Fully nonlinear integro-differential equations - Revision history

imported>Luis: /* Another definition */

2015-06-26T14:12:22Z

Another definition

← Older revision		Revision as of 09:12, 26 June 2015
Line 48:		Line 48:
	}}		}}

	{{note\|text= ~~In view of~~ the ~~nonlinear version of Courrege theorem given by Guillen and Schwab <ref name="Guillen~~-~~Schwab" />~~, ~~both definitions~~ of ~~nonlocal operators coincide~~, ~~at least when~~ the ~~operators~~ are ~~Frechet differentiable~~}}		{{note\|text= Whenever a general nonlocal operator corresponding to the first definition has an $\inf-\sup$ representation, it also fits the second definition. Any operator in the form of the second definition obviously fits the first definition. Therefore, for all practical purposes, the two definitions are practically equivalent.
			}}

	== Examples ==		== Examples ==

imported>Luis: /* Abstract definition */

2015-06-26T14:10:05Z

Abstract definition

← Older revision		Revision as of 09:10, 26 June 2015
Line 27:		Line 27:
	}}		}}

	{{note\|text=~~It is conceivable that any~~ uniformly elliptic integro-differential equation coincides with some [[Isaacs equation]] for some family of linear operators $L_{ab}$. ~~This was proved in~~ the case that the operator $I$ is Frechet differentiable <ref name="Guillen-Schwab"/>.		{{note\|text=Any uniformly elliptic integro-differential equation with respect to some class $\mathcal L$ satisfies the following $\inf-\sup$ representation
			\[ Iu(x) = \inf_{v \in C^2 \cap L^\infty} \sup_{L \in \mathcal L} \bigg\{ (Iv(x) - Lv(v)) + Lu(x) \bigg\}.\]
			Therefore, $I$ coincides with some [[Isaacs equation]] for some family of linear operators $L_{ab}$ (the index $a$ ranges over all functions $v$ and the index $b$ over all linear operators $L$).

			In the case that $I$ is not uniformly elliptic with respect to any given class, an $\inf-\sup$ formula was also obtained when the operator $I$ is Frechet differentiable <ref name="Guillen-Schwab"/>.
	}}		}}

imported>Luis: /* Abstract definition */

2015-06-24T22:05:38Z

Abstract definition

← Older revision		Revision as of 17:05, 24 June 2015
Line 7:		Line 7:
	== Abstract definition <ref name="CS"/><ref name="CS2"/> ==		== Abstract definition <ref name="CS"/><ref name="CS2"/> ==

	A nonlocal operator is any rule that assigns a value to $Iu(x)$ whenever $u$ is a bounded function in $\mathbb R^n$ that is $C^2$ around the point $x$. The most basic requirement ~~of ellipticity~~ is that whenever $u-v$ achieves a global nonnegative maximum at the point $x$, then		A nonlocal operator is any rule that assigns a value to $Iu(x)$ whenever $u$ is a bounded function in $\mathbb R^n$ that is $C^2$ around the point $x$. The most basic requirement to call $I$ '''elliptic''' is that whenever $u-v$ achieves a global nonnegative maximum at the point $x$, then
	\[ Iu(x) \leq Iv(x).\]		\[ Iu(x) \leq Iv(x).\]

imported>Luis: /* Another definition */

2015-06-24T22:03:01Z

Another definition

← Older revision		Revision as of 17:03, 24 June 2015
Line 44:		Line 44:
	}}		}}

	{{note\|text= In view of the nonlinear version of Courrege theorem given by Guillen and Schwab <ref name="Guillen-Schwab" />, both definitions of nonlocal operators coincide, at least when the ~~operator~~ are Frechet differentiable}}		{{note\|text= In view of the nonlinear version of Courrege theorem given by Guillen and Schwab <ref name="Guillen-Schwab" />, both definitions of nonlocal operators coincide, at least when the operators are Frechet differentiable}}

	== Examples ==		== Examples ==

imported>Luis: /* Abstract definition */

2015-06-24T22:02:16Z

Abstract definition

← Older revision		Revision as of 17:02, 24 June 2015
Line 27:		Line 27:
	}}		}}

	{{note\|text=It is conceivable that any uniformly elliptic integro-differential equation coincides with some [[Isaacs equation]] for some family of linear operators $L_{ab}$~~, at least in the translation invariant case~~. This was proved in the case that the operator $I$ is Frechet differentiable <ref name="Guillen-Schwab"/>.		{{note\|text=It is conceivable that any uniformly elliptic integro-differential equation coincides with some [[Isaacs equation]] for some family of linear operators $L_{ab}$. This was proved in the case that the operator $I$ is Frechet differentiable <ref name="Guillen-Schwab"/>.
	}}		}}

imported>Luis at 21:45, 24 June 2015

2015-06-24T21:45:38Z

← Older revision		Revision as of 16:45, 24 June 2015
Line 44:		Line 44:
	}}		}}

	{{note\|text= In view of the nonlinear version of Courrege theorem given by Guillen and Schwab <ref name="Guillen-Schwab">, both definitions of nonlocal operators coincide, at least when the operator are Frechet differentiable}}		{{note\|text= In view of the nonlinear version of Courrege theorem given by Guillen and Schwab <ref name="Guillen-Schwab" />, both definitions of nonlocal operators coincide, at least when the operator are Frechet differentiable}}

	== Examples ==		== Examples ==

imported>Luis: /* Another definition */

2015-06-24T21:45:10Z

Another definition

← Older revision		Revision as of 16:45, 24 June 2015
Line 43:		Line 43:
	On the other hand, the results in these papers apply to the more general definitions of fully nonlinear integro-differential equations as well. The reason for the restriction to one single integro-differential operator instead of a family $\{L_\alpha\}_\alpha$ seems to be taken only for simplicity.		On the other hand, the results in these papers apply to the more general definitions of fully nonlinear integro-differential equations as well. The reason for the restriction to one single integro-differential operator instead of a family $\{L_\alpha\}_\alpha$ seems to be taken only for simplicity.
	}}		}}

			{{note\|text= In view of the nonlinear version of Courrege theorem given by Guillen and Schwab <ref name="Guillen-Schwab">, both definitions of nonlocal operators coincide, at least when the operator are Frechet differentiable}}

	== Examples ==		== Examples ==

imported>Luis: /* Abstract definition */

2015-06-24T21:42:38Z

Abstract definition

← Older revision		Revision as of 16:42, 24 June 2015
Line 10:		Line 10:
	\[ Iu(x) \leq Iv(x).\]		\[ Iu(x) \leq Iv(x).\]

	We now proceed to define the concept of uniform ellipticity.		We now proceed to define the concept of uniform ellipticity. Given the richness of variations of nonlocal equations, we provide a flexible definition of uniform elliticity depending an arbitrary family of linear operators.

	Given a family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:		Given a family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:
Line 22:		Line 22:
	for any $x \in \Omega$.		for any $x \in \Omega$.

	A fully nonlinear elliptic equation ~~with respect to $\mathcal{L}$~~ is an equation of the form $Iu=0$ in $\Omega$ ~~with~~ $I~~$ uniformly elliptic respect to $\mathcal{L}~~$.		A fully nonlinear elliptic equation is an equation of the form $Iu=0$ in $\Omega$, for some elliptic operator $I$.

	{{note\|text= If $\mathcal L$ consists of purely second order operators of the form $\mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators. It is a ''folklore'' statement that then nonlinear operator $I$ elliptic respect to $\mathcal L$ in the sense described above must coincide with a fully nonlinear elliptic operator of the form $Iu = F(D^2u,x)$. However, this proof may have never been written anywhere.		{{note\|text= If $\mathcal L$ consists of purely second order operators of the form $\mathrm{tr} \, A \cdot D^2 u$ with $\lambda I \leq A \leq \Lambda I$, then $M^+_{\mathcal L}$ and $M^-_{\mathcal L}$ denote the usual extremal Pucci operators. It is a ''folklore'' statement that then nonlinear operator $I$ elliptic respect to $\mathcal L$ in the sense described above must coincide with a fully nonlinear elliptic operator of the form $Iu = F(D^2u,x)$. However, this proof may have never been written anywhere.

imported>Luis: /* Abstract definition */

2015-06-24T21:38:55Z

Abstract definition

← Older revision		Revision as of 16:38, 24 June 2015
Line 9:		Line 9:
	A nonlocal operator is any rule that assigns a value to $Iu(x)$ whenever $u$ is a bounded function in $\mathbb R^n$ that is $C^2$ around the point $x$. The most basic requirement of ellipticity is that whenever $u-v$ achieves a global nonnegative maximum at the point $x$, then		A nonlocal operator is any rule that assigns a value to $Iu(x)$ whenever $u$ is a bounded function in $\mathbb R^n$ that is $C^2$ around the point $x$. The most basic requirement of ellipticity is that whenever $u-v$ achieves a global nonnegative maximum at the point $x$, then
	\[ Iu(x) \leq Iv(x).\]		\[ Iu(x) \leq Iv(x).\]

			We now proceed to define the concept of uniform ellipticity.

	Given a family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:		Given a family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:

imported>Luis: /* Abstract definition */

2015-06-24T21:38:29Z

Abstract definition

← Older revision		Revision as of 16:38, 24 June 2015
Line 6:		Line 6:

	== Abstract definition <ref name="CS"/><ref name="CS2"/> ==		== Abstract definition <ref name="CS"/><ref name="CS2"/> ==

			A nonlocal operator is any rule that assigns a value to $Iu(x)$ whenever $u$ is a bounded function in $\mathbb R^n$ that is $C^2$ around the point $x$. The most basic requirement of ellipticity is that whenever $u-v$ achieves a global nonnegative maximum at the point $x$, then
			\[ Iu(x) \leq Iv(x).\]

	Given a family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:		Given a family of [[linear integro-differential operators]] $\mathcal{L}$, we define the [[extremal operators]] $M^+_\mathcal{L}$ and $M^-_\mathcal{L}$: