Proof of The Viscosity Sub- and Supersolution Properties are Local

Conventions. From the setting we use the real numbers and their order, Euclidean space with its sum, difference, dot product, norm and distance $d_{E}$, the set $\mathcal{S}(n)$ with its norm and distance $d_{\mathcal{S}(n)}$, the identity matrix $I_{n}$ and the matrix-vector product, and the notions of class $C^{2}$, gradient, Hessian, semicontinuity and local extrema. Recall from clause Second-Order Equations on Euclidean Open Sets (iii) that $\mathcal{S}(n)$ contains $aI_{n}$ for every real $a$ and is closed under sums and differences, and that $d_{\mathcal{S}(n)}(P,R)=\lVert P-R\rVert$. We write $2\varepsilon$ for $\varepsilon+\varepsilon$. Elementary order facts are those of Elementary Order Arithmetic in an Ordered Field, whose claim 1 is the compatibility of the strict order with addition, the non-strict law and the transitivity of $\le$ being axioms of the ordered field $\mathbb{R}$. We use twice that $|a|=a$ for a positive real $a$: by claim 1 of Properties of the Absolute Value in an Ordered Field the number $|a|$ is nonnegative and equals $a$ or $-a$, and in the second case $-a$ would be nonnegative, hence $a\le 0$ by claim 4 of Elementary Order Arithmetic in an Ordered Field, contrary to the positivity of $a$. In particular $|1|=1$, since $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field, and therefore $|-1|=|1|=1$ by claim 2 of Properties of the Absolute Value in an Ordered Field.

Proof of claim 1. We give the argument for subsolutions and indicate at the end the changes for supersolutions.

Being a viscosity subsolution of $F$ on $U$, the function $u$ is upper semicontinuous on $U$; hence $u|_{V}$ is upper semicontinuous on $V$ by claim 2 of Negation, Restriction, and Separated Differences of Semicontinuous Functions, applied at each point of $V$.

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 $V$. Write $p=D\varphi(x_{0})$ and $A=D^{2}\varphi(x_{0})\in\mathcal{S}(n)$. Since $F|_{V}$ agrees with $F$ at quadruples with first entry in $V$, it suffices to prove that $F(x_{0},u(x_{0}),p,A)\le 0$.

A quadratic majorant. Let $\varepsilon\in\mathbb{R}$ be positive and put $M=A+2\varepsilon I_{n}\in\mathcal{S}(n)$. Let $Q_{0}:\mathbb{R}^{n}\to\mathbb{R}$ be given by $Q_{0}(w)=\tfrac{1}{2}\,w\cdot(Mw)+p\cdot w+\varphi(x_{0})$, which by claim 1 of Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity is of class $C^{2}$ on $\mathbb{R}^{n}$ with $DQ_{0}(w)=Mw+p$ and $D^{2}Q_{0}(w)=M$ for every $w\in\mathbb{R}^{n}$. Let $Q:\mathbb{R}^{n}\to\mathbb{R}$ be given by $Q(z)=Q_{0}(z-x_{0})$; by claim 2 of Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity, applied to $Q_{0}$ on the open set $\mathbb{R}^{n}$ with the translation vector $-x_{0}$, the function $Q$ is of class $C^{2}$ on $\mathbb{R}^{n}$ with $DQ(z)=M(z-x_{0})+p$ and $D^{2}Q(z)=M$. In particular $DQ(x_{0})=p$ and $D^{2}Q(x_{0})=M$, since $M(x_{0}-x_{0})=Mx_{0}-Mx_{0}=0_{\mathbb{R}^{n}}$ by claim 3 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum; and $Q(x_{0})=Q_{0}(0_{\mathbb{R}^{n}})=\varphi(x_{0})$, because $w\cdot 0_{\mathbb{R}^{n}}=w\cdot(z-z)=w\cdot z-w\cdot z=0$ for all $w,z\in\mathbb{R}^{n}$, by the part of claim 5 of The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite concerning differences in the second argument.

By claim 2 of Basic Properties of Twice Differentiability at a Point the function $\varphi$ is twice differentiable at $x_{0}$ with first-order coefficient $p$ and Hessian $A$. Applying that definition with the positive real $\varepsilon$ gives a positive $\delta_{1}\in\mathbb{R}$ such that every $h\in\mathbb{R}^{n}$ with $\lVert h\rVert<\delta_{1}$ satisfies $x_{0}+h\in V$ and

$$\bigl|\varphi(x_{0}+h)-\varphi(x_{0})-p\cdot h-\tfrac{1}{2}\,h\cdot(Ah)\bigr|\le\varepsilon\lVert h\rVert^{2}.$$

By claim 3 of Properties of the Absolute Value in an Ordered Field a real number is at most its own absolute value, so adding $\varphi(x_{0})+p\cdot h+\tfrac{1}{2}h\cdot(Ah)$ to both sides of the resulting inequality gives

$$\varphi(x_{0}+h)\le\varphi(x_{0})+p\cdot h+\tfrac{1}{2}\,h\cdot(Ah)+\varepsilon\lVert h\rVert^{2}.$$

On the other hand claim 1 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum gives $Mh=Ah+(2\varepsilon I_{n})h$, claim 5 of The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite gives $h\cdot(Mh)=h\cdot(Ah)+h\cdot\bigl((2\varepsilon I_{n})h\bigr)$, and claim 6 of Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix gives $h\cdot\bigl((2\varepsilon I_{n})h\bigr)=2\varepsilon\lVert h\rVert^{2}$, whence $\tfrac{1}{2}h\cdot(Mh)=\tfrac{1}{2}h\cdot(Ah)+\varepsilon\lVert h\rVert^{2}$. Therefore

$$\varphi(x_{0}+h)\le\varphi(x_{0})+p\cdot h+\tfrac{1}{2}\,h\cdot(Mh)=Q_{0}(h)=Q(x_{0}+h)\qquad\text{whenever }\lVert h\rVert<\delta_{1}.$$

Transfer of the maximum. By the definition of a local maximum relative to $V$ there is a positive $\delta_{2}\in\mathbb{R}$ such that every $y\in V$ with $d_{E}(x_{0},y)<\delta_{2}$ satisfies $u(y)-\varphi(y)\le u(x_{0})-\varphi(x_{0})$. By claim 9 of Elementary Order Arithmetic in an Ordered Field there is a positive $\delta\in\mathbb{R}$ with $\delta\le\delta_{1}$ and $\delta\le\delta_{2}$. Let $y\in U$ satisfy $d_{E}(x_{0},y)<\delta$ and put $h=y-x_{0}$, so that $y=x_{0}+h$ and, by claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square together with $|-1|=1$, $\lVert h\rVert=\lVert y-x_{0}\rVert=\lVert x_{0}-y\rVert=d_{E}(x_{0},y)<\delta$. By the mixed transitivity of claim 2 of Elementary Order Arithmetic in an Ordered Field we get $\lVert h\rVert<\delta_{1}$ and $d_{E}(x_{0},y)<\delta_{2}$; hence $y\in V$, $\varphi(y)\le Q(y)$ and $u(y)-\varphi(y)\le u(x_{0})-\varphi(x_{0})$, so

$$u(y)-Q(y)\le u(y)-\varphi(y)\le u(x_{0})-\varphi(x_{0})=u(x_{0})-Q(x_{0}).$$

By claim 3 of Restriction of a Cᵏ Function to an Open Subset the restriction $Q|_{U}$ is of class $C^{2}$ on $U$, and by claim 1 of that lemma, applied to $Q$ and to its first partial derivatives, $D(Q|_{U})(x_{0})=DQ(x_{0})=p$ and $D^{2}(Q|_{U})(x_{0})=D^{2}Q(x_{0})=M$. The displayed inequality says that $u-Q|_{U}$ has a local maximum at $x_{0}$ relative to $U$, so, $u$ being a viscosity subsolution of $F$ on $U$,

$$F\bigl(x_{0},u(x_{0}),p,A+2\varepsilon I_{n}\bigr)\le 0 .$$

This holds for every positive $\varepsilon\in\mathbb{R}$.

Passing to the limit. Let $\eta\in\mathbb{R}$ be positive. Since $F$ is continuous it is continuous at $(x_{0},u(x_{0}),p,A)$, so clause Continuity of a Second-Order Equation Operator (i) gives a positive $\delta_{F}\in\mathbb{R}$ such that every quadruple whose four displayed distances to $(x_{0},u(x_{0}),p,A)$ are smaller than $\delta_{F}$ has $F$-value within $\eta$ of $F(x_{0},u(x_{0}),p,A)$. By claim 8 of Elementary Order Arithmetic in an Ordered Field the real number $\tfrac{\delta_{F}}{2}$ is positive and smaller than $\delta_{F}$, and $\varepsilon=\tfrac{1}{2}\cdot\tfrac{\delta_{F}}{2}$ is positive with $2\varepsilon=\tfrac{\delta_{F}}{2}<\delta_{F}$. For this $\varepsilon$ the quadruple $(x_{0},u(x_{0}),p,A+2\varepsilon I_{n})$ satisfies $d_{E}(x_{0},x_{0})=0<\delta_{F}$, $|u(x_{0})-u(x_{0})|=0<\delta_{F}$, $\lVert p-p\rVert=0<\delta_{F}$ and, by claim 6 of Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix,

$$d_{\mathcal{S}(n)}\bigl(A+2\varepsilon I_{n},A\bigr)=\lVert 2\varepsilon I_{n}\rVert=|2\varepsilon|=2\varepsilon<\delta_{F}.$$

Hence $\bigl|F(x_{0},u(x_{0}),p,A+2\varepsilon I_{n})-F(x_{0},u(x_{0}),p,A)\bigr|<\eta$, and claim 3 of Properties of the Absolute Value in an Ordered Field gives

$$F(x_{0},u(x_{0}),p,A)\le F\bigl(x_{0},u(x_{0}),p,A+2\varepsilon I_{n}\bigr)+\eta\le\eta .$$

As $\eta$ was an arbitrary positive real, Comparison of Real Numbers with Arbitrary Positive Slack (i) gives $F(x_{0},u(x_{0}),p,A)\le 0$, which is what had to be shown.

Supersolutions. Suppose $u$ is a viscosity supersolution of $F$ on $U$. Then $u|_{V}$ is lower semicontinuous on $V$ by claim 2 of Negation, Restriction, and Separated Differences of Semicontinuous Functions. Let $\varphi$ be of class $C^{2}$ on $V$ with $u|_{V}-\varphi$ having a local minimum at $x_{0}\in V$ relative to $V$, and keep the notation $p$, $A$. Run the argument above with $M=A-2\varepsilon I_{n}$: the same two-sided estimate, now used through the inequality $-\bigl|\varphi(x_{0}+h)-\varphi(x_{0})-p\cdot h-\tfrac{1}{2}h\cdot(Ah)\bigr|\le\varphi(x_{0}+h)-\varphi(x_{0})-p\cdot h-\tfrac{1}{2}h\cdot(Ah)$ of claim 3 of Properties of the Absolute Value in an Ordered Field, gives $Q(x_{0}+h)\le\varphi(x_{0}+h)$ whenever $\lVert h\rVert<\delta_{1}$, so that $u-Q|_{U}$ has a local minimum at $x_{0}$ relative to $U$ and $0\le F(x_{0},u(x_{0}),p,A-2\varepsilon I_{n})$ for every positive $\varepsilon$. Since $d_{\mathcal{S}(n)}(A-2\varepsilon I_{n},A)=\lVert -2\varepsilon I_{n}\rVert=|-2\varepsilon|=2\varepsilon$ by claim 6 of Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix and claim 2 of Properties of the Absolute Value in an Ordered Field, the same limiting argument together with Comparison of Real Numbers with Arbitrary Positive Slack (ii) gives $0\le F(x_{0},u(x_{0}),p,A)$.

Proof of claim 2. Assume that every $x\in U$ has an open neighbourhood $V$ with $x\in V\subseteq U$ on which $u|_{V}$ is a viscosity subsolution of $F|_{V}$.

Upper semicontinuity. Let $x\in U$ and let $V$ be such a neighbourhood. Since $V$ is open, claim 4 of The Interior is the Largest Open Subset gives $\operatorname{int}_{\mathbb{R}^{n}}(V)=V$, so claim 1 of Interior Points in the Metric Topology are Exactly the Centres of Contained Closed Balls provides a positive $r\in\mathbb{R}$ with $\{z\in\mathbb{R}^{n}:d_{E}(z,x)\le r\}\subseteq V$; in particular $\{z\in U:d_{E}(z,x)\le r\}\subseteq V$. The function $u|_{V}$ is upper semicontinuous on $V$, in particular at $x$ relative to $V$, so claim 3 of Semicontinuity and the Semicontinuous Envelopes are Local Notions, applied with $U$ in the role of the ambient subset and this $r$, shows that $u$ is upper semicontinuous at $x$ relative to $U$. As $x\in U$ was arbitrary, $u$ is upper semicontinuous on $U$.

The test inequality. Let $\varphi:U\to\mathbb{R}$ be of class $C^{2}$ on $U$ and let $x\in U$ be a point at which $u-\varphi$ has a local maximum relative to $U$, with modulus $\delta$. Let $V$ be a neighbourhood of $x$ as above. By claim 3 of Restriction of a Cᵏ Function to an Open Subset the restriction $\varphi|_{V}$ is of class $C^{2}$ on $V$, and by claim 1 of that lemma, applied to $\varphi$ and to its first partial derivatives, $D(\varphi|_{V})(x)=D\varphi(x)$ and $D^{2}(\varphi|_{V})(x)=D^{2}\varphi(x)$. Every $y\in V$ with $d_{E}(x,y)<\delta$ lies in $U$, so $u(y)-\varphi(y)\le u(x)-\varphi(x)$; hence $u|_{V}-\varphi|_{V}$ has a local maximum at $x$ relative to $V$. Since $u|_{V}$ is a viscosity subsolution of $F|_{V}$ on $V$,

$$F\bigl(x,u(x),D\varphi(x),D^{2}\varphi(x)\bigr)=F|_{V}\bigl(x,u(x),D(\varphi|_{V})(x),D^{2}(\varphi|_{V})(x)\bigr)\le 0 .$$

Thus $u$ is a viscosity subsolution of $F$ on $U$. The supersolution case is identical, with lower semicontinuity in place of upper semicontinuity, local minima in place of local maxima, and the inequality reversed. $\blacksquare$