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For$f: \mathbb R^n \to \mathbb R$ a locally integrable function, $\varepsilon \in (0, \infty)$, and $x \in \mathbb R^n$, define $I(f, \varepsilon, x)$ to be the averaged integral of $f$ over $B_{\varepsilon} (x)$, the ball of radius $\varepsilon$ around $x$.

Define $$ K(f, \varepsilon, x) := \begin{cases} 1 & \text{if }\; I(f, \varepsilon, x) > f(x),\\ -1 & \text{if }\; I(f, \varepsilon, x) < f(x),\\ 0, &\text{if }\; I(f, \varepsilon, x) = f(x).\\ \end{cases} $$

Finally, let $$ H(f, \varepsilon, x) = \dfrac{1}{\varepsilon} \int\limits_{(0, \varepsilon]} K(f, s, x) ds $$

Intuitively, H is the weighted average amount of time a function spends greater than (resp. less than) its value at a point, in an infinitesimal neighbourhood of said point.

Questions

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

2. Consider the PDE $\partial_t u(x, t)$ = $\limsup_{\varepsilon \to 0} H(u(x, t), \varepsilon, x)$.
   If (1) 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?

3. The PDE in (2) 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 $t \to \infty$?

For  $f: \mathbb R^n \to \mathbb R$ a locally integrable function, $\varepsilon \in (0, \infty)$, and $x \in \mathbb R^n$, define $I(f, \varepsilon, x)$ to be the averaged integral of $f$ over $B_{\varepsilon} (x)$, the ball of radius $\varepsilon$ around $x$.

Define $$ K(f, \varepsilon, x) := \begin{cases} 1 & \text{if }\; I(f, \varepsilon, x) > f(x),\\ -1 & \text{if }\; I(f, \varepsilon, x) < f(x),\\ 0, &\text{if }\; I(f, \varepsilon, x) = f(x).\\ \end{cases} $$

Finally, let $$ H(f, \varepsilon, x) = \dfrac{1}{\varepsilon} \int\limits_{(0, \varepsilon]} K(f, s, x) ds $$

Intuitively, H is the weighted average amount of time a function spends greater than (resp. less than) its value at a point, in an infinitesimal neighbourhood of said point.

Questions

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

2. Consider the PDE $\partial_t u(x, t)$ = $\limsup_{\varepsilon \to 0} H(u(x, t), \varepsilon, x)$.
   If (1) 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?

3. The PDE in (2) 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 $t \to \infty$?

For$f: \mathbb R^n \to \mathbb R$ a locally integrable function, $\varepsilon \in (0, \infty)$, and $x \in \mathbb R$, define $I(f, \varepsilon, x)$ to be the averaged integral of $f$ over $B_{\varepsilon} (x)$, the ball of radius $\varepsilon$ around $x$.

Define $$ K(f, \varepsilon, x) := \begin{cases} 1 & \text{if }\; I(f, \varepsilon, x) > f(x),\\ -1 & \text{if }\; I(f, \varepsilon, x) < f(x),\\ 0, &\text{if }\; I(f, \varepsilon, x) = f(x).\\ \end{cases} $$

Finally, let $$ H(f, \varepsilon, x) = \dfrac{1}{\varepsilon} \int\limits_{(0, \varepsilon]} K(f, s, x) ds $$

Intuitively, H is the weighted average amount of time a function spends greater than (resp. less than) its value at a point, in an infinitesimal neighbourhood of said point.

Questions

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

2. Consider the PDE $\partial_t u(x, t)$ = $\limsup_{\varepsilon \to 0} H(u(x, t), \varepsilon, x)$.
   If (1) 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?

3. The PDE in (2) 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 $t \to \infty$?

For$f: \mathbb R^n \to \mathbb R$ a locally integrable function, $\varepsilon \in (0, \infty)$, and $x \in \mathbb R^n$, define $I(f, \varepsilon, x)$ to be the averaged integral of $f$ over $B_{\varepsilon} (x)$, the ball of radius $\varepsilon$ around $x$.

Define $$ K(f, \varepsilon, x) := \begin{cases} 1 & \text{if }\; I(f, \varepsilon, x) > f(x),\\ -1 & \text{if }\; I(f, \varepsilon, x) < f(x),\\ 0, &\text{if }\; I(f, \varepsilon, x) = f(x).\\ \end{cases} $$

Finally, let $$ H(f, \varepsilon, x) = \dfrac{1}{\varepsilon} \int\limits_{(0, \varepsilon]} K(f, s, x) ds $$

Intuitively, H is the weighted average amount of time a function spends greater than (resp. less than) its value at a point, in an infinitesimal neighbourhood of said point.

Questions

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

2. Consider the PDE $\partial_t u(x, t)$ = $\limsup_{\varepsilon \to 0} H(u(x, t), \varepsilon, x)$.
   If (1) 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?

3. The PDE in (2) 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 $t \to \infty$?

For$f: \mathbb R^n \to \mathbb R$ a locally integrable function, $\varepsilon \in (0, \infty)$, and $x \in \mathbb R$, define $I(f, \varepsilon, x)$ to be the averaged integral of $f$ over $B_{\varepsilon} (x)$, the ball of radius $\varepsilon$ around $x$.

Define $$ K(f, \varepsilon, x) := \begin{cases} 1 & \text{if }\; I(f, \varepsilon, x) > f(x),\\ -1 & \text{if }\; I(f, \varepsilon, x) < f(x),\\ 0, &\text{if }\; I(f, \varepsilon, x) = f(x).\\ \end{cases} $$

Finally, let $$ H(f, \varepsilon, x) = \dfrac{1}{\varepsilon} \int\limits_{(0, \varepsilon]} K(f, s, x) ds $$

Intuitively, H is the weighted average amount of time a function spends greater than (resp. less than) its value at a point, in an infinitesimal neighbourhood of said point.

Questions

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

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

3. 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 $t \to \infty$?

For$f: \mathbb R^n \to \mathbb R$ a locally integrable function, $\varepsilon \in (0, \infty)$, and $x \in \mathbb R$, define $I(f, \varepsilon, x)$ to be the averaged integral of $f$ over $B_{\varepsilon} (x)$, the ball of radius $\varepsilon$ around $x$.

Define $$ K(f, \varepsilon, x) := \begin{cases} 1 & \text{if }\; I(f, \varepsilon, x) > f(x),\\ -1 & \text{if }\; I(f, \varepsilon, x) < f(x),\\ 0, &\text{if }\; I(f, \varepsilon, x) = f(x).\\ \end{cases} $$

Finally, let $$ H(f, \varepsilon, x) = \dfrac{1}{\varepsilon} \int\limits_{(0, \varepsilon]} K(f, s, x) ds $$

Intuitively, H is the weighted average amount of time a function spends greater than (resp. less than) its value at a point, in an infinitesimal neighbourhood of said point.

Questions

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

2. Consider the PDE $\partial_t u(x, t)$ = $\limsup_{\varepsilon \to 0} H(u(x, t), \varepsilon, x)$.
   If (1) 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?

3. The PDE in (2) 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 $t \to \infty$?