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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 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

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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

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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

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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

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Boundary value problems for elliptic systems Wloka, J. T. ; Rowley, B. ; Lawruk , B. | Cambridge University Press 1995 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

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

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

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

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

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

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

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