Abstract
Motivated by some questions in continuum mechanics and analysis in metric spaces, we give an intrinsic characterization of sequentially weak lower semicontinuous functionals defined on Sobolev maps with values into manifolds without embedding the target into Euclidean spaces.
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Acerbi E., Fusco N.: Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86, 125–145 (1984)
Ambrosio L.: Metric space valued functions of bounded variation. Ann. Scuola Norm. Sup. Pisa Cl. Sci 17(3), 439–478 (1990)
Ball J.M., Zarnescu A.: Orientable and non-orientable line field models for uniaxial nematic liquid crystals. Mol. Cryst. Liq. Cryst. 495, 573–585 (2008)
Brezis H., Coron J.-M., Lieb E.H.: Harmonic maps with defects. Commun. Math. Phys. 107(4), 649–705 (1986)
Capriz G.: Continua with Microstructure, volume 35 of Springer Tracts in Natural Philosophy. Springer, New York (1989)
Chiron D.: On the definitions of Sobolev and BV spaces into singular spaces and the trace problem. Commun. Contemp. Math. 9(4), 473–513 (2007)
Dacorogna B., Fonseca I., Malý J., Trivisa K.: Manifold constrained variational problems. Calc. Var. Partial Differ. Equ. 9(3), 185–206 (1999)
De Gennes P.-G.: The Physics of Liquid crystals. Clarendon Press, Oxford (1974)
De Lellis C., Focardi M., Spadaro E.N.: Lower semicontinuous functionals for Almgren’s multiple valued functions. Annales AcademiæScientiarum FennicæMathematica 36, 1–18 (2011)
De Lellis, C., Spadaro, E.N.: Q-valued functions revisited. Memoirs of the AMS 211(991) (2011)
Do Carmo, M.P.: Riemannian geometry. In: Kadison, R.V., Singer, I.M. (eds.) Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston (1992)
Ericksen J.L.: Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113(2), 97–120 (1990)
Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties Of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992)
Focardi, M., Mariano, P.M., Spadaro, E.N.: In preparation
Giaquinta, M., Modica, G., Souček, J.: Cartesian currents and variational problems for mappings into spheres. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). 16(3), 393–485 (1990), 1989
Giaquinta M., Modica G., Souček J.: Liquid crystals: relaxed energies, dipoles, singular lines and singular points. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17(3), 415–437 (1990)
Giaquinta, M., Modica, G., Souček, J.: Cartesian currents in the calculus of variations. I, volume 37 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer, Berlin (1998)
Hajłasz P.: Sobolev Mappings Between Manifolds and Metric Spaces, volume 8 of Sobolev spaces in mathematics. I, Int. Math. Ser. (N.Y.). Springer, New York (2009)
Hajłasz P., Tyson J.T.: Sobolev Peano cubes. Michigan Math. J. 56, 687–702 (2008)
Hardt R., Kinderlehrer D., Lin F.-H.: Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105(4), 547–570 (1986)
Heinonen J., Koskela P., Shanmugalingam N., Tyson J.T.: Sobolev classes of Banach space-valued functions and quasiconformal mappings. J. Anal. Math. 85, 87–139 (2001)
Jost J.: Equilibrium maps between metric spaces. Calc. Var. Partial Differ. Equ. 2(2), 173–204 (1994)
Jost J.: Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70(4), 659–673 (1995)
Jost J.: Riemannian Geometry and Geometric Analysis. Universitext, Springer, Berlin (2008)
Kirchheim B.: Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Am. Math. Soc. 121(1), 113–123 (1994)
Korevaar N., Schoen R.: Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom 1(3–4), 561–659 (1993)
Marcellini P.: Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscripta Math. 51(1–3), 1–28 (1985)
Mariano P.M.: Multifield theories in mechanics of solids. Adv. Appl. Mech. 38, 1–93 (2002)
Mariano P.M., Modica G.: Ground states in complex bodies. ESAIM Control Optim. Calc. Var. 15(2), 377–402 (2009)
Morrey C.B. Jr: Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2, 25–53 (1952)
Morrey, C.B. Jr.: Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer New York, Inc., New York (1966)
Mucci, D.: Maps into projective spaces: liquid crystals and conformal energies. Preprint (2010)
Nash J.: The imbedding problem for Riemannian manifolds. Ann. Math. (2) 63, 20–63 (1956)
Ohta S.: Cheeger type Sobolev spaces for metric space targets. Potential Anal. 20(2), 149–175 (2004)
Reshetnyak Y.G.: Sobolev classes of functions with values in a metric space. Siberian Math. J. 38(3), 567–583 (1997)
Reshetnyak Y.G.: Sobolev classes of functions with values in a metric space. ii. Siberian Math. J. 45(4), 709–721 (2004)
Reshetnyak Y.G.: On the theory of Sobolev classes of functions with values in a metric space. Siberian Math. J. 47(1), 117–134 (2006)
Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III (1993)
Virga, E.G.: Variational theories for liquid crystals, volume 8 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London (1994)
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Focardi, M., Spadaro, E. An intrinsic approach to manifold constrained variational problems. Annali di Matematica 192, 145–163 (2013). https://doi.org/10.1007/s10231-011-0216-z
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DOI: https://doi.org/10.1007/s10231-011-0216-z