Comparison principle

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The comparison principle refers to the general concept that a subsolution to an elliptic equation stays below a supersolution of the same equation. It known to hold under a great generality of assumptions.

The comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the maximum principle. The uniqueness of the solution of the equation is an immediate consequence.

Contents

General statement

The two statements below correspond to the comparison principle for elliptic and parabolic equations with Dirichlet boundary conditions. The main difference with the local case, is that for nonlocal equations the Dirichlet condition has to be taken in the whole complement of the domain $\Omega$ instead of only the boundary.

Other boundary conditions require appropriate modifications.

Elliptic case

We say that an elliptic equation $Iu=0$ satiesfies the comparison principle if the following statement is true.

Given two functions $u : \R^n \to \R$ and $v : \R^n \to \R$ such that $u$ and $v$ are upper and lower semicontinuous in $\overline \Omega$ respectively, where $\Omega$ is an open domain, $Iu \geq 0$ and $Iv \leq 0$ in the viscosity sense in $\Omega$, and $u \leq v$ in $\R^n \setminus \Omega$, then $u \leq v$ in $\Omega$ as well.

Parabolic case

We say that a parabolic equation $u_t - Iu=0$ satiesfies the comparison principle if the following statement is true.

Given two functions $u : [0,T] \times \R^n \to \R$ and $v : [0,T] \times\R^n \to \R$ such that $u$ and $v$ are upper and lower semicontinuous in $[0,T] \times \overline \Omega$ respectively, $Iu \leq 0$ and $Iv \geq 0$ in the viscosity sense in $(0,T] \times \Omega$, and $u \leq v$ in $(\{0\} \times \R^n) \cup ([0,T] \times (\R^n \setminus \Omega))$, then $u \leq v$ in $[0,T] \times \Omega$ as well.

Assumptions for which the comparison principle holds

Proper elliptic equations with $x$ dependence

A fairly general comparison principle [1] [2] is available for fully nonlinear integro-differential equations of the form \[ F(x,u,Du,D^2u,\{I_\alpha u\})=0 \] where $\{I_\alpha u\}_{\alpha \in A}$ is a family of linear integro-differential operators.

The comparison principle holds provided that the following assumptions are satisfied:

  • All integro-differential operators $I_\alpha$ are in the Levy-Ito form

\[ I_\alpha u (x)= \int_{\R^n} (u(x+j_\alpha(x,z)) - u(x) - \nabla u(x) \cdot j_\alpha(x,z) \mathbb{1}_B(z)) \mu_\alpha(dz). \]

  • The measures $\mu_\alpha$ and the functions $j_\alpha$ satisfy the following assumptions for some large constant $C$ independent of $\alpha$.

\begin{align*} \int_B |j(x,z)|^2 \mu(dz) &\leq C, \qquad \int_{\R^n \setminus B} \mu(dz) \leq C,\\ \int_{\R^n} |j(x,z)-j(y,z)|^2 \mu(dz) &\leq C|x-y|^2, \text{ and } \int_{\R^n \setminus B} |j(x,z)-j(y,z)|^2 \mu(dz) \leq C|x-y|. \end{align*}

  • There exist a $\gamma>0$ such that for all $x \in \R^n$, $u,v \in \R$, $p \in \R^n$, $X$ a symmetric matrix in $\R^{n \times n}$ and $\{i_\alpha\} \in \R^A$,

\[ F(x,u,p,X,\{i_\alpha\}) - F(x,v,p,X,\{i_\alpha\}) \geq \gamma (v-u).\]

  • One of the following two holds
    1. For any $R >0$, there exist moduli of continuity $\omega$ and $\omega_R$ such that, for any $|x|,|y|<R$, $|v|<R$, $\{i_\alpha\} \in \R^A$, and for any two symmetric matrices $X$ and $Y$ satisfying \begin{equation} \tag{1} \left[ \begin{matrix} X & 0 \\ 0 & -Y \end{matrix} \right] \leq \frac 1 \varepsilon \left[ \begin{matrix} I & -I \\ -I & I \end{matrix} \right] +r(\beta) \left[ \begin{matrix} I & 0 \\ 0 & I \end{matrix} \right] \end{equation} for some $\varepsilon>0$ and $r(\beta)\to 0$ as $\beta\to 0$, then, if $s(\beta)\to 0$ as $\beta \to 0$, we have \[ F(y,v,\varepsilon^{-1}(x-y),Y,\{i_\alpha\})-F(x,u,\varepsilon^{-1}(x-y)+s(\beta),X,\{i_\alpha\})\geq\omega(\beta) +\omega_R(|x-y|+\varepsilon^{-1}|x-y|^2)\]
    2. For any $R>0$, $F$ is uniformly continuous on $R^n\times [−R,R] \times B_R \times D_R \times R^A$ where $D_R := \{X \in \R^{n \times n}, \text{symmetric}, |X|\leq R\}$ and there exist a modulus of continuity $\omega_R$ such that, for any $x,y\in \R^n$ , $|v|\leq R$, $\{i_\alpha\}\in \R^A$ and for any $X,Y \in \R^{n\times n}$ symmetric satisfying (1) and $\varepsilon >0$, we have \[ F(y,v,\varepsilon^{-1}(x-y),Y,\{i_\alpha\})-F(x,u,\varepsilon^{-1}(x-y),X,\{i_\alpha\})\geq \omega_R(\varepsilon^{-1}|x-y|^2+|x-y|+r(\beta))\]
  • $F(x,u,p,X,\{i_\alpha\})$ is Lipschitz continuous in $\{i_\alpha\}$ uniformly respect to all other variables.

Note that the assumptions are designed so that Ishii's lemma can be reproduced in this setting.

Note. Any linear integro-differential operator which is written in Levy-Ito form can be written in the usual form (with $j(x,z)=z$ and an $x$ dependent jump measure $\mu(x,dz)$) by a change of variables. Conversely, any integro-differential operator with an $x$ dependent $\mu$ can be written in Levy-Ito form for several choices of $j$, as long as $\int_{\R^n} \mu(x,dz) = +\infty$ for all $x$. The reason for assuming that all operators $I_\alpha$ are in the Levy-Ito form is to state the right continuity condition in the second assumption. A comparison principle for a family of integro-differential operators $I_\alpha$ with measures $\mu(x,dz)$ (or kernels $K(x,y)$) which are continuous with respect to $x$, but without any continuity assumption with respect to their Levy-Ito form, is currently an open problem.

Uniformly elliptic equations without $x$ dependence

A simple comparison principle can be shown for fully nonlinear integro-differential equations which are translation invariant and uniformly elliptic respect to some family $\mathcal L$ of linear operators satisfying some assumptions [3]. In fact, if $Iu \geq 0$ and $Iv \leq 0$ in some domain $\Omega$, then $M_{\mathcal L}^+ (u-v) \geq 0$ where $M_{\mathcal L}$ is the extremal operator respect to the class $\mathcal L$.

The family $\mathcal L$ is assumed to be a family of purely integro-differential linear integro-differential operators which are translation invariant and satisfy the following assumption:

There is a constant $R_0\geq 1$ so that for every $R>R_0$, there exists a $\delta>0$ (which could depend on $R$) such that for any operator $L$ in $\mathcal L$, we have that $L \varphi > \delta$ in $B_R$, where $\varphi$ is given by $\varphi(x)=\min(1,|x|^2/R^3)$.

This assumption on the family $\mathcal L$ is a very mild form of uniform ellipticity. The strong assumption is translation invariance. The proof of this comparison principle uses inf and sup-convolutions.


References

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