Krylov-Safonov theorem and List of equations: Difference between pages

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Krylov-Safonov theorem <ref name="KS"/> <ref name="KS2"/> provides Holder estimates and a Harnack inequality for uniformly elliptic or parabolic equations of second order. It is one of the major components of regularity theory for fully nonlinear elliptic equations of second order. What makes the estimates important is that they do not require any regularity assumption on the coefficients of the equation. It just requires them to be bounded above and below. This makes it possible to apply to the linearization of fully nonlinear equations before knowing any a priori regularity estimate for the solution.
This is a list of nonlocal equations that appear in this wiki.


This theorem is the nondivergence version of [[De Giorgi-Nash-Moser theorem]].
== Linear equations ==
=== Stationary linear equations from Levy processes ===
\[ Lu = 0 \]
where $L$ is a [[linear integro-differential operator]].


For nonlocal equations, there are analogous results both for [[Holder estimates]] and the [[Harnack inequality]].
=== parabolic linear equations from Levy processes ===
\[ u_t = Lu \]
where $L$ is a [[linear integro-differential operator]].


== Elliptic case ==
=== [[Drift-diffusion equations]] ===  
=== Holder continuity ===
\[ u_t  + b \cdot \nabla u + (-\Delta)^s u = 0,\]
Given a bounded solution of the following elliptic PDE
where $b$ is a given vector field.
\[ a_{ij}(x) \partial_{ij} u (x) + b(x) \cdot \nabla u(x) = f(x) \qquad \text{in } B_1,\]
where repeated indices denotes summation and we assume
\begin{align*}
\lambda I &\leq \{a_{ij}(x)\} \leq \Lambda I \text{ for all $x$. (This is the uniform ellipticity condition)}\\
b &\in L^n(B_1), \\
f &\in L^n(B_1).
\end{align*}
Then the function $u$ is Holder continuous and for some small $\alpha>0$ it satisfies the estimate
\[ ||u||_{C^\alpha(B_{1/2})} \leq C (||u||_{L^\infty(B_1)}+||f||_{L^n(B_1)}).\]
The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.


=== Harnack inequality ===
== [[Semilinear equations]] ==
Given a nonnegative solution of the following elliptic PDE
=== Stationary equations with zeroth order nonlinearity ===
\[ a_{ij}(x) \partial_{ij} u (x) + b(x) \cdot \nabla u(x) = f(x) \qquad \text{in } B_1,\]
\[ (-\Delta)^s u = f(u). \]
Under the same assumptions as for the Holder estimates, the following Harnack inequality holds
=== Reaction diffusion equations ===
\[ \sup_{B_{1/2}} u \leq C (\inf_{B_{1/2}} u+||f||_{L^n(B_1)}).\]
\[ u_t + (-\Delta)^s u = f(u). \]
The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.
=== Burgers equation with fractional diffusion ===
\[ u_t + u \ u_x + (-\Delta)^s u = 0 \]
=== [[Surface quasi-geostrophic equation]] ===
\[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \]
=== Conservation laws with fractional diffusion ===
\[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\]
=== Hamilton-Jacobi equation with fractional diffusion ===
\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]
=== [[Keller-Segel equation]] ===
\[u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.\]


=== Viscosity solutions ===
== Quasilinear or [[fully nonlinear integro-differential equations]] ==
Both the Holder estimates and the Harnack inequality can be applied to [[viscosity solutions]] of nonlinear equations. Formally, one can replace the equation (at least when $b=0$) by
=== [[Bellman equation]] ===
\begin{align*}
\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
M^+(D^2 u) &\geq f \text{ in } B_1,\\
where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.
M^-(D^2 u) &\leq f \text{ in } B_1.
=== [[Isaacs equation]] ===
\end{align*}
\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \]
When $f$ is continuous, both inequalities above are well defined in the viscosity sense.
where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.
=== [[obstacle problem]] ===
For an elliptic operator $L$ and a function $\varphi$ (the obstacle), $u$ satisfies
\begin{align}
u &\geq \varphi \qquad \text{everywhere in the domain } D,\\
Lu &\leq 0 \qquad \text{everywhere in the domain } D,\\
Lu &= 0 \qquad \text{wherever } u > \varphi.
\end{align}


=== [[Nonlocal minimal surfaces ]] ===
The set $E$ satisfies.
\[ \int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \partial E.\]
=== [[Nonlocal porous medium equation]] ===
\[ u_t = \mathrm{div} \left ( u \nabla (-\Delta)^{-s} u \right).\]
Or
\[ u_t +(-\Delta)^{s}(u^m) = 0. \]


== Parabolic case ==
== Inviscid equations ==
The elliptic case is implied by the results in the parabolic setting.
=== [[Surface quasi-geostrophic equation|Inviscid SQG]]===
\[ \theta_t + u \cdot \nabla \theta = 0,\]
where $u = \nabla^\perp (-\Delta)^{-1/2} \theta$.


=== Holder estimates ===
=== [[Active scalar equation]] (from fluid mechanics) ===
The Holder estimate are similar in the parabolic case as in the elliptic case. Let us define the parabolic cylinder
\[ \theta_t + u \cdot \nabla \theta = 0,\]
\[ Q_r(x_0,t_0) = \{(x,t) : |x-x_0|<r \text{ and } 0 \leq t_0 - t < t^2 \}.\]
where $u = \nabla^\perp K \ast \theta$.


Given a bounded solution of the parabolic PDE
=== [[Aggregation equation]] ===
\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]
where repeated indices denotes summation and we assume
\begin{align*}
\lambda I &\leq \{a_{ij}(x,t)\} \leq \Lambda I \text{ for all $x$ and $t$. (This is the uniform ellipticity condition)}\\
f &\in L^{n+1}(Q_1).
\end{align*}
Then the function $u$ is Holder continuous and satisfies the estimate
\[ ||u||_{Q^\alpha(C_{1/2})} \leq C (||u||_{L^\infty(Q_1)}+||f||_{L^{n+1}(Q_1)}).\]
The constant $C$ depends on $\lambda$, $\Lambda$, and $n$ (dimension).


===Harnack inequality ===
\[ \theta_t + \mathrm{div}(\theta \ u) = 0,\]
In the parabolic Harnack inequality, the infimum and the maximum must be taken in cylinders which are shifted in time.
where $u = \nabla K \ast \theta$.
 
Given a nonnegative solution of the parabolic PDE
\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]
under the same assumptions as for the Holder estimates, the function $u$ satisfies the inequality
\[ \sup_{Q_{1/2}(0,-1/2)} u \leq C \left(\inf_{Q_{1/2}(0,0)} u+||f||_{L^{n+1}(Q_1)} \right).\]
The constant $C$ depends on $\lambda$, $\Lambda$ and $n$ (dimension).
 
== $C^{1,\alpha}$ estimates for fully nonlinear equations ==
 
The Holder estimates described above can be used to obtain $C^{1,\alpha}$ regularity estimates for solutions to fully nonlinear uniformly elliptic equations $F(D^2 u)=0$. Formally we can derive the equation to obtain.
\[ \frac{\partial F(D^2u)} {\partial X_{ij}} \partial_{ij} u_e = \partial_e F(D^2 u)=0. \]
The uniform ellipticity assumption on $F$ means that $a_{ij}(x) := \frac{\partial F(D^2u)} {\partial X_{ij}}$ satisfies the hypothesis of the Holder estimates, and therefore the directional derivative $u_e$ must be $C^\alpha$ for any vector $e$.
 
Exploiting the idea above, one can prove the following result. If $u$ is a bounded viscosity solution of $F(D^2 u)=0$ in $B_1$, then there exist an $\alpha>0$ such that $u \in C^{1,\alpha}$ in the interior of $B_1$ and
\[ ||u||_{C^{1,\alpha}} \leq C (||u||_{L^\infty(B_1)} + F(0)).\]
The constants $C$ and $\alpha$ depend only on $\lambda$, $\Lambda$ and $n$ (dimension), but not on any other characteristic of the function $F$.
 
== References ==
{{reflist|refs=
<ref name="KS">{{Citation | last1=Safonov | first1=M. V. | last2=Krylov | first2=N. V. | title=An estimate for the probability of a diffusion process hitting a set of positive measure | year=1979 | journal=Doklady Akademii Nauk SSSR | issn=0002-3264 | volume=245 | issue=1 | pages=18–20}}</ref>
<ref name="KS2">{{Citation | last1=Safonov | first1=M. V. | last2=Krylov | first2=N. V. | title=A property of the solutions of parabolic equations with measurable coefficients | year=1980 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=44 | issue=1 | pages=161–175}}</ref>
}}

Revision as of 19:20, 4 March 2012

This is a list of nonlocal equations that appear in this wiki.

Linear equations

Stationary linear equations from Levy processes

\[ Lu = 0 \] where $L$ is a linear integro-differential operator.

parabolic linear equations from Levy processes

\[ u_t = Lu \] where $L$ is a linear integro-differential operator.

Drift-diffusion equations

\[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0,\] where $b$ is a given vector field.

Semilinear equations

Stationary equations with zeroth order nonlinearity

\[ (-\Delta)^s u = f(u). \]

Reaction diffusion equations

\[ u_t + (-\Delta)^s u = f(u). \]

Burgers equation with fractional diffusion

\[ u_t + u \ u_x + (-\Delta)^s u = 0 \]

Surface quasi-geostrophic equation

\[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \]

Conservation laws with fractional diffusion

\[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\]

Hamilton-Jacobi equation with fractional diffusion

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

Keller-Segel equation

\[u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.\]

Quasilinear or fully nonlinear integro-differential equations

Bellman equation

\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \] where $L_a$ is some family of linear integro-differential operators indexed by an arbitrary set $\mathcal{A}$.

Isaacs equation

\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \] where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$.

obstacle problem

For an elliptic operator $L$ and a function $\varphi$ (the obstacle), $u$ satisfies \begin{align} u &\geq \varphi \qquad \text{everywhere in the domain } D,\\ Lu &\leq 0 \qquad \text{everywhere in the domain } D,\\ Lu &= 0 \qquad \text{wherever } u > \varphi. \end{align}

Nonlocal minimal surfaces

The set $E$ satisfies. \[ \int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \partial E.\]

Nonlocal porous medium equation

\[ u_t = \mathrm{div} \left ( u \nabla (-\Delta)^{-s} u \right).\] Or \[ u_t +(-\Delta)^{s}(u^m) = 0. \]

Inviscid equations

Inviscid SQG

\[ \theta_t + u \cdot \nabla \theta = 0,\] where $u = \nabla^\perp (-\Delta)^{-1/2} \theta$.

Active scalar equation (from fluid mechanics)

\[ \theta_t + u \cdot \nabla \theta = 0,\] where $u = \nabla^\perp K \ast \theta$.

Aggregation equation

\[ \theta_t + \mathrm{div}(\theta \ u) = 0,\] where $u = \nabla K \ast \theta$.