Abstract
H. Brezis and L. Nirenberg proved that if $(g_k) \subset C^0(\mathbb{S}^N, \mathbb{S}^N)$ and $g \in C^0(\mathbb{S}^N, \mathbb{S}^N)$ ($N \ge 1$) are such that $g_k \rightarrow g$ in ${\rm BMO}(\mathbb{S}^N)$, then $\mathrm{deg} \, g_k \rightarrow \mathrm{deg} \, g$. On the other hand, if $g \in C^1(\mathbb{S}^N, \mathbb{S}^N)$, then Kronecker’s formula asserts that $ \mathrm{deg} \, g = \frac{1}{|\mathbb{S}^N|} \int_{\mathbb{S}^N} \det (\nabla g) \, d \sigma$. Consequently, $\int_{\mathbb{S}^N} \det (\nabla g_k) \, d \sigma$ converges to $\int_{\mathbb{S}^N} \det (\nabla g) \, d \sigma$ provided $g_k \rightarrow g$ in ${\rm BMO}(\mathbb{S}^N)$. In the same spirit, we consider the quantity $ {\bf J}(g, \psi) := \int_{\mathbb{S}^N} \psi \det (\nabla g) \, d \sigma$, for all $\psi \in C^1(\mathbb{S}^N, \mathbb{R}) $ and study the convergence of ${\bf J}(g_k, \psi)$. In particular, we prove that ${\bf J}(g_k, \psi)$ converges to ${\bf J}(g, \psi)$ for any $\psi \in C^1(\mathbb{S}^N, \mathbb{R})$ if $g_k$ converges to $g$ in $C^{0, \alpha}(\mathbb{S}^N)$ for some $\alpha > \frac{N-1}{N}$. Surprisingly, this result is “optimal” when $N > 1$. In the case $N=1$ we prove that if $g_k \rightarrow g$ almost everywhere and $\limsup_{k \rightarrow \infty} |g_k – g|_{\rm BMO}$ is sufficiently small, then ${\bf J}(g_k, \psi) \rightarrow {\bf J}(g, \psi)$ for any $\psi \in C^1(\mathbb{S}^1, \mathbb{R})$. We also establish bounds for ${\bf J}(g, \psi)$ which are motivated by the works of J. Bourgain, H. Brezis, and H.-M. Nguyen and H.-M. Nguyen. We pay special attention to the case $N=1$.
-
[AlbertiDist] G. Alberti, "Distributional Jacobian and singularities of Sobolev maps," Ricerche Mat., vol. 54, iss. 2, pp. 375-394 (2006), 2005.
@article {AlbertiDist, MRKEY = {2289488},
AUTHOR = {Alberti, Giovanni},
TITLE = {Distributional {J}acobian and singularities of {S}obolev maps},
JOURNAL = {Ricerche Mat.},
FJOURNAL = {Ricerche di Matematica},
VOLUME = {54},
YEAR = {2005},
NUMBER = {2},
PAGES = {375--394 (2006)},
ISSN = {0035-5038},
CODEN = {RCMTAE},
MRCLASS = {49Q15 (58A25)},
MRNUMBER = {2289488},
MRREVIEWER = {Carlo Bardaro},
ZBLNUMBER = {pre05230855},
} -
[AlbertiBaldoOrlandi]
G. Alberti, S. Baldo, and G. Orlandi, "Functions with prescribed singularities," J. Eur. Math. Soc. $($JEMS$)$, vol. 5, iss. 3, pp. 275-311, 2003.
@article {AlbertiBaldoOrlandi, MRKEY = {2002215},
AUTHOR = {Alberti, Giovanni and Baldo, S. and Orlandi, G.},
TITLE = {Functions with prescribed singularities},
JOURNAL = {J. Eur. Math. Soc. $($JEMS$)$},
FJOURNAL = {Journal of the European Mathematical Society (JEMS)},
VOLUME = {5},
YEAR = {2003},
NUMBER = {3},
PAGES = {275--311},
ISSN = {1435-9855},
MRCLASS = {49Q15 (49J45 53C65 58A25)},
MRNUMBER = {2002215},
MRREVIEWER = {Doan The Hieu},
DOI = {10.1007/s10097-003-0053-5},
ZBLNUMBER = {1033.46028},
} -
[BallConvexityCond]
J. M. Ball, "Convexity conditions and existence theorems in nonlinear elasticity," Arch. Rational Mech. Anal., vol. 63, iss. 4, pp. 337-403, 1976/77.
@article {BallConvexityCond, MRKEY = {0475169},
AUTHOR = {Ball, John M.},
TITLE = {Convexity conditions and existence theorems in nonlinear elasticity},
JOURNAL = {Arch. Rational Mech. Anal.},
FJOURNAL = {Archive for Rational Mechanics and Analysis},
VOLUME = {63},
YEAR = {1976/77},
NUMBER = {4},
PAGES = {337--403},
ISSN = {0003-9527},
MRCLASS = {73.49 (73.58)},
MRNUMBER = {0475169},
MRREVIEWER = {W. Kecs},
DOI = {10.1007/BF00279992},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {0368.73040},
} -
[BallMuratQuasiconvexity]
J. M. Ball and F. Murat, "$W^{1,p}$-quasiconvexity and variational problems for multiple integrals," J. Funct. Anal., vol. 58, iss. 3, pp. 225-253, 1984.
@article {BallMuratQuasiconvexity, MRKEY = {0759098},
AUTHOR = {Ball, John M. and Murat, F.},
TITLE = {{$W\sp{1,p}$}-quasiconvexity and variational problems for multiple integrals},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {58},
YEAR = {1984},
NUMBER = {3},
PAGES = {225--253},
ISSN = {0022-1236},
CODEN = {JFUAAW},
MRCLASS = {49A50},
MRNUMBER = {0759098},
MRREVIEWER = {L. Cesari},
DOI = {10.1016/0022-1236(84)90041-7},
ZBLNUMBER = {0549.46019},
} -
[BethuelBrezisHelein] F. Bethuel, H. Brezis, and F. Hélein, Ginzburg-Landau Vortices, Boston, MA: Birkhäuser, 1994, vol. 13.
@book {BethuelBrezisHelein, MRKEY = {1269538},
AUTHOR = {Bethuel, Fabrice and Brezis, Ha{ï}m and H{é}lein, Fr{é}d{é}ric},
TITLE = {Ginzburg-{L}andau Vortices},
SERIES = {Progr. Nonlinear Differential Equations Appl.},
VOLUME={13},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1994},
PAGES = {xxviii+159},
ISBN = {0-8176-3723-0},
MRCLASS = {58E20 (35Q55 49-02 58E50 82D50)},
MRNUMBER = {1269538},
ZBLNUMBER = {0802.35142},
} -
[BethuelChiron] F. Bethuel and D. Chiron, "Some questions related to the lifting problem in Sobolev spaces," in Perspectives in Nonlinear Partial Differential Equations, Providence, RI: Amer. Math. Soc., 2007, vol. 446, pp. 125-152.
@incollection {BethuelChiron, MRKEY = {2373727},
AUTHOR = {Bethuel, Fabrice and Chiron, David},
TITLE = {Some questions related to the lifting problem in {S}obolev spaces},
BOOKTITLE = {Perspectives in {N}onlinear {P}artial {D}ifferential {E}quations},
SERIES = {Contemp. Math.},
VOLUME = {446},
PAGES = {125--152},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2007},
MRCLASS = {46E35 (55R10 57M05)},
MRNUMBER = {2373727},
ZBLNUMBER = {05831662},
} -
[BBMH1/2maps]
J. Bourgain, H. Brezis, and P. Mironescu, "$H^{1/2}$ maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation," Publ. Math. Inst. Hautes Études Sci., iss. 99, pp. 1-115, 2004.
@article {BBMH1/2maps, MRKEY = {2075883},
AUTHOR = {Bourgain, Jean and Brezis, Ha{ï}m and Mironescu, Petru},
TITLE = {{$H\sp {1/2}$} maps with values into the circle: minimal connections, lifting, and the {G}inzburg-{L}andau equation},
JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
FJOURNAL = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
NUMBER = {99},
YEAR = {2004},
PAGES = {1--115},
ISSN = {0073-8301},
MRCLASS = {58D15 (35B25 35J20 46E35 58E50)},
MRNUMBER = {2075883},
MRREVIEWER = {Robert Leon Jerrard},
DOI = {10.1007/s10240-004-0019-5},
ZBLNUMBER = {1051.49030},
} -
[BBMLift]
J. Bourgain, H. Brezis, and P. Mironescu, "Lifting, degree, and distributional Jacobian revisited," Comm. Pure Appl. Math., vol. 58, iss. 4, pp. 529-551, 2005.
@article {BBMLift, MRKEY = {2119868},
AUTHOR = {Bourgain, Jean and Brezis, Ha{ï}m and Mironescu, Petru},
TITLE = {Lifting, degree, and distributional {J}acobian revisited},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {58},
YEAR = {2005},
NUMBER = {4},
PAGES = {529--551},
ISSN = {0010-3640},
CODEN = {CPAMA},
MRCLASS = {26A99 (26A45 42A50 46E35)},
MRNUMBER = {2119868},
MRREVIEWER = {Steven George Krantz},
DOI = {10.1002/cpa.20063},
ZBLNUMBER = {1077.46023},
} -
[BBMLiftCompl] J. Bourgain, H. Brezis, and P. Mironescu, Complements to the paper: “Lifting, degree, and distributional Jacobian revisited".
@misc{BBMLiftCompl,
author={Bourgain, Jean and Brezis, Ha{ï}m and Mironescu, Petru},
TITLE = {Complements to the paper: ``{L}ifting, degree, and distributional {J}acobian revisited"},
NOTE={unpublished), posted on the website \href{http://math.univ-lyon1.fr/~mironescu/2.pdf}{http://math.univ-lyon1.fr/~mironescu/2.pdf}},
} -
[BBNg1]
J. Bourgain, H. Brezis, , and H. Nguyen, "A new estimate for the topological degree," C. R. Math. Acad. Sci. Paris, vol. 340, iss. 11, pp. 787-791, 2005.
@article {BBNg1, MRKEY = {2139888},
AUTHOR = {Bourgain, Jean and Brezis, Ha{ï}m and and Nguyen, Hoai-Minh},
TITLE = {A new estimate for the topological degree},
JOURNAL = {C. R. Math. Acad. Sci. Paris},
FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
VOLUME = {340},
YEAR = {2005},
NUMBER = {11},
PAGES = {787--791},
ISSN = {1631-073X},
MRCLASS = {55M25 (46E35 47H11)},
MRNUMBER = {2139888},
MRREVIEWER = {Christian Fenske},
DOI = {10.1016/j.crma.2005.04.007},
ZBLNUMBER = {1071.55002},
} -
[Bousquet1]
P. Bousquet, "Topological singularities in $W^{s,p}(S^N,S^1)$," J. Anal. Math., vol. 102, pp. 311-346, 2007.
@article {Bousquet1, MRKEY = {2346560},
AUTHOR = {Bousquet, Pierre},
TITLE = {Topological singularities in {$W\sp {s,p}(S\sp N,S\sp 1)$}},
JOURNAL = {J. Anal. Math.},
FJOURNAL = {Journal d'Analyse Mathématique},
VOLUME = {102},
YEAR = {2007},
PAGES = {311--346},
ISSN = {0021-7670},
CODEN = {JOAMAV},
MRCLASS = {58D15 (46E35 49Q15 58E20)},
MRNUMBER = {2346560},
MRREVIEWER = {Giovanni Alberti},
DOI = {10.1007/s11854-007-0023-z},
ZBLNUMBER = {1158.58018},
} -
[BrezisInterplay]
H. Brezis, "The interplay between analysis and topology in some nonlinear PDE problems," Bull. Amer. Math. Soc., vol. 40, iss. 2, pp. 179-201, 2003.
@article {BrezisInterplay, MRKEY = {1962295},
AUTHOR = {Brezis, Ha{ï}m},
TITLE = {The interplay between analysis and topology in some nonlinear {PDE} problems},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {American Mathematical Society. Bulletin. New Series},
VOLUME = {40},
YEAR = {2003},
NUMBER = {2},
PAGES = {179--201},
ISSN = {0273-0979},
CODEN = {BAMOAD},
MRCLASS = {35J20 (35A20 35J60 47J30 49J10 55P10 58E15)},
MRNUMBER = {1962295},
MRREVIEWER = {Alain Brillard},
DOI = {10.1090/S0273-0979-03-00976-5},
ZBLNUMBER = {1161.35354},
} -
[BrCoLieb]
H. Brezis, J. Coron, and E. H. Lieb, "Harmonic maps with defects," Comm. Math. Phys., vol. 107, iss. 4, pp. 649-705, 1986.
@article {BrCoLieb, MRKEY = {0868739},
AUTHOR = {Brezis, Ha{ï}m and Coron, Jean-Michel and Lieb, Elliott H.},
TITLE = {Harmonic maps with defects},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {107},
YEAR = {1986},
NUMBER = {4},
PAGES = {649--705},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {58E20 (81E13 82A05)},
MRNUMBER = {0868739},
MRREVIEWER = {John C. Wood},
URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104116234},
ZBLNUMBER = {0608.58016},
} -
[BrezisFuscoSbordone]
H. Brezis, N. Fusco, and C. Sbordone, "Integrability for the Jacobian of orientation preserving mappings," J. Funct. Anal., vol. 115, iss. 2, pp. 425-431, 1993.
@article {BrezisFuscoSbordone, MRKEY = {1234399},
AUTHOR = {Brezis, Ha{ï}m and Fusco, Nicola and Sbordone, Carlo},
TITLE = {Integrability for the {J}acobian of orientation preserving mappings},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {115},
YEAR = {1993},
NUMBER = {2},
PAGES = {425--431},
ISSN = {0022-1236},
CODEN = {JFUAAW},
MRCLASS = {26B10 (49Q15)},
MRNUMBER = {1234399},
MRREVIEWER = {Ludwik M. Dru{\.z}kowski},
DOI = {10.1006/jfan.1993.1098},
ZBLNUMBER = {0847.26012},
} -
[BrezisLiMironescuNirenberg] H. Brezis, Y. Li, P. Mironescu, and L. Nirenberg, "Degree and Sobolev spaces," Topol. Methods Nonlinear Anal., vol. 13, iss. 2, pp. 181-190, 1999.
@article {BrezisLiMironescuNirenberg, MRKEY = {1742219},
AUTHOR = {Brezis, Ha{ï}m and Li, Yanyan and Mironescu, Petru and Nirenberg, Louis},
TITLE = {Degree and {S}obolev spaces},
JOURNAL = {Topol. Methods Nonlinear Anal.},
FJOURNAL = {Topological Methods in Nonlinear Analysis},
VOLUME = {13},
YEAR = {1999},
NUMBER = {2},
PAGES = {181--190},
ISSN = {1230-3429},
MRCLASS = {47H11 (46E35 46N20 58D15)},
MRNUMBER = {1742219},
MRREVIEWER = {Jacobo Pejsachowicz},
ZBLNUMBER = {0956.46024},
} -
[BrMiComp]
H. Brezis and P. Mironescu, "Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces," J. Evol. Equ., vol. 1, iss. 4, pp. 387-404, 2001.
@article {BrMiComp, MRKEY = {1877265},
AUTHOR = {Brezis, Ha{ï}m and Mironescu, Petru},
TITLE = {Gagliardo-{N}irenberg, composition and products in fractional {S}obolev spaces},
JOURNAL = {J. Evol. Equ.},
FJOURNAL = {Journal of Evolution Equations},
VOLUME = {1},
YEAR = {2001},
NUMBER = {4},
PAGES = {387--404},
ISSN = {1424-3199},
MRCLASS = {46E35 (46E39)},
MRNUMBER = {1877265},
MRREVIEWER = {Alexey N. Karapetyants},
DOI = {10.1007/PL00001378},
ZBLNUMBER = {1023.46031},
} -
[BrezisMironescuPonce] H. Brezis, P. Mironescu, and A. C. Ponce, "$W^{1,1}$-maps with values into $S^1$," in Geometric Analysis of PDE and Several Complex Variables, Providence, RI: Amer. Math. Soc., 2005, vol. 368, pp. 69-100.
@incollection {BrezisMironescuPonce, MRKEY = {2126464},
AUTHOR = {Brezis, Ha{ï}m and Mironescu, Petru and Ponce, Augusto C.},
TITLE = {{$W\sp {1,1}$}-maps with values into {$S\sp 1$}},
BOOKTITLE = {Geometric {A}nalysis of {\rm PDE} and {S}everal {C}omplex {V}ariables},
SERIES = {Contemp. Math.},
VOLUME = {368},
PAGES = {69--100},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2005},
MRCLASS = {46E35 (49J45)},
MRNUMBER = {2126464},
MRREVIEWER = {Giovanni Alberti},
ZBLNUMBER = {1078.46020},
} -
[BrNgTdelta]
H. Brezis and H-M. Nguyen, "On a new class of functions related to $VMO$," C. R. Acad. Sci. Paris Sér.~ I Math., vol. 349, pp. 157-160, 2011.
@article{BrNgTdelta,
author = {Brezis, Ha{ï}m and Nguyen, H-M.},
TITLE={On a new class of functions related to {$VMO$}},
JOURNAL={C. R. Acad. Sci. Paris Sér.~{\rm I} Math.},
VOLUME={349},
YEAR={2011},
PAGES={157--160},
DOI = {10.1016/j.crma.2010.11.026},
} -
@misc{BrNgGNinequality,
author = {Brezis, Ha{ï}m and Nguyen, H-M.},
TITLE={The {J}acobian determinant revisited},
NOTE={Invent. {M}ath. (to appear)},
DOI={10.1007/s00222-010-0300-9},
} -
[BrNiDegree]
H. Brezis and L. Nirenberg, "Degree theory and BMO; Part I: Compact manifolds without boundaries," Selecta Math., vol. 1, iss. 2, pp. 197-263, 1995.
@article {BrNiDegree, MRKEY = {1354598},
AUTHOR = {Brezis, Ha{ï}m and Nirenberg, L.},
TITLE = {Degree theory and {BMO}; {P}art {I}: {C}ompact manifolds without boundaries},
JOURNAL = {Selecta Math.},
FJOURNAL = {Selecta Mathematica. New Series},
VOLUME = {1},
YEAR = {1995},
NUMBER = {2},
PAGES = {197--263},
ISSN = {1022-1824},
CODEN = {SMATF6},
MRCLASS = {58C30 (47H11 49Q20)},
MRNUMBER = {1354598},
MRREVIEWER = {Maria J. Esteban},
ZBLNUMBER = {0852.58010},
DOI = {10.1007/BF01671566},
} -
[CoifLionsMeyer] R. Coifman, P. -L. Lions, Y. Meyer, and S. Semmes, "Compensated compactness and Hardy spaces," J. Math. Pures Appl., vol. 72, iss. 3, pp. 247-286, 1993.
@article {CoifLionsMeyer, MRKEY = {1225511},
AUTHOR = {Coifman, R. and Lions, P.-L. and Meyer, Y. and Semmes, S.},
TITLE = {Compensated compactness and {H}ardy spaces},
JOURNAL = {J. Math. Pures Appl.},
FJOURNAL = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
VOLUME = {72},
YEAR = {1993},
NUMBER = {3},
PAGES = {247--286},
ISSN = {0021-7824},
CODEN = {JMPAAM},
MRCLASS = {46E99 (35S05 42B30 46F10 46N20 47G30)},
MRNUMBER = {1225511},
ZBLNUMBER = {0864.42009},
} -
[CoifmanMeyer] R. R. Coifman and Y. Meyer, "Une généralisation du théorème de Calderón sur l’intégrale de Cauchy," in Fourier Analysis, Madrid: Asoc. Mat. Espa nola, 1980, vol. 1, pp. 87-116.
@incollection {CoifmanMeyer, MRKEY = {0582251},
AUTHOR = {Coifman, R. R. and Meyer, Y.},
TITLE = {Une généralisation du théorème de {C}alderón sur l'intégrale de {C}auchy},
BOOKTITLE = {Fourier Analysis},
VENUE={{P}roc. {S}em., {E}l {E}scorial, 1979},
SERIES = {Asoc. Mat. Espa nola},
VOLUME = {1},
PAGES = {87--116},
PUBLISHER = {Asoc. Mat. Espa nola},
ADDRESS = {Madrid},
YEAR = {1980},
MRCLASS = {42B20 (30E20)},
MRNUMBER = {0582251},
MRREVIEWER = {A. B. Aleksandrov},
ZBLNUMBER = {0541.42008},
} -
[DacorognaMoser] B. Dacorogna and J. Moser, "On a partial differential equation involving the Jacobian determinant," Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 7, iss. 1, pp. 1-26, 1990.
@article {DacorognaMoser, MRKEY = {1046081},
AUTHOR = {Dacorogna, Bernard and Moser, J{ü}rgen},
TITLE = {On a partial differential equation involving the {J}acobian determinant},
JOURNAL = {Ann. Inst. H. Poincaré Anal. Non Linéaire},
FJOURNAL = {Annales de l'Institut Henri Poincaré. Analyse Non Linéaire},
VOLUME = {7},
YEAR = {1990},
NUMBER = {1},
PAGES = {1--26},
ISSN = {0294-1449},
MRCLASS = {58G20 (35J60)},
MRNUMBER = {1046081},
MRREVIEWER = {Jan Rogulski},
ZBLNUMBER = {0707.35041},
} -
[DacorognaMurat]
B. Dacorogna and F. Murat, "On the optimality of certain Sobolev exponents for the weak continuity of determinants," J. Funct. Anal., vol. 105, iss. 1, pp. 42-62, 1992.
@article {DacorognaMurat, MRKEY = {1156669},
AUTHOR = {Dacorogna, Bernard and Murat, F.},
TITLE = {On the optimality of certain {S}obolev exponents for the weak continuity of determinants},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {105},
YEAR = {1992},
NUMBER = {1},
PAGES = {42--62},
ISSN = {0022-1236},
CODEN = {JFUAAW},
MRCLASS = {49J45 (46N10)},
MRNUMBER = {1156669},
DOI = {10.1016/0022-1236(92)90071-P},
ZBLNUMBER = {0769.46025},
} -
[Carmo] M. P. do Carmo, Riemannian Geometry, Boston, MA: Birkhäuser, 1992.
@book {Carmo, MRKEY = {1138207},
AUTHOR = {do Carmo, Manfredo Perdig{ã}o},
TITLE = {Riemannian Geometry},
SERIES = {Mathematics: Theory Appl.},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1992},
PAGES = {xiv+300},
ISBN = {0-8176-3490-8},
MRCLASS = {53-01},
MRNUMBER = {1138207},
MRREVIEWER = {Bang-yen Chen},
ZBLNUMBER = {0752.53001},
} -
[Esteban]
M. J. Esteban, "A direct variational approach to Skyrme’s model for meson fields," Comm. Math. Phys., vol. 105, iss. 4, pp. 571-591, 1986.
@article {Esteban, MRKEY = {0852091},
AUTHOR = {Esteban, Maria J.},
TITLE = {A direct variational approach to {S}kyrme's model for meson fields},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {105},
YEAR = {1986},
NUMBER = {4},
PAGES = {571--591},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {58E15 (49H05 81E99)},
MRNUMBER = {0852091},
MRREVIEWER = {David Groisser},
URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104115501},
ZBLNUMBER = {0621.58035},
} -
[EstebanMuller]
M. J. Esteban and S. Müller, "Sobolev maps with integer degree and applications to Skyrme’s problem," Proc. Roy. Soc. London Ser. A, vol. 436, iss. 1896, pp. 197-201, 1992.
@article {EstebanMuller, MRKEY = {1177129},
AUTHOR = {Esteban, Maria J. and M{ü}ller, Stefan},
TITLE = {Sobolev maps with integer degree and applications to {S}kyrme's problem},
JOURNAL = {Proc. Roy. Soc. London Ser. A},
FJOURNAL = {Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences},
VOLUME = {436},
YEAR = {1992},
NUMBER = {1896},
PAGES = {197--201},
ISSN = {0962-8444},
CODEN = {PRLAAZ},
MRCLASS = {58E50 (49Q20 58E05 81V05)},
MRNUMBER = {1177129},
MRREVIEWER = {Anders Linn{é}r},
DOI = {10.1098/rspa.1992.0014},
ZBLNUMBER = {0757.49010},
} -
[FonsecaFuscoMarcelliniVar]
I. Fonseca, N. Fusco, and P. Marcellini, "On the total variation of the Jacobian," J. Funct. Anal., vol. 207, iss. 1, pp. 1-32, 2004.
@article {FonsecaFuscoMarcelliniVar, MRKEY = {2027634},
AUTHOR = {Fonseca, Irene and Fusco, Nicola and Marcellini, Paolo},
TITLE = {On the total variation of the {J}acobian},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {207},
YEAR = {2004},
NUMBER = {1},
PAGES = {1--32},
ISSN = {0022-1236},
CODEN = {JFUAAW},
MRCLASS = {49K20 (35A15 35F20 49J45 74B20)},
MRNUMBER = {2027634},
MRREVIEWER = {Martin Kru{ž}{\'ı}k},
DOI = {10.1016/S0022-1236(03)00111-3},
ZBLNUMBER = {1041.49016},
} -
[FonsecaFuscoMarcellini] I. Fonseca, N. Fusco, and P. Marcellini, "Topological degree, Jacobian determinants and relaxation," Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., vol. 8, iss. 1, pp. 187-250, 2005.
@article {FonsecaFuscoMarcellini, MRKEY = {2122983},
AUTHOR = {Fonseca, Irene and Fusco, Nicola and Marcellini, Paolo},
TITLE = {Topological degree, {J}acobian determinants and relaxation},
JOURNAL = {Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat.},
FJOURNAL = {Bollettino della Unione Matematica Italiana. Serie VIII. Sezione B. Articoli di Ricerca Matematica},
VOLUME = {8},
YEAR = {2005},
NUMBER = {1},
PAGES = {187--250},
ISSN = {0392-4041},
CODEN = {BLUMAM},
MRCLASS = {49J10 (26B10 35A15 49N60 58C30)},
MRNUMBER = {2122983},
MRREVIEWER = {Giovanni Bellettini},
ZBLNUMBER = {1177.49066},
} -
[FonsecaLeoniMaly]
I. Fonseca, G. Leoni, and J. Malý, "Weak continuity and lower semicontinuity results for determinants," Arch. Ration. Mech. Anal., vol. 178, iss. 3, pp. 411-448, 2005.
@article {FonsecaLeoniMaly, MRKEY = {2196498},
AUTHOR = {Fonseca, Irene and Leoni, Giovanni and Mal{ý},
Jan},
TITLE = {Weak continuity and lower semicontinuity results for determinants},
JOURNAL = {Arch. Ration. Mech. Anal.},
FJOURNAL = {Archive for Rational Mechanics and Analysis},
VOLUME = {178},
YEAR = {2005},
NUMBER = {3},
PAGES = {411--448},
ISSN = {0003-9527},
MRCLASS = {49J45 (15A15 35A15 35J50 46E35)},
MRNUMBER = {2196498},
MRREVIEWER = {Pietro Celada},
DOI = {10.1007/s00205-005-0377-2},
ZBLNUMBER = {1081.49013},
} -
[GiaquintaModicaSoucekDegree]
M. Giaquinta, G. Modica, and J. Souvcek, "Remarks on the degree theory," J. Funct. Anal., vol. 125, iss. 1, pp. 172-200, 1994.
@article {GiaquintaModicaSoucekDegree, MRKEY = {1297018},
AUTHOR = {Giaquinta, Mariano and Modica, G. and Sou{\v{c}}ek, J.},
TITLE = {Remarks on the degree theory},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {125},
YEAR = {1994},
NUMBER = {1},
PAGES = {172--200},
ISSN = {0022-1236},
CODEN = {JFUAAW},
MRCLASS = {58C30 (47H11 49Q20 58A25)},
MRNUMBER = {1297018},
MRREVIEWER = {Christian Fenske},
DOI = {10.1006/jfan.1994.1121},
ZBLNUMBER = {0822.55003},
} -
[GiaquintaModicaSoucek] M. Giaquinta, G. Modica, and J. Souvcek, Cartesian Currents in the Calculus of Variations. I, New York: Springer-Verlag, 1998, vol. 37.
@book {GiaquintaModicaSoucek, MRKEY = {1645086},
AUTHOR = {Giaquinta, Mariano and Modica, G. and Sou{\v{c}}ek, J.},
TITLE = {Cartesian Currents in the Calculus of Variations. {\rm I}},
SERIES = {Ergeb. Math. Grenzgeb.},
VOLUME = {37},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1998},
PAGES = {xxiv+711},
ISBN = {3-540-64009-6},
MRCLASS = {49-02 (49Jxx 49Qxx 58A25)},
MRNUMBER = {1645086},
MRREVIEWER = {Ernst C. Kuwert},
ZBLNUMBER = {0914.49001},
} -
[GiaquintaModicaSoucek2] M. Giaquinta, G. Modica, and J. Souvcek, Cartesian Currents in the Calculus of Variations. II, New York: Springer-Verlag, 1998, vol. 38.
@book {GiaquintaModicaSoucek2,
author = {Giaquinta, Mariano and Modica, G. and Sou{\v{c}}ek, J.},
TITLE = {Cartesian Currents in the Calculus of Variations. {\rm II}},
SERIES = {Ergeb. Math. Grenzgeb.},
VOLUME = {38},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1998},
MRNUMBER = {1645082},
ZBLNUMBER = {0914.49002},
} -
[GromovElisha] M. L. Gromov and Y. Eliashberg, "Construction of a smooth mapping with a prescribed Jacobian. I," Funct. Anal. Appl., vol. 7, pp. 27-33, 1973.
@article{GromovElisha,
author={Gromov, M. L. and Eliashberg, Y.},
TITLE={Construction of a smooth mapping with a prescribed {J}acobian. {I}},
JOURNAL={Funct. Anal. Appl.},
VOLUME={7},
YEAR={1973},
PAGES={27--33},
MRNUMBER = {0353357},
ZBLNUMBER = {0285.57015},
} -
[Hajlasz] P. Hajłasz, "A note on weak approximation of minors," Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 12, iss. 4, pp. 415-424, 1995.
@article {Hajlasz, MRKEY = {1341410},
AUTHOR = {Haj{\l}asz, Piotr},
TITLE = {A note on weak approximation of minors},
JOURNAL = {Ann. Inst. H. Poincaré Anal. Non Linéaire},
FJOURNAL = {Annales de l'Institut Henri Poincaré. Analyse Non Linéaire},
VOLUME = {12},
YEAR = {1995},
NUMBER = {4},
PAGES = {415--424},
ISSN = {0294-1449},
MRCLASS = {28A75 (28A05)},
MRNUMBER = {1341410},
ZBLNUMBER = {0910.49025},
} -
[HangLin]
F. Hang and F. Lin, "A remark on the Jacobians," Commun. Contemp. Math., vol. 2, iss. 1, pp. 35-46, 2000.
@article {HangLin, MRKEY = {1753137},
AUTHOR = {Hang, Fengbo and Lin, Fanghua},
TITLE = {A remark on the {J}acobians},
JOURNAL = {Commun. Contemp. Math.},
FJOURNAL = {Communications in Contemporary Mathematics},
VOLUME = {2},
YEAR = {2000},
NUMBER = {1},
PAGES = {35--46},
ISSN = {0219-1997},
MRCLASS = {49Q20 (26B10 58A25)},
MRNUMBER = {1753137},
MRREVIEWER = {Agnieszka Ka{\l}amajska},
DOI = {10.1142/S0219199700000037},
ZBLNUMBER = {1033.49047},
} -
[IwaniecLips] T. Iwaniec, Integrability theory of the Jacobians.
@misc{IwaniecLips,
author = {Iwaniec, Tadeusz},
TITLE={Integrability theory of the {J}acobians},
NOTE={Lipschitz lectures in {B}onn, preprint 36, {S}onderforschungsbereich 256, {B}onn, 1995},
} -
[IwaniecOnninen]
T. Iwaniec and J. Onninen, "$\mathcal{H}_1$-estimates of Jacobians by subdeterminants," Math. Ann., vol. 324, iss. 2, pp. 341-358, 2002.
@article {IwaniecOnninen, MRKEY = {1933861},
AUTHOR = {Iwaniec, Tadeusz and Onninen, Jani},
TITLE = {{$\mathcal{H}_1$}-estimates of {J}acobians by subdeterminants},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {324},
YEAR = {2002},
NUMBER = {2},
PAGES = {341--358},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {42B25 (26B10 46E35)},
MRNUMBER = {1933861},
MRREVIEWER = {Emmanuel Russ},
DOI = {10.1007/s00208-002-0341-5},
ZBLNUMBER = {1055.42011},
} -
[IwaniecSbordone]
T. Iwaniec and C. Sbordone, "On the integrability of the Jacobian under minimal hypotheses," Arch. Rational Mech. Anal., vol. 119, iss. 2, pp. 129-143, 1992.
@article {IwaniecSbordone, MRKEY = {1176362},
AUTHOR = {Iwaniec, Tadeusz and Sbordone, Carlo},
TITLE = {On the integrability of the {J}acobian under minimal hypotheses},
JOURNAL = {Arch. Rational Mech. Anal.},
FJOURNAL = {Archive for Rational Mechanics and Analysis},
VOLUME = {119},
YEAR = {1992},
NUMBER = {2},
PAGES = {129--143},
ISSN = {0003-9527},
CODEN = {AVRMAW},
MRCLASS = {49Q20 (26B10 46E35 49J10)},
MRNUMBER = {1176362},
MRREVIEWER = {Paolo Marcellini},
DOI = {10.1007/BF00375119},
ZBLNUMBER = {0766.46016},
} -
[JerrardSoner]
R. L. Jerrard and H. M. Soner, "Functions of bounded higher variation," Indiana Univ. Math. J., vol. 51, iss. 3, pp. 645-677, 2002.
@article {JerrardSoner, MRKEY = {1911049},
AUTHOR = {Jerrard, Robert L. and Soner, Halil Mete},
TITLE = {Functions of bounded higher variation},
JOURNAL = {Indiana Univ. Math. J.},
FJOURNAL = {Indiana University Mathematics Journal},
VOLUME = {51},
YEAR = {2002},
NUMBER = {3},
PAGES = {645--677},
ISSN = {0022-2518},
CODEN = {IUMJAB},
MRCLASS = {49Q20 (49J10)},
MRNUMBER = {1911049},
MRREVIEWER = {Giovanni Bellettini},
DOI = {10.1512/iumj.2002.51.2229},
ZBLNUMBER = {1057.49036},
} -
[JerrardSoner2]
R. L. Jerrard and H. M. Soner, "Limiting behavior of the Ginzburg-Landau functional," J. Funct. Anal., vol. 192, iss. 2, pp. 524-561, 2002.
@article {JerrardSoner2, MRKEY = {1923413},
AUTHOR = {Jerrard, Robert L. and Soner, Halil Mete},
TITLE = {Limiting behavior of the {G}inzburg-{L}andau functional},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {192},
YEAR = {2002},
NUMBER = {2},
PAGES = {524--561},
ISSN = {0022-1236},
CODEN = {JFUAAW},
MRCLASS = {35J50 (35B25 35Q60 49J45 58E50 82D55)},
MRNUMBER = {1923413},
MRREVIEWER = {Giovanni Alberti},
DOI = {10.1006/jfan.2001.3906},
ZBLNUMBER = {1028.49015},
} -
[JerrardSoner1]
R. L. Jerrard and H. M. Soner, "The Jacobian and the Ginzburg-Landau energy," Calc. Var. Partial Differential Equations, vol. 14, iss. 2, pp. 151-191, 2002.
@article {JerrardSoner1, MRKEY = {1890398},
AUTHOR = {Jerrard, Robert L. and Soner, Halil Mete},
TITLE = {The {J}acobian and the {G}inzburg-{L}andau energy},
JOURNAL = {Calc. Var. Partial Differential Equations},
FJOURNAL = {Calculus of Variations and Partial Differential Equations},
VOLUME = {14},
YEAR = {2002},
NUMBER = {2},
PAGES = {151--191},
ISSN = {0944-2669},
MRCLASS = {35J50 (35B25 35J60 35Q80 58E50)},
MRNUMBER = {1890398},
MRREVIEWER = {Stanley A. Alama},
DOI = {10.1007/s005260100093},
ZBLNUMBER = {1034.35025},
} -
[JohnNirenberg]
F. John and L. Nirenberg, "On functions of bounded mean oscillation," Comm. Pure Appl. Math., vol. 14, pp. 415-426, 1961.
@article {JohnNirenberg, MRKEY = {0131498},
AUTHOR = {John, F. and Nirenberg, L.},
TITLE = {On functions of bounded mean oscillation},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {14},
YEAR = {1961},
PAGES = {415--426},
ISSN = {0010-3640},
MRCLASS = {26.00},
MRNUMBER = {0131498},
MRREVIEWER = {L. C. Young},
DOI = {10.1002/cpa.3160140317},
ZBLNUMBER = {0102.04302},
} -
[LinYang1]
F. Lin and Y. Yang, "Existence of two-dimensional skyrmions via the concentration-compactness method," Comm. Pure Appl. Math., vol. 57, iss. 10, pp. 1332-1351, 2004.
@article {LinYang1, MRKEY = {2070206},
AUTHOR = {Lin, Fanghua and Yang, Yisong},
TITLE = {Existence of two-dimensional skyrmions via the concentration-compactness method},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {57},
YEAR = {2004},
NUMBER = {10},
PAGES = {1332--1351},
ISSN = {0010-3640},
CODEN = {CPAMA},
MRCLASS = {58E50 (35J20)},
MRNUMBER = {2070206},
MRREVIEWER = {Paul F. Bracken},
DOI = {10.1002/cpa.20038},
ZBLNUMBER = {1059.81184},
} -
[LinYangOpen] F. Lin and Y. Yang, "Analysis on Faddeev knots and Skyrme solitons: recent progress and open problems," in Perspectives in Nonlinear Partial Differential Equations, Providence, RI: Amer. Math. Soc., 2007, vol. 446, pp. 319-344.
@incollection {LinYangOpen, MRKEY = {2376667},
AUTHOR = {Lin, Fanghua and Yang, Yisong},
TITLE = {Analysis on {F}addeev knots and {S}kyrme solitons: recent progress and open problems},
BOOKTITLE = {Perspectives in Nonlinear Partial Differential Equations},
SERIES = {Contemp. Math.},
VOLUME = {446},
PAGES = {319--344},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2007},
MRCLASS = {58E50 (35Q51 81T10)},
MRNUMBER = {2376667},
MRREVIEWER = {Dave Auckly},
ZBLNUMBER = {05831673},
} -
[LinYang2]
F. Lin and Y. Yang, "Energy splitting, substantial inequality, and minimization for the Faddeev and Skyrme models," Comm. Math. Phys., vol. 269, iss. 1, pp. 137-152, 2007.
@article {LinYang2, MRKEY = {2274465},
AUTHOR = {Lin, Fanghua and Yang, Yisong},
TITLE = {Energy splitting, substantial inequality, and minimization for the {F}addeev and {S}kyrme models},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {269},
YEAR = {2007},
NUMBER = {1},
PAGES = {137--152},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {58E50 (81T10 81T13)},
MRNUMBER = {2274465},
MRREVIEWER = {Dave Auckly},
DOI = {10.1007/s00220-006-0123-0},
ZBLNUMBER = {1113.81096},
} -
[MazyaSha]
V. Maz$’$ya and T. Shaposhnikova, "On the Brezis and Mironescu conjecture concerning a Gagliardo-Nirenberg inequality for fractional Sobolev norms," J. Math. Pures Appl., vol. 81, iss. 9, pp. 877-884, 2002.
@article {MazyaSha, MRKEY = {1940371},
AUTHOR = {Maz$'$ya, Vladimir and Shaposhnikova, Tatyana},
TITLE = {On the {B}rezis and {M}ironescu conjecture concerning a {G}agliardo-{N}irenberg inequality for fractional {S}obolev norms},
JOURNAL = {J. Math. Pures Appl.},
FJOURNAL = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
VOLUME = {81},
YEAR = {2002},
NUMBER = {9},
PAGES = {877--884},
ISSN = {0021-7824},
CODEN = {JMPAAM},
MRCLASS = {46E35 (35J20)},
MRNUMBER = {1940371},
MRREVIEWER = {Luigi Greco},
DOI = {10.1016/S0021-7824(02)01262-X},
ZBLNUMBER = {1036.46026},
} -
[MironescuManifolds] P. Mironescu, "Sobolev maps on manifolds: degree, approximation, lifting," in Perspectives in Nonlinear Partial Differential Equations, Providence, RI: Amer. Math. Soc., 2007, vol. 446, pp. 413-436.
@incollection {MironescuManifolds, MRKEY = {2376670},
AUTHOR = {Mironescu, Petru},
TITLE = {Sobolev maps on manifolds: degree, approximation, lifting},
BOOKTITLE = {Perspectives in Nonlinear Partial Differential Equations},
SERIES = {Contemp. Math.},
VOLUME = {446},
PAGES = {413--436},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2007},
MRCLASS = {46E35 (47H11 55M25)},
MRNUMBER = {2376670},
ZBLNUMBER = {1201.46032},
} -
[Morrey] C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations, New York: Springer-Verlag, 1966, vol. 130.
@book {Morrey, MRKEY = {0202511},
AUTHOR = {Morrey, Jr., Charles B.},
TITLE = {Multiple Integrals in the Calculus of Variations},
SERIES = {Grundlehren Math. Wis.},
VOLUME={130},
PUBLISHER = {Springer-Verlag},
YEAR = {1966},
PAGES = {ix+506},
MRCLASS = {49.00 (00.00)},
MRNUMBER = {0202511},
MRREVIEWER = {M. Schechter},
ADDRESS = {New York},
ZBLNUMBER = {0142.38701},
} -
[Moser]
J. Moser, "On the volume elements on a manifold," Trans. Amer. Math. Soc., vol. 120, pp. 286-294, 1965.
@article {Moser, MRKEY = {0182927},
AUTHOR = {Moser, J{ü}rgen},
TITLE = {On the volume elements on a manifold},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {120},
YEAR = {1965},
PAGES = {286--294},
ISSN = {0002-9947},
MRCLASS = {53.45},
MRNUMBER = {0182927},
MRREVIEWER = {R. S. Palais},
DOI = {10.2307/1994022},
ZBLNUMBER = {0141.19407},
} -
[Muller]
S. Müller, "Higher integrability of determinants and weak convergence in $L^1$," J. Reine Angew. Math., vol. 412, pp. 20-34, 1990.
@article {Muller, MRKEY = {1078998},
AUTHOR = {M{ü}ller, Stefan},
TITLE = {Higher integrability of determinants and weak convergence in {$L\sp 1$}},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {412},
YEAR = {1990},
PAGES = {20--34},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {49J45 (58E35 73C50 73V25)},
MRNUMBER = {1078998},
MRREVIEWER = {Ricardo S. Kubrusly},
DOI = {10.1515/crll.1990.412.20},
ZBLNUMBER = {2376667},
} -
[MullerDet] S. Müller, "${ Det}={ det}$. A remark on the distributional determinant," C. R. Acad. Sci. Paris Sér. I Math., vol. 311, iss. 1, pp. 13-17, 1990.
@article {MullerDet, MRKEY = {1062920},
AUTHOR = {M{ü}ller, Stefan},
TITLE = {{${\rm Det}={\rm det}$}. {A} remark on the distributional determinant},
JOURNAL = {C. R. Acad. Sci. Paris Sér. I Math.},
FJOURNAL = {Comptes Rendus de l'Académie des Sciences. Série I. Mathématique},
VOLUME = {311},
YEAR = {1990},
NUMBER = {1},
PAGES = {13--17},
ISSN = {0764-4442},
CODEN = {CASMEI},
MRCLASS = {35J60 (58G26 73C50 73V25)},
MRNUMBER = {1062920},
MRREVIEWER = {John M. Ball},
ZBLNUMBER = {0717.46033},
} -
[MullerSupp] S. Müller, "On the singular support of the distributional determinant," Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 10, iss. 6, pp. 657-696, 1993.
@article {MullerSupp, MRKEY = {1253606},
AUTHOR = {M{ü}ller, Stefan},
TITLE = {On the singular support of the distributional determinant},
JOURNAL = {Ann. Inst. H. Poincaré Anal. Non Linéaire},
FJOURNAL = {Annales de l'Institut Henri Poincaré. Analyse Non Linéaire},
VOLUME = {10},
YEAR = {1993},
NUMBER = {6},
PAGES = {657--696},
ISSN = {0294-1449},
MRCLASS = {46E35 (46F10)},
MRNUMBER = {1253606},
MRREVIEWER = {W. P. Ziemer},
ZBLNUMBER = {0792.46027},
} -
[NgOpt]
H. Nguyen, "Optimal constant in a new estimate for the degree," J. Anal. Math., vol. 101, pp. 367-395, 2007.
@article {NgOpt, MRKEY = {2346551},
AUTHOR = {Nguyen, Hoai-Minh},
TITLE = {Optimal constant in a new estimate for the degree},
JOURNAL = {J. Anal. Math.},
FJOURNAL = {Journal d'Analyse Mathématique},
VOLUME = {101},
YEAR = {2007},
PAGES = {367--395},
ISSN = {0021-7670},
CODEN = {JOAMAV},
MRCLASS = {46E35 (58C30)},
MRNUMBER = {2346551},
MRREVIEWER = {Itai Shafrir},
DOI = {10.1007/s11854-007-0014-0},
ZBLNUMBER = {1147.47046},
} -
[NirenbergNonlinear] L. Nirenberg, Topics in Nonlinear Functional Analysis, New York: New York University Courant Institute of Mathematical Sciences, 2001, vol. 6.
@book {NirenbergNonlinear, MRKEY = {1850453},
AUTHOR = {Nirenberg, Louis},
TITLE = {Topics in Nonlinear Functional Analysis},
SERIES = {Courant Lect. Notes Math.},
VOLUME = {6},
NOTE = {Chapter 6 by E. Zehnder, Notes by R. A. Artino, Revised reprint of the 1974 original},
PUBLISHER = {New York University Courant Institute of Mathematical Sciences},
ADDRESS = {New York},
YEAR = {2001},
PAGES = {xii+145},
ISBN = {0-8218-2819-3},
MRCLASS = {47Hxx (47J05 58-01)},
MRNUMBER = {1850453},
ZBLNUMBER = {0992.47023},
} -
[Reshentnyak] J. G. Revsetnjak, "The weak convergence of completely additive vector-valued set functions," Sibirsk. Mat. \u Z., vol. 9, pp. 1386-1394, 1968.
@article {Reshentnyak, MRKEY = {0240274},
AUTHOR = {Re{š}etnjak, Ju. G.},
TITLE = {The weak convergence of completely additive vector-valued set functions},
JOURNAL = {Sibirsk. Mat. \u Z.},
FJOURNAL = {Akademija Nauk SSSR. Sibirskoe Otdelenie. SibirskiĭMatematičeskiĭŽurnal},
VOLUME = {9},
YEAR = {1968},
PAGES = {1386--1394},
ISSN = {0037-4474},
MRCLASS = {28.50},
MRNUMBER = {0240274},
MRREVIEWER = {S. Kurepa},
} -
[SteinSingular] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton, N.J.: Princeton Univ. Press, 1970, vol. 30.
@book {SteinSingular, MRKEY = {0290095},
AUTHOR = {Stein, Elias M.},
TITLE = {Singular Integrals and Differentiability Properties of Functions},
SERIES = {Princeton Math. Series},
VOLUME={30},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, N.J.},
YEAR = {1970},
PAGES = {xiv+290},
MRCLASS = {46.38 (26.00)},
MRNUMBER = {0290095},
MRREVIEWER = {R. E. Edwards},
ZBLNUMBER = {0207.13501},
} -
@book {Triebel83, MRKEY = {0781540},
AUTHOR = {Triebel, Hans},
TITLE = {Theory of Function Spaces},
SERIES = {Monogr. Math.},
VOLUME = {78},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {1983},
PAGES = {284},
ISBN = {3-7643-1381-1},
MRCLASS = {46Exx},
MRNUMBER = {0781540},
DOI = {10.1007/978-3-0346-0416-1},
ZBLNUMBER = {0546.46027},
}