The regularity problem for generalized harmonic maps into homogeneous spaces
Abstract Let ℳ be a Riemannian surface and$$\mathcal{N}$$ be a standard sphere, or more generally a Riemannian manifold on which a Lie group,Γ, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether&...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Almeida, Luís [verfasserIn] |
|---|
| Format: |
Artikel |
|---|---|
| Sprache: |
Englisch |
| Erschienen: |
1995 |
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| Anmerkung: |
© Springer-Verlag 1995 |
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| Übergeordnetes Werk: |
Enthalten in: Calculus of variations and partial differential equations - Springer-Verlag, 1993, 3(1995), 2 vom: März, Seite 193-242 |
|---|---|
| Übergeordnetes Werk: |
volume:3 ; year:1995 ; number:2 ; month:03 ; pages:193-242 |
| Links: |
|---|
| DOI / URN: |
10.1007/BF01205005 |
|---|
| Katalog-ID: |
OLC2075186553 |
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| 520 | |a Abstract Let ℳ be a Riemannian surface and$$\mathcal{N}$$ be a standard sphere, or more generally a Riemannian manifold on which a Lie group,Γ, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether's Theorem), to mapsu ε Wloc1,1(ℳ,$$\mathcal{N}$$). We prove that, under some slight technical restrictions, for 1 <-p < 2, there are generalized harmonic mapsu εW1,p(ℳ,$$\mathcal{N}$$) that are everywhere discontinuous (in particular, this solves an open problem proposed by F. Bethuel, H. Brezis and F. Hélein, in [BBH]). We also show that the natural ε-regularity condition for such maps is to require <u to belong to the Lorentz space L(2, ∞). To prove this ε-regularity result we extend a compensated compactness result of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, proved in [CLMS], to the case of Lorentz spaces in duality. | ||
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10.1007/BF01205005 doi (DE-627)OLC2075186553 (DE-He213)BF01205005-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Almeida, Luís verfasserin aut The regularity problem for generalized harmonic maps into homogeneous spaces 1995 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1995 Abstract Let ℳ be a Riemannian surface and$$\mathcal{N}$$ be a standard sphere, or more generally a Riemannian manifold on which a Lie group,Γ, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether's Theorem), to mapsu ε Wloc1,1(ℳ,$$\mathcal{N}$$). We prove that, under some slight technical restrictions, for 1 <-p < 2, there are generalized harmonic mapsu εW1,p(ℳ,$$\mathcal{N}$$) that are everywhere discontinuous (in particular, this solves an open problem proposed by F. Bethuel, H. Brezis and F. Hélein, in [BBH]). We also show that the natural ε-regularity condition for such maps is to require <u to belong to the Lorentz space L(2, ∞). To prove this ε-regularity result we extend a compensated compactness result of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, proved in [CLMS], to the case of Lorentz spaces in duality. Enthalten in Calculus of variations and partial differential equations Springer-Verlag, 1993 3(1995), 2 vom: März, Seite 193-242 (DE-627)165669977 (DE-600)1144181-1 (DE-576)033045690 0944-2669 nnns volume:3 year:1995 number:2 month:03 pages:193-242 https://doi.org/10.1007/BF01205005 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_32 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4012 GBV_ILN_4036 GBV_ILN_4103 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4325 AR 3 1995 2 03 193-242 |
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10.1007/BF01205005 doi (DE-627)OLC2075186553 (DE-He213)BF01205005-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Almeida, Luís verfasserin aut The regularity problem for generalized harmonic maps into homogeneous spaces 1995 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1995 Abstract Let ℳ be a Riemannian surface and$$\mathcal{N}$$ be a standard sphere, or more generally a Riemannian manifold on which a Lie group,Γ, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether's Theorem), to mapsu ε Wloc1,1(ℳ,$$\mathcal{N}$$). We prove that, under some slight technical restrictions, for 1 <-p < 2, there are generalized harmonic mapsu εW1,p(ℳ,$$\mathcal{N}$$) that are everywhere discontinuous (in particular, this solves an open problem proposed by F. Bethuel, H. Brezis and F. Hélein, in [BBH]). We also show that the natural ε-regularity condition for such maps is to require <u to belong to the Lorentz space L(2, ∞). To prove this ε-regularity result we extend a compensated compactness result of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, proved in [CLMS], to the case of Lorentz spaces in duality. Enthalten in Calculus of variations and partial differential equations Springer-Verlag, 1993 3(1995), 2 vom: März, Seite 193-242 (DE-627)165669977 (DE-600)1144181-1 (DE-576)033045690 0944-2669 nnns volume:3 year:1995 number:2 month:03 pages:193-242 https://doi.org/10.1007/BF01205005 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_32 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4012 GBV_ILN_4036 GBV_ILN_4103 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4325 AR 3 1995 2 03 193-242 |
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10.1007/BF01205005 doi (DE-627)OLC2075186553 (DE-He213)BF01205005-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Almeida, Luís verfasserin aut The regularity problem for generalized harmonic maps into homogeneous spaces 1995 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1995 Abstract Let ℳ be a Riemannian surface and$$\mathcal{N}$$ be a standard sphere, or more generally a Riemannian manifold on which a Lie group,Γ, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether's Theorem), to mapsu ε Wloc1,1(ℳ,$$\mathcal{N}$$). We prove that, under some slight technical restrictions, for 1 <-p < 2, there are generalized harmonic mapsu εW1,p(ℳ,$$\mathcal{N}$$) that are everywhere discontinuous (in particular, this solves an open problem proposed by F. Bethuel, H. Brezis and F. Hélein, in [BBH]). We also show that the natural ε-regularity condition for such maps is to require <u to belong to the Lorentz space L(2, ∞). To prove this ε-regularity result we extend a compensated compactness result of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, proved in [CLMS], to the case of Lorentz spaces in duality. Enthalten in Calculus of variations and partial differential equations Springer-Verlag, 1993 3(1995), 2 vom: März, Seite 193-242 (DE-627)165669977 (DE-600)1144181-1 (DE-576)033045690 0944-2669 nnns volume:3 year:1995 number:2 month:03 pages:193-242 https://doi.org/10.1007/BF01205005 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_32 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4012 GBV_ILN_4036 GBV_ILN_4103 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4325 AR 3 1995 2 03 193-242 |
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10.1007/BF01205005 doi (DE-627)OLC2075186553 (DE-He213)BF01205005-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Almeida, Luís verfasserin aut The regularity problem for generalized harmonic maps into homogeneous spaces 1995 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1995 Abstract Let ℳ be a Riemannian surface and$$\mathcal{N}$$ be a standard sphere, or more generally a Riemannian manifold on which a Lie group,Γ, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether's Theorem), to mapsu ε Wloc1,1(ℳ,$$\mathcal{N}$$). We prove that, under some slight technical restrictions, for 1 <-p < 2, there are generalized harmonic mapsu εW1,p(ℳ,$$\mathcal{N}$$) that are everywhere discontinuous (in particular, this solves an open problem proposed by F. Bethuel, H. Brezis and F. Hélein, in [BBH]). We also show that the natural ε-regularity condition for such maps is to require <u to belong to the Lorentz space L(2, ∞). To prove this ε-regularity result we extend a compensated compactness result of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, proved in [CLMS], to the case of Lorentz spaces in duality. Enthalten in Calculus of variations and partial differential equations Springer-Verlag, 1993 3(1995), 2 vom: März, Seite 193-242 (DE-627)165669977 (DE-600)1144181-1 (DE-576)033045690 0944-2669 nnns volume:3 year:1995 number:2 month:03 pages:193-242 https://doi.org/10.1007/BF01205005 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_32 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4012 GBV_ILN_4036 GBV_ILN_4103 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4325 AR 3 1995 2 03 193-242 |
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10.1007/BF01205005 doi (DE-627)OLC2075186553 (DE-He213)BF01205005-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Almeida, Luís verfasserin aut The regularity problem for generalized harmonic maps into homogeneous spaces 1995 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1995 Abstract Let ℳ be a Riemannian surface and$$\mathcal{N}$$ be a standard sphere, or more generally a Riemannian manifold on which a Lie group,Γ, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether's Theorem), to mapsu ε Wloc1,1(ℳ,$$\mathcal{N}$$). We prove that, under some slight technical restrictions, for 1 <-p < 2, there are generalized harmonic mapsu εW1,p(ℳ,$$\mathcal{N}$$) that are everywhere discontinuous (in particular, this solves an open problem proposed by F. Bethuel, H. Brezis and F. Hélein, in [BBH]). We also show that the natural ε-regularity condition for such maps is to require <u to belong to the Lorentz space L(2, ∞). To prove this ε-regularity result we extend a compensated compactness result of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, proved in [CLMS], to the case of Lorentz spaces in duality. Enthalten in Calculus of variations and partial differential equations Springer-Verlag, 1993 3(1995), 2 vom: März, Seite 193-242 (DE-627)165669977 (DE-600)1144181-1 (DE-576)033045690 0944-2669 nnns volume:3 year:1995 number:2 month:03 pages:193-242 https://doi.org/10.1007/BF01205005 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_32 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4012 GBV_ILN_4036 GBV_ILN_4103 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4325 AR 3 1995 2 03 193-242 |
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510 VZ 17,1 ssgn The regularity problem for generalized harmonic maps into homogeneous spaces |
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Calculus of variations and partial differential equations |
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The regularity problem for generalized harmonic maps into homogeneous spaces |
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The regularity problem for generalized harmonic maps into homogeneous spaces |
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Almeida, Luís |
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Calculus of variations and partial differential equations |
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1995 |
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the regularity problem for generalized harmonic maps into homogeneous spaces |
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The regularity problem for generalized harmonic maps into homogeneous spaces |
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Abstract Let ℳ be a Riemannian surface and$$\mathcal{N}$$ be a standard sphere, or more generally a Riemannian manifold on which a Lie group,Γ, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether's Theorem), to mapsu ε Wloc1,1(ℳ,$$\mathcal{N}$$). We prove that, under some slight technical restrictions, for 1 <-p < 2, there are generalized harmonic mapsu εW1,p(ℳ,$$\mathcal{N}$$) that are everywhere discontinuous (in particular, this solves an open problem proposed by F. Bethuel, H. Brezis and F. Hélein, in [BBH]). We also show that the natural ε-regularity condition for such maps is to require <u to belong to the Lorentz space L(2, ∞). To prove this ε-regularity result we extend a compensated compactness result of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, proved in [CLMS], to the case of Lorentz spaces in duality. © Springer-Verlag 1995 |
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Abstract Let ℳ be a Riemannian surface and$$\mathcal{N}$$ be a standard sphere, or more generally a Riemannian manifold on which a Lie group,Γ, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether's Theorem), to mapsu ε Wloc1,1(ℳ,$$\mathcal{N}$$). We prove that, under some slight technical restrictions, for 1 <-p < 2, there are generalized harmonic mapsu εW1,p(ℳ,$$\mathcal{N}$$) that are everywhere discontinuous (in particular, this solves an open problem proposed by F. Bethuel, H. Brezis and F. Hélein, in [BBH]). We also show that the natural ε-regularity condition for such maps is to require <u to belong to the Lorentz space L(2, ∞). To prove this ε-regularity result we extend a compensated compactness result of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, proved in [CLMS], to the case of Lorentz spaces in duality. © Springer-Verlag 1995 |
| abstract_unstemmed |
Abstract Let ℳ be a Riemannian surface and$$\mathcal{N}$$ be a standard sphere, or more generally a Riemannian manifold on which a Lie group,Γ, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether's Theorem), to mapsu ε Wloc1,1(ℳ,$$\mathcal{N}$$). We prove that, under some slight technical restrictions, for 1 <-p < 2, there are generalized harmonic mapsu εW1,p(ℳ,$$\mathcal{N}$$) that are everywhere discontinuous (in particular, this solves an open problem proposed by F. Bethuel, H. Brezis and F. Hélein, in [BBH]). We also show that the natural ε-regularity condition for such maps is to require <u to belong to the Lorentz space L(2, ∞). To prove this ε-regularity result we extend a compensated compactness result of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, proved in [CLMS], to the case of Lorentz spaces in duality. © Springer-Verlag 1995 |
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The regularity problem for generalized harmonic maps into homogeneous spaces |
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