For$f: \mathbb R \to \mathbb R$ a locally integrable function, $e \in (0, \infty)$, and $x \in \mathbb R$, define $I(f, e, x)$ to be the integral of $f$ over $B_e (x)$, the ball of radius $e$ around $x$.

Define $K(f, e, x) :=$

$1$ if $\frac{1}{2e} I(f, e, x) > f(x)$,

$-1$ if $\frac{1}{2e} I(f, e, x) < f(x)$,

$0$, if $\frac{1}{2e} I(f, e, x) = f(x)$.

Finally, let $H(f, e, x) = (1/e) \int_{(0, e]} K(f, s, x) ds$

i) Is it true that any $C^2$ function $f$ satisfies $\limsup_{e \to 0} H(f, e, x) = \liminf_{e \to 0} H(f, e, x)$ for all $x \in \mathbb R$?

ii) Consider the PDE $\partial_t u(x, t)$ = $\limsup_{e \to 0} H(u(x, t), e, x)$.

If (i) is true, then the limsup may be replaced by a limit, so that no arbitrary choice between limsup and liminf must be made.

The PDE is meant in a strong sense, to be solved over functions $u: \mathbb R \times [0, \infty) \to \mathbb R$; denoted $u(x, t)$ that are $C^2$ in $x$ for each fixed $t$, and $C^1$ in $t$ for each fixed $x$; with initial condition $u(x, 0) = f(x)$ for arbitrary $f \in C^2$. Do strong solutions exist? Are they unique?

iii) The PDE in (ii) is solvable by $u(x, t) := f(x)$ if the initial condition $f$ is a harmonic function, since harmonic functions satisfy the mean value property. Suppose the PDE in (ii) is uniquely solvable for some initial condition $f \in C^2$. Denoting by $u$ the solution, is it true that the functions $u(., t)$ converge pointwise to a harmonic function $u_\infty$ as $t \to \infty$?