An epiperimetric inequality for the thin obstacle problem (Q5962997)

In this paper, the authors prove an epiperimetric inequality for the thin obstacle problem, which provides the means to study the rate of convergence of the rescaled solutions to their limits, and consequently the regularity properties of the free boundary.NEWLINENEWLINETo describe the Signorini problem, one considers the minimizers of the Dirichlet energy NEWLINE\[NEWLINE \mathcal{E}(u):= \int_{B_1^+}|\nabla u|^2 NEWLINE\]NEWLINE among functions in \(\mathcal{A}_w:=\{ u \in H^1(B_1^+) \;: \;u \geq 0 \text{ on } B_1', \;u=w \text{ on } (\partial B_1)^+\}\), where \(B_1^+=B_1\cap \{x_n >0\}\) and \(B_1'=\partial (B_1^+)\cap \{x_n=0\}\). The function \(u\) is extended by even symmetry to \(B_1\).NEWLINENEWLINEThe coincidence set is defined as \(\Lambda(u):=\{(\hat{ x},0)\in B_1' : u(\hat{x},0)=0\}\), and its topological boundary in the relative topology of \(B_1'\) is the free boundary, \(\Gamma(u)\).NEWLINENEWLINEBy analysis of a frequency function, points in the free boundary can be classified: given \(x_0\in \Gamma(u)\), Almgren's frequency NEWLINE\[NEWLINE (0,1-|x_0|) \ni r \mapsto N^{x_0}(r,u):=\frac{ r\int_{B_r(x_0)}|\nabla u|^2dx}{\int_{\partial D_r(x_0)}u^2 d\mathcal{H}^{n-1}} NEWLINE\]NEWLINE is nondecreasing and \(N^{x_0}(0+,u)\in [3/2, \infty)\).NEWLINENEWLINEFollowing works by \textit{G. S. Weiss} [Commun. Partial Differ. Equations 23, No. 3--4, 439--455 (1998; Zbl 0897.35017); Invent. Math. 138, No. 1, 23--50 (1999; Zbl 0940.35102)], \textit{N. Garofalo} and \textit{A. Petrosyan} [Invent. Math. 177, No. 2, 415--461 (2009; Zbl 1175.35154)] and \textit{N. Garofalo}, \textit{A. Petrosyan} and \textit{M. Smit Vega Garcia}, who studied this problem in the context of variable Lipschitz coefficients in [``An epiperimetric inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients'', J. Math. Pures Appl. (to appear), \url{arXiv:1501.06498}], a family of Weiss monotonicity formulas can be introduced, which leads to regularity properties of the free boundary. In the context of the problem at hand, given \(x_0\in \Gamma(u)\), the family of boundary adjusted energies is defined as NEWLINE\[NEWLINE W_{\lambda}^{x_0}(r,u):=\frac{1}{r^{n+1}}\int_{B_r(x_0)}|\nabla u|^2dx - \frac{\lambda}{r^{n+2}}\int_{\partial B_r(x_0)}u^2 d\mathcal{H}^{n-1}. NEWLINE\]NEWLINE The authors are able to prove that, analogously to the case of the classical obstacle problem, there are classes of points \(x_0 \in \Gamma(u)\) where the monotonicity of \(W_{\lambda}^{x_0}\) can be explicitly quantified, i.e., there exist \(\gamma ,r_0, C>0\) such that NEWLINE\[NEWLINE W_{\lambda}^{x_0}(r,u)\leq Cr^{\gamma} \text{ for all } r\in (0,r_0), NEWLINE\]NEWLINE which is the key estimate to obtain the regularity of the free boundary near points of least frequency.NEWLINENEWLINEThe main focus of this paper is how to prove the above estimate, which is done by proving a Weiss type epiperimetric inequality. To illustrate the statement in the case of lowest frequency, \(\lambda=3/2\), the epiperimetric inequality states that there exist dimensional constants \(\kappa, \delta>0\) such that if \(c\in H^1(B_1)\) is a 3/2-homogeneous function with positive trace on \(B_1'\) that is \(\delta\)-close to the cone of \(3/2\)-homogeneous global solutions, then there exists another function \(v\) with the same boundary values of \(c\) such that NEWLINE\[NEWLINE W_{3/2}^0(1,v)\leq (1-\kappa)W_{3/2}^0(1,c). NEWLINE\]NEWLINE The main result of this paper is the proof of this result for points of least frequency \(3/2\) and for isolated points of the free boundary with frequency \(2m\), \(m\in \mathbb{N}\setminus\{0\}\). This is achieved by a contradiction argument based on two variational principles.NEWLINENEWLINEAs a consequence of this result, the authors prove the \(C^{1,\alpha}\) regularity of the part of the free boundary with least frequency.