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Boundary Regularity in Variational Problems

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Abstract

We prove that, if \({u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}\) is a solution to the Dirichlet variational problem

$$\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega,$$

involving a regular boundary datum (u 0, ∂Ω) and a regular integrand F(x, w, Dw) strongly convex in Dw and satisfying suitable growth conditions, then \({{\mathcal H}^{n-1}}\) -almost every boundary point is regular for u in the sense that Du is Hölder continuous in a relative neighborhood of the point. The existence of even one such regular boundary point was previously not known except for some very special cases treated by Jost & Meier (Math Ann 262:549–561, 1983). Our results are consequences of new up-to-the-boundary higher differentiability results that we establish for minima of the functionals in question. The methods also allow us to improve the known boundary regularity results for solutions to non-linear elliptic systems, and, in some cases, to improve the known interior singular sets estimates for minimizers. Moreover, our approach allows for a treatment of systems and functionals with “rough” coefficients belonging to suitable Sobolev spaces of fractional order.

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References

  1. Acerbi, E., Fusco, N.: An approximation lemma for W 1,p functions. Material Instabilities in Continuum Mechanics (Edinburgh, 1985–1986), pp. 1–5. Oxford Science Publication, Oxford University Press, New York, 1988

  2. Acerbi E., Fusco N.: A regularity theorem for minimizers of quasiconvex integrals. Arch. Rational Mech. Anal. 99, 261–281 (1987)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Acerbi E., Mingione G.: Gradient estimates for the p(x)-Laplacean system. J. Reine Angew. Math. (Crelles J.) 584, 117–148 (2005)

    MATH  MathSciNet  Google Scholar 

  4. Acerbi E., Mingione G.: Gradient estimates for a class of parabolic systems. Duke Math. J. 136, 285–320 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  6. Arkhipova A.: Partial regularity up to the boundary for weak solutions to elliptic systems with nonlinearity q greater than two. J. Math. Sci. 115, 2735–2746 (2003)

    Article  MathSciNet  Google Scholar 

  7. Ball J.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63, 337–403 (1976/1977)

    Article  Google Scholar 

  8. Beck, L.: Boundary regularity results for weak solutions of subquadratic elliptic systems. Ph.D. Thesis, Universität Erlangen-Nuremberg, 2008

  9. Beck, L.: Boundary regularity results for variational integrals. Quart. J. Math. (to appear)

  10. Bensoussan, A., Frehse, J.: Regularity results for nonlinear elliptic systems and applications. Applied Mathematical Sciences, vol. 151. Springer, Berlin, 2002

  11. Bögelein V., Duzaar F., Mingione G.: The boundary regularity of non-linear parabolic systems I. Ann. Inst. H. Poincaré Anal. Non Linèaire. 27, 201–255 (2010)

    Article  MATH  ADS  Google Scholar 

  12. Bögelein V., Duzaar F., Mingione G.: The boundary regularity of non-linear parabolic systems II. Ann. Inst. H. Poincaré Anal. Non Linèaire. 27, 145–200 (2010)

    Article  MATH  ADS  Google Scholar 

  13. Bojarski B., Hajlasz P.: Pointwise inequalities for Sobolev functions and some applications. Stud. Math. 106, 77–92 (1993)

    MATH  MathSciNet  Google Scholar 

  14. Caffarelli L.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. (II) 130, 189–213 (1989)

    Article  MathSciNet  Google Scholar 

  15. Caffarelli L., Peral I.: On W 1,p estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51, 1–21 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Campanato S.: Elliptic systems with non-linearity q greater than or equal to two. Regularity of the solution of the Dirichlet problem. Ann. Math. Pura Appl. (IV) 147, 117–150 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  17. Canfora A., Zecca P.: An elliptic equation in \({\mathbb{R}^3}\) with discontinuous nondifferentiable coefficients. Ann. Math. Pura Appl. (IV) 141, 369–384 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  18. Carozza M., Fusco N., Mingione G.: Partial regularity of minimizers of quasiconvex integrals with subquadratic growth. Ann. Math. Pura Appl. (IV) 175, 141–164 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Cho S., Yan X.: On the singular set for Lipschitzian critical points of polyconvex functionals. J. Math. Anal. Appl. 336(1), 372–398 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Dacorogna, B.: Direct methods in the calculus of variations. Applied Mathematical Sciences, vol. 78, 2nd edn. Springer, Berlin, 2008

  21. DeVore, R.A., Sharpley, R.C.: Maximal functions measuring smoothness. Mem. Am. Math. Soc. 47(293) (1984)

  22. Dolzmann G., Kristensen J.: Higher integrability of minimizing Young measures. Calc. Var. Partial Differ. Equ. 22, 283–301 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Duzaar F., Gastel A.: Nonlinear elliptic systems with Dini continuous coefficients. Arch. Math. (Basel) 78, 58–73 (2002)

    MATH  MathSciNet  Google Scholar 

  24. Duzaar F., Grotowski J.F., Kronz M.: Partial and full boundary regularity for minimizers of functionals with nonquadratic growth. J. Convex Anal. 11, 437–476 (2004)

    MATH  MathSciNet  Google Scholar 

  25. Duzaar F., Kristensen J., Mingione G.: The existence of regular boundary points for non-linear elliptic systems. J. Reine Angew. Math. (Crelles J.) 602, 17–58 (2007)

    MATH  MathSciNet  Google Scholar 

  26. Duzaar F., Steffen K.: Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals. J. Reine Angew. Math. (Crelles J.) 546, 73–138 (2002)

    MATH  MathSciNet  Google Scholar 

  27. Escauriaza L.: Weak type (1,1) inequalitites and regularity properties of adjoint and normalized adjoint solutions to linear nondivergence form operators with VMO coefficients. Duke Math. J. 74, 177–201 (2008)

    Article  MathSciNet  Google Scholar 

  28. Evans L.C.: Quasiconvexity and partial regularity in the calculus of variations. Arch. Ration. Mech. Anal. 95, 227–252 (1986)

    Article  MATH  Google Scholar 

  29. Foss M.: Global regularity for almost minimizers of nonconvex variational problems. Ann. Math. Pura Appl. (IV) 187, 263–321 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Franciosi M., Fusco N.: A W 2,2 bound for a class of elliptic equations with discontinuous coefficients. Ricerche di Matematica 31, 207–218 (1982)

    MATH  MathSciNet  Google Scholar 

  31. Giaquinta, M. Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton, 1983

  32. Giaquinta M., Giusti E.: Sharp estimates for the derivatives of local minima of variational integrals. Boll. UMI 3A, 239–248 (1984)

    MathSciNet  Google Scholar 

  33. Giusti, E.: Direct Methods in the Calculus of variations. World Scientific Publishing Co., Inc., River Edge, 2003

  34. Grotowski J.F.: Boundary regularity for nonlinear elliptic systems. Calc. Var. Partial Differ. Equ. 15, 353–388 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  35. Grotowski J.F.: Boundary regularity for quasilinear elliptic systems. Commun. Partial Differ. Equ. 27, 2491–2512 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  36. Habermann J.: Calderón-Zygmund estimates for higher order systems with p(x)-growth. Math. Z. 258, 427–462 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  37. Hamburger C.: Regularity of differential forms minimizing degenerate elliptic functionals. J. Reine Angew. Math. (Crelles J.) 431, 7–64 (1992)

    MATH  MathSciNet  Google Scholar 

  38. Hamburger C.: Partial boundary regularity of solutions of nonlinear superelliptic systems. Boll. UMI. (B) (8) 10, 63–81 (2007)

    MATH  MathSciNet  Google Scholar 

  39. Hildebrandt S., Widman K.O.: Some regularity results for quasilinear elliptic systems of second order. Math. Z. 142, 67–86 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  40. Iwaniec T.: p-harmonic tensors and quasiregular mappings. Ann. Math. (II) 136, 589–624 (1992)

    Article  MathSciNet  Google Scholar 

  41. Jost J., Meier M.: Boundary regularity for minima of certain quadratic functionals. Math. Ann. 262, 549–561 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  42. Koshelev, A.: Regularity problem for quasilinear elliptic and parabolic systems. Lecture Notes in Mathematics, vol. 1614. Springer, Berlin, 1995

  43. Krantz, S.G., Parks, H.R.: The geometry of domains in the space. Advanced Texts, Birkhäuser, Basel, 1999

  44. Kristensen J., Melcher C.: Regularity in oscillatory nonlinear elliptic systems. Math. Z. 260, 813–847 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  45. Kristensen J., Mingione G.: The singular set of ω-minima. Arch. Rational Mech. Anal. 177, 93–114 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  46. Kristensen J., Mingione G.: Non-differentiable functionals and singular sets of minima. C. R. Acad. Sci. Paris Ser. I 340, 93–98 (2005)

    MATH  MathSciNet  Google Scholar 

  47. Kristensen J., Mingione G.: The singular set of minima of integral functionals. Arch. Rational Mech. Anal. 180, 331–398 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  48. Kristensen J., Mingione G.: The singular set of Lipschitzian minima of multiple integrals. Arch. Rational Mech. Anal. 184, 341–369 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  49. Kristensen J., Mingione G.: Boundary regularity of minima. Rendiconti Lincei Matematica e Applicazioni 19, 265–277 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  50. Kristensen J., Taheri A.: Partial regularity of strong local minimizers in the multi-dimensional calculus of variations. Arch. Rational Mech. Anal. 170, 63–89 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  51. Kronz M.: Boundary regularity for almost minimizers of quasiconvex variational problems. Nonlinear Diff. Equ. Appl. 12, 351–382 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  52. Mingione G.: The singular set of solutions to non-differentiable elliptic systems. Arch. Rational Mech. Anal. 166, 287–301 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  53. Mingione G.: Bounds for the singular set of solutions to non linear elliptic systems. Calc. Var. Partial Differ. Equ. 18, 373–400 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  54. Mingione G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51, 355–425 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  55. Mingione G.: The Calderón-Zygmund theory for elliptic problems with measure data. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (V) 6, 195–261 (2007)

    MATH  MathSciNet  Google Scholar 

  56. Mingione G.: Gradient estimates below the duality exponent. Math. Ann. 346, 571–627 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  57. Miranda C.: Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui. Ann. Math. Pura Appl. (IV) 31, 353–386 (1963)

    Article  MathSciNet  Google Scholar 

  58. Morrey C.B.: Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2, 25–53 (1952)

    MATH  MathSciNet  Google Scholar 

  59. Müller S., Šverák V.: Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. Math. (II) 157, 715–742 (2003)

    Article  MATH  Google Scholar 

  60. Nečas, J.: Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity. Theory of Nonlinear Operators. Proceedings of Fourth International Summer School, Academy of Science, Berlin, pp. 197–206, 1975

  61. Scheven C., Schmidt T.: Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure. Ann. Scu. Norm. Sup. Pisa Cl. Sci. (V) 8, 469–507 (2009)

    MATH  MathSciNet  Google Scholar 

  62. Schmidt T.: Regularity of relaxed minimizers of quasiconvex variational integrals with (p,q)-growth. Arch. Rational Mech. Anal. 193, 311–337 (2009)

    Article  MATH  ADS  Google Scholar 

  63. Stein E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  64. Šverák V., Yan X.: A singular minimizer of a smooth strongly convex functional in three dimensions. Calc. Var. Partial Differ. Equ. 10, 213–221 (2000)

    Article  MATH  Google Scholar 

  65. Šverák V., Yan X.: Non-Lipschitz minimizers of smooth uniformly convex variational integrals. Proc. Natl. Acad. Sci. USA 99/24, 15269–15276 (2002)

    ADS  Google Scholar 

  66. Uhlenbeck K.: Regularity for a class of non-linear elliptic systems. Acta Math. 138, 219–240 (1977)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24-29 St. Giles’, Oxford, OX1 3LB, UK

    Jan Kristensen

  2. Dipartimento di Matematica, Università di Parma, Viale Usberti 53/a, 43100, Parma, taly

    Giuseppe Mingione

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  1. Jan Kristensen
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  2. Giuseppe Mingione
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Correspondence to Jan Kristensen.

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Communicated by V. Šverák

To Sir John M. Ball on his 60th birthday.

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Kristensen, J., Mingione, G. Boundary Regularity in Variational Problems. Arch Rational Mech Anal 198, 369–455 (2010). https://doi.org/10.1007/s00205-010-0294-x

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