Nonlocal minimal surfaces - Revision history

imported>Luis at 20:24, 25 February 2012

2012-02-25T20:24:52Z

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	It can be checked easily that this agrees (save for a factor of $2$) with norm of the characteristic function $\chi_E$ in the homogenous Sobolev space $\dot{H}^{\frac{s}{2}}$. The dimensional constant $C_{n,s}$ blows up as $s \to 1^-$, in which case (at least when the boundary of $E$ is smooth enough) one can check that $J_s(E)$ converges to the perimeter of $E$.		It can be checked easily that this agrees (save for a factor of $2$) with norm of the characteristic function $\chi_E$ in the homogenous Sobolev space $\dot{H}^{\frac{s}{2}}$. The dimensional constant $C_{n,s}$ blows up as $s \to 1^-$, in which case (at least when the boundary of $E$ is smooth enough) one can check that $J_s(E)$ converges to the perimeter of $E$.

	Classically, [[minimal surfaces]] (or generally [[surfaces of constant mean curvature]] ) arise in physical situations where one has two phases interacting (eg. water-air, water-ice ) and the energy of interaction is proportional to the area of the interface, which is due to the interaction between particles/agents in both phases being negligible when they are far apart.		Classically, [[minimal surfaces]] (or generally surfaces of constant mean curvature) arise in physical situations where one has two phases interacting (eg. water-air, water-ice ) and the energy of interaction is proportional to the area of the interface, which is due to the interaction between particles/agents in both phases being negligible when they are far apart.

	Nonlocal minimal surfaces then describe physical phenomena where the interaction potential does not decay fast enough as particles get farther and farther apart, so that two particles on different phases contribute a non-trivial amount to the total interaction energy even if they are away from the interface. In particular, one may consider much more general energy functionals corresponding to different interaction potentials		Nonlocal minimal surfaces then describe physical phenomena where the interaction potential does not decay fast enough as particles get farther and farther apart, so that two particles on different phases contribute a non-trivial amount to the total interaction energy even if they are away from the interface. In particular, one may consider much more general energy functionals corresponding to different interaction potentials

imported>Luis at 06:11, 21 June 2011

2011-06-21T06:11:15Z

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	<div style="background:#DDEEFF;">		<div style="background:#DDEEFF;">
	<blockquote>		<blockquote>
	Open question: The dimensional estimate for the singular set $n-2$ is not expected to be sharp (at least when $s$ is not too small). It has been shown that if $s$ is close to $1$ then the estimate becomes $n-7$, which is the same (sharp) bound known for classical minimal surfaces. However, for nonlocal minimal surfaces it is not known what the sharp bound is. Even more, there are no known explicit examples of singular cones which are nonlocal minimal surfaces.		Open question: The dimensional estimate for the singular set $n-2$ is not expected to be sharp (at least when $s$ is not too small). It has been shown that if $s$ is close to $1$ then the estimate becomes $n-8$, which is the same (sharp) bound known for classical minimal surfaces. However, for nonlocal minimal surfaces it is not known what the sharp bound is. Even more, there are no known explicit examples of singular cones which are nonlocal minimal surfaces. Even in two dimensions, the possibility of a singularity is not ruled out.
	</blockquote>		</blockquote>
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imported>Nestor at 06:22, 2 June 2011

2011-06-02T06:22:27Z

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	[[Category:Minimal surfaces]] [[Category:Quasilinear equations]]		[[Category:Minimal surfaces]] [[Category:Quasilinear equations]] [[Category:Open problems]]

imported>Nestor at 06:10, 2 June 2011

2011-06-02T06:10:44Z

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	<ref name="CRS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Roquejoffre \| first2=Jean Michel \|last3= Savin \| first3= Ovidiu \| title= Nonlocal Minimal Surfaces \| url=http://onlinelibrary.wiley.com/doi/10.1002/cpa.20331/abstract \| doi=10.1002/cpa.20331 \| year=2010 \| journal=[[Communications on Pure and Applied Mathematics]] \| issn=0003-486X \| volume=63 \| issue=9 \| pages=1111–1144}}</ref>		<ref name="CRS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Roquejoffre \| first2=Jean Michel \|last3= Savin \| first3= Ovidiu \| title= Nonlocal Minimal Surfaces \| url=http://onlinelibrary.wiley.com/doi/10.1002/cpa.20331/abstract \| doi=10.1002/cpa.20331 \| year=2010 \| journal=[[Communications on Pure and Applied Mathematics]] \| issn=0003-486X \| volume=63 \| issue=9 \| pages=1111–1144}}</ref>
	}}		}}


			[[Category:Minimal surfaces]] [[Category:Quasilinear equations]]

imported>Nestor at 06:03, 2 June 2011

2011-06-02T06:03:40Z

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	== References ==		== References ==
	{{reflist\|refs=		{{reflist\|refs=
	<ref name="CRS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Roquejoffre \| first2=Jean Michel \|last3= Savin \| first3= Ovidiu \| title=~~Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation~~ \| url=http://dx.doi~~.org~~/10.~~4007~~/~~annals~~.~~2010.171.1903~~ \| doi=10.~~4007~~/~~annals.2010.171~~.~~1903~~ \| year=2010 \| journal=[[~~Annals of Mathematics\|Annals of~~ Mathematics~~. Second Series~~]] \| issn=0003-486X \| volume=~~171~~ \| issue=3 \| pages=~~1903–1930~~}}</ref>		<ref name="CRS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Roquejoffre \| first2=Jean Michel \|last3= Savin \| first3= Ovidiu \| title= Nonlocal Minimal Surfaces \| url=http://onlinelibrary.wiley.com/doi/10.1002/cpa.20331/abstract \| doi=10.1002/cpa.20331 \| year=2010 \| journal=[[Communications on Pure and Applied Mathematics]] \| issn=0003-486X \| volume=63 \| issue=9 \| pages=1111–1144}}</ref>
	}}		}}

imported>Nestor at 05:52, 2 June 2011

2011-06-02T05:52:28Z

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	== The Caffarelli-Roquejoffre-Savin Regularity Theorem==		== The Caffarelli-Roquejoffre-Savin Regularity Theorem==

	A regularity theory for solutions of the above variational problem has been developed recently < ref name = "CRS"/>. It draws inspiration from geometric measure theory, as developed by De Giorgi, Federer, Almgren and many others. In particular, the $C^{1,\alpha}$ regularity of minimal surfaces near flat points is proved ("improvement of flatness") as well as a first estimate on the dimensional set. The main result is as follows:		A regularity theory for solutions of the above variational problem has been developed recently <ref name = "CRS"/>. It draws inspiration from geometric measure theory, as developed by De Giorgi, Federer, Almgren and many others. In particular, the $C^{1,\alpha}$ regularity of minimal surfaces near flat points is proved ("improvement of flatness") as well as a first estimate on the dimensional set. The main result is as follows:

	"Let $E$ be a minimal surface in $\Omega$, then there is a $C^{1,\alpha}$ submanifold $\partial^* E$ such that $\partial^E \subset \partial E$ and $\partial E \setminus \partial^ E$ is a closed set of Hausdorff dimension at most $n-2$."		"Let $E$ be a minimal surface in $\Omega$, then there is a $C^{1,\alpha}$ submanifold $\partial^* E$ such that $\partial^E \subset \partial E$ and $\partial E \setminus \partial^ E$ is a closed set of Hausdorff dimension at most $n-2$."

imported>Nestor at 05:49, 2 June 2011

2011-06-02T05:49:11Z

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	{{reflist\|refs=		{{reflist\|refs=
	<ref name="CRS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Roquejoffre \| first2=Jean Michel \|last3= Savin \| first3= Ovidiu \| title=Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation \| url=http://dx.doi.org/10.4007/annals.2010.171.1903 \| doi=10.4007/annals.2010.171.1903 \| year=2010 \| journal=[[Annals of Mathematics\|Annals of Mathematics. Second Series]] \| issn=0003-486X \| volume=171 \| issue=3 \| pages=1903–1930}}</ref>		<ref name="CRS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Roquejoffre \| first2=Jean Michel \|last3= Savin \| first3= Ovidiu \| title=Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation \| url=http://dx.doi.org/10.4007/annals.2010.171.1903 \| doi=10.4007/annals.2010.171.1903 \| year=2010 \| journal=[[Annals of Mathematics\|Annals of Mathematics. Second Series]] \| issn=0003-486X \| volume=171 \| issue=3 \| pages=1903–1930}}</ref>
			}}

imported>Nestor at 05:48, 2 June 2011

2011-06-02T05:48:11Z

← Older revision		Revision as of 00:48, 2 June 2011
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	== The Caffarelli-Roquejoffre-Savin Regularity Theorem==		== The Caffarelli-Roquejoffre-Savin Regularity Theorem==

	~~In (reference), a~~ regularity theory for solutions of the above variational problem ~~was~~ developed. It draws inspiration from geometric measure theory, as developed by De Giorgi, Federer, Almgren and many others. In particular, the $C^{1,\alpha}$ regularity of minimal surfaces near flat points is proved ("improvement of flatness") as well as a first estimate on the dimensional set. The main result is as follows:		A regularity theory for solutions of the above variational problem has been developed recently < ref name = "CRS"/>. It draws inspiration from geometric measure theory, as developed by De Giorgi, Federer, Almgren and many others. In particular, the $C^{1,\alpha}$ regularity of minimal surfaces near flat points is proved ("improvement of flatness") as well as a first estimate on the dimensional set. The main result is as follows:

	"Let $E$ be a minimal surface in $\Omega$, then there is a $C^{1,\alpha}$ submanifold $\partial^* E$ such that $\partial^E \subset \partial E$ and $\partial E \setminus \partial^ E$ is a closed set of Hausdorff dimension at most $n-2$."		"Let $E$ be a minimal surface in $\Omega$, then there is a $C^{1,\alpha}$ submanifold $\partial^* E$ such that $\partial^E \subset \partial E$ and $\partial E \setminus \partial^ E$ is a closed set of Hausdorff dimension at most $n-2$."
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	</blockquote>		</blockquote>
	</div>		</div>


			== References ==
			{{reflist\|refs=
			<ref name="CRS">{{Citation \| last1=Caffarelli \| first1=Luis A. \| last2=Roquejoffre \| first2=Jean Michel \|last3= Savin \| first3= Ovidiu \| title=Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation \| url=http://dx.doi.org/10.4007/annals.2010.171.1903 \| doi=10.4007/annals.2010.171.1903 \| year=2010 \| journal=[[Annals of Mathematics\|Annals of Mathematics. Second Series]] \| issn=0003-486X \| volume=171 \| issue=3 \| pages=1903–1930}}</ref>

imported>Luis at 19:52, 1 June 2011

2011-06-01T19:52:20Z

← Older revision		Revision as of 14:52, 1 June 2011
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	<div style="background:#DDEEFF;">		<div style="background:#DDEEFF;">
	<blockquote>		<blockquote>
	Open question: The dimensional estimate for the singular set $n-2$ is not expected to be sharp (at least when $s$ is not too small). It has been shown that if $s$ is close to $1$ then the estimate becomes $n-7$, which is the same (sharp) bound known for classical minimal surfaces. However, for nonlocal minimal surfaces it is known what the sharp bound is. Even more, there are no known explicit examples of singular cones which are nonlocal minimal surfaces.		Open question: The dimensional estimate for the singular set $n-2$ is not expected to be sharp (at least when $s$ is not too small). It has been shown that if $s$ is close to $1$ then the estimate becomes $n-7$, which is the same (sharp) bound known for classical minimal surfaces. However, for nonlocal minimal surfaces it is not known what the sharp bound is. Even more, there are no known explicit examples of singular cones which are nonlocal minimal surfaces.
	</blockquote>		</blockquote>
	</div>		</div>

imported>Nestor: /* Surfaces minimizing non-local energy functionals */

2011-06-01T16:21:28Z

Surfaces minimizing non-local energy functionals

← Older revision		Revision as of 11:21, 1 June 2011
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	That there exists a unique minimizer is always true by the Sobolev embedding and the lower-semicontinuity of $J_s$ with respect to $L^1$ convergence. Of course to carry the argument one needs to assume that there is at least on set $E$ such that $\chi_E \in H^s$ and $E$ agrees with $E_0$ outside $\Omega$ (namely $E \Delta E_0 \subset \Omega$).		That there exists a unique minimizer is always true by the Sobolev embedding and the lower-semicontinuity of $J_s$ with respect to $L^1$ convergence. Of course to carry the argument one needs to assume that there is at least on set $E$ such that $\chi_E \in H^s$ and $E$ agrees with $E_0$ outside $\Omega$ (namely $E \Delta E_0 \subset \Omega$).

			In contrast to the perimeter functional, which is local, the functional Js(E) has the remarkable feature of behaving like a quadratic form. In particular, one has the following identity

			\[ J_s(E) = L(E,E^c) \]

			where for any pair of measurable sets $A,B$ we define

			\[ L(A,B) := C_{n,s}\int_A \int_B \frac{1}{\|x-y\|^{n+s}}dxdy = L(B,A) \]

			Therefore, if we denote $A^- = E \setminus F, A^+= F\setminus E$ then

			\[ J_s(F) - J_s(E) = L(F,F^c) - L(E,E^c ) \]

			\[ = \left [ L(A^-,E \setminus A^-) -L(A^-,E^c) \right ]- \left [ L(A^+,E)-L(A^+,(E \cup A^+)^c) \right ] +2 L(A^-,A^+) \geq 0 \]

	== The Caffarelli-Roquejoffre-Savin Regularity Theorem==		== The Caffarelli-Roquejoffre-Savin Regularity Theorem==