Equivalence of alternative definitions of 'viscosity solution'

Question

Consider the first-order Hamilton-Jacobi equation (HJ):

$$H(x,u,\nabla u) = 0 \quad \text{ on } \ \Omega,$$ where $\Omega$ is an open set of $\mathbb{R}^n$, $u:\Omega \to \mathbb{R}$, and $H:\Omega \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ (Hamiltonian) is continuous.

Definition 1 (Crandall-Lions-Evans): We say that $u \in C(\Omega)$ is a viscosity solution of (HJ) iff, $\forall \phi \in C^1(\Omega)$,

$\forall x_0$ point of local maximum of $u-\phi$, $\ H(x_0, u(x_0), \nabla\phi(x_0)) \le 0$;

$\forall x_0$ point of local minimum of $u-\phi$, $\ H(x_0, u(x_0), \nabla\phi(x_0)) \ge 0$.

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Q1: Where can I find a detailed proof that we can replace

• "local maximum" by "strict local maximum" or "global maximum" or "strict global maximum";
• "local minimum" by "strict local minimum" or "global minimum" or "strict global minimum"

and obtain an equivalent definition? (Or anyway how would that proof go?)

Q2: Is it true that we can replace $C^1$ by $C^k$ or $C^k_{\text{comp}}$, with $1 < k \le \infty$, and obtain an equivalent definition?

Answer 1

Only the subsolution case is proven, as the supersolution case is identical.

Q1

Suppose $u$ is a subsolution under the definition with strict extremum. Let $\phi$ be a test function such that $u-\phi$ has a possibly nonstrict maximum at $x_0$. Let $$\psi(x)=\phi(x)+|x-x_0|^2$$ so that $x_0$ is a strict maximum of $u-\psi$. Moreover, $$H(x_0,u,\nabla \phi(x_0)) = H(x_0,u,\nabla \psi(x_0)) \leq 0.$$

Q2

Suppose $u$ is a subsolution under the definition with compact test functions and $1 < k < \infty$. Let $\phi$ be a noncompact test function such that $u-\phi$ has a local maximum at $x_0$. Let $B_r$ denote the Euclidean ball of radius $r$ around $x_0$. Without loss of generality, we assume that $B_3 \subset \Omega$ (otherwise, perform some scaling to fix things). Now, let \begin{align*} \psi(x)&=\phi(x) 1_{B_1}(x) + \zeta(x) \phi(\hat{x}) 1_{B_2 \setminus B_1}(x),\\ \zeta(x)&=\exp(1-1/(1-|x-\hat{x}|^{2k})), \end{align*} and $\hat{x}$ is the unique closest point to $x$ in $\operatorname{cl}B_1$. $\psi$ inherits all the local properties of $\phi$ at $x_0$ and hence $$H(x_0,u,\nabla \phi(x_0)) = H(x_0,u,\nabla \psi(x_0)) \leq 0.$$ The supersolution case is identical.