Global Large-α Constraint in Parabolic Equations

• Global Large-α constraint is defined as an upper bound α₀ < 1 for the Hölder regularity exponent in parabolic boundary value problems, ensuring compatibility with interior and boundary barrier constructions.
• It is established through a combination of interior oscillation decay and dedicated barrier estimates at the initial, lateral, and corner domains, rigorously restricting regularity to 0 < α < α₀.
• The constraint is essential in regularity theory for degenerate, non-Euclidean parabolic PDEs, guiding both existence results and optimal boundary regularity frameworks.

A global Large-α constraint refers to a structural limitation imposed on the allowable Hölder regularity exponents for solutions to boundary value problems involving parabolic (or elliptic–parabolic) partial differential equations with degenerate or non-Euclidean geometries. Specifically, in the context of regularity theory for the linearized parabolic Monge–Ampère equation, the “global Large-α constraint” prescribes an upper bound α₀ < 1 for the Hölder regularity exponent α in global estimates of the form $$u \in C^{1+\alpha,\frac{1+\alpha}{2}}(\overline{\Omega}\times[0,T])$$, reflecting nontrivial limitations arising from the interplay of interior and boundary barrier constructions, and characteristic degeneracies at the parabolic boundary.

1. Definition and Main Theorem

For the linearized parabolic Monge–Ampère equation

$$\mathcal{L}_\phi u(x,t) := -u_t\,\mathrm{det}D^2\phi(x) + \mathrm{tr}[\Phi(x) D^2 u] = f(x,t)$$

with φ a strictly convex solution to $\det D^2\phi = g(x)$, the global regularity result establishes that, under appropriate geometric and coefficient conditions, for every $0<\alpha<\alpha_0$ (where $\alpha_0$ is determined by the domain and equation coefficients), any solution u with compatible initial and boundary data admits the estimate

$$\|u\|_{C^{1+\alpha,\tfrac{1+\alpha}{2}}(\overline\Omega\times[0,T])} \le C \Big( \|u\|_{L^\infty} + \|f\|_{C^{\alpha,\alpha/2}} + \|\psi\|_{C^{2,1}} \Big),$$

where $C=C(n,\lambda,\Lambda,p,T,[g]_{C^0})$ is uniform. The critical point is that the global exponent must satisfy $0 < \alpha < \alpha_0 < 1$, with $\alpha_0$ defined in terms of several geometric and analytic thresholds arising in the proof.

2. Geometric and Boundary Requirements

The bounded domain $\Omega \subset \mathbb{R}^n$ must satisfy an interior-ball condition of uniform radius p at every boundary point; i.e., for every $x_0\in\partial\Omega$, there exists a ball of radius p tangent to ∂Ω at $x_0$. The Monge–Ampère potential φ must be strictly convex with quadratic separation from tangent planes: $$p|x-x_0|^2 \leq \phi(x) - \phi(x_0) - \nabla\phi(x_0)\cdot(x-x_0) \leq p^{-1}|x-x_0|^2,$$ for all $x,x_0\in\partial\Omega$. The Monge–Ampère measure $g(x)$ must be bounded: $\lambda \leq g(x) \leq \Lambda$.

3. Breakdown of Regularity Exponent Restrictions

The constraint $\alpha < \alpha_0$ does not result from a single obstruction, but from the compounding effect of four distinct a priori estimates, each derived on a different “face” of the space–time domain:

• Interior regularity: Interior oscillation decay (Harnack inequality and intrinsic parabolic geometry) provides an exponent $\beta(n,\lambda,\Lambda)\in(0,1)$.
• Initial time boundary $t=0$: Barrier arguments for data at $t=0$ provide an exponent $b(n)\in(0,1)$.
• Lateral (spatial) boundary: A combination of weak Harnack estimates and construction of sub- and super-solutions gives an exponent $\tau(n,\lambda,\Lambda,p)\in(0,1)$.
• Spatio-temporal corner ($\partial\Omega\times\{0\}$): The “corner” estimate, matching the space and time boundary, forces $\alpha<1$ so that barriers constructed for maximum principles decay appropriately.

The global Large-α constraint then is

$\alpha_0 = \min\{\beta,\,b,\,\tau,\,1\},$

and all interpolations and bootstrapping proceed within the range $0<\alpha<\alpha_0$.

4. Explicit A Priori Estimates and Representative Lemmas

Several intermediate lemmas establish the behavior of u near each sub-boundary:

Subdomain	Exponent Constraint	Key Estimate or Lemma
Interior	$\beta$	$$\operatorname{osc}_{Q_\phi(z_0,r)}u \leq C (r/R)^\alpha[\sup_{Q_\phi(z_0,R)}|u| + \dots]$$
$t=0$	$b$	$|u(x,t)-\ell_{x_0,0}(x)|\leq C[\phi_{x_0}(x)+t]^b$
Lateral	$\tau$	Parabolic iteration yields $u\in C^{1+\tau,\frac{1+\tau}{2}}$ under sandwiching by barrier functions
Corner	$<1$	$$|u(x,t)-\ell_{0,0}(x)|\leq C(|x|^2+t)^{(1+\alpha)/2}$$, necessarily requiring $\alpha<1$

Each domain-specific restriction limits the permissible global exponent, irrespective of the local smoothness of coefficients or data.

5. Comparison to Interior Regularity and the Elliptic Case

In the elliptic linearized Monge–Ampère equation, global $C^{1+\alpha}(\overline\Omega)$ bounds hold as soon as φ and g are regular, and ∂Ω is sufficiently smooth, with any loss in regularity coming from boundary-localization phenomena. By contrast, the parabolic problem’s multi-faced boundary—comprising spatial, temporal, and spatio-temporal “corner” elements—necessitates a strictly subunit global exponent, reflecting the sharper degeneracy and interaction of geometric barriers in time-dependent settings. A plausible implication is that even with arbitrarily high interior regularity, global extensions cannot cross the α₀ threshold set by the combination of these barriers.

6. Interpretation and Broader Context

The Large-α regularity constraint is not an artifact of proof technique, but a genuine feature of parabolic boundary geometry for degenerate–non-Euclidean operators. Attempts to exceed α₀ globally encounter barrier failure and nonclosure of comparison principles. This limitation is similar in spirit to what occurs in other geometric/homogeneous PDEs (e.g., subelliptic equations), where “corners” or singular submanifolds force global exponents strictly below those available in the interior.

Furthermore, the precise construction of barriers, the use of the intrinsic Monge–Ampère geometry, and the coupling with the initial data at $t=0$ are essential in arriving at the global constraint. Thus, any program aiming to upgrade the global regularity must address—not circumvent—the sharpness of the exponent restriction imposed by the geometry of both the equation and the domain.

7. Significance for Regularity Theory

The global Large-α constraint exemplifies how boundary geometry and operator degeneracy dictate the optimal regularity of PDE solutions in global Hölder spaces. Its role in the theory of Monge–Ampère and related nonlinear parabolic equations is foundational: it sets the attainable regularity for solutions with general geometric data, structuring both existence theorems and boundary regularity programs. The constraint shapes the structure of further refinements, such as inhomogeneous data, non-Euclidean reflection arguments, or higher codimension subboundary analysis. These features indicate that the Large-α constraint is an intrinsic and necessary component of parabolic Monge–Ampère regularity theory, delineating the boundary of what can be achieved globally in Hölder scales for such degenerate PDEs (Tang et al., 2018).