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Carriero, M.; Leaci, A.

\(S^ k\)-valued maps minimizing the \(L^ p\) norm of the gradient with free discontinuities. (English) Zbl 0753.49018

Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 18, No. 3, 321-352 (1991).
We prove existence and partial regularity of a solution of a free discontinuity problem in the vector case with a constraint. More precisely for every \(p>1\) and \(q\geq 1\) we prove the existence of a minimizing pair for the functional defined for every closed set \(K\subset{\mathbf R}^ n\) and for every \(u\in C^ 1(\Omega\setminus K; S^ k)\) by \[ \int_{\Omega\setminus K} |\nabla u|^ p dy+\int_{\Omega\setminus K}| u-g|^ q dy+{\mathcal H}^{n- 1}(K\cap\Omega), \] where \(n\geq 2\), \(\Omega\subset{\mathbf R}^ n\) is a bounded open set, \(k\in N\) and \(S^ k=\{z\in{\mathbf R}^{k+1}; | z|=1\}\), \(g\in L^ \infty(\Omega; {\mathbf R}^{k+1})\) and \({\mathcal H}^{n-1}\) is the \((n-1)\)-dimensional Hausdorff measure. Moreover for an optimal closed set \(K\) an estimate of the lower \(n\)-dimensional density is given. Analogous results hold under Dirichlet type conditions. The interest in considering the constraint \(u(x)\in S^ k\) is connected to the recent studies of minima of non-convex functionals and on possible applications of these studies to the static theory of liquid crystals.
Reviewer: M.Carriero

Cited in 16 Documents

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
26A45 Functions of bounded variation, generalizations

Keywords:

BV functions; harmonic maps; existence; partial regularity; free discontinuity problem
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References:

[1] E. Acerbi - N. Fusco , Regularity for minimizers of non-quadratic functionals: the case 1 <p<2 , J. Math. Anal. Appl. , 140 ( 1989 ), 115 - 135 . MR 997847 | Zbl 0686.49004 · Zbl 0686.49004 · doi:10.1016/0022-247X(89)90098-X
[2] F.J. Almgren Jr. - E.H. Lieb , Singularities of energy minimizing maps from the ball to the sphere: Examples, counterexamples, and bounds , Ann. of Math. , 128 ( 1988 ), 483 - 530 . MR 970609 | Zbl 0673.58013 · Zbl 0673.58013 · doi:10.2307/1971434
[3] L. Ambrosio , A compactness theorem for a special class of functions of bounded variation , Boll. Un. Mat. Ital. , 3-B ( 1989 ), 857 - 881 . MR 1032614 | Zbl 0767.49001 · Zbl 0767.49001
[4] L. Ambrosio , Existence theory for a new class of variational problems , Arch. Rational Mech. Anal. , 111 ( 1990 ), 291 - 322 . MR 1068374 | Zbl 0711.49064 · Zbl 0711.49064 · doi:10.1007/BF00376024
[5] L. Ambrosio - G. Dal Maso , The chain rule for the distributional derivatives , Proc. Amer. Math. Soc. , to appear. MR 969514 · Zbl 0685.49027
[6] L. Ambrosio - V.M. Tortorelli , Approximation of functionals depending on jumps by elliptic functionals via r-convergence , Comm. Pure Appl. Math. , 43 ( 1990 ), 999 - 1036 . MR 1075076 | Zbl 0722.49020 · Zbl 0722.49020 · doi:10.1002/cpa.3160430805
[7] F. Bethuel - H. Brézis - J.M. Coron , Relaxed energies for harmonic maps , Preprint 1989 . MR 1205144 · Zbl 0793.58011
[8] H. Brézis - J.M. Coron - E.H. Lieb , Harmonic Maps with Defects , Comm. Math. Phys ., 107 ( 1986 ), 679 - 705 . Article | MR 868739 | Zbl 0608.58016 · Zbl 0608.58016 · doi:10.1007/BF01205490
[9] M. Carriero - A. Leaci , Existence theorem for a Dirichlet problem with free discontinuity set , Nonlinear Anal. TMA , 15 ( 1990 ), 661 - 677 . MR 1073957 | Zbl 0713.49003 · Zbl 0713.49003 · doi:10.1016/0362-546X(90)90006-3
[10] S. Chandrasekhar , Liquid Crystals . Cambridge University Press , Cambridge , 1977 . · Zbl 0127.41403
[11] E. De Giorgi , Free Discontinuity Problems in Calculus of Variations, Proc. Int. Meeting in J.L. Lions’s honour , Paris , June 6-10, 1988 , to appear. Zbl 0758.49002 · Zbl 0758.49002
[12] E. De Giorgi - L. Ambrosio , Un nuovo tipo di funzionale del Calcolo delle Variazioni , Atti Accad. Naz. Lincei , 82 ( 1988 ), 199 - 210 . Zbl 0715.49014 · Zbl 0715.49014
[13] [13) E. De Giorgi - M. Carriero - A. Leaci , Existence theorem for a minimum problem with free discontinuity set , Arch. Rational Mech. Anal ., 108 ( 1989 ), 195 - 218 . MR 1012174 | Zbl 0682.49002 · Zbl 0682.49002 · doi:10.1007/BF01052971
[14] J. Eells - L. Lemaire , Another report on harmonic maps , Bull. London Math. Soc. , 20 ( 1988 ), 385 - 524 . MR 956352 | Zbl 0669.58009 · Zbl 0669.58009 · doi:10.1112/blms/20.5.385
[15] J.L. Ericksen , Equilibrium theory of liquid crystals , Adv. Liq. Cryst. , Vol. 2 , G.H. Brown Ed., Academic Press , New York , 1976 , 233 - 299 .
[16] H. Federer , Geometric Measure Theory , Springer Verlag , Berlin , 1969 . MR 257325 | Zbl 0176.00801 · Zbl 0176.00801
[17] M. Giaquinta - E. Giusti , The singular set of the minima of certain quadratic functionals , Ann. Scuola Norm. Sup. Pisa Cl. Sci. , 11 ( 1984 ), 45 - 55 . Numdam | MR 752579 | Zbl 0543.49018 · Zbl 0543.49018
[18] M. Giaquinta - G. Modica , Remarks on the regularity of the minimizers of certain degenerate functionals , Manuscripta Math. , 57 ( 1986 ), 55 - 100 . Article | MR 866406 | Zbl 0607.49003 · Zbl 0607.49003 · doi:10.1007/BF01172492
[19] M. Giaquinta - G. Modica - J. Sou , Cartesian currents and variational problems for mappings into spheres , Ann. Scuola Norm. Sup. Pisa Cl. Sci. , 16 ( 1989 ), 393 - 485 . Numdam | MR 1050333 | Zbl 0713.49014 · Zbl 0713.49014
[20] M. Giaquinta - G. Modica - J. SOUčEK, The Dirichlet energy of mappings with values into the sphere , Manuscripta Math. , 65 ( 1989 ), 489 - 507 . Article | MR 1019705 | Zbl 0678.49006 · Zbl 0678.49006 · doi:10.1007/BF01172794
[21] E. Giusti , Minimal Surfaces and Functions of Bounded Variation . Birkhäuser Verlag , Boston , 1984 . MR 775682 | Zbl 0545.49018 · Zbl 0545.49018
[22] R. Hardt - D. Kinderlehrer - F.-H. Lin , Existence and partial regularity of static liquid crystal configurations , Comm. Math. Phys. , 105 ( 1986 ), 547 - 570 . Article | MR 852090 | Zbl 0611.35077 · Zbl 0611.35077 · doi:10.1007/BF01238933
[23] R. Hardt - F.-H. Lin , Mappings minimizing the LP norm of the gradient , Comm. Pure Appl. Math. , 40 ( 1987 ), 555 - 588 . MR 896767 | Zbl 0646.49007 · Zbl 0646.49007 · doi:10.1002/cpa.3160400503
[24] R. Hardt - F.-H. Lin , A remark on H1 mappings , Manuscripta Math. , 56 ( 1986 ), 1 - 10 . Article | MR 846982 | Zbl 0618.58015 · Zbl 0618.58015 · doi:10.1007/BF01171029
[25] U. Massari - M. Miranda , Minimal Surfaces of Codimension One , North-Holland , Amsterdam , 1984 . MR 795963 | Zbl 0565.49030 · Zbl 0565.49030
[26] C.B. Morrey , Multiple Integrals in the Calculus of Variations , Springer Verlag , New York , 1966 . MR 202511 | Zbl 0142.38701 · Zbl 0142.38701
[27] R. Schoen - K. Uhlenbeck , A regularity theory for harmonic maps , J. Differential Geom. , 18 ( 1982 ), 307 - 335 . MR 664498 | Zbl 0521.58021 · Zbl 0521.58021
[28] R. Schoen - K. Uhlenbeck , Boundary regularity and the Dirichlet problem of harmonic maps , J. Differential Geom. , 18 ( 1983 ), 253 - 268 . MR 710054 | Zbl 0547.58020 · Zbl 0547.58020
[29] P. Tolksdorff , Everywhere regularity for some quasilinear systems with a lack of ellipticity , Ann. Mat. Pura Appl. , 134 ( 1983 ), 241 - 266 . MR 736742 | Zbl 0538.35034 · Zbl 0538.35034 · doi:10.1007/BF01773507
[30] E. Virga , Drops of nematic liquid crystals , Arch. Rational Mech. Anal. , 107 ( 1989 ), 371 - 390 . MR 1004716 | Zbl 0688.76074 · Zbl 0688.76074 · doi:10.1007/BF00251555
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