Proof of Restriction of Viscosity Sub- and Supersolutions to an Open Subset

Each result cited is universally quantified over the data in its own statement. Balls are taken in $(\mathbb{R}^{n},d_{E})$, and local maxima and minima are those of Local Maximum of a Function Relative to a Subset of a Metric Space, as fixed by Second-Order Equations on Euclidean Open Sets §extrema.

Proof of claim 1. By claim 4 of Semicontinuity and Continuity Under Composition with a Continuous Map, $u|_{V}$ is upper semicontinuous on $V$. Let $\varphi:V\to\mathbb{R}$ be of class $C^{2}$ on $V$ and let $x_{0}\in V$ be a point at which $u|_{V}-\varphi$ has a local maximum relative to $V$: there is a positive $\rho$ with

We must show $F(x_{0},u(x_{0}),D\varphi(x_{0}),D^{2}\varphi(x_{0}))\le0$, which is the subsolution inequality for $F|_{V}$ at $x_{0}$, the two operators agreeing at points of $V$.

Radii. Since $V$ is open (Second-Order Equations on Euclidean Open Sets §space), there is a positive $\rho'$ such that every $y\in\mathbb{R}^{n}$ with $d_{E}(y,x_{0})<\rho'$ lies in $V$. Let $s$ be half the least of $\rho$ and $\rho'$ and let $r$ be half of $s$ (claims 8 and 9 of Elementary Order Arithmetic in an Ordered Field); then $0<r<s<\rho'$ and $s<\rho$.

The cut-off test function. By Existence of Smooth Bump Functions on Euclidean Space there is a smooth $\chi:\mathbb{R}^{n}\to\mathbb{R}$ with $\chi(y)=1$ when $d_{E}(y,x_{0})\le r$ and $\chi(y)=0$ when $d_{E}(y,x_{0})\ge s$. Its partial derivatives of every order exist and are continuous at every point by Smooth Map on an Open Subset of Euclidean Space, so $\chi$ is of class $C^{2}$ on $\mathbb{R}^{n}$ by C^k Maps on a Euclidean Open Set, and $\chi|_{V}$ is of class $C^{2}$ on $V$ by claim 3 of Restriction of a $C^k$ Map to an Open Subset. Define $\psi:U\to\mathbb{R}$ by $\psi(y)=\chi(y)\varphi(y)$ for $y\in V$ and $\psi(y)=0$ for $y\in U\setminus V$.

(a) $\psi$ vanishes far from $x_{0}$: if $y\in U$ and $s\le d_{E}(y,x_{0})$, then $\psi(y)=0$, because either $y\notin V$, or $y\in V$ and $\chi(y)=0$.

(b) $\psi$ equals $\varphi$ near $x_{0}$: the open ball $B$ of radius $r$ about $x_{0}$ lies in $V$ (as $r<\rho'$) and $\psi=\varphi$ on $B$, because $\chi=1$ there.

(c) $\psi$ is of class $C^{2}$ on $U$. Let $y\in U$. We produce an open ball $W\subseteq U$ about $y$ and a function $h$, of class $C^{2}$ on an open set containing $W$, with $\psi=h$ on $W$. If $d_{E}(y,x_{0})<\rho'$, then $y\in V$; take $W$ an open ball about $y$ contained in $V$ and $h=(\chi|_{V})\varphi$, which is of class $C^{2}$ on $V$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. If $\rho'\le d_{E}(y,x_{0})$, then $s<d_{E}(y,x_{0})$; take $W$ an open ball about $y$ contained in $U$ of radius at most $d_{E}(y,x_{0})-s$ and $h=0$ on $U$, which is of class $C^{2}$ by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set; every $z\in W$ satisfies $s<d_{E}(z,x_{0})$ by the triangle inequality of the metric axioms, so $\psi(z)=0=h(z)$ by (a). In either case, claim 1 of Restriction of a $C^k$ Map to an Open Subset, applied to $\psi|_{W}=h|_{W}$ and then to the first partial derivatives of both, shows that the first and second partial derivatives of $\psi$ exist at every point of $W$ and coincide there with those of $h$; since those of $h$, and $h$ itself, are continuous at $y$ and agree with those of $\psi$ on the open ball $W$ about $y$, the corresponding functions for $\psi$ are continuous at $y$. As $y\in U$ was arbitrary, $\psi$ is of class $C^{2}$ on $U$ by C^k Maps on a Euclidean Open Set.

(d) Same test data at $x_{0}$. By (b) and claim 1 of Restriction of a $C^k$ Map to an Open Subset, applied on $B$ to $\psi$ and $\varphi$ and then to their first partial derivatives, $D\psi(x_{0})=D\varphi(x_{0})$ and $D^{2}\psi(x_{0})=D^{2}\varphi(x_{0})$.

Conclusion. Let $y\in U$ with $d_{E}(x_{0},y)<r$. Then $y\in B\subseteq V$, $\psi(y)=\varphi(y)$ by (b), and $d_{E}(x_{0},y)<\rho$, so

Thus $u-\psi$ has a local maximum at $x_{0}$ relative to $U$, and since $u$ is a viscosity subsolution of $F$ on $U$ and $\psi$ is of class $C^{2}$ on $U$ by (c), $F(x_{0},u(x_{0}),D\psi(x_{0}),D^{2}\psi(x_{0}))\le0$. By (d) this is the required inequality.

Proof of claim 2. The same argument applies with lower semicontinuity (claim 4 of Semicontinuity and Continuity Under Composition with a Continuous Map), local minima, the inequality $u(y)-\varphi(y)\ge u(x_{0})-\varphi(x_{0})$ near $x_{0}$, and the supersolution inequality $0\le F(x_{0},u(x_{0}),D\psi(x_{0}),D^{2}\psi(x_{0}))$ in place of their counterparts; the construction of $\psi$ and steps (a)–(d) are unchanged.