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| Let <math>\Omega</math> be an open domain and <math> u </math> a solution of an elliptic equation in <math> \Omega </math>. The following theorems say that <math> u </math> satisfies some regularity estimates in the interior of <math> \Omega </math> (but not necessarily up the the boundary).
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| | The Boltzmann equation is an evolution equation used to describe the configuration of particles in a gas, but only statistically. Specifically, if the probability that a particle in the gas lies in some region $A$ of phase space $\mathbb{R}^d\times \mathbb{R}^d$ at time $t$ is given by |
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| == Linear equations ==
| | \begin{equation*} |
| | \int_A f(x,v,t)dxdy |
| | \end{equation*} |
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| Regularity results for linear equations are applicable to nonlinear equations as well through the linearization of the equation. However, this process requires some initial regularity knowledge on the solution (since the coefficients of the linearization depend on the solution itself). Therefore, the less regularity required for the coefficients, the more useful the theorem is.
| | then $f(x,v,t)$ solves the non-local equation |
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| All regularity results that require some modulus of continuity or smallness condition for the coefficients rely on the idea that the solution is locally close to a solution to an equation with constant coefficients. The proof is based on an estimate on how far these two solutions are at small scales. These type of arguments are often called [[perturbation methods]].
| | \begin{equation*} |
| | \partial_t f + v \cdot \nabla_x f = Q(f,f) |
| | \end{equation*} |
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| From the results below for linear equations, [[De Giorgi-Nash-Moser]] and [[Krylov-Safonov]] are the only non perturbative results. Their assumptions are scale invariant in the sense that a rescaling of the solution ($u_r(x) = u(rx)$) would solve an elliptic equation with the same bounds as the original.
| | where $Q(f,f)$ is the Boltzmann collision operator, given by |
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| * [[De Giorgi-Nash-Moser]] | | \begin{equation*} |
| <math> {\rm div \,} A(x) Du + b(x) \cdot \nabla u = 0 </math>
| | Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(v-v_*,e) (f(v')f(v'_*)-f(v)f(v_*) d\sigma(e) dv_* |
| | \end{equation*} |
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| then <math>u</math> is Holder continuous if <math>A</math> is just uniformly elliptic and <math>b</math>
| | where $d\sigma$ denotes the Hausdorff measure on $\mathbb{S}^{d-1}$, and given $v,v_* \in \mathbb{R}^d$ and $e \in \mathbb{S}^{d-1}$ we write |
| is in <math>L^n</math> (or <math>BMO^{-1}</math> if <math>{\rm div \,} b=0</math>).
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| * [[Krylov-Safonov]] | | \begin{align*} |
| <math>a_{ij}(x) u_{ij} + b \cdot \nabla u = f </math>
| | v' & = v-(v-v_*,e)e\\ |
| | v'_* & = v_*+(v-v_*,e)e |
| | \end{align*} |
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| with <math>a_{ij}</math> unif elliptic, <math>b \in L^n</math> and <math>f \in L^n</math>, then the
| | where $B$, known as the Boltzmann collision kernel, measures the strength of collisions in different directions. |
| solution is <math>C^\alpha</math>
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| * [[Calderon-zygmund]]
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| <math>a_{ij}(x) u_{ij} = f</math>
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| with <math>a_{ij}</math> close enough to the identity and <math>f \in L^p</math>, then <math>u</math> is in <math>W^{2,p}</math>.
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| * [[Cordes-Nirenberg]]
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| <math>a_{ij}(x) u_{ij} = f</math>
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| with <math>a_{ij}</math> close enough to the identity uniformly and f in L^infty,
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| then $u$ is in $C^{1,\alpha}$
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| * [[Cordes-Nirenberg improved]] (corollary of work of Caffarelli for nonlinear equations)
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| <math>a_{ij}(x) u_{ij} = f</math>
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| with $a_{ij}$ close enough to the identity in a scale invariant Morrey
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| norm in terms of $L^n$ and $f \in L^n$, then $u$ is in $C^{1,\alpha}$.
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| ($a_{ij} \in VMO$ is a particular case of this)
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| *[[Schauder]]
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| <math>a_{ij}(x) u_{ij} = f</math>
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| with $a_{ij}$ in $C^\alpha$ and $f \in C^\alpha$, then $u$ is in $C^{2,\alpha}$
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| == Non linear equations ==
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| * [[De Giorgi-Nash-Moser]]
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| For any smooth strictly convex Lagrangian $L$, minimizers of functionals
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| $ \int_D L(\nabla u) \ dx $
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| are smooth (analytic if $L$ is analytic).
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| * [[Krylov-Safonov|Krylov-Safonov-Caffarelli]]
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| Any continuous function $u$ such that
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| $M^+(D^2 u) \geq 0 \geq M^-(D^2 u)$
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| in the viscosity sense (where $M^+$ and $M^-$ are the Pucci operators), is Holder continuous.
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| (Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)
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| * [[Ishii-Lions]]
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| If $u$ solves a fully nonlinear equation
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| $F(D^2 u, Du, u, x) = 0$
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| which is degenerate elliptic but satisfies some structure conditions and some smoothness assumptions respect to $x$, then $u$ is Lipschitz.
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| (The proof of this is based on the uniqueness technique for viscosity solutions)
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| * [[Lin]]
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| Any continuous function $u$ such that
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| $0 \geq M^-(D^2 u)$
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| in the viscosity sense, is twice differentiable almost everywhere and $D^2 u \in L^\varepsilon$. | |
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| (Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)
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| * [[Krylov-Safonov|Krylov-Safonov-Caffarelli]]
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| If $u$ solves a fully nonlinear equation
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| $F(D^2 u, x) = 0$
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| which is uniformly elliptic and continuous respect to $x$ ($VMO$ is actually enough), then $u \in C^{1,\alpha}$.
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| * [[Evans-Krylov]]
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| If $u$ solves a convex (or concave) fully nonlinear equation
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| $F(D^2 u, x) = 0$
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| which is uniformly elliptic and $C^\alpha$ respect to $x$, then $u \in C^{2,\alpha}$.
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This article is a stub. You can help this nonlocal wiki by expanding it.
The Boltzmann equation is an evolution equation used to describe the configuration of particles in a gas, but only statistically. Specifically, if the probability that a particle in the gas lies in some region $A$ of phase space $\mathbb{R}^d\times \mathbb{R}^d$ at time $t$ is given by
\begin{equation*}
\int_A f(x,v,t)dxdy
\end{equation*}
then $f(x,v,t)$ solves the non-local equation
\begin{equation*}
\partial_t f + v \cdot \nabla_x f = Q(f,f)
\end{equation*}
where $Q(f,f)$ is the Boltzmann collision operator, given by
\begin{equation*}
Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(v-v_*,e) (f(v')f(v'_*)-f(v)f(v_*) d\sigma(e) dv_*
\end{equation*}
where $d\sigma$ denotes the Hausdorff measure on $\mathbb{S}^{d-1}$, and given $v,v_* \in \mathbb{R}^d$ and $e \in \mathbb{S}^{d-1}$ we write
\begin{align*}
v' & = v-(v-v_*,e)e\\
v'_* & = v_*+(v-v_*,e)e
\end{align*}
where $B$, known as the Boltzmann collision kernel, measures the strength of collisions in different directions.