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.



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