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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.
| | This is a generic technique which is used to show that solutions to some equations are Holder continuous. The technique consist in proving iteratively that the oscillation of the function in a ball of radius $r$ (for elliptic equations) or a parabolic cylinder (for parabolic equations) has a certain decay as $r \to 0$ and then obtain a Holder continuity result from it. |
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| = Statement of the problem = | | = Main scaling assumption = |
| 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
| | In order for the technique to work, we need to start with a solution to a scale invariant equation or class of equations. That is, we have a function $u : B_1 \to \R$ such that, the scaled functions |
| \begin{align*}
| | \[ u_r(x) = \lambda u(rx),\] |
| &Lu \leq 0 \text{ in } \Omega, \ \ (\text{supersolution in the whole domain})\\
| | satisfy some convenient equation for all $r<1$ and $\lambda>0$. The equation can depend on $r$, as long as the assumptions on it do not deteriorate as $r \to 0$. |
| &u \geq \varphi \text{ on } S, \ \ (\text{constrained to remain above the obstacle})\\
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| &Lu = 0 \text{ in } \Omega \setminus (S \cap \{u=\varphi\}). \ \ (\text{a solution wherever it does not touch the obstacle})
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| \end{align*} | |
| Normally, the equation would be accompanied by a boundary condition on $\partial \Omega$.
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| 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
| | = What we need to prove = |
| \begin{align*}
| | == Main lemma == |
| &Lu = 0 \text{ in } \Omega \cap \{x_1>0\}, \ \ (\text{solution on one side})\\
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| &u \geq \varphi \text{ on } \Omega \cap \{x_1=0\}, \ \ (\text{constrained to remain above the obstacle})\\
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| &\frac{\partial u}{\partial x_1} \leq 0 \text{ on } \Omega \cap \{x_1=0\}, \\
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| &\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 $\{x_1=0\}$})
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| \end{align*}
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| = Relationship with the [[fractional Laplacian]] =
| | The main step is to prove that there exists a radius $\rho>0$ and $\delta>0$ so that for any solution $u$ such that $\textrm{osc}_{B_1} u \leq 1$ then $\textrm{osc}_{B_\rho} u \leq 1-\delta$. |
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| 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.
| | Alternatively, for parabolic equations, we would have to prove that if |
| | \[ \textrm{osc}_{B_1 \times [-1,0]} u \leq 1 \ \text{ then } \ \textrm{osc}_{B_\rho \times [-\rho^2,0]} u \leq 1-\delta.\] |
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| 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
| | == How it works == |
| \begin{align*}
| | Iterating a scaled version of the main lemma mentioned above, we get that for all integers $k>0$, |
| & \mathrm{div}(x_1^{1-2s} \nabla u) = 0 \text{ in } \{x_1>0\},\\
| | \[ \textrm{osc}_{B_{\rho^k}} u \leq (1-\delta)^k.\] |
| &u \geq \varphi \text{ on } \{x_1=0\}, \\
| | This implies that $u$ is $C^\alpha$ at the origin for $\alpha = \log(1-\delta)/\log(\rho)$. |
| &\lim_{x_1 \to 0^+} \frac{u(x_1,x')}{x_1^{2s}} \leq 0 \text{ on } \{x_1=0\}, \\
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| &\lim_{x_1 \to 0^+} \frac{u(x_1,x')}{x_1^{2s}} = 0 \text{ on } \{x_1=0\} \cap \{u>\varphi\}.
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| \end{align*}
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| is equivalent after the restriction $\tilde u(x) = u(0,x)$ to the [[obstacle problem for the fractional Laplacian]] in $\R^d$.
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| \begin{align*}
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| & (-\Delta)^s u \leq 0 \text{ in } \R^d,\\
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| & (-\Delta)^s u = 0 \text{ in } \R^d \cap \{u>\varphi\},\\
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| & u \geq \varphi \text{ in } \R^d.
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| \end{align*} | |
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| = Regularity results =
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| == Optimal regularity of the solution ==
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| 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.
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| For the classical thin obstacle problem with $L = \Delta$, the solution is as regular as the obstacle up to $C^{1,1/2}$ <ref name="MR2120184" /> <ref name="MR2367025" />. The proof is significantly harder than for the usual [[obstacle problem]] and requires the use of nontrivial monotonicity formulas.
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| 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 $C^{1,s}$ <ref name="MR2367025" />.
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| == Regularity of the free boundary ==
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| 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 <ref name="MR2405165" /> <ref name="MR2367025" />. On the other hand, the singular points of the free boundary are contained inside a differentiable surface. <ref name="MR2511747" />
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| = Bibliography =
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| == References ==
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| {{reflist|refs=
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| <ref name="MR2120184">{{Citation | last1=Athanasopoulos | first1= I. | last2=Caffarelli | first2= L. A. | title=Optimal regularity of lower dimensional obstacle problems | url=http://dx.doi.org/10.1007/s10958-005-0496-1 | journal=Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) | issn=0373-2703 | year=2004 | volume=310 | pages=49--66, 226 | doi=10.1007/s10958-005-0496-1}}</ref>
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| <ref name="MR2367025">{{Citation | last1=Caffarelli | first1= Luis A. | last2=Salsa | first2= Sandro | last3=Silvestre | first3= Luis | title=Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian | url=http://dx.doi.org/10.1007/s00222-007-0086-6 | journal=Invent. Math. | issn=0020-9910 | year=2008 | volume=171 | pages=425--461 | doi=10.1007/s00222-007-0086-6}}</ref>
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| <ref name="MR2405165">{{Citation | last1=Athanasopoulos | first1= I. | last2=Caffarelli | first2= L. A. | last3=Salsa | first3= S. | title=The structure of the free boundary for lower dimensional obstacle problems | url=http://dx.doi.org/10.1353/ajm.2008.0016 | journal=Amer. J. Math. | issn=0002-9327 | year=2008 | volume=130 | pages=485--498 | doi=10.1353/ajm.2008.0016}}</ref>
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| <ref name="MR2511747">{{Citation | last1=Garofalo | first1= Nicola | last2=Petrosyan | first2= Arshak | title=Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem | url=http://dx.doi.org/10.1007/s00222-009-0188-4 | journal=Invent. Math. | issn=0020-9910 | year=2009 | volume=177 | pages=415--461 | doi=10.1007/s00222-009-0188-4}}</ref>
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| }}
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This is a generic technique which is used to show that solutions to some equations are Holder continuous. The technique consist in proving iteratively that the oscillation of the function in a ball of radius $r$ (for elliptic equations) or a parabolic cylinder (for parabolic equations) has a certain decay as $r \to 0$ and then obtain a Holder continuity result from it.
Main scaling assumption
In order for the technique to work, we need to start with a solution to a scale invariant equation or class of equations. That is, we have a function $u : B_1 \to \R$ such that, the scaled functions
\[ u_r(x) = \lambda u(rx),\]
satisfy some convenient equation for all $r<1$ and $\lambda>0$. The equation can depend on $r$, as long as the assumptions on it do not deteriorate as $r \to 0$.
What we need to prove
Main lemma
The main step is to prove that there exists a radius $\rho>0$ and $\delta>0$ so that for any solution $u$ such that $\textrm{osc}_{B_1} u \leq 1$ then $\textrm{osc}_{B_\rho} u \leq 1-\delta$.
Alternatively, for parabolic equations, we would have to prove that if
\[ \textrm{osc}_{B_1 \times [-1,0]} u \leq 1 \ \text{ then } \ \textrm{osc}_{B_\rho \times [-\rho^2,0]} u \leq 1-\delta.\]
How it works
Iterating a scaled version of the main lemma mentioned above, we get that for all integers $k>0$,
\[ \textrm{osc}_{B_{\rho^k}} u \leq (1-\delta)^k.\]
This implies that $u$ is $C^\alpha$ at the origin for $\alpha = \log(1-\delta)/\log(\rho)$.