Boundary Harnack inequality and List of equations: Difference between pages

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The '''Boundary Harnack Inequality''' is a name given to two related statements for nonnegative functions $u$ which are solutions of elliptic equations.  
This is a list of nonlocal equations that appear in this wiki.


The first result, also known as '''Carleson's estimate''', says that for non-negative solutions, their values in a neighborhood of the (suitably smooth) boundary are bounded in terms of the value at some interior point. Let $u$ be a non-negative solution of an elliptic equation $Lu = 0$ on some domain $\Omega \subset \mathbb{R}^n$, such that $u = 0$ on $B_r(x_0) \cap \partial \Omega$, where $x_0$ lies on the boundary $\partial \Omega$, and $x'$ is some other point lying within $B_\frac{r}{2}(x_0) \cap \Omega$. Then, inside $B_\frac{r}{2}(x_0) \cap \Omega$, there exists a constant $M > 0$ such that $u(x) \leq M u(x')$.  
== Linear equations ==
=== Stationary linear equations from Levy processes ===
\[ Lu = 0 \]
where $L$ is a [[linear integro-differential operator]].


The second result, also known as the '''boundary comparison estimate''', says that two non-negative solutions which are zero on some portion of the boundary, have a Holder continuous ratio with respect to each other in some neighborhood of the boundary. That is, let $ Lu = Lv = 0$ inside some domain $\Omega$ with smooth boundary, with $u,v \geq 0$, and $u = v = 0$ along $B_r(x_0) \cap \partial \Omega$ for some $x_0 \in \partial \Omega$. Then the ratio $\frac{u}{v}$ lies in the Holder class $C^\alpha (B_\frac{r}{2}(x_0))$.
=== parabolic linear equations from Levy processes ===
\[ u_t = Lu \]
where $L$ is a [[linear integro-differential operator]].


==Further reading==
=== [[Drift-diffusion equations]] ===
*{{Citation | last1=Caffarelli | first1=Luis | last2=Salsa | first2=Sandro | title=A geometric approach to free boundary problems | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-3784-9 | year=2005 | volume=68}}, Chapter 11
\[ u_t  + b \cdot \nabla u + (-\Delta)^s u = 0,\]
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 ==
=== [[Surface quasi-geostrophic equation|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 $u = \nabla K \ast \theta$.

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 $u = \nabla K \ast \theta$.