Continuity of critical points for 1-dimensional non-local energies

Technical Summary: Continuity of Critical Points for 1-Dimensional Non-Local Energies

Problem Statement
This paper investigates the regularity of bounded critical points for a class of non-local energies defined on probability measures in one dimension. The energy functional is of the attractive-repulsive type, given by:
$$E(\mu) = \iint g(x-y) d\mu(x) d\mu(y)$$
where $g: \mathbb{R} \to [0, +\infty]$ is a symmetric kernel. The authors focus on measures $\mu = f \mathcal{L}^1$ (absolutely continuous with respect to the Lebesgue measure) that satisfy the Euler-Lagrange conditions for local minimizers, specifically that the induced potential $\psi_f = g * f$ is constant almost everywhere on the support of $f$. While existence results and $L^\infty$ bounds for minimizers are known under certain conditions, the internal regularity (specifically continuity) of these densities within their support remains a central question. The paper addresses whether bounded critical points are continuous inside their support, assuming the kernel $g$ satisfies specific growth and convexity conditions near the origin.

Methodology
The authors employ a proof by contradiction centered on the behavior of the second derivative of the potential. The core strategy involves the following steps:

1. Regularization: The density $f$ is regularized via convolution with a smooth mollifier $\rho_\delta$ to obtain a smooth function $f_\delta$. Since the potential $\psi_f$ is constant on the support, $\psi_{f_\delta}$ is also constant on a slightly smaller interval.
2. Critical Point Analysis: The authors assume $f$ is discontinuous at a point (specifically, that essential limits do not coincide). This implies that for sufficiently small $\delta$, the regularized function $f_\delta$ must exhibit oscillations (local maxima and minima) near that point.
3. Second Derivative Estimates: The proof relies on analyzing the second derivative of the potential, $\psi''_{f_\delta}$, at the critical points of $f_\delta$. Since $\psi_{f_\delta}$ is constant, $\psi''_{f_\delta}$ must vanish at these points.
4. Cancellation Lemmas: A significant portion of the technical work involves deriving integral inequalities (Lemmas 2.2, 2.3, and 2.4) that estimate the contribution of the kernel's derivatives to the second derivative of the potential. These lemmas exploit the convexity of $g$ and the concavity of $g'$ in a neighborhood of the origin. They establish that if $f_\delta$ oscillates between values corresponding to a jump in $f$, the integral terms cannot cancel out to zero, provided the kernel satisfies a specific growth condition involving the ratio of derivatives at $x/2$ and $x$.
5. Selection of "Good" Critical Points: The authors construct a sequence of alternating local minima and maxima for $f_\delta$ near the discontinuity. They select a "good couple" of consecutive critical points where the distance between them satisfies specific geometric constraints relative to their neighbors.
6. Symmetry Reduction: To handle general discontinuities, the problem is decomposed into symmetric (even) and antisymmetric (odd) parts of the density. The authors prove that if the essential limits exist for both the even and odd parts, they must be equal, implying the existence of a unique essential limit for $f$.

Key Contributions and Results
The primary result is Theorem A, which states:
Let $g$ satisfy hypotheses (i) and (ii) (symmetry, $C^1$ away from 0, decreasing, convex, and a specific growth condition on $g'$ near 0 involving a constant $\Lambda > 1$). If $f \in L^\infty(\mathbb{R})$ is a finite measure with support $[a, b]$ and its potential $\psi_f$ is constant almost everywhere in $[a, b]$, then $f$ is continuous in the open interval $(a, b)$.

• Hypotheses on the Kernel: The kernel must be convex and have a concave derivative near the origin. Crucially, the ratio $\liminf_{x \to 0} |g'(x/2)|/|g'(x)|$ must be strictly greater than 1. This condition is satisfied by prototypical attractive-repulsive kernels of the form $g(x) = \frac{|x|^\alpha}{\alpha} - \frac{|x|^\lambda}{\lambda}$ for appropriate $\alpha, \lambda$.
• Boundedness Assumption: The theorem assumes the measure has a bounded density ($f \in L^\infty$). The authors note this is a mild assumption, as $L^\infty$ bounds are known to hold for minimizers under slightly stronger conditions on the kernel.
• Internal Regularity: The result guarantees continuity strictly inside the support. The paper explicitly acknowledges that boundary behavior (near $a$ and $b$) is not addressed and may differ, as minimizers can be unbounded or singular at the boundary in certain cases.

Significance and Claims
The paper claims to establish the internal continuity of bounded critical points for a broad class of 1D non-local energies. The significance lies in bridging the gap between the known existence and $L^\infty$ bounds of minimizers and their pointwise continuity.

The authors emphasize that their approach is distinct from previous works that utilize PDE techniques related to the fractional Laplacian (e.g., [8]). Instead, this work relies on direct analysis of the convolution structure and the specific singularity properties of the kernel. The paper does not claim to solve the regularity problem for unbounded minimizers or in higher dimensions, nor does it provide explicit formulas for minimizers. It strictly establishes that, under the stated assumptions on the kernel and the boundedness of the density, the density cannot have jump discontinuities within its support. The proof demonstrates that the specific singularity of the kernel (governed by $\Lambda > 1$) is sufficient to force the cancellation of oscillations in the potential, thereby enforcing continuity.