Interior regularity results (local) - Revision history

imported>Luis at 17:19, 29 January 2012

2012-01-29T17:19:26Z

← Older revision		Revision as of 12:19, 29 January 2012
Line 55:		Line 55:
	are smooth (analytic if $L$ is analytic).		are smooth (analytic if $L$ is analytic).

	* [[Krylov-Safonov\|Krylov-Safonov]]		* [[Krylov-Safonov\|Krylov-Safonov-Caffarelli]]

	Any continuous function $u$ such that		Any continuous function $u$ such that

imported>Luis: /* Non linear equations */

2012-01-29T17:14:33Z

Non linear equations

← Older revision		Revision as of 12:14, 29 January 2012
Line 55:		Line 55:
	are smooth (analytic if $L$ is analytic).		are smooth (analytic if $L$ is analytic).

	* [[Krylov-Safonov\|Krylov-Safonov~~-Caffarelli~~]]		* [[Krylov-Safonov\|Krylov-Safonov]]

	Any continuous function $u$ such that		Any continuous function $u$ such that

imported>Nestor at 06:18, 2 June 2011

2011-06-02T06:18:48Z

← Older revision		Revision as of 01:18, 2 June 2011
Line 98:		Line 98:

	which is uniformly elliptic and $C^\alpha$ respect to $x$, then $u \in C^{2,\alpha}$.		which is uniformly elliptic and $C^\alpha$ respect to $x$, then $u \in C^{2,\alpha}$.

	~~[[Category:Mini second order elliptic wiki]]~~

imported>Nestor at 06:18, 2 June 2011

2011-06-02T06:18:25Z

← Older revision		Revision as of 01:18, 2 June 2011
Line 98:		Line 98:

	which is uniformly elliptic and $C^\alpha$ respect to $x$, then $u \in C^{2,\alpha}$.		which is uniformly elliptic and $C^\alpha$ respect to $x$, then $u \in C^{2,\alpha}$.

			[[Category:Mini second order elliptic wiki]]

imported>Luis at 00:36, 30 May 2011

2011-05-30T00:36:50Z

← Older revision		Revision as of 19:36, 29 May 2011
Line 16:		Line 16:
	is in <math>L^n</math> (or <math>BMO^{-1}</math> if <math>{\rm div \,} b=0</math>).		is in <math>L^n</math> (or <math>BMO^{-1}</math> if <math>{\rm div \,} b=0</math>).

	* [[Krylov-Safonov]]		* [[Krylov-Safonov theorem]]
	<math>a_{ij}(x) u_{ij} + b \cdot \nabla u = f </math>		<math>a_{ij}(x) u_{ij} + b \cdot \nabla u = f </math>

Line 22:		Line 22:
	solution is <math>C^\alpha</math>		solution is <math>C^\alpha</math>

	* [[Calderon-zygmund]]		* [[Calderon-zygmund estimate]]
	<math>a_{ij}(x) u_{ij} = f</math>		<math>a_{ij}(x) u_{ij} = f</math>

	with <math>a_{ij}</math> close enough to the identity (or continuous) and <math>f \in L^p</math>, then <math>u</math> is in <math>W^{2,p}</math>.		with <math>a_{ij}</math> close enough to the identity (or continuous) and <math>f \in L^p</math>, then <math>u</math> is in <math>W^{2,p}</math>.

	* [[Cordes-Nirenberg]]		* [[Cordes-Nirenberg estimate]]
	<math>a_{ij}(x) u_{ij} = f</math>		<math>a_{ij}(x) u_{ij} = f</math>

Line 33:		Line 33:
	then $u$ is in $C^{1,\alpha}$		then $u$ is in $C^{1,\alpha}$

	* [[Cordes-Nirenberg improved]] (corollary of work of Caffarelli for nonlinear equations)		* [[Cordes-Nirenberg estimate improved]] (corollary of work of Caffarelli for nonlinear equations)
	<math>a_{ij}(x) u_{ij} = f</math>		<math>a_{ij}(x) u_{ij} = f</math>

Line 41:		Line 41:
	($a_{ij} \in VMO$ is a particular case of this)		($a_{ij} \in VMO$ is a particular case of this)

	*[[Schauder]]		*[[Schauder estimate]]
	<math>a_{ij}(x) u_{ij} = f</math>		<math>a_{ij}(x) u_{ij} = f</math>

Line 65:		Line 65:
	(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)		(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)

	* [[Ishii-Lions]]		* [[Ishii-Lions Lipschitz estimate]]

	If $u$ solves a fully nonlinear equation		If $u$ solves a fully nonlinear equation
Line 75:		Line 75:
	(The proof of this is based on the uniqueness technique for viscosity solutions)		(The proof of this is based on the uniqueness technique for viscosity solutions)

	* [[Lin]]		* [[Lin theorem]]

	Any continuous function $u$ such that		Any continuous function $u$ such that
Line 92:		Line 92:
	which is uniformly elliptic and continuous respect to $x$ ($VMO$ is actually enough), then $u \in C^{1,\alpha}$.		which is uniformly elliptic and continuous respect to $x$ ($VMO$ is actually enough), then $u \in C^{1,\alpha}$.

	* [[Evans-Krylov]]		* [[Evans-Krylov theorem]]
	If $u$ solves a convex (or concave) fully nonlinear equation		If $u$ solves a convex (or concave) fully nonlinear equation

imported>Luis: moved Interior regularity results to Interior regularity results (local)

2011-05-29T18:07:57Z

moved Interior regularity results to Interior regularity results (local)

← Older revision	Revision as of 13:07, 29 May 2011
(No difference)

173.61.107.148: /* Linear equations */ notation

2011-05-25T01:07:17Z

Linear equations: notation

← Older revision		Revision as of 20:07, 24 May 2011
Line 30:		Line 30:
	<math>a_{ij}(x) u_{ij} = f</math>		<math>a_{ij}(x) u_{ij} = f</math>

	with <math>a_{ij}</math> close enough to the identity uniformly and f in L^infty,		with <math>a_{ij}</math> close enough to the identity uniformly and $f \in L^\infty$,
	then $u$ is in $C^{1,\alpha}$		then $u$ is in $C^{1,\alpha}$

imported>Luis at 22:09, 18 May 2011

2011-05-18T22:09:47Z

← Older revision		Revision as of 17:09, 18 May 2011
Line 25:		Line 25:
	<math>a_{ij}(x) u_{ij} = f</math>		<math>a_{ij}(x) u_{ij} = f</math>

	with <math>a_{ij}</math> close enough to the identity and <math>f \in L^p</math>, then <math>u</math> is in <math>W^{2,p}</math>.		with <math>a_{ij}</math> close enough to the identity (or continuous) and <math>f \in L^p</math>, then <math>u</math> is in <math>W^{2,p}</math>.

	* [[Cordes-Nirenberg]]		* [[Cordes-Nirenberg]]

imported>Luis at 22:08, 18 May 2011

2011-05-18T22:08:55Z

← Older revision		Revision as of 17:08, 18 May 2011
Line 55:		Line 55:
	are smooth (analytic if $L$ is analytic).		are smooth (analytic if $L$ is analytic).

	* [[Krylov-Safonov]] ~~(maybe Caffarelli?)~~		* [[Krylov-Safonov\|Krylov-Safonov-Caffarelli]]

	Any continuous function $u$ such that		Any continuous function $u$ such that
Line 85:		Line 85:
	(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)		(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)

	* [[Krylov-Safonov]] ~~(should we add Caffarelli?)~~		* [[Krylov-Safonov\|Krylov-Safonov-Caffarelli]]
	If $u$ solves a fully nonlinear equation		If $u$ solves a fully nonlinear equation

imported>Luis at 21:53, 18 May 2011

2011-05-18T21:53:11Z

← Older revision		Revision as of 16:53, 18 May 2011
Line 1:		Line 1:
	~~'''~~The math ~~formulas do~~ not ~~work!!!'''~~		Let <math>\Omega</math> be an open domain and <math> u </math> a solution of an elliptic equation in <math> \Omega </math>. The following theorems say that <math> u </math> satisfies some regularity estimates in the interior of <math> \Omega </math> (but not necessarily up the the boundary).


			== Linear equations ==

			Regularity results for linear equations are applicable to nonlinear equations as well through the linearization of the equation. However, this process requires some initial regularity knowledge on the solution (since the coefficients of the linearization depend on the solution itself). Therefore, the less regularity required for the coefficients, the more useful the theorem is.

			All regularity results that require some modulus of continuity or smallness condition for the coefficients rely on the idea that the solution is locally close to a solution to an equation with constant coefficients. The proof is based on an estimate on how far these two solutions are at small scales. These type of arguments are often called [[perturbation methods]].

			From the results below for linear equations, [[De Giorgi-Nash-Moser]] and [[Krylov-Safonov]] are the only non perturbative results. Their assumptions are scale invariant in the sense that a rescaling of the solution ($u_r(x) = u(rx)$) would solve an elliptic equation with the same bounds as the original.

	~~Let~~ <math>\~~Omega</math> be an open domain and <math> u </math> a solution of an elliptic equation in <math>~~ \~~Omega </math>. The following theorems say that <math>~~ u </math> ~~satisfies some regularity estimates in the interior of <math> \Omega </math> (but not necessarily up the the boundary).~~		* [[De Giorgi-Nash-Moser]]
			<math> {\rm div \,} A(x) Du + b(x) \cdot \nabla u = 0 </math>

	* [[De Giorgi - Nash - Moser]]
	~~<math> div A Du + b Du = 0 </math>~~
	then <math>u</math> is Holder continuous if <math>A</math> is just uniformly elliptic and <math>b</math>		then <math>u</math> is Holder continuous if <math>A</math> is just uniformly elliptic and <math>b</math>
	is in <math>L^n</math> (or <math>BMO^{-1}</math> if <math>div b=0</math>).		is in <math>L^n</math> (or <math>BMO^{-1}</math> if <math>{\rm div \,} b=0</math>).

	* [[Krylov-Safonov]]		* [[Krylov-Safonov]]
	<math>~~a_ij~~(x) ~~u_ij~~ + b Du = f</math>		<math>a_{ij}(x) u_{ij} + b \cdot \nabla u = f </math>
	with <math>a_{ij}</math> unif elliptic ~~(L^infty)~~, b in L^n and f in L^n, then the
	solution is C^alpha		with <math>a_{ij}</math> unif elliptic, <math>b \in L^n</math> and <math>f \in L^n</math>, then the
			solution is <math>C^\alpha</math>

	* [[Calderon-zygmund]]		* [[Calderon-zygmund]]
	~~a_ij~~(x) ~~u_ij~~ = f		<math>a_{ij}(x) u_{ij} = f</math>
	with ~~a_ij~~ close enough to the identity and f in L^p, then w is in W^{2,p}.
	~~I am not sure how lower order terms affect the result. It should be~~		with <math>a_{ij}</math> close enough to the identity and <math>f \in L^p</math>, then <math>u</math> is in <math>W^{2,p}</math>.
	~~somewhere in [GT]~~

	* [[Cordes-Nirenberg]]		* [[Cordes-Nirenberg]]
	~~a_ij~~(x) ~~u_ij~~ = f		<math>a_{ij}(x) u_{ij} = f</math>
	with ~~a_ij~~ close enough to the identity uniformly and f in L^infty,
	then w is in C^{1,alpha}		with <math>a_{ij}</math> close enough to the identity uniformly and f in L^infty,
	~~I am pretty sure you can add first order terms as long as the~~		then $u$ is in $C^{1,\alpha}$
	~~coefficients are small enough in L^infyt~~

	* [[Cordes-Nirenberg improved]] (corollary of work of Caffarelli for nonlinear equations)		* [[Cordes-Nirenberg improved]] (corollary of work of Caffarelli for nonlinear equations)
	<math>~~a_ij~~(x) ~~u_ij~~ = f</math>		<math>a_{ij}(x) u_{ij} = f</math>
	with ~~a_ij~~ close enough to the identity in a scale invariant Morrey
	norm in terms of L^n and f in L^n, then w is in C^{1,alpha}		with $a_{ij}$ close enough to the identity in a scale invariant Morrey
	(VMO is a particular case of this)		norm in terms of $L^n$ and $f \in L^n$, then $u$ is in $C^{1,\alpha}$.

			($a_{ij} \in VMO$ is a particular case of this)

	*[[Schauder]]		*[[Schauder]]
	<math>~~a_ij~~(x) ~~u_ij~~ = f</math>		<math>a_{ij}(x) u_{ij} = f</math>
	with ~~a_ij~~ in C^alpha and f in C^alpha, then u is in C^{2,alpha}
	~~I don't remember~~ the ~~hypothesis for~~ the ~~first order terms~~, but this is		with $a_{ij}$ in $C^\alpha$ and $f \in C^\alpha$, then $u$ is in $C^{2,\alpha}$
	in [GT] ~~for sure~~.
			== Non linear equations ==

			* [[De Giorgi-Nash-Moser]]
			For any smooth strictly convex Lagrangian $L$, minimizers of functionals

			$ \int_D L(\nabla u) \ dx $

			are smooth (analytic if $L$ is analytic).

			* [[Krylov-Safonov]] (maybe Caffarelli?)

			Any continuous function $u$ such that

			$M^+(D^2 u) \geq 0 \geq M^-(D^2 u)$

			in the viscosity sense (where $M^+$ and $M^-$ are the Pucci operators), is Holder continuous.

			(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)

			* [[Ishii-Lions]]

			If $u$ solves a fully nonlinear equation

			$F(D^2 u, Du, u, x) = 0$

			which is degenerate elliptic but satisfies some structure conditions and some smoothness assumptions respect to $x$, then $u$ is Lipschitz.

			(The proof of this is based on the uniqueness technique for viscosity solutions)

			* [[Lin]]

			Any continuous function $u$ such that

			$0 \geq M^-(D^2 u)$

			in the viscosity sense, is twice differentiable almost everywhere and $D^2 u \in L^\varepsilon$.

			(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)

			* [[Krylov-Safonov]] (should we add Caffarelli?)
			If $u$ solves a fully nonlinear equation

			$F(D^2 u, x) = 0$

			which is uniformly elliptic and continuous respect to $x$ ($VMO$ is actually enough), then $u \in C^{1,\alpha}$.

			* [[Evans-Krylov]]
			If $u$ solves a convex (or concave) fully nonlinear equation

			$F(D^2 u, x) = 0$

			which is uniformly elliptic and $C^\alpha$ respect to $x$, then $u \in C^{2,\alpha}$.