Jensen's Approximation
Introduction
Given a continuous function u \in C (\mathbb T ^n), consider \newcommand{\xo}{x _0} \newcommand{\ue}{u ^\e} \newcommand{\xos}{x _0 ^*} \newcommand{\Pd}{P ^\delta} \newcommand{\ve}{v _\e} \begin{align*} \ue (\xo) := \sup _x \set{ u (x) - \frac1\e \abs{x - \xo} ^2 } = \sup _h \set{ u (\xo - h) - \frac1\e \abs h ^2. }. \end{align*} Think each of the function inside \set{} as a function of \xo, there are two intepretations:
- \ue is a sup of a family of paraboloids (concave of opening \frac2\e). (Left)
- \ue is a sup of a family of translations of u (h horizontally, \frac1\e\abs h ^2 downwards). (Right)
It has the following properties.
- \ue (\xo) = u (\xos) - \frac1\e \abs{x - \xos} ^2 for some \xos. \ue (\xo) = u (\xo - h ^*) - \frac1\e \abs{h ^*} ^2.
This is because the sup is attainable in a compact set.
- \ue (\xo) \ge u(\xo).
By definition, set x = \xo.
- \ue is \frac2\e \diam(\mathbb T)-Lipschitz.
This is because for every x, u (x) - \frac1\e\abs{x - \xo} ^2 is \frac2\e \diam(\mathbb T)-Lipschitz in \xo.
- \ue decreases as \e \to 0.
Because u (x) - \frac1\e\abs{x - \xo} ^2 decreases as \e \to 0.
- \abs{\xos - \xo} ^2 \le \e \osc u.
Because u (\xo) \le u (\xos) - \frac1\e\abs{\xos - \xo} ^2.
- 0 \le \ue (\xo) - u (\xo) \le u (\xos) - u (\xo).
Because 1. and 2.
Lemma.
- \ue \in C (\mathbb T ^n).
- \ue \downarrow u uniformly as \e \to 0.
- For every \xo there exists a concave paraboloid of opening 2/\e that touches \ue by below at \xo. Hence \ue is C ^{1,1} by below. In particular, \ue is punctually second order differentiable almost everywhere.
Proof.
- We even have Lipschitz continuity from Property 3.
- Property 4. implies monotonocity. Uniformily can be seen from \begin{align*} \ue (\xo) - u (\xo) \le u (\xos) - u(\xo) \to 0 \end{align*} uniformly since \xos \to \xo uniformly and u is uniform continuous.
- Because Property 1. asserts that the sup is attained by \xos, \ue is touched by a paraboloid by below.
Jensenn’s Approximate Solutions
Theorem. Suppose u is a viscosity subsolution of F (D ^2 u) = 0, we have that \ue is a viscosity subsolution of F (D ^2 u) = 0. In particular F (D ^2 \ue) \ge 0 a.e.
Proof.
Let P (x) touches \ue at \xo by above. Then it touches a translation of u by above (using the second intepretation mentioned at the beginning): P (\xo) = \ue (\xo) = u (\xo - h ^*) - \frac1\e \abs{h ^*} ^2.
Consider \begin{align*} Q (x) = P (x + h ^*) + \frac1\e \abs{h ^\ast} ^2. \end{align*} Then Q touches u by above at \xos. Since F (D ^2 u) \ge 0 in the viscosity sense, \begin{align*} 0 \le F (D ^2 Q) = F(D ^2 P). \end{align*} Therefore \ue is a viscosity subsolution.
Uniqueness
Theorem. Let u be a viscosity subsolution and v be a viscosity supersolution to F (D ^2 w) = 0 in \Omega. Then \begin{align*} u - v \in \underline S (\lambda/n, \Lambda). \end{align*}
Proof.
It suffices to show f := u ^\e - \ve \in \underline S (\lambda/n, \Lambda) in the interior because S is closed under uniform convergence. Let P be a paraboloid that touches f from above at \xo.
By Approximation theorem, \ue and \ve are again viscosity sub/supersolutions, and punctually second order differentiable almost everywhere. In particular, if they are both punctually second order differentiable at \xo, then \begin{align*} 0 \le F (D ^2 \ue (\xo)) - F (D ^2 \ve (\xo)) \le \mathcal M ^+ (D ^2 f (\xo); \lambda/n, \Lambda) \le \mathcal M ^+ (D ^2 P (\xo); \lambda/n, \Lambda) \end{align*} which finishes the proof. The problem is that we don’t know if \ue and \ve are punctually second order differentiable at \xo.
To overcome this, we fix a small r > 0 and define \begin{align*} \Pd (x) = P (x) + \delta \abs{x - \xo} ^2 - \delta r ^2. \end{align*} That is, we pinch \Pd a little such that it bends more downward without changing its value on \partial B _r (\xo). Then we still have \Pd > f on \partial B _r (\xo) but \Pd < f at \xo. Define w = \Pd - f. Let \Gamma _w be the convex envelope of -w ^- \ind{B _r (\xo)}, then since w can be touched from above by a paraboloid with uniform opening at any given point (\ue can be touched by below, \ve can be touched from above by a paraboloid of opening 2/\e, and \Pd is a paraboloid), by Lemma 3.5 we know that the contact set \set{w = \Gamma _w} has positive measure.
Let x _1 be a point in B _r intersect \set{w = \Gamma _w} intersect the punctual second order differentiability set of \ue - \ve. Then at this point, D ^2 w (x _1) = D ^2 \Gamma _w (x _1) is nonnegative, so \begin{align*} 0 &\le F (D ^2 \ue (x _1)) - F (D ^2 \ve (x _1)) \newline &\le F (D ^2 \ue (x _1) + D ^2 w (x _1)) - F (D ^2 \ve (x _1)) \newline &\le \mathcal M ^+ (D ^2 w (x _1) + D ^2 f (x _1); \lambda/n, \Lambda) \newline &\le \mathcal M ^+ (D ^2 \Pd; \lambda/n, \Lambda) \newline &\le \mathcal M ^+ (D ^2 P; \lambda/n, \Lambda) + 2 \Lambda \delta. \end{align*} Finally take \delta \to 0.
Corollary. The Dirichlet problem \begin{align*} \begin{cases} F (D ^2 u) = 0 & \inn \Omega, \newline u = \vp & \onn \partial \Omega \end{cases} \end{align*} has at most one viscosity solution u \in C (\bar \Omega).