Radially Symmetric Minimization Problem

• Radially symmetric minimization problems are optimization challenges involving nonlinear functionals defined on radially invariant domains with degenerate weights.
• The methodology features shell-wise decomposition, auxiliary weight construction, and tailored weighted Poincaré inequalities to ensure coercivity and proper relaxation.
• Key outcomes include establishing existence, uniqueness, and radiality of minimizers via rigorous Euler–Lagrange equations and comparison principles.

A radially symmetric minimization problem concerns the optimization of functionals—typically nonlinear and possibly degenerate, often arising in variational calculus or PDEs—under the restriction or emergence of radial symmetry, usually in a Lebesgue-class or Sobolev-class setting. Such problems are characterized both by their analytical reduction (often via rearrangement inequalities, symmetrization, or optimality arguments) and by the critical role of the weight or geometry admitting singularities, vanishing, or lack of regularity. These frameworks generalize classical variational methods and extend to degenerate, weighted, or non-doubling settings where standard techniques fail.

1. Functional Framework and Admissible Classes

Consider a degenerate nonlinear integral functional of the form

$F(u) = \int_\Omega |\nabla u|^p\, w(x)\, dx$

where $\Omega \subset \mathbb{R}^d$ is a rotation-invariant (e.g., annular or ball-shaped) domain and $w(x) = \eta(|x|)$ is a radial, potentially degenerate weight without global doubling or Muckenhoupt conditions. The admissible class is defined via the radial absolutely continuous space

$$AC^d_r(\Omega) = \{u(x) = v(|x|):\ v \in AC(\operatorname{supp}\,\eta)\},$$

where $v$ is absolutely continuous and $\operatorname{supp}\, \eta \subset (a, b)$. In this setting, $u$ is almost everywhere differentiable, with the gradient $\nabla u(x) = v'(|x|) x/|x|$.

Crucially, the function space for relaxation is tailored to the degeneracy of the weight, leading to a Banach space

$$W = \{u \in AC^d_r(\Omega) \cap L^p(\Omega, (\hat w_p)^{p-1}): \bar F(u) < +\infty\}$$

with norm

$$\|u\|_W^p = \int_\Omega |\nabla u|^p w(x)\, dx + \int_\Omega |u|^p (\hat w_p(x))^{p-1}\, dx.$$

The relaxed functional $\bar F$ coincides with the original integral on its effective domain, and is $+\infty$ otherwise, demonstrating the precise relaxation property in the generalized setting (Piat et al., 28 Jul 2025).

2. Degenerate Weight Structure and Auxiliary Weights

Unlike classical theories that assume doubling or Muckenhoupt properties for weights, the degenerate setting admits weights with arbitrary vanishing or blow-up near endpoints. The central hypothesis is

$$I_{p, \operatorname{supp} \eta} = \bigcup_{i=1}^{N_\eta} (a_i, b_i)$$

where $\eta(r)^{-1/(p-1)} \in L^1_{\text{loc}}$ only in finitely many intervals. This partition into finitely many shells $(a_i, b_i)$ enables the analysis to proceed shell-wise, despite the possibility of $w(r)$ vanishing or becoming singular at endpoints. For each shell, an auxiliary weight is constructed: $$\hat \eta_p(r) = \left( \int_r^{m_i} (s^{d-1} \eta(s))^{-1/(p-1)} ds \right)^{-1}$$ for $a_i < r < m_i = (a_i + b_i)/2$, and analogously for the upper shell. This produces an auxiliary radial weight

$\hat w_p(x) = \omega_d^{-1} \hat \eta_p(|x|)$

with $\omega_d$ the surface measure of the unit sphere, used to compensate for degeneracy in the Poincaré-type inequalities (Piat et al., 28 Jul 2025).

3. Weighted Poincaré Inequality and Energy Relaxation

A central advancement is the establishment of a weighted Poincaré inequality tailored to the degenerate structure: $$\sum_{i=1}^{N_\eta} \frac{\omega_d^{p-1}}{b_i-a_i}\int_{I_{a_i, b_i}} |u(\zeta) - u(x_i)|^p (\hat w_p(\zeta))^{p-1} d\zeta \leq \int_\Omega |\nabla u|^p w(x)\, dx$$ where $x_i$ (the midradius) is chosen in each shell. This inequality guarantees coercivity when testing with radial functions, even though the original weight may collapse at shell endpoints. As a result, the relaxation of the functional in the $L^p(\Omega, (\hat w_p)^{p-1})$ topology coincides exactly with its naive form on the defined domain, with no extra singular term (Piat et al., 28 Jul 2025).

Existence and uniqueness of minimizers for the regularized functional

$$H(u) = \bar F(u) + \|u - g\|_{L^p(\Omega, (\hat w_p)^{p-1})}^p$$

where $g$ is a given radial datum, are then established via standard direct methods in the reflexive Banach space $W$.

4. Euler–Lagrange Equations in Radial Reduction

Within each shell $(a_i, b_i)$, the minimization reduces to a 1D variational problem for radial profiles,

$$\min \int_{a_i}^{b_i} r^{d-1} \left\{ \eta(r) |u'(r)|^p + |u(r) - g(r)|^p (\hat \eta_p(r))^{p-1} \right\} dr.$$

The corresponding Euler–Lagrange ODE is

$$-\frac{d}{dr}\left( r^{d-1} \eta(r) |u'|^{p-2} u' \right) + r^{d-1} (\hat \eta_p(r))^{p-1} |u - g|^{p-2}(u - g) = 0$$

with natural boundary conditions reflecting finite energy and possibly vanishing of the weighted flux at degeneracy points. The ODE is solved separately in each shell, with coupling achieved through regularity at endpoints as needed for matching (Piat et al., 28 Jul 2025).

5. Symmetry and Comparison Principle for Minimizers

A key structural result asserts that minimizers of such degenerate, radially weighted functionals are necessarily radial, even if the minimization is performed in the entire (possibly non-radial) admissible class $L^p(\Omega, w(x))$ with general boundary conditions. The argument uses a shell-wise comparison: any test function $z$ is compared with its radial rearrangement $z_{\text{rad}}$, and by convexity and appropriate monotonicity hypotheses, one proves

$H(z_{\text{rad}}) \leq H(z)$

implying that minimizing sequences can be replaced by radial ones without increasing the energy. The uniqueness argument then follows from strict convexity, showing that every minimizer coincides (up to a null set) with a radial profile (Piat et al., 28 Jul 2025).

6. Illustrative Examples and Generalizations

Explicit instances include:

• For $w(r) \equiv 1$, the theory recovers the standard radial Sobolev case, with $\hat w_p \simeq 1$ and classical relaxation.
• Highly singular weights $\eta(r)$ with vanishing or blow-up in countably many points (but finitely many shells of integrability for $\eta^{-1/(p-1)}$) are encompassed.
• For $d \geq 2$, $p=2$, this setting extends the classical Dirichlet-form approach to cases in which density of smooth functions fails, as observed in the work of Hamza, Fusco, and Moscariello.

A plausible implication is that the radially symmetric reduction is a robust mechanism even for degenerate problems far from the scope of standard elliptic regularity or functional analytic density theorems.

7. Impact and Resolution of the General Problem

The methodology capitalizes on the radial structure by constructing a double-weighted Sobolev space permitting optimal inequalities and relaxation, despite severe degeneracy in the original weight. The careful shell decomposition, auxiliary weight design, and comparison arguments guarantee the existence, uniqueness, and radiality of minimizers. As a result, the general class of degenerate, weighted, nonlinear functionals in radial domains—regardless of the weight's non-doubling, non-Muckenhoupt nature—is completely characterized regarding minimizer symmetry and associated Euler–Lagrange equations (Piat et al., 28 Jul 2025).