Optimal regularity and free boundary regularity for the Signorini problem

Andersson, J.

doi:10.1090/S1061-0022-2013-01244-1

Optimal regularity and free boundary regularity for the Signorini problem
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by J. Andersson

St. Petersburg Math. J. 24 (2013), 371-386

DOI: https://doi.org/10.1090/S1061-0022-2013-01244-1

Published electronically: March 21, 2013

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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 \[ J(\textbf {u})=\int _{B_1^+}|\nabla \textbf {u}+\nabla ^\bot \textbf {u}|^2+\lambda \big (\operatorname {div}(\textbf {u})\big )^2 \] in the convex set \begin{align*} K=\big \{ \textbf {u} =(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\mathbb {R}^3);\; u^3\ge 0 \ \text { on } \ \Pi ,& \\ \textbf {u} =f\in C^\infty (\partial B_1) \ \text { on }\ (\partial B_1)^+ \big \},& \end{align*} 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

John Andersson, Henrik Shahgholian, and Georg S. Weiss, Regularity below the $C^2$ threshold for a torsion problem, based on regularity for Hamilton-Jacobi equations, Nonlinear partial differential equations and related topics, Amer. Math. Soc. Transl. Ser. 2, vol. 229, Amer. Math. Soc., Providence, RI, 2010, pp. 1–14. MR 2667629, DOI 10.1090/trans2/229/01
D. E. Apushkinskaya, H. Shahgholian, and N. N. Uraltseva, Boundary estimates for solutions of a parabolic free boundary problem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 31, 39–55, 313 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 115 (2003), no. 6, 2720–2730. MR 1810607, DOI 10.1023/A:1023357416587
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), no. Kraev. Zadachi Mat. Fiz. i Smezhnye Vopr. Teor. Funktsiĭ18, 5–17, 181 (Russian, with English summary); English transl., J. Soviet Math. 40 (1988), no. 5, 591–598. MR 869237, DOI 10.1007/BF01094182
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), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34], 49–66, 226 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 132 (2006), no. 3, 274–284. MR 2120184, DOI 10.1007/s10958-005-0496-1
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, DOI 10.1353/ajm.2008.0016
Ioannis Athanasopoulos and Luis 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, DOI 10.1002/cpa.3160380503
Michael Benedicks, Positive harmonic functions vanishing on the boundary of certain domains in $\textbf {R}^{n}$, Ark. Mat. 18 (1980), no. 1, 53–72. MR 608327, DOI 10.1007/BF02384681
Haïm Brézis and David Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J. 23 (1973/74), 831–844. MR 361436, DOI 10.1512/iumj.1974.23.23069
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, DOI 10.1002/cpa.3160350604
Luis A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), no. 3-4, 155–184. MR 454350, DOI 10.1007/BF02392236
Luis A. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Differential Equations 5 (1980), no. 4, 427–448. MR 567780, DOI 10.1080/0360530800882144
Luis A. Caffarelli, 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, DOI 10.4171/RMI/47
Luis A. Caffarelli, 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
Luis A. Caffarelli, 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, DOI 10.1002/cpa.3160420105
Lawrence C. Evans, Quasiconvexity and partial regularity in the calculus of variations, Arch. Rational Mech. Anal. 95 (1986), no. 3, 227–252. MR 853966, DOI 10.1007/BF00251360
Gaetano Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 34 (1963), 138–142 (Italian). MR 176661
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 412442, DOI 10.1007/BF02568147
Martin 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, DOI 10.1080/03605308908820641
Enrico Giusti, Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 1962933, DOI 10.1142/9789812795557
David 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
Hans Lewy and Guido Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math. 22 (1969), 153–188. MR 247551, DOI 10.1002/cpa.3160220203
J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493–519. MR 216344, DOI 10.1002/cpa.3160200302
Rainer Schumann, Regularity for Signorini’s problem in linear elasticity, Manuscripta Math. 63 (1989), no. 3, 255–291. MR 986184, DOI 10.1007/BF01168371
Henrik Shahgholian and Nina 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, DOI 10.1215/S0012-7094-03-11611-7
A. Signorini, Sopra alcune questioni di elasticit, Soc. Italiana per il Progr. delle Sci., 1933.