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Optimal regularity and free boundary regularity for the Signorini problem


Author: J. Andersson
Original publication: Algebra i Analiz, tom 24 (2012), nomer 3.
Journal: St. Petersburg Math. J. 24 (2013), 371-386
MSC (2010): Primary 49J40, 49N60
DOI: https://doi.org/10.1090/S1061-0022-2013-01244-1
Published electronically: March 21, 2013
MathSciNet review: 3014126
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Abstract | References | Similar Articles | Additional Information

Abstract: A proof of the optimal regularity and free boundary regularity is announced and informally discussed for the Signorini problem for the Lamé system. The result, which is the first of its kind for a system of equations, states that if $ \textbf {u}=(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\mathbb{R}^3)$ minimizes

$\displaystyle J(\textbf {u})=\int _{B_1^+}\vert\nabla \textbf {u}+\nabla ^\bot \textbf {u}\vert^2+\lambda \big (\operatorname {div}(\textbf {u})\big )^2 $

in the convex set

$\displaystyle K=\big \{ \textbf {u} =(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\mathbb{R}^3);\; u^3\ge 0 \ $$\displaystyle \text { on } \ \Pi ,$      
$\displaystyle \textbf {u} =f\in C^\infty (\partial B_1) \ $$\displaystyle \text { on }\ (\partial B_1)^+ \big \},$      

where, say, $ \lambda \ge 0$, then $ \textbf {u}\in C^{1,1/2}(B_{1/2}^+)$. Moreover, the free boundary, given by $ \Gamma _\textbf {u}=\partial \{x;\;u^3(x)=0,\; x_3=0\}\cap B_{1}, $ will be a $ C^{1,\alpha }$-graph close to points where $ \textbf {u}$ is nondegenerate. Historically, the problem is of some interest in that it is the first formulation of a variational inequality. A detailed version of this paper will appear in the near future.

References [Enhancements On Off] (What's this?)

  • 1. J. Andersson, H. Shahgholian, and G. S. Weiss, On the singularities of a free boundary through Fourier series expansion (to appear in Invent. Math.). MR 2667629 (2012f:46166)
  • 2. D. E. Apushkinskaya, H. Shahgholian, and N. N. Ural'tseva, Boundary estimates for solutions of a parabolic free boundary problem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000), 39-55; English transl., J. Math. Sci. (N. Y.) 115 (2003), no. 6, 2720-2730. MR 1810607 (2002b:35205)
  • 3. A. A. Arkhipova and N. N. Ural'tseva, Regularity of solutions of diagonal elliptic systems under convex constraints on the boundary of the domain, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 152 (1986), 5-17; English transl., J. Soviet Math. 40 (1988), no. 5, 591-598. MR 869237 (87m:35107)
  • 4. I. Athanasopoulos and L. A. Caffarelli, Optimal regularity of lower dimensional obstacle problems, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), 49-66; English transl., J. Math. Sci. (N. Y.) 132 (2006), no. 3, 274-284. MR 2120184 (2006i:35053)
  • 5. I. Athanasopoulos, L. A. Caffarelli, and S. Salsa, The structure of the free boundary for lower dimensional obstacle problems, Amer. J. Math. 130 (2008), no. 2, 485-498. MR 2405165 (2009g:35345)
  • 6. I. Athanasopoulos and L. A. Caffarelli, A theorem of real analysis and its application to free boundary problems, Comm. Pure Appl. Math. 38 (1985), no. 5, 499-502. MR 803243 (86j:49062)
  • 7. M. Benedicks, Positive harmonic functions vanishing on the boundary of certain domains in $ {\mathbf R}^n$, Ark. Mat. 18 (1980), no. 1, 53-72. MR 608327 (82h:31004)
  • 8. H. Brézis and D. Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J. 23 (1973/74), 831-844. MR 0361436 (50:13881)
  • 9. L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), no. 6, 771-831. MR 673830 (84m:35097)
  • 10. L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), no. 3-4, 155-184. MR 0454350 (56:12601)
  • 11. -, Compactness methods in free boundary problems, Comm. Partial Differential Equations 5 (1980), no. 4, 427-448. MR 567780 (81e:35121)
  • 12. -, A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are $ C^{1,\alpha }$, Rev. Mat. Iberoamericana 3 (1987), no. 2, 139-162. MR 990856 (90d:35306)
  • 13. -, A Harnack inequality approach to the regularity of free boundaries. III. Existence theory, compactness, and dependence on $ X$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 4, 583-602 (1989). MR 1029856 (91a:35170)
  • 14. -, A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math. 42 (1989), no. 1, 55-78. MR 973745 (90b:35246)
  • 15. L. C. Evans, Quasiconvexity and partial regularity in the calculus of variations, Arch. Rational Mech. Anal. 95 (1986), no. 3, 227-252. MR 853966 (88a:49007)
  • 16. G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 34 (1963), 138-142. MR 0176661 (31:933)
  • 17. S. Friedland and W. K. Hayman, Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions, Comment. Math. Helv. 51 (1976), no. 2, 133-161. MR 0412442 (54:568)
  • 18. M. Fuchs, The smoothness of the free boundary for a class of vector-valued problems, Comm. Partial Differential Equations 14 (1989), no. 8-9, 1027-1041. MR 1017061 (91d:35240)
  • 19. E. Giusti, Direct methods in the calculus of variations, World Sci. Publ. Co., Inc., River Edge, NJ, 2003, pp. viii+403. MR 1962933 (2004g:49003)
  • 20. D. Kinderlehrer, Remarks about Signorini's problem in linear elasticity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 4, 605-645. MR 656002 (83h:73005)
  • 21. H. Lewy and G. Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math. 22 (1969), 153-188. MR 0247551 (40:816)
  • 22. J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493-519. MR 0216344 (35:7178)
  • 23. R. Schumann, Regularity for Signorini's problem in linear elasticity, Manuscripta Math. 63 (1989), no. 3, 255-291. MR 986184 (90c:35098)
  • 24. H. Shahgholian and N. Uraltseva, Regularity properties of a free boundary near contact points with the fixed boundary, Duke Math. J. 116 (2003), no. 1, 1-34. MR 1950478 (2003m:35253)
  • 25. A. Signorini, Sopra alcune questioni di elasticit, Soc. Italiana per il Progr. delle Sci., 1933.

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Additional Information

J. Andersson
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: j.e.andersson@warwick.ac.uk

DOI: https://doi.org/10.1090/S1061-0022-2013-01244-1
Keywords: Free boundary regularity, Signorini problem, optimal regularity, system of equations
Received by editor(s): November 1, 2011
Published electronically: March 21, 2013
Article copyright: © Copyright 2013 American Mathematical Society

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