# Minimizers of variational problems with less symmetry

In am interested in understanding the so-called Hartree ground states, namely, minimizers of variational problems of the form $$\inf_{\phi\in H^1,~\|\phi\|_2=1}\left\{\int|\nabla\phi(x)|^2dx -\int|\phi(x)|^2K(x-y)|\phi(y)|^2dxdy\right\}\tag{1}$$ for some kernel $$K\geq0$$.

In the special case of the Riesz kernel ($$K(x)=|x|^{-\alpha}$$ for an appropriate choice of $$\alpha>0$$), it is well understood that minimizers of $$(1)$$ (up to translations) have the same "symmetry" and "monotonicity" behaviour as the kernel $$K$$. That is, rotationally invariant and radially decreasing.

As I understand it, the key argument in such results lies in the realization that the Dirichlet form $$\int|\nabla\phi(x)|^2dx$$ and the integral $$-\int|\phi(x)|^2K(x-y)|\phi(y)|^2dxdy$$ both decrease if we replace $$\phi$$ by its symmetric decreasing rearrangement $$\phi^*$$. Most crucially the latter integral becomes strictly smaller unless $$\phi$$ is already equal to a translate of its symmetric decreasing rearrangement (e.g., the equality cases of the Riesz rearrangement theorem).

With this background out of the way, here is my question:

Question Part 1. Suppose that we consider the minimizers of $$(1)$$ where the kernel $$K$$ has less symmetry than $$K(x)=|x|^{-\alpha}$$. For instance, perhaps $$K$$ is permutation-invariant ($$K(\Pi x)=K(x)$$ for every permutation matrix $$\Pi$$) instead of rotation-invariant, or maybe $$K(x)=\prod_{i=1}^nK_i(x_i)$$, where each $$K_i$$ is a one-dimensional kernel that is symmetric decreasing.

In both examples I mentioned above, it seems natural to expect (or at least hope) that minimizers would once again have the same symmetry as $$K$$, namely, permutation-invariant in the first case, and a tensor product of symmetric decreasing functions in the second case. However, I cannot find any literature on such problems.

Given how rearrangements are so key to the case of the Riesz kernel:

Question Part 2. Is there a well-developed theory of "rearrangement inequalities" similar to that of symmetric decreasing rearrangements but for less symmetric situations? E.g., rearrangements that turn a function into a permutation-invariant or tensor product version of itself, and how such rearrangements interact with the Dirichlet form and convolution integrals against $$K$$.

The only such thing I've found so far are Steiner symmetrizations, which do not seem to help for the kinds of symmetries I'm interested in.