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.
- $\ue \in C (\mathbb T ^n)$.
- $\ue \downarrow u$ 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
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
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 $\delta \to 0$.