Higher regularity of the free boundary in the elliptic Signorini problem.(English)Zbl 1329.35362
In this very interesting paper, the authors study the higher regularity of the free boundary for the elliptic Signorini problem. This consists of minimizing the Dirichlet functional \(J(v):=\int_{B_R^+}|\nabla v|^2dx\) over \[\mathcal{K}=\{v\in W^{1,2}(B_R^+) \;: \;v\geq 0\text{ on } B_R'\}, \] where \(B_R^+=B_R\cap\{x_n>0\}\) and \(B_R'=B_R\cap\{x_n=0\}\).
The coincidence set is denoted by \(\Lambda_u=\{x\in B_R' \;: \;u(x)=0\}\) and the free boundary by \(\Gamma_u=\partial_{B_R'}\{x\in B_R' \;: \;u(x)>0\}\). By using a partial hodograph-Legendre transformation, the authors show that the regular part of the free boundary (that is, the set \(\mathcal{R}_u\) of points \(x_0\in \Gamma_u\) such that the blowups of \(u\) at \(x_0\) have the form \(c_n\text{Re}(x_{n-1}+ix_n)^{3/2}\) after a possible rotation of coordinate axes in \(\mathbb{R}^{n-1}\)) is real analytic. It is assumed that all free boundary points are regular and that there exists \(f\in C^{1,\alpha}(\overline{B_R'\cap \{x_{n-1}=0\}})\) with \(f(0)=|\nabla_{x''}f(0)|=0\) such that \[ \Gamma_u=\{(x'',x_{n-1})\in B_R' \;: \;x_{n-1}=f(x'')\}, \]
\[\Lambda_u=\{(x'',x_{n-1})\in B_R' \;: \;x_{n-1}\leq f(x'')\}. \]
The main tool used to improve on the known regularity of the free boundary is a partial hodograph-Legendre transformation. The goal is to straighten the free boundary and then apply the boundary regularity of the solution to the transformed elliptic PDE. The invertibility of the hodograph transform (which is only \(C^{0,1/2}\) regular) is overcome by studying the precise asymptotic behavior of the solutions near regular free boundary points. The difficulty stemming from the fact that the equation satisfied by the Legendre transform is degenerate is tackled by observing that it has a subelliptic structure, which can be viewed as a perturbation of the Baouendi-Grushin operator. The \(L^p\) theory available for that operator and a bootstrapping argument lead to the real analyticity of the free boundary.
The coincidence set is denoted by \(\Lambda_u=\{x\in B_R' \;: \;u(x)=0\}\) and the free boundary by \(\Gamma_u=\partial_{B_R'}\{x\in B_R' \;: \;u(x)>0\}\). By using a partial hodograph-Legendre transformation, the authors show that the regular part of the free boundary (that is, the set \(\mathcal{R}_u\) of points \(x_0\in \Gamma_u\) such that the blowups of \(u\) at \(x_0\) have the form \(c_n\text{Re}(x_{n-1}+ix_n)^{3/2}\) after a possible rotation of coordinate axes in \(\mathbb{R}^{n-1}\)) is real analytic. It is assumed that all free boundary points are regular and that there exists \(f\in C^{1,\alpha}(\overline{B_R'\cap \{x_{n-1}=0\}})\) with \(f(0)=|\nabla_{x''}f(0)|=0\) such that \[ \Gamma_u=\{(x'',x_{n-1})\in B_R' \;: \;x_{n-1}=f(x'')\}, \]
\[\Lambda_u=\{(x'',x_{n-1})\in B_R' \;: \;x_{n-1}\leq f(x'')\}. \]
The main tool used to improve on the known regularity of the free boundary is a partial hodograph-Legendre transformation. The goal is to straighten the free boundary and then apply the boundary regularity of the solution to the transformed elliptic PDE. The invertibility of the hodograph transform (which is only \(C^{0,1/2}\) regular) is overcome by studying the precise asymptotic behavior of the solutions near regular free boundary points. The difficulty stemming from the fact that the equation satisfied by the Legendre transform is degenerate is tackled by observing that it has a subelliptic structure, which can be viewed as a perturbation of the Baouendi-Grushin operator. The \(L^p\) theory available for that operator and a bootstrapping argument lead to the real analyticity of the free boundary.
Reviewer: Mariana Vega Smit (Essen)
Keywords:
Signorini problem;thin obstacle problem;thin free boundary;higher regularity;smoothness;real analyticity;Almgren’s frequency’ partial hodograph-Legendre transform;subelliptic equations’ Baouendi-Grushin operatorCite
References:
| [1] | Almgren, F. J., Almgren’s big regularity paper, (World Scientific Monograph Series in Mathematics, vol. 1 (2000), World Scientific Publishing Co. Inc.: World Scientific Publishing Co. Inc. River Edge, NJ), \(Q\)-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2; With a preface by Jean E. Taylor and Vladimir Scheffer. MR1777737 (2003d:49001) ·Zbl 0985.49001 |
| [2] | Athanasopoulos, I.; Caffarelli, L. A., 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. http://dx.doi.org/10.1007/s10958-005-0496-1 (English, with English and Russian summaries); Eng- lish transl., J. Math. Sci. (N. Y.) 132(3) (2006), 274-284. MR2120184 (2006i:35053) ·Zbl 1108.35038 |
| [3] | Athanasopoulos, I.; Caffarelli, L. A.; Salsa, S., The structure of the free boundary for lower dimensional obstacle problems, Amer. J. Math., 130, 2, 485-498 (2008), MR2405165 (2009g:35345) ·Zbl 1185.35339 |
| [4] | Caffarelli, L. A.; Salsa, S.; Silvestre, L., Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171, 2, 425-461 (2008), MR2367025 (2009g:35347) ·Zbl 1148.35097 |
| [5] | Caffarelli, L. A.; Roquejoffre, J.-M.; Sire, Y., Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12, 5, 1151-1179 (2010), MR2677613 (2011f:49024) ·Zbl 1221.35453 |
| [6] | Caffarelli, L.; Silvestre, L., An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32, 7-9, 1245-1260 (2007), MR2354493 (2009k:35096) ·Zbl 1143.26002 |
| [8] | De Silva, D.; Savin, O., \(C^{2, \alpha}\) regularity of flat free boundaries for the thin one-phase problem, J. Differential Equations, 253, 8, 2420-2459 (2012), MR2950457 ·Zbl 1248.35238 |
| [13] | Folland, G. B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13, 2, 161-207 (1975) ·Zbl 0312.35026 |
| [14] | Garofalo, N.; Petrosyan, A., Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem, Invent. Math., 177, 2, 415-461 (2009), MR2511747 (2010m:35574) ·Zbl 1175.35154 |
| [15] | Heinonen, J., (Lectures on analysis on metric spaces. Lectures on analysis on metric spaces, Universitext (2001), Springer-Verlag: Springer-Verlag New York), MR1800917 (2002c:30028) ·Zbl 0985.46008 |
| [16] | Hewitt, E.; Ross, K. A., (Abstract Harmonic Analysis, Vol. I. Abstract Harmonic Analysis, Vol. I, Second Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115 (1979), Springer-Verlag: Springer-Verlag Berlin), Structure of topological groups, integration theory, group representations ·Zbl 0416.43001 |
| [17] | Kato, K., New idea for proof of analyticity of solutions to analytic nonlinear elliptic equations, SUT J. Math., 32, 2, 157-161 (1996) ·Zbl 0894.35036 |
| [18] | Kinderlehrer, D.; Nirenberg, L., Regularity in free boundary problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 4, 2, 373-391 (1977), MR0440187 (55 #13066) ·Zbl 0352.35023 |
| [19] | Lewy, H., On the coincidence set in variational inequalities, J. Differential Geom., 6, 497-501 (1972), Collection of articles dedicated to S. S. Chern and D. C. Spencer on their sixtieth birthdays. MR0320343 (47 #8882) ·Zbl 0255.31002 |
| [21] | Petrosyan, A.; Shahgholian, H.; Uraltseva, N., Regularity of free boundaries in obstacle-type problems, (Graduate Studies in Mathematics, vol. 136 (2012), American Mathematical Society: American Mathematical Society Providence, RI), MR2962060 ·Zbl 1254.35001 |
| [22] | Reiter, H., Über den Satz von Wiener und lokalkompakte Gruppen, Comment. Math. Helv., 49, 333-364 (1974) ·Zbl 0292.43007 |
| [23] | Sánchez-Calle, A., \(L^p\) estimates for degenerate elliptic equations, Rev. Mat. Iberoam., 4, 1, 177-185 (1988) ·Zbl 0711.35021 |
| [24] | Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, (Murphy, Timothy S., Monographs in Harmonic Analysis, III. Monographs in Harmonic Analysis, III, Princeton Mathematical Series, vol. 43 (1993), Princeton University Press: Princeton University Press Princeton, NJ) ·Zbl 0821.42001 |
| [25] | C. J., Xu, Subelliptic variational problems, Bull. Soc. Math. France, 118, 2, 147-169 (1990) ·Zbl 0717.49004 |
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