I'm not an expert in analysis whatsoever, so I might be posing a well-established question, or even an unanswerable one. Also, any suggestion on changes that might make the problem better are welcome.

We are working with interval-valued real functions over $\mathbb{R}^2$ (i.e., $[0,1]^{\mathbb{R}^2}$). We have on our hands a kernel $k$ which is a functional over non-negative functions on a circle $\mathcal{B}_r$ around the origin with radius $r$. A couple of assumptions on $k$:

1. It is isotropic; i.e. it's value is stable when composing its argument with rotations ($k(f) = k(f \circ R_\theta)$).

2. It's monotonic; i.e., it's value is monotonic in relation to the norm (integral) of the argument function ($\int_{\mathcal{B}_r} f \geq \int_{\mathcal{B}_r} f' \implies k(f) \geq k(f')$).

The problem is that of maximizing the following integral:

$$ \int_U k(f \circ \tau_x) dx $$

over some nice region $U$ under the restriction that the integral of $f$ is some constant

$$ \int_U f(x) dx = K $$

where $\tau_x$ is just a translation by $x$:

\begin{align*} \tau_x &: \mathcal{B}_r \to \mathbb{R}^2 \\ \tau_x &: u \mapsto u + x \end{align*}

Someone pointed to me the similarity to the problems in the calculus of variations, so that's where I went. But the problem is that, e.g., in the Euler-Lagrange equation, the thing under the integral

$$ \int L(t, q(t), q'(t)) dt $$

is a function $\mathbb{R}^3 \to \mathbb{R}$ -- i.e., it's strictly local, in that its value depends on the values of its argument and its derivatives on a single point, while my kernel $k$ depends on a whole neighborhood of values of the point.

I think this is my problem. Perhaps something is not clear, and the problem can be reduced to something more standard, by I can't see it.

The question is, more explicitly: how can I maximize the given integral expression under the given restriction? The goal is to have the most general assumptions on $k$ possible.

Where did this come from?

We're trying to model how vegetation distribution affects so-called ecological services around an urban area. The functions on $U$ represent the density of vegetation, and $k$ represents the way the vegetation affects, e.g., the drop in temperature around it. We assume the effect is isotropic and local (all covered by the formalism). The question is: given a fixed amount of vegetation, what's the better way of distributing it around the city? Clumped together? In parks? In squares? All over the streets? One big forest? Etc.

I'm not an expert in analysis whatsoever, so I might be posing a well-established question, or even an unanswerable one. Also, any suggestion on changes that might make the problem better are welcome.

We are working with interval-valued real functions over $\mathbb{R}^2$ (i.e., $[0,1]^{\mathbb{R}^2}$). We have on our hands a kernel $k$ which is a functional over non-negative functions on a ball $\mathcal{B}_r$ around the origin with radius $r$. A couple of assumptions on $k$:

1. It is isotropic; i.e. it's value is stable when composing its argument with rotations ($k(f) = k(f \circ R_\theta)$).

2. It's monotonic; i.e., it's value is monotonic in relation to the norm (integral) of the argument function ($\int_{\mathcal{B}_r} f \geq \int_{\mathcal{B}_r} f' \implies k(f) \geq k(f')$).

The problem is that of maximizing the following integral:

$$ \int_U k(f \circ \tau_x) dx $$

over some nice region $U$ under the restriction that the integral of $f$ is some constant

$$ \int_U f(x) dx = K $$

where $\tau_x$ is just a translation by $x$:

\begin{align*} \tau_x &: \mathcal{B}_r \to \mathbb{R}^2 \\ \tau_x &: u \mapsto u + x \end{align*}

Someone pointed to me the similarity to the problems in the calculus of variations, so that's where I went. But the problem is that, e.g., in the Euler-Lagrange equation, the thing under the integral

$$ \int L(t, q(t), q'(t)) dt $$

is a function $\mathbb{R}^3 \to \mathbb{R}$ -- i.e., it's strictly local, in that its value depends on the values of its argument and its derivatives on a single point, while my kernel $k$ depends on a whole neighborhood of values of the point.

I think this is my problem. Perhaps something is not clear, and the problem can be reduced to something more standard, by I can't see it.

The question is, more explicitly: how can I maximize the given integral expression under the given restriction? The goal is to have the most general assumptions on $k$ possible.

Where did this come from?

We're trying to model how vegetation distribution affects so-called ecological services around an urban area. The functions on $U$ represent the density of vegetation, and $k$ represents the way the vegetation affects, e.g., the drop in temperature around it. We assume the effect is isotropic and local (all covered by the formalism). The question is: given a fixed amount of vegetation, what's the better way of distributing it around the city? Clumped together? In parks? In squares? All over the streets? One big forest? Etc.

I'm not an expert in analysis whatsoever, so I might be posing a well-established question, or even an unanswerable one.

We are working with non-negative real functions over a sufficiently nice region $U$ of $\mathbb{R}^2$. We have on our hands a kernel $k$ which is a functional over non-negative functions on a circle $\mathcal{B}_r$ around the origin with radius $r$ (presumably small when compared to $U$). A couple of assumptions on $k$:

1. It is isotropic; i.e. it's value is stable when composing its argument with rotations ($k(f) = k(f \circ R_\theta)$).

2. It's monotonic; i.e., it's value is monotonic in relation to the norm (integral) of the argument function ($\int_U f \geq \int_U f' \implies k(f) \geq k(f')$).

The problem is that of maximizing the following integral:

$$ \int_U k(f \circ \tau_x) dx $$

under the restriction that the integral of $f$ is some constant

$$ \int_U f(x) dx = K $$

where $\tau_x$ is just a translation by $x$:

\begin{align*} \tau_x &: \mathcal{B}_r \to \mathbb{R}^2 \\ \tau_x &: u \mapsto u + x \end{align*}

Someone pointed to me the similarity to the problems in the calculus of variations, so that's where I went. But the problem is that, e.g., in the Euler-Lagrange equation, the thing under the integral

$$ \int L(t, q(t), q'(t)) dt $$

is a function $\mathbb{R}^3 \to \mathbb{R}$ -- i.e., it's strictly local, in that its value depends on the values of its argument and its derivatives on a single point, while my kernel $k$ depends on a whole neighborhood of values of the point.

I think this is my problem. Perhaps something is not clear, and the problem can be reduced to something more standard, by I can't see it.

The question is, more explicitly: how can I maximize the given integral expression under the given restriction? The goal is to have the most general assumptions on $k$ possible.

Where did this come from?

We're trying to model how vegetation distribution affects so-called ecological services around an urban area. The functions on $U$ represent the density of vegetation, and $k$ represents the way the vegetation affects, e.g., the drop in temperature around it. We assume the effect is isotropic and local (all covered by the formalism). The question is: given a fixed amount of vegetation, what's the better way of distributing it around the city? Clumped together? In parks? In squares? All over the streets? One big forest? Etc.

I'm not an expert in analysis whatsoever, so I might be posing a well-established question, or even an unanswerable one. Also, any suggestion on changes that might make the problem better are welcome.

We are working with interval-valued real functions over $\mathbb{R}^2$ (i.e., $[0,1]^{\mathbb{R}^2}$). We have on our hands a kernel $k$ which is a functional over non-negative functions on a circle $\mathcal{B}_r$ around the origin with radius $r$. A couple of assumptions on $k$:

1. It is isotropic; i.e. it's value is stable when composing its argument with rotations ($k(f) = k(f \circ R_\theta)$).

2. It's monotonic; i.e., it's value is monotonic in relation to the norm (integral) of the argument function ($\int_{\mathcal{B}_r} f \geq \int_{\mathcal{B}_r} f' \implies k(f) \geq k(f')$).

The problem is that of maximizing the following integral:

$$ \int_U k(f \circ \tau_x) dx $$

over some nice region $U$ under the restriction that the integral of $f$ is some constant

$$ \int_U f(x) dx = K $$

where $\tau_x$ is just a translation by $x$:

\begin{align*} \tau_x &: \mathcal{B}_r \to \mathbb{R}^2 \\ \tau_x &: u \mapsto u + x \end{align*}

Someone pointed to me the similarity to the problems in the calculus of variations, so that's where I went. But the problem is that, e.g., in the Euler-Lagrange equation, the thing under the integral

$$ \int L(t, q(t), q'(t)) dt $$

is a function $\mathbb{R}^3 \to \mathbb{R}$ -- i.e., it's strictly local, in that its value depends on the values of its argument and its derivatives on a single point, while my kernel $k$ depends on a whole neighborhood of values of the point.

I think this is my problem. Perhaps something is not clear, and the problem can be reduced to something more standard, by I can't see it.

The question is, more explicitly: how can I maximize the given integral expression under the given restriction? The goal is to have the most general assumptions on $k$ possible.

Where did this come from?

We're trying to model how vegetation distribution affects so-called ecological services around an urban area. The functions on $U$ represent the density of vegetation, and $k$ represents the way the vegetation affects, e.g., the drop in temperature around it. We assume the effect is isotropic and local (all covered by the formalism). The question is: given a fixed amount of vegetation, what's the better way of distributing it around the city? Clumped together? In parks? In squares? All over the streets? One big forest? Etc.