Monneau-Type Monotonicity Formulas

• Monneau-type monotonicity formulas are scale-invariant identities that measure deviations from model solutions in elliptic and parabolic PDEs.
• They provide robust analytical tools to derive rigidity results, uniqueness of tangent cones, and quantitative stability in free boundary and singularity analysis.
• These formulas underpin applications ranging from sharp heat kernel estimates to geometric compactness and stratification in nonlinear PDE settings.

Monneau-type monotonicity formulas form a family of scale-invariant analytic identities that play a central role in the quantitative analysis of solutions to elliptic and parabolic partial differential equations, particularly in geometric analysis and the paper of blow-up phenomena. While originally introduced in the context of obstacle and free boundary problems, the essential mechanism—identifying a functional whose derivative is controlled by nonnegative geometric or analytic quantities, such that equality signals a model solution—extends to a broad range of settings encompassing elliptic, parabolic, and geometric flows.

1. Monotonicity Formulas in Parabolic and Elliptic Settings

In parabolic settings, a prototypical example is the monotonicity of the $W$-functional for the normalized heat equation. For a positive solution $u(x,t)$ with $\int u(x,t)\,dx=1$ for all $t$, define:

• Entropy: $S(t) = -\int_{M} u(x,t)\log u(x,t)\,dx$
• Fisher information: $F(t) = t \int_{M} |\nabla \log u(x,t)|^2 u(x,t) dx$
• Combined Monotone Functional: $W(t) = S(t) + F(t)$

The evolution is given by

$$W'(t) = -2t \int_{M} \left( \left| \mathrm{Hess}(\log u) - \frac{1}{2t} g \right|^2 + \mathrm{Ric}(\nabla \log u, \nabla \log u) \right) u\,dx,$$

so $W(t)$ is nonincreasing when $\mathrm{Ric} \geq 0$. Rigidity holds: $W'(t)=0$ iff $u$ is the Euclidean heat kernel.

In elliptic settings, for $u > 0$ harmonic in a punctured neighborhood and $u(x) = |x|^{2-n}$ in $\mathbb{R}^n$, set $b^{2-n}=u$, so $b = |x|$ (Euclidean), and define

$A(r) = r^{1-n} \int_{b=r} |\nabla b|^3.$

A computation yields

$$A'(r) = -C \cdot r^{1-n} \int_{b=r} \left( \left| \mathrm{Hess}(b^2) - \frac{\Delta b^2}{n} g \right|^2 + \mathrm{Ric}(\nabla b, \nabla b) \right),$$

again guaranteeing monotonicity and rigidity in the model case (Euclidean space).

2. Model-Versus-Perturbation Philosophy and Geometric Interpretation

The defining characteristic of Monneau-type formulas is their "model-versus-perturbation" nature: the monotone quantity is strictly constant for the reference model (e.g., Euclidean space, the heat kernel, or the Green's function), and the rate of decrease (or vanishing of the derivative) quantifies deviation from this model. The analytic integrand (such as $|\mathrm{Hess}(\log u) - (1/2t)g|^2$ or $|\mathrm{Hess}(b^2) - (\Delta b^2/n)g|^2$) captures precise geometric or analytic invariants—measuring whether the manifold is conical (in the harmonic case) or the solution is an extremal kernel (in the heat case).

Consequently, these functionals yield:

• Rigidity statements: Equality in the monotonicity formula implies the solution and space coincide with the model.
• Quantitative stability: The deficit in the derivative provides a scale-invariant measure of flexibility in the geometry or function space.

3. Analytic and Geometric Implications

The interplay between monotonicity and geometry produces strong consequences:

• Sharp Analytic Bounds: For the parabolic case, monotonicity of $W(t)$ directly leads to Li–Yau gradient estimates and Harnack inequalities. The function space (e.g., heat semigroup solutions) reflects the underlying geometry via Ricci curvature’s role in the error terms.
• Geometric Compactness and Rigidity: In the elliptic case, monotonicity of $A(r)$ provides a mechanism for identifying tangent cones at singularities or at infinity: if $A'(r)\rightarrow 0$ sufficiently rapidly, blow-ups converge to metric cones, and uniqueness of tangent cones can often be deduced via quantitative decay (see inequalities such as $-A'(r) \geq C O_2(r)$).
• Functional Inequalities as Consequences: Classical inequalities such as the logarithmic Sobolev inequality and the Bishop–Gromov volume comparison are encompassed by such monotonicity principles, highlighting a unifying framework.

4. Structural Overview of Key Formulas

The following table summarizes the main monotone functionals and their derivative representations:

Setting	Monotone Functional	Derivative Formula / Deficit
Parabolic (heat)	$W(t) = S(t) + F(t)$	$$W'(t) = -2t \int [\cdots]^2 + \mathrm{Ric}(\cdots)]u$$
Elliptic (harmonic/Green)	$A(r)$	$$A'(r) = -C \int_{b=r} [|\mathrm{Hess}(b^2) - (\Delta b^2/n)g|^2 + \mathrm{Ric}(\nabla b, \nabla b)]$$

Where the deficit terms vanish exactly in the reference model spaces.

5. Methodological Consequences: Uniqueness and Stratification

The tight control afforded by the monotonicity is crucial to singularity analysis:

• Blow-up Analysis: Since the scale-invariant excess decays, blow-up sequences (under appropriate normalization) converge to the model profile—a measure of "homogeneity" at all scales.
• Uniqueness of Tangent Cones: Strong bounds on the decay of the deficit (such as via $Q(r) \sim -r A'(r)$) ensure uniqueness of the asymptotic (tangent) objects, guaranteeing stratification and regularity of the free boundary or singular set.
• Singular Set Stratification: Application of Whitney’s extension theorem and further geometric analysis ensures the singular set is contained in countable unions of smooth manifolds of varying dimensions, with precise geometric characterization.

6. Unified Perspective and Applications

The broader significance of Monneau-type monotonicity formulas lies in their ability to unify disparate branches of geometric analysis by a common analytic mechanism, including:

• Uniform treatment of parabolic and elliptic PDEs leading to analytic-to-geometric transfer principles.
• Bridging function space properties (e.g., sharp pointwise or integral estimates) with global and asymptotic geometry (rigidity, cone structure, regularity).
• Providing new insights into the regularity and stratification of singularities in nonlinear elliptic and parabolic equations, seen, for example, in free boundary problems, Ricci and Einstein geometry, and heat kernel analysis.

7. Summary and Outlook

Monneau-type monotonicity formulas, as demonstrated in both the analytic and geometric settings, provide both a diagnostic of rigidity and a quantitative bridge between the analytic structure of solutions and the global geometry of the underlying space. Their derivatives, written as integrals of positive-definite quantities, vanish only when the solution aligns with the canonical model. This paradigm has produced a suite of sharp results across PDE, geometric analysis, and probability (e.g., heat kernel estimates, uniqueness of tangent cones, compactness theorems), with the function space–geometry interplay made explicit by the monotonicity identities. These tools now form part of the core analytic machinery for analyzing geometric PDE and continue to inspire generalizations to broader settings, including non-smooth spaces and nonlinear flows.