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.