On the measure and the structure of the free boundary of the lower dimensional obstacle problem.(English)Zbl 1405.35257

Arch. Ration. Mech. Anal. 230, No. 1, 125-184 (2018); correction ibid. 230, No. 2, 783-784 (2018).
The authors deal with three properties of a solution to the following nonlinear thin obstacle problem. \left\{\begin{aligned} &u(x',0)\geq 0\;\text{ for }(x',0)\in B'_R,\\ &u(x',x_{n+1})=u(x',-x_{n+1})\; \text{ for }x=(x',x_{n+1})\in B_R,\\ &\text{div}(|x_{n+1}|^a\nabla u(x))=0\;\text{ for }x\in B_R\setminus\{(x',0):u(x',0)=0\},\\ &\text{div}(|x_{n+1}|^a\nabla u(x))\leq 0\;\text{ in the sense of distribution in }B_R,\\ &u(x)=g(x)\;\text{ for }x\in\partial B_R, \end{aligned}\right.{(*)_a}
$$g\in\mathcal{A}_R=\{v\in H^1(B_R,|x_{n+1}|^a\mathcal{L}^{n+1}):v(x',x_{n+1})=v(x',-x_{n+1})\text{ and }v(x',0)\geq 0\}$$ such that $$\mathcal{L}^{n+1}$$ is the Lebesgue measure, $$B_R,B'_R$$ are open balls of radius $$R$$ in $$\mathbb R^{n+1}$$ and $$\mathbb R^n$$, respectively, $$\partial B_R$$ represents the boundary of $$B_R$$, and $$|a|<1$$.
The particular case $$(*)_0$$ provides the Signorini problem in elasticity.
Let $$\Lambda(u)=\{(x',0)\in B'_R:u(x',0)=0\}$$ be the coincidence set and we denote by $$\Gamma(u)$$ the free boundary of $$\Lambda(u)$$, i.e., the topological boundary of $$\Lambda(u)$$. The first property states that the free boundary of a solution $$u$$ of $$(*)_a$$ has locally finite $$(n-1)$$-dimensional Minkowski content, i.e., for $$K$$ a relative compact in $$B'_1$$, the Lebesgue measure of $$\mathcal{T}_r(K\cap \Gamma(u))$$ is bounded, up to a multiplicative constant relying on $$K$$, by $$r^2$$, where $$\mathcal{T}_r(E)=\text{dist}(\cdot,E)^{-1}(0,r)\subset \mathbb R^{n+1}$$ such that $$E$$ is a subset of $$\mathbb R^{n}$$ (Theorem 1.1). The second property states that there exists $$(M_i)_{i\in\mathbb N}$$, a family comprised by at most countable $$C^1$$-regular $$(n-1)$$-dimensional submanifolds in $$\mathbb R^{n+1}$$, such that the $$(n-1)$$-dimensional Hausdorff measure of the complement of $$\bigcup_{i\in\mathbb N} M_i$$ in the free boundary of a solution of $$(*)_a$$ is zero (Theorem 1.2). The third property focuses on the blow-up and the frequency of a solution of $$(*)_a$$ at a point of the free boundary (Theorem 1.3). Thanks to a counter-example, the authors declared in [Arch. Ration. Mech. Anal. 230, No. 2, 783–784 (2018; Zbl 1466.35376)] that Proposition 8.2, regarding the classification of homogeneous solutions, is not correct.

MSC:

 35R35 Free boundary problems for PDEs 28A78 Hausdorff and packing measures 35R06 PDEs with measure

Zbl 1466.35376

DLMF
Full Text:

References:

 [1] Almgren, Jr. F.J.: Almgren’s Big Regularity Paper. World Scientific Monograph Series in Mathematics, 1. World Scientific Publishing Co., Inc., River Edge, NJ, 2000 · Zbl 06695663 [2] Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000) · Zbl 0957.49001 [3] Andersson, J, Optimal regularity for the Signorini problem and its free boundary, Invent. Math., 204, 1-82, (2016) · Zbl 1339.35345 [4] Athanasopoulos, I., Caffarelli, L.A.: Optimal regularity of lower dimensional obstacle problems. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310, 49-66, 226 (2004) (translation in J. Math. Sci. (N. Y.), 132 2006, no. 3, 274-284). · Zbl 1108.35038 [5] Athanasopoulos, I; Caffarelli, LA; Salsa, S, The structure of the free boundary for lower dimensional obstacle problems, Am. J. Math., 130, 485-498, (2008) · Zbl 1185.35339 [6] Azzam, J; Tolsa, X, Characterization of $$n$$-rectifiability in terms of Jones’ square function: part II, Geom. Funct. Anal., 25, 1371-1412, (2015) · Zbl 1334.28010 [7] Barrios, B., Figalli, A., Ros-Oton, X.: Global regularity for the free boundary in the obstacle problem for the fractional Laplacian. Am. J. Math. (2017) (in press) · Zbl 1387.35631 [8] Caffarelli, LA, The regularity of free boundaries in higher dimensions, Acta Math., 139, 155-184, (1977) · Zbl 0386.35046 [9] Caffarelli, LA, Further regularity for the Signorini problem, Commun. Partial Differ. Equ., 4, 1067-1075, (1979) · Zbl 0427.35019 [10] Caffarelli, LA, The obstacle problem revisited, J. Fourier Anal. Appl., 4, 383-402, (1998) · Zbl 0928.49030 [11] Caffarelli, LA; Silvestre, L, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32, 1245-1260, (2007) · Zbl 1143.26002 [12] Caffarelli, LA; Salsa, S; Silvestre, L, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171, 425-461, (2008) · Zbl 1148.35097 [13] David, G., Toro, T.: Reifenberg parameterizations for sets with holes. Mem. Am. Math. Soc. 215(1012), vi+102 (2012) · Zbl 1236.28002 [14] Silva, D; Savin, O, $$C^∞$$ regularity of certain thin free boundaries, Indiana Univ. Math. J., 64, 1575-1608, (2015) · Zbl 1331.35397 [15] De Lellis, C., Spadaro, E.: Regularity of area minimizing currents III: blow-up. Ann. Math. (2) 183(2), 577-617 (2016) · Zbl 1345.49053 [16] De Lellis, C., Marchese, A., Spadaro, E., Valtorta, D.: Rectifiability and Upper Minkowski Bounds for Singularities of Harmonic Q-Valued Maps. Comment. Math. Helv. (To appear) · Zbl 1408.49045 [17] Fabes, E; Kenig, C; Serapioni, R, The local regularity of solutions of degenerate elliptic equations, Commun. Partial Differ. Equ., 7, 77-116, (1982) · Zbl 0498.35042 [18] Focardi, M; Gelli, MS; Spadaro, E, Monotonicity formulas for obstacle problems with Lipschitz coefficients, Calc. Var. Partial Differ. Equ., 54, 1547-1573, (2015) · Zbl 1328.35332 [19] Focardi, M; Geraci, F; Spadaro, E, The classical obstacle problem for nonlinear variational energies, Nonlinear Anal., 154, 7187, (2017) · Zbl 06695663 [20] Focardi, M., Geraci, F., Spadaro, E.: Quasi-monotonicity formulas for classical obstacle problems with Sobolev coefficients and applications (To appear) · Zbl 06695663 [21] Focardi, M; Marchese, A; Spadaro, E, Improved estimate of the singular set of dir-minimizing Q-valued functions via an abstract regularity result, J. Funct. Anal., 268, 3290-3325, (2015) · Zbl 1330.49045 [22] Focardi, M; Spadaro, E, An epiperimetric inequality for the fractional obstacle problem, Adv. Differ. Equ., 21, 153-200, (2016) · Zbl 1336.35370 [23] Focardi,M., Spadaro, E.: How a minimal surface leaves a thin obstacle (To appear) · Zbl 1336.35370 [24] Focardi, M., Spadaro, E.: Rectifiability of the free boundary for the fractional obstacle problem (To appear) · Zbl 1336.35370 [25] Frehse, J, Two-dimensional variational problems with thin obstacles, Math. Z., 143, 279-288, (1975) · Zbl 0295.49003 [26] Frehse, J, On Signorini’s problem and variational problems with thin obstacles, Ann. Scuola Norm. Sup. Pisa, 4, 343-362, (1977) · Zbl 0353.49020 [27] Garofalo, N; Petrosyan, A, Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem, Invent. Math., 177, 415-461, (2009) · Zbl 1175.35154 [28] Garofalo, N., Petrosyan, A., Pop, C., Smit Vega Garcia, M.: Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift. Ann. Inst. H. Poincar Anal. Non Linéaire, 34(3), 533-570 (2017) · Zbl 1365.35230 [29] Garofalo, N., Ros-Oton,X.: Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian. arXiv:1704.00097 · Zbl 1328.35332 [30] Geraci, F.: The Classical Obstacle Problem for Nonlinear Variational Energies and Related Problems. Ph.D. thesis, University of Firenze, 2016 · Zbl 0498.35042 [31] Geraci, F.: An Epiperimetric Inequality for the Lower Dimensional Obstacle Problem. arXiv:1709.00996. [32] Hopf, E, Über den funktionalen, insbesondere den analytischen charakter der Lösungen elliptischer differentialgleichungen zweiter ordnung. (German), Math. Z., 34, 194-233, (1932) · JFM 57.0556.01 [33] Jones, P, Rectifiable sets and the traveling salesman problem, Invent. Math., 102, 1-15, (1990) · Zbl 0731.30018 [34] Kinderlehrer, D, The smoothness of the solution of the boundary obstacle problem, J. Math. Pures Appl., 60, 193-212, (1981) · Zbl 0459.35092 [35] Koch, H; Petrosyan, A; Shi, W, Higher regularity of the free boundary in the elliptic Signorini problem, Nonlinear Anal., 126, 3-44, (2015) · Zbl 1329.35362 [36] Krummel, B., Wickramasekera, N.: Fine Properties of Branch Point Singularities: Two-Valued Harmonic Functions. arXiv:1311.0923 · Zbl 1143.26002 [37] Krummel, B., Wickramasekera, N.: Fine Properties of Branch Point Singularities: Dirichlet Energy Minimizing Multi-Valued Functions. arXiv:1711.06222 · Zbl 0928.49030 [38] Monneau, R, On the number of singularities for the obstacle problem in two dimensions, J. Geom. Anal., 13, 359-389, (2003) · Zbl 1041.35093 [39] Naber, A., Valtorta, D.: Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps. Ann. Math. (2) 185(1), 131-227 (2017) · Zbl 1393.58009 [40] Naber, A., Valtorta, D.: The Singular Structure and Regularity of Stationary and Minimizing Varifolds. arXiv:1505.03428 · Zbl 1393.58009 [41] Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V. (eds.): NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.14 of 2016-12-21 · Zbl 1334.28010 [42] Ros-Oton, X; Serra, J, Boundary regularity for fully nonlinear integro-differential equations, Duke Math. J., 165, 2079-2154, (2016) · Zbl 1351.35245 [43] Silvestre, L, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60, 67-112, (2007) · Zbl 1141.49035 [44] Uraltseva, NN, Hölder continuity of gradients of solutions of parabolic equations with boundary conditions of Signorini type, Dokl. Akad. Nauk SSSR, 280, 563-565, (1985) [45] Uraltseva, N.N.: On the regularity of solutions of variational inequalities. (Russian) Uspekhi Mat. Nauk 42(6(258)), 151-174, 248 (1987) [46] Weiss, GS, A homogeneity improvement approach to the obstacle problem, Invent. Math., 138, 23-50, (1999) · Zbl 0940.35102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.