Abstract

We consider regularity at the boundary for minimizers of variational integrals whose integrands have nonquadratic growth in the gradient. Under relatively mild assumptions on the coefficients we obtain a partial regularity result. For coefficients of a more particular type, namely those satifying a particular splitting condition, we obtain full boundary regularity. The results are new for the situation under consideration. The key ingredients are a new version of the usual Gehring-type lemma, and a careful adaptation of the technique of dimension-reduction to the current setting

Author information

Contact details are reproduced from the original publication and may be historical.

Frank Duzaar

Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054 Erlangen,
Germany

duzaar@mi.uni-erlangen.de

Joseph F. Grotowski

Dept. of Mathematics, City College of New York, City University of New York, Convent Avenue at
138th Street, New York, NY 10031, U.S.A.

grotow@sci.ccny.cuny.edu

Manfred Kronz

Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054 Erlangen,
Germany

kronz@mi.uni-erlangen.de

Mathematics Subject Classification

49J45, 58E20

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.https://doi.org/10.1016/0022-247X(89)90098-XGoogle Scholar
  2. R. R. Coifman, G. Weiss: Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, Lecture Notes in Mathematics 242, Springer, Berlin (1971).https://doi.org/10.1007/BFb0058946Google Scholar
  3. H. Federer: Geometric Measure Theory, Springer, New York (1969).Google Scholar
  4. H. Federer: The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Am. Math. Soc. 76 (1970) 767–771.https://doi.org/10.1090/S0002-9904-1970-12542-3Google Scholar
  5. M. Fuchs: p-harmonic obstacle problems. I: Partial regularity theory, Annali Mat. Pura Applicata 156 (1990) 127–158.https://doi.org/10.1007/BF01766976Google Scholar
  6. M. Fuchs: p-harmonic obstacle problems. II: Extensions of maps and applications, Manus. Math. 63 (1989) 381–419.https://doi.org/10.1007/BF01171756Google Scholar
  7. M. Fuchs: p-harmonic obstacle problems. III: Boundary regularity, Ann. Mat. Pura Appl. 156(4) (1990) 159–180.https://doi.org/10.1007/BF01766977Google Scholar
  8. N. Fusco, J. Hutchinson: C 1,α partial regularity of functions minimising quasiconvex integrals, Manus. Math. 54 (1986) 121–143.https://doi.org/10.1007/BF01171703Google Scholar
  9. N. Fusco, J. Hutchinson: Partial regularity for minimisers of certain functionals having nonquadratic growth, Ann. Mat. Pura Appl., IV. Ser. 155 (1989) 1–24.https://doi.org/10.1007/BF01765932Google Scholar
  10. F. W. Gehring: The Lp –integrability of the partial derivatives of a quasiconformal map, Acta Math. 130 (1973) 265–277.https://doi.org/10.1007/BF02392268Google Scholar
  11. M. Giaquinta: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton (1983).https://doi.org/10.1515/9781400881628Google Scholar
  12. M. Giaquinta: Introduction to Regularity Theory for Nonlinear Elliptic Systems, Birkhäuser, Basel (1993).Google Scholar
  13. M. Giaquinta, E. Giusti: On the regularity of the minima of variational integrals, Acta Math. 148 (1982) 31–46.https://doi.org/10.1007/BF02392725Google Scholar
  14. M. Giaquinta, E. Giusti: The singular set of the minima of certain quadratic functionals, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11 (1984) 45–55.Google Scholar
  15. M. Giaquinta, G. Modica: Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré, Anal. Non Linéaire 3 (1986) 185–208.https://doi.org/10.1016/s0294-1449(16)30385-7Google Scholar
  16. M. Giaquinta, G. Modica: Remarks on the regularity of minimizers of certain degenerate functionals, Manus. Math. 57 (1986) 55–99.https://doi.org/10.1007/BF01172492Google Scholar
  17. E. Giusti, M. Miranda: Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni, Boll. U.M.I. Ser. IV 1 (1968) 219–226.Google Scholar
  18. R. Hardt, F. H. Lin: Mappings minimizing the Lp norm of the gradient, Comm. Pure Appl. Math. 40 (1987) 555–588.https://doi.org/10.1002/cpa.3160400503Google Scholar
  19. R. Hardt, F. H. Lin, L. Mou: Strong convergence of p-harmonic mappings, in: Progress in Partial Differential Equations: The Metz Surveys 3, M. Chipot et al. (eds.), Pitman Res. Notes Math. Ser. 314, Harlow, Longman Scientific & Technical (1994) 58–64.Google Scholar
  20. M. Kronz: Some Function Spaces on Spaces of Homogeneous Type, Manus. Math. 106 (2001) 219–248.https://doi.org/10.1007/s002290100205Google Scholar
  21. J. Jost, M. Meier: Boundary regularity for minima of certain quadratic functionals, Math. Ann. 262 (1983) 549–561.https://doi.org/10.1007/BF01456068Google Scholar
  22. S. Luckhaus: Partial Hölder continuity for minima of certain energies among maps into a Riemannian manifold, Indiana Univ. Math. J. 37 (1988) 349–367.https://doi.org/10.1512/iumj.1988.37.37017Google Scholar
  23. R. Schoen, K. Uhlenbeck: A regularity theory for harmonic maps, J. Differ. Geom. 17 (1982) 307–335.https://doi.org/10.4310/jdg/1214436923Google Scholar
  24. R. Schoen, K. Uhlenbeck: Boundary regularity and the Dirichlet problem for harmonic maps, J. Differ. Geom. 18 (1983) 253–268.https://doi.org/10.4310/jdg/1214437663Google Scholar
  25. L. Simon: Theorems on Regularity and Singularity of Energy Minimizing Maps, Birkhäuser, Basel (1996).https://doi.org/10.1007/978-3-0348-9193-6Google Scholar
  26. E. W. Stredulinsky: Higher integrability from reverse Hölder inequalities, Indiana Univ. Math. J. 29 (1980) 407–413.https://doi.org/10.1512/iumj.1980.29.29029Google Scholar
  27. P. Tolksdorf: Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl., IV. Ser. 134 (1986) 241–266.https://doi.org/10.1007/BF01773507Google Scholar
  28. K. Uhlenbeck: Regularity for a class of non-linear systems, Acta Math. 138 (1977) 219–240.https://doi.org/10.1007/BF02392316Google Scholar
  29. W. P. Ziemer: Weakly Differentiable Functions, Springer, Berlin (1989).https://doi.org/10.1007/978-1-4612-1015-3Google Scholar

Suggested citation

F. Duzaar, J. F. Grotowski, M. Kronz. “Partial and Full Boundary Regularity for Minimizers of Functionals with Nonquadratic Growth.” Journal of Convex Analysis 11 (2004), No. 2, 437–476. https://doi.org/10.68381/jca11027

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