Mathematical Society 2016 Nouveau

Reduced basis methods: approximation of PDE's, interpolation and a posteriori estimate Maday, Yvon | CIRM H Nouveau

Tutorial with Freefem++ Hecht, Frédéric | CIRM H Nouveau

Approximation of elliptic boundary value problems Aubin, Jean-Pierre | Robert E. Krieger Publishing Co. 1980 Nouveau

Integral operators in potential theory Kral, Josef | Springer-Verlag 1985 Nouveau

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Numerical analysis lancaster 1984 :
proceedings of the S.E.R.C. summer school held in lancaster#July 15 - Aug. 3 Turner, P. R. | Springer-Verlag 1985 Nouveau

Calculus of variations and partial differential equations
proceedings of a conference held in trento,italy,june 16-21,1986 Hildebrandt, S. ; Kinderlehrer, D. ; Miranda, M. | Springer-Verlag 1988 Nouveau

Calculus of variations and partial differential equations
proceedings of a conference held in trento,italy,june 16-21,1986 Hildebrandt, S. ; Kinderlehrer, D. ; Miranda, M. | Springer-Verlag 1988 Nouveau

Nonlinear analysis and its applications to differential equations :
papers from the Autumn School on ... held at the University of Lisbon#Sept.14 - Oct. 23 Grossinho, M. R. ; Ramos, M. ; Rebelo, C. | Birkhäuser 2001 Nouveau

Actes du colloque de théorie de Hodge :
Luminy#Juin 1 - 6 | Société Mathématique de France 1989 Nouveau

Control and optimization with PDE constraints.
Based on the international workshop on control and optimization of PDEs
Mariatrost # october 10-14, 2011 Bredies, Kristian ; Clason, Christian ; Kunisch, Karl ; von Winckel, Gregory | Birkhäuser 2013 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

New trends in shape optimization.
Based on a workshop held at the Friedrich-Alexander University Erlangen-Nürnberg
Erlangen # September 2013
Pratelli, Aldo ; Leugering, Günter | Birkhäuser 2015 Nouveau

Imaging, multi-scale and high contrast partial differential equations.
Seoul ICM 2014 satellite conference
Daejeon # August 7-9, 2014 Ammari, Habib ; Capdeboscq, Yves ; Kang, Hyeonbae ; Sim, Imbo | American Mathematical Society 2016 Nouveau