Optimal regularity for the thin obstacle problem with $C^{0,\alpha }$ coefficients

Abstract

In this article we study solutions to the (interior) thin obstacle problem under low regularity assumptions on the coefficients, the obstacle and the underlying manifold. Combining the linearization method of Andersson (Invent Math 204(1):1–82, 2016. doi:10.1007/s00222-015-0608-6) and the epiperimetric inequality from Focardi and Spadaro (Adv Differ Equ 21(1–2):153–200, 2016), Garofalo, Petrosyan and Smit Vega Garcia (J Math Pures Appl 105(6):745–787, 2016. doi:10.1016/j.matpur.2015.11.013), we prove the optimal $C^{1,\min \{\alpha ,1/2\}}$ regularity of solutions in the presence of $C^{0,\alpha }$ coefficients $a^{ij}$ and $C^{1,\alpha }$ obstacles $\phi$. Moreover we investigate the regularity of the regular free boundary and show that it has the structure of a $C^{1,\gamma }$ manifold for some $\gamma \in (0,1)$.

Appendix

Last but not least, in this section we present a sketch of the proof of Lemma 2 following the ideas of Andersson (c.f. [2], Lemma 6.2).

Proof of Lemma 2

We argue in two steps and first discuss the behavior with respect to the tangential variables $x_1,\ldots ,x_{n-1}$, which then, in a second step, allows us to reduce the problem to a problem in the two variables $x_n, x_{n+1}$ only.

Step 1: Tangential variables. We note that the countable set of linearly independent eigenfunctions $\{l_k\}_{k\in \mathbb {N}}$ of the restriction of (8) onto the sphere $\partial B_1 =: S^n$ forms an orthonormal basis for $L^2(S^{n})$. As a consequence any solution w to (8) can be decomposed into

$$\begin{aligned} w(r,\theta ) = \sum \limits _{k=1}^{\infty } \alpha _k(r) l_k(\theta ), \end{aligned}$$

where $\theta \in S^n, r\in \mathbb {R}_+$. Orthogonality, implies that $\alpha _k(r) = \tilde{\alpha }_k r^{\kappa (k)}$ with $\tilde{\alpha }_k\in \mathbb {R}$ being independent of r and $\kappa (k)\in \mathbb {R}$. Thus, without loss of generality, we may assume that

$$\begin{aligned} w(x) = \sum \limits _{k=1}^{\infty } \tilde{a}_k q_k(x), \end{aligned}$$

where $q_k$ are homogeneous functions solving (8) with some still unknown homogeneity which we seek to determine in the sequel.

Let hence q be such a function with homogeneity $\kappa$. We differentiate q k-times in an arbitrary combination of the directions $e_1,\ldots ,e_{n-1}$ and denote this by $D^{\alpha } q$ for some multi-index $\alpha \in \mathbb {N}^{n+1}$ with $|\alpha |=k$, $\alpha _n =0=\alpha _{n+1}$. Thus, choosing $k>\kappa$, we infer that $D^{\alpha } q$ is homogeneous of order $\kappa -k<0$. By a difference quotient argument, regularity theory for the Dirichlet and Neumann problems and the regularity of our domain, we obtain that $D^{\alpha }q\in W^{1,2}(B_1)\cap L^{\infty }(B_1)$. Combining the boundedness with the negative homogeneity then implies that $D^{\alpha }q = 0$.

Integrating this and using that $\alpha \in \mathbb {N}^{n+1}$ (with $|\alpha |=k$ and $\alpha _n =0=\alpha _{n+1}$) was arbitrary in its first $(n-1)$ components, yields that $\kappa$ can be expressed as the sum of an integer and the homogeneity of an arbitrary two-dimensional solution of (8) in the variables $x_n, x_{n+1}$. Hence it remains to investigate the two-dimensional problem, in order to determine the full set of possible inhomogeneities.

Step 2: The two-dimensional problem. A simple calculation in 2d polar coordinates shows that all solutions to (8) are of the form

$$\begin{aligned} v(x_n,x_{n+1})&= x_{n+1},\\ v(x_n,x_{n+1})&= \hbox {Re}(x_n + i|x_n|)^{\kappa } \text{ with } \kappa \in \left\{ \frac{2n+1}{2}: n \in \mathbb {N}\right\} ,\\ v(x_n,x_{n+1})&= \hbox {Im}(x_n + i x_n)^{\kappa } \text{ with } \kappa \in \mathbb {N}. \end{aligned}$$

Combining this with the previous discussion concludes the proof. $\square$

Remark 6

Going through the proof carefully, shows that it reveals more information than stated in Lemma 2: The proof allows us to conclude that the only solution of homogeneity 1 / 2 is given by $\hbox {Re}(x_n + i|x_n|)^{1/2}$, it shows that the solutions of homogeneity one are only linear polynomials and it reveals the possible structure of solutions of homogeneity 3 / 2 as being of the form

$$\begin{aligned} v(x) = \hbox {Re}(x_n + i|x_n|)^{1/2}\sum \limits _{j=1}^{n-1}a_j x_j + b \hbox {Re}(x_n + i|x_n|)^{3/2}. \end{aligned}$$

We use this more detailed knowledge in Step 3b in the proof of Lemma 12.