Thin obstacle problem

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The thin obstacle problem refers to a classical free boundary problem which is a variation of the obstacle problem in which the obstacle provides a constraint on a surface of co-dimension one only.

Statement of the problem

Given an elliptic operator $L$ (for example $L = \Delta$), a surface $S \subset \Omega$ and a smooth function $\varphi:S \to \R$, a solution to the thin obstacle problem is a function $u: \Omega \to \R$ such that \begin{align*} &Lu \leq 0 \text{ in } \Omega, \ \ (\text{supersolution in the whole domain})\\ &u \geq \varphi \text{ on } S, \ \ (\text{constrained to remain above the obstacle})\\ &Lu = 0 \text{ in } \Omega \setminus (S \cap \{u=\varphi\}). \ \ (\text{a solution wherever it does not touch the obstacle}) \end{align*} Normally, the equation would be accompanied by a boundary condition on $\partial \Omega$.

In the case that $\Omega$ is a symmetric domain along the plane $S=\{x_1=0\}$, we may concentrate our study on functions $u$ which are even respect to $x_1$. In that case, the problem can be reformulated as \begin{align*} &Lu = 0 \text{ in } \Omega \cap \{x_1>0\}, \ \ (\text{solution on one side})\\ &u \geq \varphi \text{ on } \Omega \cap \{x_1=0\}, \ \ (\text{constrained to remain above the obstacle})\\ &\frac{\partial u}{\partial x_1} \leq 0 \text{ on } \Omega \cap \{x_1=0\}, \\ &\frac{\partial u}{\partial x_1} = 0 \text{ on } \Omega \cap \{x_1=0\} \cap \{u>\varphi\}. \ \ (\text{the Neumann condition would make the even reflection a solution across $\{x_1=0\}$}) \end{align*}

Relationship with the fractional Laplacian

If we study solutions of the thin obstacle problem in the full space $\Omega = \R^{d+1}$, which are even in $x_1$, and have a sufficiently fast decay at infinity, then the restriction to $\{y_1=0\}$: $\tilde u(x_2,\dots,x_{d+1}) = u(0,x_2,\dots,x_{d+1})$ is a solution to the obstacle problem for the fractional Laplacian in the case $s=1/2$ (half Laplacian). This is a simple consequence of the fact that the Dirichlet to Neumann map for the Laplace equation in the upper half space coincides with the square root of the Laplacian.

For other powers of the Laplacian, we can achieve a similar construction replacing $L = \Delta$ by a degenerate elliptic operator. This is a consequence of the extension technique. The thin obstacle problem \begin{align*} & \mathrm{div}(x_1^{1-2s} \nabla u) = 0 \text{ in } \{x_1>0\},\\ &u \geq \varphi \text{ on } \{x_1=0\}, \\ &\lim_{x_1 \to 0^+} \frac{u(x_1,x')}{x_1^{2s}} \leq 0 \text{ on } \{x_1=0\}, \\ &\lim_{x_1 \to 0^+} \frac{u(x_1,x')}{x_1^{2s}} = 0 \text{ on } \{x_1=0\} \cap \{u>\varphi\}. \end{align*} is equivalent after the restriction $\tilde u(x) = u(0,x)$ to the obstacle problem for the fractional Laplacian in $\R^d$. \begin{align*} & (-\Delta)^s u \leq 0 \text{ in } \R^d,\\ & (-\Delta)^s u = 0 \text{ in } \R^d \cap \{u>\varphi\},\\ & u \geq \varphi \text{ in } \R^d. \end{align*}

Regularity results

Optimal regularity of the solution

The solution will always have a jump on its derivatives across the surface $S$. However, it is more regular if we restricted to $S$, or if we focus our attention to one side of $S$ only. This is how we understand the optimal regularity of the solution.

For the classical thin obstacle problem with $L = \Delta$, the solution is as regular as the obstacle up to $C^{1,1/2}$ [1] [2]. The proof is significantly harder than for the usual obstacle problem and requires the use of nontrivial monotonicity formulas.

For degenerate equations of the form $L = \mathrm{div}(x_1^{1-2s} \nabla \cdot)$, the solution is as regular as the obstacle up to $C^{1,s}$ [2].

Regularity of the free boundary

The study of free boundary regularity is similar to the classical obstacle problem. The free boundary is $C^{1,\alpha}$ smooth, for some $\alpha>0$, wherever the free boundary satisfies some generic regularity conditions [3] [2]. On the other hand, the singular points of the free boundary are contained inside a differentiable surface. [4]