Krylov-Safonov theorem

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Krylov-Safonov theorem [1] [2] 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 theorem is the nondivergence version of De Giorgi-Nash-Moser theorem.

For nonlocal equations, there are analogous results both for Holder estimates and the Harnack inequality.

Contents

Elliptic case

Holder continuity

Given a bounded solution of the following elliptic PDE \[ a_{ij}(x) \partial_{ij} u (x) + b(x) \cdot \nabla u(x) = f(x) \qquad \text{in } B_1,\] 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

Given a nonnegative solution of the following elliptic PDE \[ a_{ij}(x) \partial_{ij} u (x) + b(x) \cdot \nabla u(x) = f(x) \qquad \text{in } B_1,\] 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

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 \begin{align*} 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.


Parabolic case

The elliptic case is implied by the results in the parabolic setting.

Holder estimates

The Holder estimate are similar in the parabolic case as in the elliptic case. Let us define the parabolic cylinder \[ Q_r(x_0,t_0) = \{(x,t) : |x-x_0|<r \text{ and } 0 \leq t_0 - t < t^2 \}.\]

Given a bounded solution of the parabolic PDE \[ u_t(x,t) - a_{ij}(x,t) \partial_{ij} u (x,t) = f(x,t) \qquad \text{in } Q_1(0,0),\] 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

In the parabolic Harnack inequality, the infimum and the maximum must be taken in cylinders which are shifted in time.

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 $e$ (some previous argument is needed to get that $u$ is Lipchitz).

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

  1. Safonov, M. V.; Krylov, N. V. (1979), "An estimate for the probability of a diffusion process hitting a set of positive measure", Doklady Akademii Nauk SSSR 245 (1): 18–20, ISSN 0002-3264 
  2. Safonov, M. V.; Krylov, N. V. (1980), "A property of the solutions of parabolic equations with measurable coefficients", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 44 (1): 161–175, ISSN 0373-2436 
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