Boundary Harnack inequality and Fractional obstacle problem: Difference between pages

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The '''Boundary Harnack Inequality''' is a name given to two related statements for nonnegative functions $u$ which are solutions of elliptic equations.  
The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$ if $\Omega$ is, say, all of $\mathbb{R}^n$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$.  


The first result, also known as '''Carleson's estimate''', says that for non-negative solutions, their values in a neighborhood of the (suitably smooth) boundary are bounded in terms of the value at some interior point. Let $u$ be a non-negative solution of an elliptic equation $Lu = 0$ on some domain $\Omega \subset \mathbb{R}^n$, such that $u = 0$ on $B_r(x_0) \cap \partial \Omega$, where $x_0$ lies on the boundary $\partial \Omega$, and $x'$ is some other point lying within $B_\frac{r}{2}(x_0) \cap \Omega$. Then, inside $B_\frac{r}{2}(x_0) \cap \Omega$, there exists a constant $M > 0$ such that $u(x) \leq M u(x')$.  
Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the [[Almgren frequency formula]].<ref name="S"/><ref name="CSS"/><ref name="CS"/>


The second result, also known as the '''boundary comparison estimate''', says that two non-negative solutions which are zero on some portion of the boundary, have a Holder continuous ratio with respect to each other in some neighborhood of the boundary. That is, let $ Lu = Lv = 0$ inside some domain $\Omega$ with smooth boundary, with $u,v \geq 0$, and $u = v = 0$ along $B_r(x_0) \cap \partial \Omega$ for some $x_0 \in \partial \Omega$. Then the ratio $\frac{u}{v}$ lies in the Holder class $C^\alpha (B_\frac{r}{2}(x_0))$.
==References==
 
{{reflist|refs=
==Further reading==
<ref name="CSS">{{Citation | last1=Caffarelli | first1=Luis | 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 | doi=10.1007/s00222-007-0086-6 | year=2008 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=171 | issue=2 | pages=425–461}}</ref>
*{{Citation | last1=Caffarelli | first1=Luis | last2=Salsa | first2=Sandro | title=A geometric approach to free boundary problems | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-3784-9 | year=2005 | volume=68}}, Chapter 11
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=An extension problem related to the fractional Laplacian | url=http://dx.doi.org.ezproxy.lib.utexas.edu/10.1080/03605300600987306 | doi=10.1080/03605300600987306 | year=2007 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=32 | issue=7 | pages=1245–1260}}</ref>
 
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref>
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Revision as of 01:16, 2 June 2011

The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$ if $\Omega$ is, say, all of $\mathbb{R}^n$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$.

Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the Almgren frequency formula.[1][2][3]

References

  1. ↑ Silvestre, Luis (2007), "Regularity of the obstacle problem for a fractional power of the Laplace operator", Communications on Pure and Applied Mathematics 60 (1): 67–112, doi:10.1002/cpa.20153, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20153 
  2. ↑ Caffarelli, Luis; Salsa, Sandro; Silvestre, Luis (2008), "Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian", Inventiones Mathematicae 171 (2): 425–461, doi:10.1007/s00222-007-0086-6, ISSN 0020-9910, http://dx.doi.org/10.1007/s00222-007-0086-6 
  3. ↑ Caffarelli, Luis; Silvestre, Luis (2007), "An extension problem related to the fractional Laplacian", Communications in Partial Differential Equations 32 (7): 1245–1260, doi:10.1080/03605300600987306, ISSN 0360-5302, http://dx.doi.org.ezproxy.lib.utexas.edu/10.1080/03605300600987306 
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