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FREE BOUNDARY REGULARITY for ALMOST EVERY SOLUTION to the SIGNORINI PROBLEM 1. Introduction the Signorini Problem (Also Known As

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FREE BOUNDARY REGULARITY for ALMOST EVERY SOLUTION to the SIGNORINI PROBLEM 1. Introduction the Signorini Problem (Also Known As FREE BOUNDARY REGULARITY FOR ALMOST EVERY SOLUTION TO THE SIGNORINI PROBLEM XAVIER FERNANDEZ-REAL´ AND XAVIER ROS-OTON Abstract. We investigate the regularity of the free boundary for the Signorini problem in n+1 1 R . It is known that regular points are (n − 1)-dimensional and C . However, even for C1 obstacles ', the set of non-regular (or degenerate) points could be very large | e.g. with infinite Hn−1 measure. The only two assumptions under which a nice structure result for degenerate points has been established are: when ' is analytic, and when ∆' < 0. However, even in these cases, the set of degenerate points is in general (n − 1)-dimensional | as large as the set of regular points. In this work, we show for the first time that, \usually", the set of degenerate points is small. Namely, we prove that, given any C1 obstacle, for almost every solution the non-regular part of the free boundary is at most (n − 2)-dimensional. This is the first result in this direction for the Signorini problem. Furthermore, we prove analogous results for the obstacle problem for the fractional Laplacian (−∆)s, and for the parabolic Signorini problem. In the parabolic Signorini problem, our main result establishes that the non-regular part of the free boundary is (n − 1 − α◦)-dimensional for almost all times t, for some α◦ > 0. Finally, we construct some new examples of free boundaries with degenerate points. 1. Introduction The Signorini problem (also known as the thin or boundary obstacle problem) is a classical free boundary problem that was originally studied by Antonio Signorini in connection with linear elasticity [Sig33, Sig59, KO88]. The problem gained further attention in the seventies due to its connection to mechanics, biology, and even finance | see [DL76], [Mer76, CT04], and [Ros18] |, and since then it has been widely studied in the mathematical community; see [Caf79, AC04, CS07, ACS08, GP09, PSU12, KPS15, KRS19, DGPT17, FS18, CSV19, JN17, FJ18, Shi18] and references therein. The main goal of this work is to better understand the size and structure of the non-regular part of the free boundary for such problem. In particular, our goal is to prove for the first time that, for almost every solution (see Re- mark 1.2), the set of non-regular points is small. As explained in detail below, this is completely new even when the obstacle ' is analytic or when it satisfies ∆' < 0. 0 n + 1.1. The Signorini problem. Let us denote x = (x ; xn+1) 2 R × R and B1 = B1 \ fxn+1 > 1 + 0g. We say that u 2 H (B1 ) is a solution to the Signorini problem with a smooth obstacle ' 2010 Mathematics Subject Classification. 35R35. Key words and phrases. Thin obstacle problem; Signorini problem; free boundary; fractional obstacle problem. This work has received funding from the European Research Council (ERC) under the Grant Agreements No 721675 and No 801867. In addition, X.R. was supported by the Swiss National Science Foundation and by MINECO grant MTM2017-84214-C2-1-P.. 1 2 XAVIER FERNANDEZ-REAL´ AND XAVIER ROS-OTON 0 defined on B1 := B1 \ fxn+1 = 0g if u solves ∆u = 0 in B+ 1 (1.1) min{−@xn+1 u; u − 'g = 0 on B1 \ fxn+1 = 0g; 0 in the weak sense, for some boundary data g 2 C (@B1 \fxn+1 ≥ 0g). Solutions to the Signorini problem are minimizers of the Dirichlet energy Z jruj2; + B1 under the constrain u ≥ ' on fxn+1 = 0g, and with boundary conditions u = g on @B1 \fxn+1 > 0g. Problem (1.1) is a free boundary problem, i.e., the unknowns of the problem are the solution itself, and the contact set 0 n 0 0 n+1 Λ(u) := x 2 R : u(x ; 0) = '(x ) × f0g ⊂ R ; n whose topological boundary in the relative topology of R , which we denote Γ(u) = @Λ(u) = 0 n 0 0 @fx 2 R : u(x ; 0) = '(x )g × f0g, is known as the free boundary. 1; 1 Solutions to (1.1) are known to be C 2 (see [AC04]), and this is optimal. 1.2. The free boundary. While the optimal regularity of the solution is already known, the structure and regularity of the free boundary is still not completely understood. The main known results are the following. The free boundary can be divided into two sets, Γ(u) = Reg(u) [ Deg(u); the set of regular points, ( ) 0 3=2 3=2 Reg(u) := x = (x ; 0) 2 Γ(u) : 0 < cr ≤ sup (u − ') ≤ Cr ; 8r 2 (0; r◦) ; 0 0 Br(x ) and the set of non-regular points or degenerate points ( ) 0 2 Deg(u) := x = (x ; 0) 2 Γ(u) : 0 ≤ sup (u − ') ≤ Cr ; 8r 2 (0; r◦) ; (1.2) 0 0 Br(x ) (see [ACS08]). Alternatively, each of the subsets can be defined according to the order of the 3 blow-up at that point. Namely, the set of regular points are those whose blow-up is of order 2 , and the set of degenerate points are those whose blow-up is of order κ for some κ 2 [2; 1]. Let us denote Γκ the set of free boundary points of order κ. That is, those points whose blow-up is homogeneous of order κ (we will be more precise about it later on, in Section 2; the definition of Γ1 is slightly different). Then, it is well known that the free boundary can be divided as Γ(u) = Γ3=2 [ Γeven [ Γodd [ Γhalf [ Γ∗ [ Γ1; (1.3) where: • Γ3=2 = Reg(u) is the set of regular points. They are an open (n − 1)-dimensional subset of Γ(u), and it is C1 (see [ACS08, KPS15, DS16]). S • Γeven = m≥1 Γ2m(u) denotes the set of points whose blow-ups have even homogeneity. Equivalently, they can also be characterised as those points of the free boundary where the contact set has zero density, and they are often called singular points. They are contained in the countable union of C1 (n − 1)-dimensional manifolds; see [GP09]. FREE BOUNDARY REGULARITY FOR A.E. SOLUTION TO THE SIGNORINI PROBLEM 3 S • Γodd = m≥1 Γ2m+1(u) is, a priori, also an at most (n − 1)-dimensional subset of the free boundary and it is (n − 1)-rectifiable (see [FS18, KW13, FS19, FRS19]), although it is not actually known whether it exists. S 7 11 • Γhalf = m≥1 Γ2m+3=2(u) corresponds to those points with blow-up of order 2 , 2 , etc. They are much less understood than regular points. The set Γhalf is an (n − 1)-dimensional subset of the free boundary and it is (n − 1)-rectifiable (see [FS18, KW13, FS19]). 1 • Γ∗ is the set of all points with homogeneities κ 2 (2; 1), with κ2 = N and κ2 = 2N − 2 . This set has Hausdorff dimension at most n − 2, so it is always small, see [FS18, KW13, FS19]. • Γ1 is the set of points with infinite order (namely, those points at which u − ' vanishes at infinite order, see (2.11)). For general C1 obstacles it could be a huge set, even a fractal set of infinite perimeter with dimension exceeding n − 1. When ' is analytic, instead, Γ1 is empty. Overall, we see that, for general C1 obstacles, the free boundary could be really irregular. The only two assumptions under which a better regularity is known are: 0 ◦ ∆' < 0 on B1 and u = 0 on @B1 \ fxn+1 > 0g. In this case, Γ(u) = Γ3=2 [ Γ2 and the set of degenerate points is locally contained in a C1 manifold; see [BFR18]. ◦ ' is analytic. In this case, Γ1 = ? and Γ is (n − 1)-rectifiable, in the sense that it is contained in a countable union of C1 manifolds, up to a set of zero Hn−1-measure, see [FS18, KW13]. The goal of this paper is to show that, actually, for most solutions, all the sets Γeven,Γodd, Γhalf , and Γ1 are small, namely, of dimension at most n − 2. This is new even in case that ' is analytic and ∆' < 0. 1.3. Our results. We will prove here that, even if degenerate points could potentially constitute a large part of the free boundary (of the same dimension as the regular part, or even higher), they are not common. More precisely, for almost every obstacle (or for almost every boundary datum), the set of degenerate points is small. This is the first result in this direction for the Signorini problem, even for zero obstacle. 0 Let gλ 2 C (@B1) for λ 2 [0; 1], and let us denote by uλ the family of solutions to (1.1), satisfying uλ = gλ; on @B1 \ fxn+1 > 0g; (1.4) with gλ satisfying gλ+" ≥ gλ; on @B1 \ fxn+1 > 0g 1 (1.5) gλ+" ≥ gλ + " on @B1 \ fxn+1 ≥ 2 g; for all λ 2 [0; 1), " 2 (0; 1 − λ). Our main result reads as follows. Theorem 1.1. Let uλ be any family of solutions of (1.1) satisfying (1.4)-(1.5), for some obstacle ' 2 C1. Then, we have dimH Deg(uλ) ≤ n − 2 for a.e. λ 2 [0; 1]; where Deg(uλ) is defined by (1.2). 1 In other words, for a.e. λ 2 [0; 1], the free boundary Γ(uλ) is a C (n − 1)-dimensional manifold, up to a closed subset of Hausdorff dimension n − 2. This result is completely new even for analytic obstacles, or for ' = 0. No result of this type was known for the Signorini problem.
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