Epiperimetric Inequality Overview

• Epiperimetric inequality is a quantitative variational estimate that provides improved control over energy functionals in free-boundary, obstacle, and geometric measure theory problems.
• It employs spectral decomposition and explicit competitor constructions to achieve optimal regularity, frequency gap results, and stratification of singular sets.
• Its applications span classical, fractional, and parabolic settings, underpinning convergence rates, uniqueness of blow-ups, and the detailed structure of singularities.

An epiperimetric inequality is a quantitative variational estimate that provides improved control over the energy functional associated with free boundary, obstacle, and geometric measure theory problems. By yielding uniform decay (or improvement) rates for nearly homogeneous, conical configurations, epiperimetric inequalities play a central role in free-boundary regularity theory, the analysis of singularities, and the stratification of solution spaces. They underlie optimal regularity results, quantitative stratification of singular sets, frequency gap phenomena, convergence rates of blow-up sequences, and uniqueness of tangent objects in classical, thin, fractional, parabolic, and geometric settings.

1. Foundational Concepts and Definitions

The epiperimetric inequality is fundamentally a comparison principle for certain scale-invariant, often "Weiss-type," energy functionals. For a solution $u$ (typically normalized to vanish or meet the obstacle on a constraint set), and a boundary trace $c$ on $\partial B_1$, one considers the homogeneous extension $z$—for example, $z(x) = |x|^\lambda c(x/|x|)$ for homogeneity $\lambda$. The epiperimetric inequality asserts the existence of a competitor $\zeta$ with the same boundary values and admissibility (e.g., nonnegativity on a constraint set) such that: $$W_\lambda(\zeta) \leq (1-\varepsilon) W_\lambda(z) \quad \text{or} \quad W_\lambda(\zeta) \leq (1+\varepsilon) W_\lambda(z)$$ depending on the sign of the energy. Here $W_\lambda$ is a Weiss-type functional, for instance,

$$W_\lambda(u) = \int_{B_1} |\nabla u|^2 dx - \lambda \int_{\partial B_1} u^2 d\mathcal H^n$$

for the classical obstacle or thin obstacle (Signorini) problems (Carducci, 2023, Focardi et al., 2015, Colombo et al., 2017, Carducci, 2023).

Logarithmic epiperimetric inequalities instead produce an improvement governed by a power less than one,

$$W_\lambda(\zeta) \leq W_\lambda(z) - |W_\lambda(z)|^{1+\gamma},$$

with $\gamma \in (0,1)$, and are sharp in the setting of stratified, non-integrable singularities (Engelstein et al., 2018, Colombo et al., 2017, Colombo et al., 2017).

These inequalities can be generalized to signed or "symmetric" versions, which yield two-sided decay/growth control and are applicable even when the sign of the energy is not known a priori (Edelen et al., 2023).

2. Construction and Proof Strategies

Proofs of epiperimetric inequalities proceed via detailed spectral decompositions and variational constructions in the underlying functional spaces. Typically, the trace $c$ is decomposed via spherical harmonics or, more generally, into eigenmodes of the Laplace-Beltrami (or analogous) operator on the constraint manifold. The expansion separates lower modes (corresponding to tangential directions of the model cone or minimal solution) from higher modes, exploiting the spectral gap between them.

A canonical example, used for the thin obstacle at frequency $3/2$, is: $c = C\,h_e + c_0\,u_0 + \phi,$ where $h_e$ spans the $3/2$-homogeneous blow-ups, $u_0$ is a constant-mode, and $\phi$ is orthogonal to all lower modes (Carducci, 2023). The competitor $\zeta$ is constructed by adjusting the lower-modes' homogeneity (e.g., setting $c_0$ to degree 1 rather than $3/2$), leaving higher modes at the original homogeneity or raising them to a larger degree to leverage the eigenvalue gap for energy descent.

When multiple modes lead to partial cancellations or neutral directions (kernel), variational Lyapunov-Schmidt reductions and finite-dimensional gradient flows, possibly combined with Łojasiewicz inequalities, are invoked to obtain polynomial (if integrable) or logarithmic (if non-integrable) energy contraction (Engelstein et al., 2018, Engelstein et al., 2018). This is particularly delicate in geometric problems involving area-minimizing surfaces and almost-minimizers.

In the fractional/weighted setting, the construction relies on the spectral structure of the extended (Caffarelli–Silvestre) operator and its associated spherical eigenfunctions (Carducci, 2023, Geraci, 2017).

3. Consequences: Regularity, Frequency Gaps, and Stratification

Epiperimetric inequalities are a primary variational tool to derive optimal regularity of the solution and the free boundary, uniqueness and quantitative convergence of blow-ups, and the structure and rectifiability of singular sets.

Frequency Gap and Admissible Homogeneities

A central consequence is the exclusion of certain intervals of possible homogeneities for blow-up solutions. For example, in the thin obstacle problem, no nontrivial global homogeneous solution exists with $1 < \lambda < 3/2$, ensuring that the Almgren frequency at any free boundary point satisfies $N^{x_0}(0^+,u) \geq 3/2$ (Carducci, 2023, Focardi et al., 2015, Colombo et al., 2017). In the fractional obstacle problem, the allowed frequencies are similarly discrete, e.g., $\{1+s\} \cup \{2m\}$ for $s \in (0,1)$ (Carducci, 2023).

Regularity and Modulus of Continuity

The regular set, typically characterized by the minimal admissible frequency (e.g., $3/2$ for the thin obstacle, $1+s$ for the fractional case), is shown to be a $C^{1,\alpha}$ manifold, with explicit rates derived from the ODEs associated to the monotonicity and epiperimetric inequalities: $W_\lambda(r,u) \leq C r^\gamma,$ which, combined with boundary growth/decay estimates, yields $C^{1,\beta}$ regularity of the solution (Carducci, 2023, Carducci, 2023, Garofalo et al., 2015).

For singular points (higher frequency), logarithmic epiperimetric inequalities yield at best $C^{1,\log}$ structure on the relevant strata, matching sharp counterexamples (Colombo et al., 2017, Engelstein et al., 2018, Colombo et al., 2017).

Blow-up Uniqueness and Quantitative Convergence

The functional decay ensures the uniqueness of the blow-up limit at each point and provides explicit convergence rates (algebraic for polynomial epiperimetric inequalities, logarithmic for logarithmic ones). In multidimensional and geometric contexts, this extends to uniqueness and graph structure near isolated singularities of area-minimizing currents and minimal surfaces (Engelstein et al., 2018, Engelstein et al., 2018).

Strata and Rectifiability

The singular set admits a stratified structure, where each stratum $\Gamma_{2m}^j$ is contained in a countable union of $C^{1,\log}$ $j$-dimensional manifolds, with $C^{1,\alpha}$ regularity possible only in special or integrable cases (Carducci, 2023, Engelstein et al., 2018, Colombo et al., 2017).

4. Extensions: Parabolic, Fractional, and Odd-Frequency Settings

Epiperimetric inequalities have been adapted to:

• Parabolic thin obstacle problems, using time-dependent Weiss-type energies in self-similar coordinates, providing convergence to asymptotic profiles and frequency gap arguments in the time-dependent setting (Shi, 2018).
• Fractional obstacle problems, incorporating weighted Sobolev spaces and the Caffarelli–Silvestre extension, yielding analogues of the frequency gap, regularity, and blow-up classification in nonlocal problems (Carducci, 2023, Geraci, 2017).
• Odd-frequency thin obstacle points, most recently an epiperimetric inequality for frequencies $\lambda=2m+1$, ensuring uniqueness rates and structure of the corresponding contact sets and providing new frequency gap theorems (Carducci et al., 18 Sep 2024).
• Almost area-minimizing currents and geometric measure theory, in which (log-)epiperimetric inequalities are established for cones with isolated singularities, yielding uniqueness and modulus of continuity results for tangent cones and singularities (Engelstein et al., 2018, Engelstein et al., 2018, Edelen et al., 2023).

5. Methodological Innovations and Connections

Key developments include:

• Direct approaches using explicit competitor constructions for all admissible traces without closeness assumptions (Colombo et al., 2017, Spolaor et al., 2016).
• Symmetric (log-)epiperimetric inequalities, which control both signs of the energy and enable decay-growth estimates and log-convexity of energy, useful even when monotonicity is two-sided or in annular domains (Edelen et al., 2023).
• Variational and analytic reduction via Lyapunov–Schmidt and Łojasiewicz techniques, essential when the linearized operator admits a nontrivial kernel, as in non-integrable cones or higher codimensional singularities (Engelstein et al., 2018, Engelstein et al., 2018).
• Spectral techniques and spherical harmonic analysis for weighted problems, especially in the presence of thin or fractional operators (Carducci, 2023, Geraci, 2017).

6. Impact, Comparisons, and Open Directions

The epiperimetric inequality is a central quantitative variational tool in free boundary and geometric regularity theory, distinguished from prior approaches (e.g., improvement of flatness, frequency monotonicity) by its ability to yield explicit, dimensionally robust, and stratified regularity and uniqueness results without requiring a priori closeness or restrictive geometric assumptions (Carducci, 2023, Colombo et al., 2017, Carducci, 2023).

It has unified the treatment of regular and singular free boundary points, enabled generalizations to fractional, parabolic, and geometric contexts, and provided the foundation for modern stratification and quantitative regularity theory (Engelstein et al., 2018, Edelen et al., 2023, Shi, 2018).

The development of logarithmic and symmetric forms extends regularity insight to settings with non-integrable singularities and signed energies. Nonetheless, challenges remain in fully characterizing all possible singularities in higher dimensions, minimal surface analogues, stability of frequency gaps in non-classical regimes, and the optimal exponents and constants related to the epiperimetric contraction in various settings.

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Selected Central References:

Paper	Main Context	Key Results
(Carducci, 2023)	Thin obstacle, negative energy	$3/2$-epiperimetric inequality, frequency gap, $C^{1,1/2}$ regularity
(Carducci, 2023)	Fractional obstacle	Epiperimetric and logarithmic inequalities, frequency gap, stratification
(Colombo et al., 2017)	Thin obstacle, all dimensions	Direct epiperimetric and log-epiperimetric inequalities, regularity of singular set
(Engelstein et al., 2018)	Area-minimizing currents	Log-epiperimetric inequality for cones, $\varepsilon$-regularity at singularities
(Edelen et al., 2023)	Symmetric/log-epiperimetric	Unified decay-growth, log-convexity, graphicality propagation
(Carducci et al., 18 Sep 2024)	Odd-frequency thin obstacle	Epiperimetric inequality at odd frequencies, frequency gap

Each of these pivotal works underlines the broad influence and technical centrality of epiperimetric inequalities across advanced regularity theory, geometric analysis, and applied variational frameworks.