Linear integro-differential operator and Quasilinear equations: Difference between pages

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The linear integro-differential operators that we consider ''in this wiki'' are the generators of [[Levy processes]]. According to the Levy-Kintchine formula, they have the general form
Quasilinear equations are those which are linear in all terms except for the  highest order derivatives  (whether they are of fractional order or not).


\[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + d(x) + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]
For instance, the following equations are all quasilinear (and the first two are NOT semilinear)
where $A(x)$ is a nonnegative matrix for all $x$, and $\mu_x$ is a nonnegative measure for all $x$ satisfying
\[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]


The above definition is very general. Many theorems, and in particular regularity theorems, require extra assumptions in the kernels $K$. These assumptions restrict the study to certain sub-classes of linear operators. The simplest of all is the [[fractional Laplacian]]. We list below several extra assumptions that are usually made.
\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]


== Absolutely continuous measure ==
<center> [[Mean curvature flow]] </center>


In most cases, the nonnegative measure $\mu$ is assumed to be absolutely continuous: $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$.
\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]


We keep this assumption in all the examples below.
<center> [[Nonlocal porous medium equation]] </center>


== Purely integro-differential operator ==
\[ u_t  + (-\Delta)^s u + H(x,t,u,\nabla u)= 0.\]


In this case we neglect the local part of the operator
<center> Hamilton-Jacobi with fractional diffusion </center>
\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(x,y) \mathrm d y. \]


== Symmetric kernels ==
Equations which are NOT quasilinear, and thus involve no linearity assumption of any sort, are called [[Fully nonlinear equations]], they include for instance the [[Monge Ampére Equation]] and [[Fully nonlinear integro-differential equations]]. Note that all [[Semilinear equations]] are automatically quasilinear.
If the kernel is symmetric $K(x,y) = K(x,-y)$, then we can remove the gradient term from the integral and replace the difference by a second order quotient.


In the purely integro-differentiable case, it reads as
\[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \, K(x,y) \mathrm d y. \]


The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x+y)-2u(x))$.
{{stub}}
 
== Translation invariant operators ==
In this case, all coefficients are independent of $x$.
\[ Lu(x) = \mathrm{tr} \, A \cdot D^2 u + b \cdot \nabla u + c u + d + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(y) \mathrm{d}y. \]
 
== The fractional Laplacian ==
 
The [[fractional Laplacian]] is the simplest and most common purely integro-differential operator. It corresponds to a translation invariant operator for which $K(y)$ is radially symmetric and homogeneous.
 
\[ -(-\Delta)^{s/2} u(x) = C_{n,s} \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \frac{1}{|y|^{n+s}} \mathrm d y. \]
 
== Stable operators ==
 
These are the operators whose kernel is homogeneous in $y$
\[ K(y)=\frac{a(y/|y|)}{|y|^{n+s}}\qquad\textrm{or}\qquad K(x,y)=\frac{a(x,y/|y|)}{|y|^{n+s}}.\]
They are the generators of stable Lévy processes. The function $a$ cound be any $L^1$ function on $S^{n-1}$, or even any measure.
 
== Uniformly elliptic of order $s$ ==
 
This corresponds to the assumption that the kernel is comparable to the one of the fractional Laplacian of the same order.
\[ \frac {(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac {(2-s)\Lambda}{|y|^{n+s}}. \]
 
The normalizing factor $(2-s)$ is a normalizing factor which is only important when $s$ approaches two.
 
An operator of variable order can be either one for which $s$ depends on $x$, or one for which there are two values $s_1<s_2$, one for the left hand side and another for the right hand side.
 
== Smoothness class $k$ of order $s$ ==
This class (sometimes denoted as $\mathcal L_k^s$) corresponds to kernels that are uniformly elliptic of order $s$ and, moreover, their derivatives are also bounded
\[ |\partial_y^r K(x,y)| \leq \frac {\Lambda}{|y|^{n+s+r}} \ \ \text{for all } r\leq k. \]
 
== Order strictly below one ==
 
If a non symmetric kernel $K$ satisfies the extra local integrability assumption
\[ \int_{\R^n} \min(|y|,1) K(x,y) \mathrm d y < +\infty, \]
then the extra gradient term is not necessary in order to define the operator.
 
\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x)) \, K(x,y) \mathrm d y. \]
 
The modification in the integro-differential part of the operator becomes an extra drift term.
 
A uniformly elliptic operator of order $s<1$ satisfies this condition.
 
== Order strictly above one ==
 
If a non symmetric kernel $K$ satisfies the extra integrability assumption on its tail.
\[ \int_{\R^n} \min(|y|^2,|y|) K(x,y) \mathrm d y < +\infty, \]
then the gradient term in the integral can be taken global instead of being cut off in the unit ball.
 
\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) \, K(x,y) \mathrm d y. \]
 
The modification in the integro-differential part of the operator becomes an extra drift term.
 
A uniformly elliptic operator of order $s>1$ satisfies this condition.
 
== More singular/irregular kernels ==
 
The concept of uniform ellipticity can be relaxed in various ways. The following family of operators was considered in the paper of Silvestre and Schwab.<ref name="schwab2014regularity" />. The integro-differential operators have the form
\begin{align*}
\int_{\R^d} \big(u(x+y) - u(x) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha < 1, \\
\int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x)\chi_{B_1}(y) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha = 1, \\
\int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha > 1.
\end{align*}
 
For $\lambda$, $\Lambda$, $\mu$ and $\alpha$ given, the kernel $K: \Omega \times \R^d \to \R$ is assumed to satisfy the following assumptions for all $x \in \Omega$,
* $K(x,h) \geq 0$ for all $h\in\R^d$.
This is a basic assumption for the integral operator to be a legit diffusion operator.
 
* For every $r>0$, \[ \int_{B_{2r} \setminus B_r} K(x,h) \mathrm{d} h \leq (2-\alpha) \Lambda r^{-\alpha}\]
This assumption is more general than $K(x,h) \leq (2-\alpha) \Lambda |h|^{-n-\alpha}$. Indeed, it only requires the upper bound on average.
 
* For every $r>0$, there exists a set $A_r$ such that
** $A_r \subset B_{2r} \setminus B_r$.
** $A_r$ is symmetric in the sense that $A_r = -A_r$.
** $|A_r| \geq \mu |B_{2r} \setminus B_r|$.
** $K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha}$ in $A_r$.
Equivalently \[ \left\vert \left\{y \in B_{2r} \setminus B_r: K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha} \text{ and } K(x,-h) \geq (2-\alpha) \lambda r^{-d-\alpha} \right\} \right\vert \geq \mu |B_{2r} \setminus B_r|.
\]
 
This assumption says that the lower bound $K(x,h) \geq (2-\alpha) \lambda |h|^{-n-\alpha}$ takes place on sets of positive density in certain scale invariant sense.
 
* For all $r>0$, \[ \left\vert \int_{B_{2r} \setminus B_r} h K(h) \mathrm{d} h \right\vert \leq \Lambda |1-\alpha| r^{1-\alpha}. \]
 
This last assumption is a technical restriction which measures the contribution of the non-symmetric part of $K$ and is only relevant for the limit $\alpha \to 1$.
 
== Indexed by a matrix ==
In some cases, it is interesting to study a family of kernels $K$ that are indexed by a matrix. For example, given the matrix $A$, one can consider the kernel of order $s$:
\[ K_A(y) =  \frac{(2-s) \langle y , Ay \rangle}{|y|^{n+2+s}}. \]
This family of kernels has the outstanding property that the corresponding linear operator $L$ coincides with $Lu(x) = a_{ij} \partial_{ij}\left[(-\Delta)^{(s-2)/2} u \right] (x)$ for some coefficients $a_{ij}$.
 
== Second order elliptic operators as limits of purely integro-differential ones ==
 
Given any bounded, even, positive function $a: \mathbb{R}^n\to \mathbb{R}$, the family of operators
 
\[L_\sigma u(x) = (2-\sigma) \int_{\mathbb{R}^n} (u(x+y)+u(x-y)-2u(x))\frac{a(y)}{|y|^{n+\sigma}}dy,\;\; \sigma \in (0,2), \]
 
define in the limit $\sigma \to 2^-$ a second order linear elliptic operator (possibly degenerate). This can be checked for any fixed $C^2$ function $u$ by a straightforward computation using the second order Taylor expansion. A class of kernels that is big enough to recover all translation invariant elliptic operators of the form  $Lu(x) = Tr ( A \cdot D^2u(x) )$ is given by the kernels
 
\[ K_A(y) = (2-\sigma) \frac{1}{|Ay|^{n+\sigma}},\]
 
where $A$ is an invertible symmetric matrix.
 
== Characterization via global maximum principle ==
 
A bounded linear operator
 
\[ L: C^2_0(\mathbb{R}^n) \to C(\mathbb{R}^n) \]
 
is said to satisfy the global maximum principle if given any $u \in C^2_0(\mathbb{R}^n)$ with a global maximum at some point $x_0$ we have
 
\[ (Lu)(x_0) \leq 0 \]
 
It turns out this property imposes strong restrictions on the operator $L$, and we have the following theorem due to Courrège <ref name="C65"/> <ref name="C64"/>: if $L$ satisfies the global maximum principle then it has the form
 
\[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]
 
where again $A(x)$ is a nonnegative matrix for all $x$, $c(x)\leq 0$ and $\mu_x$ is a nonnegative measure for all $x$ satisfying
 
\[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]
 
and $A(x),c(x)$ and $b(x)$ are bounded.
 
 
== See also ==
 
* [[Fractional Laplacian]]
* [[Levy processes]]
* [[Dirichlet form]]
 
 
== References ==
{{reflist|refs=
<ref name="C64">{{Citation | last1=Courrège | first1=Philippe | title=Générateur infinitésimal d'un semi-groupe de convolution sur $R^n$, et formule de Lévy-Khinchine | year=1964 | journal=Bulletin des Sciences Mathématiques. 2e Série | issn=0007-4497 | volume=88 | pages=3–30}}</ref>
<ref name="C65">{{Citation | last1=Courrège | first1=P. | title=Sur la forme intégro-différentielle des opéateurs de  $C_k^\infty(\mathbb{R}^n)$  dans $C(\mathbb{R}^n)$ satisfaisant au principe du maximum | journal=Sém. Théorie du potentiel (1965/66) Exposé | volume=2}}</ref>
<ref name="schwab2014regularity">{{Citation | last1=Schwab | first1= Russell W | last2=Silvestre | first2= Luis | title=Regularity for parabolic integro-differential equations with very irregular kernels | journal=arXiv preprint arXiv:1412.3790}}</ref>
}}

Revision as of 21:35, 5 February 2012

Quasilinear equations are those which are linear in all terms except for the highest order derivatives (whether they are of fractional order or not).

For instance, the following equations are all quasilinear (and the first two are NOT semilinear)

\[u_t-\mbox{div} \left ( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right ) = 0 \]

Mean curvature flow

\[ u_t = \mbox{div} \left ( u \nabla \mathcal{K_\alpha} u\right ),\;\;\; \mathcal{K_\alpha} u = u * |x|^{-n+\alpha} \]

Nonlocal porous medium equation

\[ u_t + (-\Delta)^s u + H(x,t,u,\nabla u)= 0.\]

Hamilton-Jacobi with fractional diffusion

Equations which are NOT quasilinear, and thus involve no linearity assumption of any sort, are called Fully nonlinear equations, they include for instance the Monge Ampére Equation and Fully nonlinear integro-differential equations. Note that all Semilinear equations are automatically quasilinear.


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