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

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

\[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \] 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 $u = \nabla K \ast \theta$.

@@ Line 1: / Line 1: @@
-Krylov-Safonov theorem 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.
-== Elliptic case ==
+== Linear equations ==
-=== Holder continuity ===
+=== Stationary linear equations from Levy processes ===
-Given a bounded solution of the following elliptic PDE
+\[ Lu = 0 \]
-\[ a_{ij}(x) \partial_{ij} u (x) + b(x) \cdot \nabla u(x) = f(x) \qquad \text{in } B_1,\]
+where $L$ is a [[linear integro-differential operator]].
-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 ===
+=== parabolic linear equations from Levy processes ===
-Given a nonnegative solution of the following elliptic PDE
+\[ u_t = Lu \]
-\[ a_{ij}(x) \partial_{ij} u (x) + b(x) \cdot \nabla u(x) = f(x) \qquad \text{in } B_1,\]
+where $L$ is a [[linear integro-differential operator]].
-Under the same assumptions as for the Holder estimates, the following Harnack inequality holds
-\[ \sup_{B_{1/2}} u \leq C (\inf_{B_{1/2}} u+||f||_{L^n(B_1)}).\]
-The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.
-=== Viscosity solutions ===
+=== [[Drift-diffusion 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
+\[ u_t  + b \cdot \nabla u + (-\Delta)^s u = 0,\]
-\begin{align*}
+where $b$ is a given vector field.
-M^+(D^2 u) &\geq f \text{ in } B_1,\\
-M^-(D^2 u) &\leq f \text{ in } B_1.
-\end{align*}
-When $f$ is continuous, both inequalities above are well defined in the viscosity sense.
+== [[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]] ===
+\[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0, \]
+where $u = R^\perp \theta := \nabla^\perp (-\Delta)^{-1/2} \theta$.
-== Parabolic case ==
+=== Conservation laws with fractional diffusion ===
-The elliptic case is implied by the results in the parabolic setting.
+\[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\]
+=== Hamilton-Jacobi equation with fractional diffusion ===
+\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]
+=== [[Keller-Segel equation]] ===
+\[u_t + \mathrm{div} \left( u \, \nabla (-\Delta)^{-1} u \right) - \Delta u = 0.\]
-=== Holder estimates ===
+== Quasilinear or [[fully nonlinear integro-differential equations]] ==
-The Holder estimate are similar in the parabolic case as in the elliptic case. Let us define the parabolic cylinder
+=== [[Bellman equation]] ===
-\[ Q_r(x_0,t_0) = \{(x,t) : |x-x_0|<r \text{ and } 0 \leq t_0 - t < t^2 \}.\]
+\[ \sup_{a \in \mathcal{A}} \, L_a u(x) = f(x), \]
+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}
-Given a bounded solution of the parabolic PDE
+=== [[Nonlocal minimal surfaces ]] ===
-\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) + b(x,t) \cdot \nabla u(x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]
+The set $E$ satisfies.
-where repeated indices denotes summation and we assume
+\[ \int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \partial E.\]
-\begin{align*}
+=== [[Nonlocal porous medium equation]] ===
-\lambda I &\leq \{a_{ij}(x,t)\} \leq \Lambda I \text{ for all $x$ and $t$. (This is the uniform ellipticity condition)}\\
+\[ u_t = \mathrm{div} \left ( u \nabla (-\Delta)^{-s} u \right).\]
-b &\in L^n(Q_1), \\
+Or
-f &\in L^n(Q_1).
+\[ u_t +(-\Delta)^{s}(u^m) = 0. \]
-\end{align*}
-Then the function $u$ is Holder continuous and satisfies the estimate
-\[ ||u||_{Q^\alpha(C_{1/2})} \leq C (||u||_{L^\infty(Q_1)}+||f||_{L^n(Q_1)}).\]
-The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.
-===Harnack inequality ===
+== Inviscid equations ==
-In the parabolic Harnack inequality, the infimum and the maximum must be taken in cylinders which are shifted in time.
+=== [[Surface quasi-geostrophic equation|Inviscid SQG]]===
+\[ \theta_t + u \cdot \nabla \theta = 0,\]
+where $u = \nabla^\perp (-\Delta)^{-1/2} \theta$.
-Given a nonnegative solution of the parabolic PDE
+=== [[Active scalar equation]] (from fluid mechanics) ===
-\[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) + b(x,t) \cdot \nabla u(x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\]
+\[ \theta_t + u \cdot \nabla \theta = 0,\]
-under the same assumptions as for the Holder estimates, the function $u$ satisfies the inequality
+where $u = \nabla^\perp K \ast \theta$.
-\[ \sup_{Q_{1/2}(0,-1/2)} u \leq C \left(\inf_{Q_{1/2}(0,0)} u+||f||_{L^n(Q_1)} \right).\]
-The constant $C$ depends on $\lambda$, $\Lambda$, $n$ (dimension) and $||b||_{L^n}$.
-== $C^{1,\alpha}$ estimates for fully nonlinear equations ==
+=== [[Aggregation equation]] ===
-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.
+\[ \theta_t + \mathrm{div}(\theta \ u) = 0,\]
-\[ \frac{\partial F(D^2u)} {\partial X_{ij}} \partial_{ij} u_e = \partial_e F(D^2 u)=0. \]
+where $u = \nabla K \ast \theta$.
-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 $F$.

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

Revision as of 09:48, 6 March 2012

Contents

Linear equations

Stationary linear equations from Levy processes

parabolic linear equations from Levy processes

Drift-diffusion equations

Semilinear equations

Stationary equations with zeroth order nonlinearity

Reaction diffusion equations

Burgers equation with fractional diffusion

Surface quasi-geostrophic equation

Conservation laws with fractional diffusion

Hamilton-Jacobi equation with fractional diffusion

Keller-Segel equation

Quasilinear or fully nonlinear integro-differential equations

Bellman equation

Isaacs equation