Hölder Continuous Solutions

• Hölder continuous solutions are functions satisfying |u(x) - u(y)| ≤ C|x - y|^α, quantifying scale-invariant regularity in nonlinear and degenerate PDEs.
• Research shows that under conditions like uniform ellipticity, capacity domination, and Morrey space integrability, unique global Hölder solutions arise in elliptic, Monge–Ampère, and p(x)-Laplace equations.
• Advanced methods such as convex integration and stochastic analysis demonstrate how fine regularity thresholds lead to nonuniqueness phenomena in fluid dynamics and transport models.

Hölder continuous solutions are functions solving partial differential equations or dynamical problems while enjoying quantitative, scale-invariant continuity: for a given exponent $0 < \alpha \leq 1$, $u$ is Hölder continuous of exponent $\alpha$ on a domain $\Omega$ if $|u(x)-u(y)| \leq C|x-y|^\alpha$ for all $x,y\in\Omega$. The theory of Hölder continuous solutions forms a cornerstone in modern analysis of nonlinear and degenerate PDEs, optimal transport, geometric flows, convex integration, and stochastic PDEs, with a central objective: to characterize how equations transmit (or create) fine regularity on their solutions in the presence of rough data or singularities.

1. Classical and Nonlinear Elliptic Equations: Regularity Paradigms

The prototype for elliptic and parabolic regularity is the solution $u$ to equations of the form $$-\operatorname{div}(A(x)\nabla u) + b(x)\cdot\nabla u + \mu u = \nu$$ or fully nonlinear equations such as the Monge–Ampère or Hessian equations. For equations in divergence form with bounded measurable coefficients, local Hölder continuity is guaranteed under uniform ellipticity (De Giorgi–Nash–Moser), but nonuniform or degenerate coefficients demand alternative frameworks.

In "Global Hölder solvability of second order elliptic equations with locally integrable lower-order coefficients" (Hara, 2 Mar 2024), Hara constructs globally Hölder continuous solutions under precise control of lower-order coefficients and data in Morrey-type spaces, even when the coefficients may behave singularly near the boundary. The main result asserts: for $A$ uniformly elliptic and $|b|^2m$ and $\mu$ in critical Morrey spaces (allowing local behavior $\sim \delta(x)^{\beta-1}$ near $\partial\Omega$), every measure datum and Hölder boundary datum induces a unique globally Hölder continuous solution, with the Hölder exponent sharply dependent on dimension, $L^p$-integrability, the capacity-density condition on the domain, and the data regularity.

Similarly, variable exponent $p(x)$-Laplace equations $\Delta_{p(x)}u = g + \operatorname{div}\mathbf{F}$ possess locally Hölder continuous solutions under suitable integrability on $g$ and $\mathbf{F}$ (Lyaghfouri, 2020). The optimal exponent is determined by the infimum $p_-$ of $p(x)$, and the Lebesgue exponents for $g$ and $\mathbf{F}$. This framework encompasses degenerate and inhomogeneous models in the calculus of variations and geometric PDE.

2. Pluripotential Theory: Complex and Quaternionic Monge–Ampère Equations

The regularity of solutions to complex Monge–Ampère equations is central on compact Kähler and Hermitian manifolds as well as on bounded pseudoconvex domains. For $(X,\omega)$ a compact Kähler manifold, the fundamental result of Kołodziej guarantees boundedness of $\omega$-psh solutions to $(\omega+dd^c\phi)^n = f\omega^n$ with $f\in L^p(X)$, $p>1$. Demailly–Guédj–Kołodziej–Zeriahi et al. sharpen this: for every $\alpha<2/(nq+1)$ ($q=p/(p-1)$), the solution is $\alpha$-Hölder continuous globally, with the exact estimate

$$\|\phi\|_{C^\alpha(X)} \leq C(\|f\|_{L^p},X,\omega)$$

(Demailly et al., 2011). The proof critically combines capacity estimates (Kołodziej), pluripotential envelopes, a stability theorem for $L^1$-distances, and a Demailly–Kiselman regularization process. The theory is robust, extending to big classes, where local Hölder continuity is proven on the "ample locus" for singular classes (Demailly et al., 2011). For the non-Kähler, Hermitian setting, the equivalence between existence of Hölder continuous solutions and domination of the data measure by (locally) a Monge–Ampère measure of a Hölder continuous $\omega$-psh potential is proved (Kolodziej et al., 2017), with an explicit exponent $\beta = \min\left\{\alpha,1/(1+(n+2)(n+1))\right\}$.

Further generalizations address non-smooth geometric scenarios: in non-smooth pseudoconvex domains (of plurisubharmonic type $m$), bounded solutions to the Monge–Ampère equation with $L^p$ right-hand side and Hölder data achieve $C^\gamma$ regularity for any $\gamma < \min\{\alpha/(2m),1/(1+n[1-1/p(n+1)])\}$ (Hong et al., 2016). For the complex Hessian equations $(dd^cu)^m\wedge \beta^{n-m}=f\,\beta^n$ in $m$-pseudoconvex domains, explicit Hölder regularity holds for data $f\in L^p$, $p>n/m$, under additional boundary and near-boundary assumptions (Nguyen, 2013, Kolodziej et al., 18 Jul 2024).

In the quaternionic context, Boukhari (Boukhari, 2019) establishes sharp Hölder regularity for the quaternionic Monge–Ampère equation $(\Delta u)^n = f\,dV$, where $f\in L^p(\Omega)$, $p>2$, and the exponent is $\alpha<\min\{1/2,2/(nq+2)\}$, $q$ being the dual of $p$.

3. Parabolic, Transport, and Fluid Equations: Propagation and Creation of Hölder Regularity

Regularity mechanisms for parabolic and transport-type PDEs are nontrivially tied to the structure of the equation. For the kinematic dynamo system (induction equation for 3D MHD with prescribed velocity), initial Hölder regularity propagates exactly for the magnetic field $B$, provided the drift is in the critical scaling space $L_t^pW_x^{1,q}$ (Friedlander et al., 2014). No improvement (gain) of the Hölder exponent is expected.

For nonlocal and drift-diffusion equations, e.g.,

$$\partial_tu + B\cdot \nabla u + \mathcal{L}_t u = g(x,t),$$

with $\mathcal{L}_t$ a nonlocal operator of order $2s\leq 1$, local Hölder regularity is enforced under minimal ellipticity and drift $B$ in critical spaces ($BMO$ or $C^{1-2s}$), with explicit exponent $\alpha$ reaching the De Giorgi threshold $1-(d+1)/(2sq)$ (Nguyen et al., 2020).

Balance laws of the form $u_t + f(u)_x = g$ with $f$ polynomial nonlinearity of degree $\ell\geq 2$ and $g\in L^\infty$ admit unique continuous solutions $u\in C^{0,1/\ell}_{\mathrm{loc}}$ (Caravenna et al., 2023), with the exponent $1/\ell$ reflecting the polynomial degeneracy—a result mirrored in the analysis of intrinsic Lipschitz graphs in sub-Riemannian geometry.

For nonlinear SPDEs, as in the stochastic equations arising from one-dimensional superprocesses, solutions attain optimal spatial Hölder $1/2$ and temporal $1/4$ regularity, established via Malliavin calculus and integration-by-parts for stochastic neural kernels (Hu et al., 2011).

4. Convex Integration and Nonuniqueness Phenomena: Weak Solutions Below Onsager Thresholds

The last decade has seen the emergence of convex integration as a tool for constructing highly irregular (often wild or nonunique) solutions with prescribed regularity for fluid dynamics systems, including the incompressible Euler, the 3D Prandtl, and ideal MHD equations. In the Euler setting, it is possible to construct weak solutions $v\in C^{1/5-\epsilon}_{t,x}$ compactly supported in time and indistinguishable from any smooth solution on a subdomain (Isett, 2012). The method iterates Reynolds-correcting perturbations, based on high-frequency Beltrami waves, to drive the stress error to zero while controlling the regularity via delicate frequency-energy bookkeeping.

Major implications include the realization (in principle) of the Onsager conjecture (dissipation of energy for Hӧlder exponents $<1/3$), and the flexibility to glue (up to $C^{1/5-\epsilon}$) any smooth flow segment to zero (Isett, 2012).

The method extends to other settings: for the 3D Prandtl equations, nonuniqueness of weak $C^{0,\alpha,\beta}$-solutions is established without monotonicity or analytic structure (Luo et al., 2018). In ideal MHD, recent results demonstrate the existence of solutions $(v,B)\in C^{\alpha}$ for $\alpha=10^{-8}$ which conserve magnetic helicity but not energy or cross-helicity (Enciso et al., 31 Jul 2025), thereby explicitly separating which conservation laws persist at various regularity thresholds.

In stochastic fluid systems, stochastic convex integration delivers infinitely many Hölder continuous solutions (in the sense of probability law or pathwise) up to exponents $\vartheta<\frac{5}{7}\beta$ with $\beta<1/24$ (Kinra et al., 25 Jul 2024, Lü, 29 Jul 2024), showing that stochastic forcing does not enforce uniqueness above the Onsager threshold.

5. Hölder Spaces, Solution Maps, and Continuity of Flows

Hölder spaces (classically $C^\alpha$; "little Hölder" $c^\gamma$) and their Zygmund analogues ($\Lambda^*$, $\lambda^*$) are central for expressing the regularity of solutions and for analyzing the stability and robustness of solution operators. For nonlinear active scalar equations with transport by convolution, the solution map from initial data in the little Hölder class $c^\gamma$ to the solution remains continuous as long as $\gamma\in(0,1)$, while the criticality or ill-posedness at $\gamma=1$ is open (Magaña, 24 Oct 2024).

The framework encompasses vorticity formulations of 2D Euler and SQG, and shows well-posedness in the Hadamard sense for initial data in $c^\gamma$ or $\lambda^*$, with local-in-time estimates but global-in-time extensions under additional conservation or dissipativity structure.

6. Capacity Theory, Stability, and Intrinsic Characterizations

Capacity domination, a notion quantifying the concentration properties of admissible measures relative to analytic capacities (e.g., Bedford–Taylor or Hessian capacity), underlies most global Hölder regularity results. Central is the equivalence between existence of global Hölder continuous solutions and domination of the data measure by the Monge–Ampère (or Hessian) measure of a local Hölder continuous plurisubharmonic function. Key stability estimates link the $L^1$-distance of solution approximations to oscillation or modulus-of-continuity estimates, with precise exponents dictated by capacity conditions, duality, and dimensional parameters (Demailly et al., 2011, Kolodziej et al., 2017, Nguyen, 2013, Kolodziej et al., 18 Jul 2024).

This leads to precise characterizations for the range of the Monge--Ampère operator (for measures that arise as Monge–Ampère of a Hölder potential), convexity and the $L^p$-property of this class, and explicit descriptions for radial and torically symmetric measures.

7. Applications and Broader Implications

Sharp Hölder regularity estimates enable progress on:

• Local and global well-posedness, uniqueness, and mixing in nonlinear PDEs;
• Nonuniqueness, compact-support gluing, and Onsager-critical constructions in hydrodynamics, Prandtl, and MHD;
• Regularity of optimal transport maps and complex geodesic flows on Kähler manifolds;
• Analysis of singular structures in geometric PDEs and rigidity questions for translation flows (via interval exchange maps);
• PDE-based proofs of geometric measure-theoretic results, as for intrinsic Lipschitz graphs in the Heisenberg group (Caravenna et al., 2023).

These findings clarify the mechanisms for both propagation and breakdown of regularity, the structural roles of capacities and singular measures, and the interplay between analytic regularity and dynamical flexibility. They also reveal precise thresholds for the preservation of physical conservation laws and the subtlety of stochastic effects on regularity, uniqueness, and pathwise robustness.