The Quartic Bump: Making a Local Maximum Strict without Changing the Test Data

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets, whose notation, including that of Euclidean Space and Real Matrices: Standing Notation on which it rests, is in force in the dimension $n$, a natural number with $1\le n$. In addition we abbreviate $\lVert w\rVert^{4}=\lVert w\rVert^{2}\lVert w\rVert^{2}$ for $w\in\mathbb{R}^{n}$.

Let $V\subseteq\mathbb{R}^{n}$ be open, let $z\in V$, and let $\beta:V\to\mathbb{R}$ be the function given by

$$\beta(x)=\lVert x-z\rVert^{4}\qquad\text{for }x\in V .$$

Then the following hold.

1. (Regularity)(i) $\beta$ is of class $C^{2}$ on $V$, and

$$D\beta(z)=0_{\mathbb{R}^{n}},\qquad D^{2}\beta(z)=0_{n}.$$

2. (Positivity)(ii) $0\le\beta(x)$ for every $x\in V$, and $\beta(x)=0$ if and only if $x=z$.

3. (Strict maximum)(iii) Let $S\subseteq V$ with $z\in S$, let $h:S\to\mathbb{R}$, and let $\varphi:V\to\mathbb{R}$ be of class $C^{2}$ on $V$. Suppose that the function $S\to\mathbb{R}$ whose value at $x$ is $h(x)-\varphi(x)$ has a local maximum at $z$ relative to $S$. Let $\psi:V\to\mathbb{R}$ be given by $\psi(x)=\varphi(x)+\beta(x)$. Then $\psi$ is of class $C^{2}$ on $V$,

$$D\psi(z)=D\varphi(z),\qquad D^{2}\psi(z)=D^{2}\varphi(z),$$

and the function $S\to\mathbb{R}$ whose value at $x$ is $h(x)-\psi(x)$ has a strict local maximum at $z$ relative to $S$.