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

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 and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation on which it rests, is in force in a dimension $n$, a natural number with $1\le n$.

Let $\Omega\subseteq\mathbb{R}^{n}$ be open and nonempty, let $F$ be a second-order equation operator on $\Omega$ that is continuous and degenerate elliptic, and let $u:\Omega\to\mathbb{R}$ be a viscosity subsolution of $F$ on $\Omega$ that is bounded below near each point of $\Omega$, so that its lower semicontinuous envelope $u_{*}$ is defined.

Suppose that $u_{*}$ fails the supersolution inequality at a point $\hat x\in\Omega$, in the following sense: there is a function $\varphi:\Omega\to\mathbb{R}$ of class $C^{2}$ on $\Omega$ such that $u_{*}-\varphi$ has a local minimum at $\hat x$ relative to $\Omega$ and

$$F\bigl(\hat x,u_{*}(\hat x),D\varphi(\hat x),D^{2}\varphi(\hat x)\bigr)<0 .$$

Then for every positive $\kappa\in\mathbb{R}$ there is a function $U_{\kappa}:\Omega\to\mathbb{R}$ such that the following hold.

1. (It is a subsolution)(i) $U_{\kappa}$ is a viscosity subsolution of $F$ on $\Omega$.

2. (It lies above $u$, strictly somewhere)(ii) $u(x)\le U_{\kappa}(x)$ for every $x\in\Omega$, and there is $x\in\Omega$ with $u(x)<U_{\kappa}(x)$.

3. (It is unchanged away from $\hat x$)(iii) $U_{\kappa}(x)=u(x)$ for every $x\in\Omega$ with $\kappa\le d_{E}(x,\hat x)$.