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The singular set of minima of integral functionals. (English) Zbl 1116.49010

The authors provide upper bounds for Hausdorff dimension of the singular set of minima of general variational integrals \[ \int_\Omega F(x, v, Dv)dx, \] where \(F\) is suitably convex with respect to \(Dv\) and Hölder continuous with respect to \((x, v).\) In particular, we prove that the Hausdorff dimension of the singular set is always strictly less than \(n\), where \(\Omega \subseteq \mathbb{R} ^n.\)

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
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[1] Acerbi, J. Math. Anal. Appl., 140, 115 (1989) · Zbl 0686.49004
[2] Acerbi, J. Reine Ang. Math. (Crelles J.), 584, 117 (2005) · Zbl 1093.76003
[3] Adams, R.A.: Sobolev Spaces. Academic Press, New York, 1975 · Zbl 0314.46030
[4] Bojarski, Ann. Acad. Sci. Fenn. Ser. A I Math., 8, 257 (1983) · Zbl 0548.30016
[5] Caffarelli, L., Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2), 130, 189-213 (1989) · Zbl 0692.35017
[6] Caffarelli, Comm. Pure Appl. Math., 51, 1 (1998) · Zbl 0924.35197
[7] Campanato, S., Differentiability of the solutions of nonlinear elliptic systems with natural growth, Ann. Mat. Pura Appl, 131, 4, 75-106 (1982) · Zbl 0493.35022
[8] Campanato, Adv. Math., 48, 16 (1983) · Zbl 0519.35027
[9] Campanato, S., Elliptic systems with non-linearity q greater than or equal to two. Regularity of the solution of the Dirichlet problem, Ann. Mat. Pura Appl, 147, 4, 117-150 (1987) · Zbl 0635.35038
[10] Campanato, S.: Sistemi ellittici in forma divergenza. Regolarità all’interno. Quaderni SNS di Pisa, 1980 · Zbl 0453.35026
[11] Giorgi, E., Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital, 1, 4, 135-137 (1968) · Zbl 0155.17603
[12] DiBenedetto, Amer. J. Math., 115, 1107 (1993) · Zbl 0805.35037
[13] Duzaar, J. Convex Analysis, 11, 437 (2004) · Zbl 1066.49022
[14] Evans, Arch. Ration. Mech. Anal., 95, 227 (1986) · Zbl 0627.49006
[15] Fonseca, I.; Fusco, N., Regularity results for anisotropic image segmentation models, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24, 4, 463-499 (1997) · Zbl 0899.49018
[16] Fusco, N.; Hutchinson, J. E., Partial regularity for minimisers of certain functionals having nonquadratic growth, Ann. Mat. Pura Appl, 155, 4, 1-24 (1989) · Zbl 0698.49001
[17] Gehring, Acta Math., 130, 265 (1973) · Zbl 0258.30021
[18] Giaquinta, M.: Introduction to regularity theory for nonlinear elliptic systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993 · Zbl 0786.35001
[19] Giaquinta, Acta Math., 148, 31 (1982) · Zbl 0494.49031
[20] Giaquinta, Invent. Math., 72, 285 (1983) · Zbl 0513.49003
[21] Giusti, E., singolarità delle soluzioni deboli di sistemi ellittici non lineari, Boll. Un. Mat. Ital, 2, 4, 71-76 (1969) · Zbl 0175.40103
[22] Giusti, E.: Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003 · Zbl 1028.49001
[23] Giusti, E.; Miranda, M., Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni, Boll. Un. Mat. Ital, 1, 4, 219-226 (1968) · Zbl 0155.44501
[24] Giusti, E.; Miranda, M., Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasi-lineari, Arch. Ration. Mech. Anal., 31, 173-184 (1968) · Zbl 0167.10703
[25] Hamburger, J. Reine Angew. Math. (Crelles J.), 431, 7 (1992) · Zbl 0776.35006
[26] Hamburger, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13, 255 (1996) · Zbl 0863.35022
[27] Hildebrandt, Acta Math., 138, 1 (1977) · Zbl 0356.53015
[28] Hildebrandt, Math. Z., 142, 67 (1975) · Zbl 0317.35040
[29] Hildebrandt, S.; Widman, K. O., On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4, 4, 145-178 (1977) · Zbl 0353.35013
[30] Ivert, P.A.: Partial regularity of vector valued functions minimizing variational integrals. Preprint Univ. Bonn, 1982
[31] Ivert, P. A., Regularittsuntersuchungen von Lösungen elliptischer Systeme von quasilinearen Differentialgleichungen zweiter Ordnung, Manuscripta Math., 30, 53-88 (1979) · Zbl 0429.35033
[32] Iwaniec, Studia Math., 75, 293 (1983) · Zbl 0552.35034
[33] Iwaniec, T., p-harmonic tensors and quasiregular mappings, Ann. Math. (2), 136, 589-624 (1992) · Zbl 0785.30009
[34] Kristensen, Arch. Ration. Mech. Anal., 177, 93 (2005) · Zbl 1082.49036
[35] Kristensen, C. R. Acad. Sci. Paris Ser. I, 340, 93 (2005) · Zbl 1058.49012
[36] Kristensen, J., Mingione, G.: Integral functionals, fractional differentiability and boudary regularity of minima. In preparation. · Zbl 1116.49010
[37] Meyers, Proc. Amer. Math. Soc., 15, 717 (1964) · Zbl 0129.04002
[38] Mingione, Arch. Ration. Mech. Anal., 166, 287 (2003) · Zbl 1142.35391
[39] Mingione, Calc. Var. Partial Differential Equations, 18, 373 (2003) · Zbl 1045.35024
[40] Morrey, C.B.: Multiple integrals in the calculus of variations. Springer-Verlag, New York, 1966 · Zbl 0142.38701
[41] Nečas, J.: Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity. Theory of nonlinear operators (Proc. Fourth Internat. Summer School, Acad. Sci., Berlin, 1975), pp. 197-206
[42] Nirenberg, Comm. Pure. Appl. Math., 8, 649 (1955) · Zbl 0067.07602
[43] Rivière, Acta Math., 175, 197 (1995) · Zbl 0898.58011
[44] Schoen, J. Differential Geom., 17, 307 (1983)
[45] Shiffman, M., Differentiability and analyticity of double integral variational problems, Ann. Math. (2), 48, 274-284 (1947) · Zbl 0029.26702
[46] Simon, L.: Theorems on regularity and singularity of energy minimizing maps. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1996 · Zbl 0864.58015
[47] Stein, E.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton, NJ, 1993 · Zbl 0821.42001
[48] Šverák, Calc. Var. Partial Differential Equations, 10, 213 (2000) · Zbl 1013.49027
[49] Šverák, Proc. Natl. Acad. Sci. USA, 99, 15269 (2002) · Zbl 1106.49046
[50] Uhlenbeck, Acta Math., 138, 219 (1977) · Zbl 0372.35030
[51] Wang, Acta Math. Sin. (Engl. Ser.), 19, 381 (2003) · Zbl 1054.60063
[52] Widman, Manuscripta Math., 5, 299 (1971) · Zbl 0223.35044
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