Proof of The Bump Construction: Raising a Subsolution Whose Lower Envelope Fails the Supersolution Inequality

Conventions. From the setting we use the real numbers with their order and absolute value, Euclidean space with its sum, difference, dot product, norm and distance $d_{E}$, the set $\mathcal{S}(n)$ with its norm and distance, the identity matrix $I_{n}$, and the notions of class $C^{2}$, gradient, Hessian, semicontinuity and local extrema. 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 being an axiom of the ordered field $\mathbb{R}$, whose order $\le$ is a total order; a strict inequality implies the corresponding non-strict one. We use repeatedly that a finite list of positive reals admits a positive lower bound, obtained by iterating claim 9 of Elementary Order Arithmetic in an Ordered Field, and that $\tfrac{s}{2}$ is positive and smaller than $s$ for positive $s$, by claim 8 of that lemma; $\tfrac{s}{4}$ abbreviates $\tfrac{1}{2}\cdot\tfrac{s}{2}$. Multiplying an inequality by a nonnegative real is claim 5 of Elementary Arithmetic in an Ordered Field.

Write $a=u_{*}(\hat x)$, $p=D\varphi(\hat x)$, $A=D^{2}\varphi(\hat x)$ and $\eta=-F(\hat x,a,p,A)$, which is positive by claim 4 of Elementary Order Arithmetic in an Ordered Field. Let $\kappa\in\mathbb{R}$ be positive.

Step 1: the data of the construction. By the definition of a local minimum there is a positive $r_{1}$ such that

$$(1)\qquad u_{*}(\hat x)-\varphi(\hat x)\le u_{*}(x)-\varphi(x)\qquad\text{for every }x\in\Omega\text{ with }d_{E}(\hat x,x)<r_{1}.$$

Since $\Omega$ is open, claim 4 of The Interior is the Largest Open Subset and claim 1 of Interior Points in the Metric Topology are Exactly the Centres of Contained Closed Balls give a positive $r_{2}$ with $\{x\in\mathbb{R}^{n}:d_{E}(x,\hat x)\le r_{2}\}\subseteq\Omega$.

Since $F$ is continuous it is continuous at $(\hat x,a,p,A)$, so clause Continuity of a Second-Order Equation Operator (i), applied with the positive real $\eta$, provides a positive $\theta$ such that all $y\in\Omega$, $s\in\mathbb{R}$, $q\in\mathbb{R}^{n}$ and $Y\in\mathcal{S}(n)$ with

$$d_{E}(y,\hat x)<\theta,\qquad|s-a|<\theta,\qquad\lVert q-p\rVert<\theta,\qquad d_{\mathcal{S}(n)}(Y,A)<\theta$$

satisfy $|F(y,s,q,Y)-F(\hat x,a,p,A)|<\eta$ and therefore, by claim 3 of Properties of the Absolute Value in an Ordered Field,

$$(2)\qquad F(y,s,q,Y)<F(\hat x,a,p,A)+\eta=0 .$$

By claims 1, 2 and 3 of Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function the maps $x\mapsto\varphi(x)$, $x\mapsto D\varphi(x)$ and $x\mapsto D^{2}\varphi(x)$ are continuous at $\hat x$ relative to $\Omega$, so there are positive $\rho_{1},\rho_{2},\rho_{3}$ such that every $y\in\Omega$ with $d_{E}(\hat x,y)$ smaller than the corresponding radius satisfies, respectively,

$$|\varphi(y)-\varphi(\hat x)|<\tfrac{\theta}{4},\qquad\lVert D\varphi(y)-p\rVert<\tfrac{\theta}{2},\qquad d_{\mathcal{S}(n)}\bigl(D^{2}\varphi(y),A\bigr)<\tfrac{\theta}{2}.$$

Choose a positive $r$ with

$$r\le\rho_{1},\quad r\le\rho_{2},\quad r\le\rho_{3},\quad r\le r_{2},\quad r\le 1,\quad r<r_{1},\quad r<\theta,\quad r<\kappa ,$$

which is possible by the convention above (take a positive lower bound of the eight numbers $\rho_{1},\rho_{2},\rho_{3},r_{2},1,\tfrac{r_{1}}{2},\tfrac{\theta}{2},\tfrac{\kappa}{2}$). Put $\gamma=\tfrac{\theta}{8}$, a positive real, and let $\delta$ be a positive real with $\delta\le\tfrac{\theta}{4}$ and $\delta\le\tfrac{\gamma r^{2}}{8}$, where $r^{2}=r\cdot r$.

Since $0<r\le1$, multiplying $r\le 1$ by the nonnegative $r$ gives $r^{2}\le r\le 1$, so

$$(3)\qquad 2\gamma=\tfrac{\theta}{4}\le\tfrac{\theta}{2},\qquad 2\gamma r\le 2\gamma\le\tfrac{\theta}{2},\qquad \gamma r^{2}\le\gamma\le\tfrac{\theta}{4}.$$

Step 2: the tilted test function. Let $Q_{0}:\mathbb{R}^{n}\to\mathbb{R}$ be the quadratic function given by $Q_{0}(w)=\tfrac{1}{2}\,w\cdot\bigl((2\gamma I_{n})w\bigr)+0_{\mathbb{R}^{n}}\cdot w+\bigl(\varphi(\hat x)-a-\delta\bigr)$. Since $0_{\mathbb{R}^{n}}=w-w$ and claim 3 of The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite gives $(w-w)\cdot w=w\cdot w-w\cdot w=0$, and since claim 6 of Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix gives $w\cdot\bigl((2\gamma I_{n})w\bigr)=2\gamma\lVert w\rVert^{2}$, we have $Q_{0}(w)=\gamma\lVert w\rVert^{2}+\varphi(\hat x)-a-\delta$; and by claim 1 of Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity the function $Q_{0}$ is of class $C^{2}$ on $\mathbb{R}^{n}$ with $DQ_{0}(w)=2\gamma w$ and $D^{2}Q_{0}(w)=2\gamma I_{n}$. By claim 2 of Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity, applied with the translation vector $-\hat x$, the function $Q(x)=Q_{0}(x-\hat x)=\gamma\lVert x-\hat x\rVert^{2}+\varphi(\hat x)-a-\delta$ is of class $C^{2}$ on $\mathbb{R}^{n}$ with $DQ(x)=2\gamma(x-\hat x)$ and $D^{2}Q(x)=2\gamma I_{n}$; its restriction to $\Omega$ is of class $C^{2}$ there with the same gradient and Hessian, by claims 3 and 1 of Restriction of a Cᵏ Function to an Open Subset. Let $g:\Omega\to\mathbb{R}$ be given by

$$g(x)=\varphi(x)-\varphi(\hat x)+a+\delta-\gamma\lVert x-\hat x\rVert^{2}.$$

Then $g(x)=\varphi(x)-Q(x)$ for every $x\in\Omega$, so, sums and scalar multiples of functions of class $C^{2}$ being of class $C^{2}$ by claim 3 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, the function $g$ is of class $C^{2}$ on $\Omega$; and by claim 1 of that lemma, applied to the first partial derivatives and then to their partial derivatives,

$$(4)\qquad Dg(x)=D\varphi(x)-2\gamma(x-\hat x),\qquad D^{2}g(x)=D^{2}\varphi(x)-2\gamma I_{n}\qquad\text{for }x\in\Omega .$$

Note also $g(\hat x)=a+\delta$.

Step 3: $g$ is a classical subsolution on the ball of radius $r$. Put $V_{1}=\{x\in\mathbb{R}^{n}:d_{E}(x,\hat x)<r\}$, an open ball, hence open, and contained in $\Omega$ because $r\le r_{2}$. Let $y\in V_{1}$ and write $h=y-\hat x$, so that $\lVert h\rVert=d_{E}(y,\hat x)<r$ 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. Then $d_{E}(y,\hat x)<r<\theta$. Next,

$$|g(y)-a|=\bigl|\varphi(y)-\varphi(\hat x)+\delta-\gamma\lVert h\rVert^{2}\bigr|\le|\varphi(y)-\varphi(\hat x)|+\delta+\gamma\lVert h\rVert^{2}<\tfrac{\theta}{4}+\tfrac{\theta}{4}+\tfrac{\theta}{4}<\theta$$

by the triangle inequality of claim 5 of Properties of the Absolute Value in an Ordered Field (applied twice), by claim 1 of that lemma for the nonnegative terms $\delta$ and $\gamma\lVert h\rVert^{2}$, by the choice of $\delta$, and by $\gamma\lVert h\rVert^{2}\le\gamma r^{2}\le\tfrac{\theta}{4}$, which uses claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and (3). Similarly, by (4), the triangle inequality of claim 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, claim 5 of that lemma and (3),

$$\lVert Dg(y)-p\rVert\le\lVert D\varphi(y)-p\rVert+2\gamma\lVert h\rVert<\tfrac{\theta}{2}+2\gamma r\le\tfrac{\theta}{2}+\tfrac{\theta}{2}$$

is at most $\theta$, and in fact smaller, since the first inequality is strict; and, using claim 5 of Properties of the Norm of a Symmetric Real Matrix for the triangle inequality of the matrix norm and claim 6 of Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix for $\lVert 2\gamma I_{n}\rVert=2\gamma$,

$$d_{\mathcal{S}(n)}\bigl(D^{2}g(y),A\bigr)=\lVert D^{2}\varphi(y)-2\gamma I_{n}-A\rVert\le\lVert D^{2}\varphi(y)-A\rVert+\lVert 2\gamma I_{n}\rVert<\tfrac{\theta}{2}+\tfrac{\theta}{2}=\theta .$$

Hence (2) applies with $(y,g(y),Dg(y),D^{2}g(y))$ and gives $F(y,g(y),Dg(y),D^{2}g(y))<0$, so in particular this quantity is at most $0$. As $y\in V_{1}$ was arbitrary, and since by claims 3 and 1 of Restriction of a Cᵏ Function to an Open Subset the restriction of $g$ to the open set $V_{1}$ is of class $C^{2}$ there, with the same gradient and Hessian at every point of $V_{1}$, that restriction is a classical subsolution of the restricted operator $F|_{V_{1}}$ on $V_{1}$, which is a second-order equation operator on $V_{1}$, degenerate elliptic and continuous, by claims 1, 2 and 3 of Restriction of a Second-Order Equation Operator to an Open Subset. By claim 1 of Classical Sub- and Supersolutions of a Degenerate Elliptic Operator are Viscosity Sub- and Supersolutions it is a viscosity subsolution of $F|_{V_{1}}$ on $V_{1}$.

Step 4: $u$ dominates $g$ on the outer part of the ball. Let $x\in\Omega$ satisfy $\tfrac{r}{2}\le d_{E}(x,\hat x)<r$. Then $d_{E}(\hat x,x)<r<r_{1}$, so (1) gives $a-\varphi(\hat x)\le u_{*}(x)-\varphi(x)$, and claim 2 of Properties of the Lower Semicontinuous Envelope, by Duality gives $u_{*}(x)\le u(x)$; hence

$$u(x)\ge a+\varphi(x)-\varphi(\hat x)=g(x)-\delta+\gamma\lVert x-\hat x\rVert^{2}.$$

Since $\tfrac{r}{2}\le\lVert x-\hat x\rVert$ and both sides are nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\tfrac{r^{2}}{4}\le\lVert x-\hat x\rVert^{2}$, so $\gamma\tfrac{r^{2}}{4}\le\gamma\lVert x-\hat x\rVert^{2}$; and $\delta\le\tfrac{\gamma r^{2}}{8}<\gamma\tfrac{r^{2}}{4}$ by claim 8 of Elementary Order Arithmetic in an Ordered Field. Therefore $-\delta+\gamma\lVert x-\hat x\rVert^{2}$ is positive and

$$(5)\qquad g(x)<u(x)\qquad\text{for every }x\in\Omega\text{ with }\tfrac{r}{2}\le d_{E}(x,\hat x)<r .$$

Step 5: the glued function. Define $U_{\kappa}:\Omega\to\mathbb{R}$ by

$$U_{\kappa}(x)=\max\{u(x),g(x)\}\ \text{ if }d_{E}(x,\hat x)<r,\qquad U_{\kappa}(x)=u(x)\ \text{ otherwise},$$

the maximum being that of Maximum of Two Elements of a Totally Ordered Set. By claim 1 of Elementary Properties of the Maximum of Two Elements we have $u(x)\le U_{\kappa}(x)$ for every $x\in\Omega$, which is the first half of claim 2 of the statement.

$U_{\kappa}$ agrees with $u$ outside the inner half-ball. Let $x\in\Omega$ with $\tfrac{r}{2}\le d_{E}(x,\hat x)$. If $d_{E}(x,\hat x)<r$ then $g(x)<u(x)$ by (5), so $\max\{u(x),g(x)\}=u(x)$ by the definition of the maximum together with the fact that $u(x)\le g(x)$ fails; otherwise $U_{\kappa}(x)=u(x)$ by definition. In particular, if $\kappa\le d_{E}(x,\hat x)$ then, since $\tfrac{r}{2}<r<\kappa$, we get $\tfrac{r}{2}\le d_{E}(x,\hat x)$ and hence $U_{\kappa}(x)=u(x)$, which is claim 3 of the statement.

$U_{\kappa}$ exceeds $u$ somewhere. By claim 1 of Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function and the argument of Step 2, $g$ is continuous at $\hat x$ relative to $\Omega$, so there is a positive $\sigma$ such that every $x\in\Omega$ with $d_{E}(\hat x,x)<\sigma$ satisfies $|g(x)-g(\hat x)|<\tfrac{\delta}{4}$. Choose a positive $\varepsilon_{0}$ with $\varepsilon_{0}<\tfrac{r}{2}$, $\varepsilon_{0}<\sigma$ and $\varepsilon_{0}\le\tfrac{\delta}{4}$. By claim 6 of Properties of the Lower Semicontinuous Envelope, by Duality there is a sequence $(z_{k})_{k\in\mathbb{N}}$ in $\Omega$ converging to $\hat x$ in $(\mathbb{R}^{n},d_{E})$ such that $(u(z_{k}))_{k\in\mathbb{N}}$ converges to $u_{*}(\hat x)=a$ in $(\mathbb{R},d_{\mathbb{R}})$, the metric $d_{\mathbb{R}}$ being that of The Absolute Value Metric on the Real Numbers. Applying the definition of convergence to each of these two sequences with the positive real $\varepsilon_{0}$, and taking for $k$ the maximum of the two resulting indices, which dominates both by claim 1 of Elementary Properties of the Maximum of Two Elements, we obtain a point $z=z_{k}\in\Omega$ with $d_{E}(z,\hat x)<\varepsilon_{0}$, hence $d_{E}(z,\hat x)\le\varepsilon_{0}$, and $|u(z)-a|<\varepsilon_{0}$. Then $d_{E}(\hat x,z)\le\varepsilon_{0}<\sigma$, so $|g(z)-a-\delta|<\tfrac{\delta}{4}$, and $|u(z)-a|<\tfrac{\delta}{4}$. By claim 9 of Properties of the Absolute Value in an Ordered Field these give $u(z)<a+\tfrac{\delta}{4}$ and $a+\delta-\tfrac{\delta}{4}<g(z)$, whence

$$u(z)<a+\tfrac{\delta}{4}\le a+\delta-\tfrac{\delta}{2}+\tfrac{\delta}{4}\quad\text{is at most}\quad a+\delta-\tfrac{\delta}{4}<g(z),$$

using $\tfrac{\delta}{4}+\tfrac{\delta}{4}\le\tfrac{\delta}{2}$ and claim 8 of Elementary Order Arithmetic in an Ordered Field; so $u(z)<g(z)$. Moreover $d_{E}(z,\hat x)\le\varepsilon_{0}<\tfrac{r}{2}<r$, so $U_{\kappa}(z)=\max\{u(z),g(z)\}=g(z)$ by the definition of the maximum, and therefore $u(z)<U_{\kappa}(z)$. This completes claim 2 of the statement.

Step 6: $U_{\kappa}$ is a viscosity subsolution. Put $V_{2}=\{x\in\Omega:\tfrac{r}{2}<d_{E}(x,\hat x)\}$. The set $\{x\in\mathbb{R}^{n}:d_{E}(x,\hat x)\le\tfrac{r}{2}\}$ is a closed ball, hence closed by claim 3 of Elementary Properties of the Closed Ball in a Metric Space, so its complement is open, and $V_{2}$, being the intersection of that complement with $\Omega$, is open by The Open Subsets of a Metric Space Form a Topology. Every $x\in\Omega$ lies in $V_{1}$ or in $V_{2}$: by totality either $d_{E}(x,\hat x)<r$, and then $x\in V_{1}$, or $r\le d_{E}(x,\hat x)$, and then $\tfrac{r}{2}<r\le d_{E}(x,\hat x)$, so $x\in V_{2}$.

On $V_{1}$ the function $U_{\kappa}$ agrees with the pointwise maximum of the restrictions of $u$ and $g$. The restriction of $u$ is a viscosity subsolution of $F|_{V_{1}}$ on $V_{1}$ by claim 1 of The Viscosity Sub- and Supersolution Properties are Local, and the restriction of $g$ is one by Step 3; since $F|_{V_{1}}$ is continuous by claim 3 of Restriction of a Second-Order Equation Operator to an Open Subset, The Maximum of Two Viscosity Subsolutions is a Viscosity Subsolution shows that $U_{\kappa}$ restricted to $V_{1}$ is a viscosity subsolution of $F|_{V_{1}}$ on $V_{1}$.

On $V_{2}$ the function $U_{\kappa}$ agrees with $u$, by the second paragraph of Step 5, and the restriction of $u$ to $V_{2}$ is a viscosity subsolution of $F|_{V_{2}}$ on $V_{2}$ by claim 1 of The Viscosity Sub- and Supersolution Properties are Local.

Thus every point of $\Omega$ has an open neighbourhood contained in $\Omega$ on which $U_{\kappa}$ restricts to a viscosity subsolution of the restricted operator, so claim 2 of The Viscosity Sub- and Supersolution Properties are Local shows that $U_{\kappa}$ is a viscosity subsolution of $F$ on $\Omega$. This is claim 1 of the statement. $\blacksquare$