Global Hölder Continuity in PDEs

• Global Hölder continuous solutions are functions that satisfy uniform Hölder estimates across entire domains, ensuring robust control up to boundaries.
• The approach utilizes capacity density conditions, De Giorgi–Nash–Moser iterations, and barrier techniques to derive sharp regularity in elliptic and parabolic equations.
• Applications span nonlinear PDEs, stochastic models, and dynamical systems, bridging qualitative existence results with quantitative regularity insights.

A globally Hölder continuous solution is a function satisfying a prescribed PDE or functional equation for which a global Hölder estimate holds uniformly across the entire domain. This property is central in the modern theory of partial differential equations, dynamical systems, and geometric analysis. Such solutions arise in elliptic and parabolic regularity theory, nonlinear transport equations, stochastic PDEs, and dynamical systems (e.g., cohomological equations on manifolds), providing a critical bridge between qualitative existence/uniqueness results and quantitative regularity. The global nature ensures uniform control up to the boundary (or at all points in the phase space), not just locally.

1. Definition and Core Properties

Let $u:\Omega\to\mathbb{R}$, $\Omega \subset \mathbb{R}^n$ (or a manifold), be a bounded function. For $0<\alpha\le 1$, $u$ is globally $\alpha$-Hölder continuous (notation: $u \in C^{0,\alpha}(\bar\Omega)$) if

$$\|u\|_{C^{0,\alpha}(\bar\Omega)} := \sup_{x\in\Omega} |u(x)| + \sup_{x\neq y\in\Omega} \frac{|u(x) - u(y)|}{|x-y|^{\alpha}} < \infty.$$

In the context of parabolic or time-dependent equations, the global Hölder norm is typically parabolic,

$$[u]_{C^{\alpha,\alpha/2}(\Omega_T)} := \sup_{(x,t)\neq (y,s)} \frac{|u(x,t) - u(y,s)|}{\max\{ |x-y|, |t-s|^{1/2}\}^{\alpha}}.$$

The global Hölder exponent of $u$ is the maximal $\alpha$ such that $u\in C^{0,\alpha}(\bar\Omega)$; equivalently,

$$h(u) := \inf_{x} \liminf_{y\to x} \frac{\log|u(x)-u(y)|}{\log|x-y|},$$

with $u\in C^{0,\alpha}$ iff $h(u)\ge \alpha$ (Todorov, 2015).

Global Hölder continuity ensures robust pointwise control, uniform up to the boundary (Dirichlet, Neumann, or complex geometric boundaries), and is distinct from local Hölder control, which may fail at singularities or near boundary points.

2. Sharp Global Hölder Regularity in Elliptic and Parabolic Equations

Classical elliptic and parabolic equations in divergence form, as well as their quasilinear and nonlinear extensions, admit global Hölder continuous solutions under quantitative geometric and analytic hypotheses. Two seminal advances are the use of capacity density conditions (CDC) on domains and Morrey-typified data.

• Elliptic Equations: For

$$-\text{div}(A(x)\nabla u) = f(x), \quad u|_{\partial\Omega} = g(x), \quad A \text{ uniformly elliptic}$$

on a CDC-domain (uniform lower bound on variational capacity of boundary layers), if $f\in L^p(\Omega),\ p>n/2$, and $g\in C^{0,\beta}(\partial\Omega)$, there exists $\alpha>0$ (depending explicitly on $n$, $\lambda/\Lambda$, CDC-constant, $p$, $\beta$) so that $u\in C^{0,\alpha}(\bar\Omega)$, with the estimate (Hara, 2024)

$$\|u\|_{C^{0,\alpha}(\bar\Omega)} \le C \left( \|f\|_{L^p(\Omega)} + \|g\|_{C^{0,\beta}(\partial\Omega)} \right).$$

The exponent $\alpha$ is determined via interior De Giorgi–Nash–Moser theory, boundary-barrier decay from CDC, and data integrability.

• Parabolic Equations: For

$$u_t - \text{div}(A(x,t)\nabla u) = f(x,t) \text{ in } \Omega_T = \Omega \times (0,T)$$

on a CDC-domain, with $A$ uniformly parabolic, and $f$ satisfying $|f(x,t)|\le C\delta(x)^{-2+\alpha}$ (with $\delta(x) = \text{dist}(x,\partial\Omega)$), the solution $u$ is globally Hölder continuous in $C^{\alpha_*,\alpha_*/2}(\bar\Omega_T)$, for some $\alpha_*\in (0,1]$ depending only on problem parameters (Hara, 6 Jan 2026).

• Quasilinear & Morrey Data: For nonlinear $p$-Laplacean/Poisson equations and quasilinear divergence-form equations driven by Morrey-class measures ($L^{p,\lambda}$ or $M_{\partial\Omega}^q$), global Hölder regularity follows if the domain is CDC or $m$-thick, and the data satisfy Morrey-type integrability, with explicit exponents derived from the data's "gaps" $(p(m-1)+\lambda-n, mq+\mu-n, m+\omega-n)$ (Byun et al., 2015, Hara, 2023, Hara, 2024). The global exponent is always the minimum of: interior Hölder exponent arising from higher integrability, boundary decay exponent from barrier construction, and the threshold set by data regularity.

Equation type	Domain Condition	Data Class	Exponent Formula	Reference
Elliptic, divergence-form linear	Capacity density	$L^p$, $p>n/2$	$$\alpha = \min\{\alpha_{\mathrm{int}}, \alpha_{\mathrm{bdry}}, 2-n/p, \beta\}$$	(Hara, 2024)
Parabolic, divergence-form linear	Capacity density	$|f(x,t)|\le C\delta^{-2+\alpha}$	$\alpha_* = \min\{\alpha_0, \alpha_H\}$	(Hara, 6 Jan 2026)
Quasilinear/p-Laplacian, Morrey data	$m$-thick (CDC)	Morrey $L^{p,\lambda}$	$\alpha = \min\{\alpha_{\mathrm{int}},\tau\}$ (from data gaps)	(Byun et al., 2015, Hara, 2023)

3. Dynamical Systems and Nonlinear Weierstrass-Type Solutions

Global Hölder continuous solutions also manifest in dynamical systems, especially in the context of cohomological equations for smooth expanding maps. For expanding $C^2$ maps $f$ on $S^1$, and $\theta\in (0,1)$, consider the $\theta$-twisted cohomological equation: $u(f(x)) - (f'(x))^{\theta}u(x) = \phi(x)$ with $\phi\in C^{1+}(S^1)$. There exists a unique bounded solution admitted by the explicit Weierstrass-like series (Todorov, 2015): $$u(x) = - \sum_{n=0}^\infty \frac{\phi(f^n(x))}{\prod_{k=0}^n (f'(f^k(x)))^{\theta}}$$ This solution is globally $\theta$-Hölder continuous on $S^1$ unless it is smooth, and, generically, cannot be improved to any $C^{\theta+\gamma}$ at almost every point. The proof combines symbolic (box-dimension) and distortion/ergodicity arguments, showing the optimality of the exponent both globally and pointwise almost everywhere. These results generalize the classical (nonlinear) Weierstrass functions.

4. Stochastic and Convex Integration Construction of Hölder Solutions

In fluid dynamics and dispersive/hyperbolic equations, convex integration (and its stochastic variants) yields global Hölder continuous, yet highly nonunique, weak solutions with prescribed (often low) exponents.

• Deterministic and Stochastic Euler/NS (Convex Integration): For the 3D incompressible Euler equations (deterministic or stochastic), convex integration produces infinitely many global solutions (even stationary) in $C(\mathbb{R}; C^{\vartheta}(\mathbb{T}^3))$ with small $\vartheta>0$; constraints on the exponent are dictated by transport, commutator, and amplitude-frequency scaling—e.g., $\vartheta<\frac{5}{7}\beta$, $0<\beta<\frac{1}{24}$ for stochastic flows (Kinra et al., 2024, Lü, 2024, Enciso et al., 31 Jul 2025). These solutions may strictly dissipate energy (violating conservation laws despite the lack of viscosity), or preserve other invariants (e.g., magnetic helicity in MHD at $\alpha=10^{-8}$ (Enciso et al., 31 Jul 2025)).
• Negative-Order Dispersive Equations: For certain active scalar equations, one-sided global Hölder bounds with explicit time-decaying coefficients hold for entropy solutions, controlling both the modulus of continuity and the maximal lifespan of classical solutions (Maehlen et al., 2021). For $s\in[0,1]$ and kernel $K$,

$u(t,x) - u(t,y) \le a(t) (x-y)^{\frac{1+s}{2}}$

with $a(t)$ decaying explicitly, becomes a key regularity tool.

5. Hölder Regularity in Nonlinear Geometric and Complex Equations

In complex geometric analysis, the Dirichlet problem for the complex Hessian equations

$$(dd^c u)^m \wedge \omega^{n-m} = \mu \quad \text{in}\ \Omega,\quad u|_{\partial\Omega} = \varphi$$

admits a unique global Hölder continuous solution as soon as a Hölder continuous subsolution exists with $\mu \le (dd^c\psi)^m\wedge\omega^{n-m}$, without any assumption on the finite mass of $\mu$ (Kolodziej et al., 2024). The exponent $\alpha'$ depends on the ambient and subsolution exponents and the capacity domination, and remains strictly positive: $u\in C^{0,\alpha'}(\bar\Omega)$ even for unbounded right-hand side measures, provided capacity domination via a Hölder subsolution.

6. Applications, Extensions, and Optimality

Global Hölder regularity is foundational for:

• Analytic Well-posedness: Justification of uniqueness, stability, and a priori estimates in nonlinear PDEs, including fully nonlinear and non-divergence form equations with rough data, as well as anisotropic and vectorial elliptic systems (e.g., Maxwell's equations (Yu et al., 2024)).
• Homogenization: Uniform regularity yields quantitative convergence rates (typically, Hölder exponent over two) between oscillatory and effective solutions (Hara, 2024).
• Active Scalar Flows and Transport: Sharp propagation and continuity modulus in Hölder (and Zygmund) spaces for scalar transport equations with singular convolution velocities, ensuring persistent regularity and well-posedness (Magaña, 2024).
• Sharpness and Limiting Exponents: The CDC or $m$-thickness conditions are sharp; relaxing them destroys the possibility of global supremum estimates or boundary regularity. Exponents are minimal in examples; for instance, for nonlinear Weierstrass constructions and certain "optimal" a priori inequalities, pointwise and global exponents coincide and are unimprovable (Todorov, 2015).

7. Methodological and Technical Frameworks

A unified technical backbone across these results consists of:

• Capacity Density and Barrier Techniques: Quantitative CDC (exterior thickening), Hardy inequalities, and global barrier functions enable boundary comparison and maximum principle arguments (Hara, 2024, Byun et al., 2015).
• De Giorgi–Nash–Moser and Stampacchia Iterations: Oscillation and maximum principle techniques, tailored to Morrey or measure data and non-smooth domains (Byun et al., 2015, Hara, 2024).
• Symbolic Dynamics and Ergodic Arguments: For solutions to cohomological equations/Weierstrass-type series, symbolic dynamics, box dimension, and distortion estimates precisely yield sharp exponents (Todorov, 2015).
• Convex Integration: Iterative construction of nonsmooth solutions with prescribed modulus, amplitude, and frequency selection, ensuring Cauchy convergence in Hölder spaces, for Euler, MHD, and related systems (Enciso et al., 31 Jul 2025, Kinra et al., 2024, Lellis et al., 2020, Isett, 2012).
• Capacity Domination for Complex Monge–Ampère/Hessian Equations: Capacity estimates, Perron method, and Hölder barrier comparison control regularity in complex geometry (Kolodziej et al., 2024).

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The theory of globally Hölder continuous solutions has evolved into a quantitative, flexible framework encompassing a large class of analytic, probabilistic, and geometric PDEs, underpinned by precise domain geometry, measure/data structure, and robust iterative or comparison schemes. The optimal exponents are sharply characterized by the nature of the data, domain, and dynamical structure of the system under consideration.