Radial Energy Functionals

• Radial energy functionals are variational objects whose integrands depend explicitly on the radius, reducing multidimensional problems to a one-dimensional analysis.
• They are pivotal in studying nonlinear PDEs, geometric variational challenges, and physical models, revealing phenomena like minimizer existence and symmetry breaking.
• Applications span weighted p-Laplacian, Ginzburg–Landau models, and atomic electronic structure, with computational integration schemes exploiting radial grid reductions.

A radial energy functional is a variational object in which the integrand, weighting, or potential terms depend explicitly on the radius $r=|x|$—frequently leveraging radial symmetry or reducing a multidimensional optimization problem to one dimension. Radial energy functionals underlie a substantial body of modern analysis concerning nonlinear PDEs, geometric variational problems, and models in mathematical physics, particularly where symmetry leads to dimension reduction or qualitative simplification. Their rigorous study encompasses existence and symmetry of minimizers, classification of optimizers in weighted settings, sensitivity to the degeneracy or monotonicity of the weights, and connections to physical or geometric stability.

1. General Structure and Classes of Radial Energy Functionals

A general radial energy functional in $\mathbb{R}^d$ is formulated as

$$E[u] = \int_{\mathbb{R}^d} F(|x|,u(x),|\nabla u(x)|)\,dx,$$

with $F$ possibly depending nontrivially on $|x|$ ("radial weight"), $u$, and its derivatives. Upon restricting to radially symmetric $u=u(r)$, this reduces to a one-dimensional integral, e.g.

$$E[u] = \int_0^\infty L(r, u(r), u'(r))\, w(r)\, dr,$$

where $w(r)$ may encode Jacobian factors (such as $r^{d-1}$) and/or additional degeneracy or weight structure. Examples include degenerate weighted $p$-Laplacian energies, free energy functionals for densities with radial potentials, Riesz kernel energies with radial confinement, and radial Ginzburg–Landau or Kohn–Sham functionals.

Minimizers of such functionals can be profoundly affected by the regularity and monotonicity properties of $w(r)$—in particular, existence and symmetry-breaking phenomena arise when $w$ does not satisfy classical regularity conditions, such as doubling or Muckenhoupt properties (Piat et al., 28 Jul 2025). Nonlinearities with $p$-growth, or functionals with degenerate or singular weights, require the development of specialized compactness and weighted inequality tools for analysis.

2. Existence and Symmetry of Minimizers in the Degenerate Weighted Case

The minimization of nonlinear functionals with degenerate radial weights $w(r)$, especially under $p$-growth, is characterized by several analytical subtleties. When $w$ fails the doubling or Muckenhoupt conditions, classical compactness and Poincaré inequalities are not immediately available. The theory developed by De Cicco and collaborators (Piat et al., 28 Jul 2025) constructs an auxiliary weight suited to the degenerate $w$ and establishes the validity of a weighted Poincaré inequality for a specifically defined functional class. This foundational step is indispensable for showing the existence of minimizers.

The minimization problem

$$I[u] = \int_{\mathbb{R}^d} w(|x|)\, |\nabla u(x)|^p\, dx,$$

with $w$ possibly vanishing or singular on sets of positive measure, admits minimizers subject to suitable constraints, and—crucially—the minimizers are shown to necessarily be radially symmetric in a wide class, despite $w$ itself lacking full regularity. The methodology closely follows and generalizes the framework introduced by Chiadò Piat, De Cicco, and Melchor Hernandez, extending the scope of prior results where $w$ was required to be doubling or in the Muckenhoupt $A_p$ class.

This suggests that the presence of radial symmetry in both the weight and the domain, even in highly singular settings, generically enforces radiality of minimizers when the energy is $p$-convex in the gradient variable.

3. Explicit Examples Across Mathematical Physics and PDEs

Radial energy functionals are fundamental to a variety of concrete models:

• Radially Weighted Free Energies: For sets $A\subset\mathbb{R}^n$, with smooth radially-dependent density $f(x)=f(r)$ and potential $g(x)=g(r)$, the “free energy” is

$$E(A) = \int_{\partial^*A} f(r)\, d\mathcal{H}^{n-1} + \int_A g(r) f(r)\, dx,$$

with $r=|x|$. Such functionals arise in isoperimetric and geometric analysis under weighted measures and potentials and admit sharp classification results for global minimizers—centered balls are optimal if $f$ and $g$ are both monotone increasing, but not in general (Aryan et al., 5 Dec 2024).

• Radial Ginzburg–Landau Functionals: In superconductivity and vortex physics, the planar Ginzburg–Landau energy reduces to

$$E_\varepsilon(f) = 2\pi\int_0^1 \left[(f'(r))^2 + \frac{f(r)^2}{r^2} + \frac{(f(r)^2 - 1)^2}{2\varepsilon^2}\right] r\, dr,$$

with minimizers $f_\varepsilon(r)$ representing radial vortex profiles (Brandolini et al., 2013, Almog et al., 2010). Their monotonicity, regularity, and sharp interface properties in the large-$p$ limit have been rigorously established.

• Schrödinger–Poisson and Riesz Energy Functionals: Models with nonlocal interactions (e.g., Schrödinger-Poisson systems with cubic-type nonlinearities, or Riesz energies with radial external fields) admit variational structures where both local and nonlocal terms exhibit radial dependence (Murcia et al., 2018, Chafaï et al., 30 Apr 2024).
• Radial Kohn–Sham Functionals: For spherically symmetric atoms,

$$E_{\text{tot}} = T_s + E_{\text{ext}} + E_H + E_{xc}^{\text{L}} + E_x^{\text{NL}},$$

all energy contributions can be recast as one-dimensional radial integrals, facilitating highly efficient atomic electronic calculations (Užulis et al., 2022).

4. Classification, Stability, and Uniqueness of Radial Minimizers

Sharp characterization results have been obtained for several prominent classes of radial functionals:

• Weighted Isoperimetric and Free Energy Problems: For functionals combining radially weighted perimeter and bulk potential terms, existence, uniqueness, and explicit form of minimizers depend sensitively on monotonicity of the weights. If $f$ and $g$ are both strictly increasing, centered balls are uniquely optimal for all prescribed weighted volume. However, lack of monotonicity, even under $\ln(f)''+g'\geq0$, can yield non-symmetric or off-center minimizers for certain volume regimes, as demonstrated by explicit counterexamples (Aryan et al., 5 Dec 2024).
• Ginzburg–Landau Type Functionals: Strict convexity and regularity of the reduced one-dimensional functional, together with spectral analysis of the second variation, yields not just uniqueness but also stability of the radially symmetric minimizer. For $2<p\leq 4$, local stability holds for the degree-one Ginzburg–Landau vortex (Almog et al., 2010).
• Total Energy of Radial Mappings: For orientation-preserving diffeomorphisms between annuli, the total energy functional reduces to a variational problem in the radial profile, with existence and uniqueness of the minimizing diffeomorphism ensured via shooting arguments and careful ODE analysis, relying on strict convexity of the reduced Lagrangian (Chen et al., 2017).

5. Computational Aspects and Integration Schemes

Numerical and analytical tractability of radial energy functionals often exploits the one-dimensional reduction:

• In atomic electronic structure, high-accuracy evaluation of total energy and its components relies on integrating radial energy densities over a discretized radial grid. Atom-centered Lebedev-radial grids (SG-1 and SG-2) achieve errors down to the microhartree regime, with substantial error cancellation in correlation energy densities (Awad et al., 16 Sep 2025).
• For range-separated hybrid density functionals, convolution with the complementary error function kernel is reduced to sums of Yukawa-type potentials, permitting rapid and accurate radial integration (Užulis et al., 2022).
• Radial splines and kernel methods for interpolating or approximating radial data employ minimization of higher-order energies (e.g., Beppo Levi energy), with explicit $L^p$-approximation order proofs and basis function dilation representations (Bejancu, 2014).

6. Specialized and Probabilistic Radial Energy Potentials

Radial energy functionals appear beyond classical PDEs and potential theory:

• Loewner and Onsager–Machlup Functionals: In random conformal geometry (SLE processes), the Onsager–Machlup or Loewner energy functional for radial SLE paths takes the explicit form

$$S_{\text{OM}}(\gamma) = \frac{c(\kappa)}{24}\int_0^\infty (\dot U(t))^2\, dt,$$

where $U(t)$ is the Loewner driving function and $c(\kappa)$ the SLE central charge. This functional quantifies large deviations in the law of SLE and connects to the Liouville action in conformal field theory (Fan, 9 Aug 2025).

• Atomic Correlation Energy Densities: The radial decomposition of the correlation energy in atomic systems details the shell structure and spin-dependence in open-shell atoms, describing how each energy contribution varies radially, and thus providing a powerful diagnostic for electron correlation effects (Awad et al., 16 Sep 2025).

7. Open Questions and Outlook

The study of radial energy functionals continues to evolve in several directions:

• Sharp conditions for dimension reduction (support on concentric spheres/shells) have been obtained for Riesz energies with radial confinement, with explicit dependence on exponents and field parameters (Chafaï et al., 30 Apr 2024).
• Classification in degenerate regimes or for strongly non-monotonic weights remains delicate, and optimal compactness techniques for minimizers in singularly weighted spaces are still the focus of ongoing research (Piat et al., 28 Jul 2025).
• Connections to gradient flows, Wasserstein geometry, and constrained optimization provide additional structural context and motivation for the further understanding of radial functional minimization and associated stability/rigidity phenomena.

The unifying feature across applications is the leverage of symmetry and dimensional reduction—radial energy functionals distill high-dimensional variational problems into analytically and computationally tractable forms, enabling both rigorous theory and physically meaningful computation.