Interior regularity results (local)

From Mwiki

(Redirected from Interior regularity results)
Jump to: navigation, search

Let \(\Omega\) be an open domain and \( u \) a solution of an elliptic equation in \( \Omega \). The following theorems say that \( u \) satisfies some regularity estimates in the interior of \( \Omega \) (but not necessarily up the the boundary).


Linear equations

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.

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.

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.

\( {\rm div \,} A(x) Du + b(x) \cdot \nabla u = 0 \)

then \(u\) is Holder continuous if \(A\) is just uniformly elliptic and \(b\) is in \(L^n\) (or \(BMO^{-1}\) if \({\rm div \,} b=0\)).

\(a_{ij}(x) u_{ij} + b \cdot \nabla u = f \)

with \(a_{ij}\) unif elliptic, \(b \in L^n\) and \(f \in L^n\), then the solution is \(C^\alpha\)

\(a_{ij}(x) u_{ij} = f\)

with \(a_{ij}\) close enough to the identity (or continuous) and \(f \in L^p\), then \(u\) is in \(W^{2,p}\).

\(a_{ij}(x) u_{ij} = f\)

with \(a_{ij}\) close enough to the identity uniformly and $f \in L^\infty$, then $u$ is in $C^{1,\alpha}$

\(a_{ij}(x) u_{ij} = f\)

with $a_{ij}$ close enough to the identity in a scale invariant Morrey norm in terms of $L^n$ and $f \in L^n$, then $u$ is in $C^{1,\alpha}$.

($a_{ij} \in VMO$ is a particular case of this)

\(a_{ij}(x) u_{ij} = f\)

with $a_{ij}$ in $C^\alpha$ and $f \in C^\alpha$, then $u$ is in $C^{2,\alpha}$

Non linear equations

For any smooth strictly convex Lagrangian $L$, minimizers of functionals

$ \int_D L(\nabla u) \ dx $

are smooth (analytic if $L$ is analytic).

Any continuous function $u$ such that

$M^+(D^2 u) \geq 0 \geq M^-(D^2 u)$

in the viscosity sense (where $M^+$ and $M^-$ are the Pucci operators), is Holder continuous.

(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)

If $u$ solves a fully nonlinear equation

$F(D^2 u, Du, u, x) = 0$

which is degenerate elliptic but satisfies some structure conditions and some smoothness assumptions respect to $x$, then $u$ is Lipschitz.

(The proof of this is based on the uniqueness technique for viscosity solutions)

Any continuous function $u$ such that

$0 \geq M^-(D^2 u)$

in the viscosity sense, is twice differentiable almost everywhere and $D^2 u \in L^\varepsilon$.

(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)

If $u$ solves a fully nonlinear equation

$F(D^2 u, x) = 0$

which is uniformly elliptic and continuous respect to $x$ ($VMO$ is actually enough), then $u \in C^{1,\alpha}$.

If $u$ solves a convex (or concave) fully nonlinear equation

$F(D^2 u, x) = 0$

which is uniformly elliptic and $C^\alpha$ respect to $x$, then $u \in C^{2,\alpha}$.

Personal tools