Singular-Degenerate Coefficients in PDEs

• Singular-degenerate coefficients are weights in PDEs that vanish or become unbounded, critically influencing solution behavior and boundary regularity.
• They are analyzed via weighted Lebesgue and Sobolev spaces, employing methods such as weighted Calderón–Zygmund and Schauder estimates.
• This framework underpins applications in fractional diffusion, stochastic processes, and geometric analysis by extending classical regularity theories.

Singular-degenerate coefficients arise in linear and nonlinear partial differential equations (PDEs) when the principal part of the operator involves weights or diffusion coefficients that either vanish (degenerate) or become unbounded (singular) in parts of the domain, often near the boundary or along interior sets. Such coefficients are prevalent in regular and fractional elliptic/parabolic theory, stochastic fast-diffusion, control theory, and models of anomalous transport. The analysis of PDEs with singular-degenerate coefficients has given rise to a robust theory integrating weighted functional spaces, sharp regularity estimates, and modern perturbative techniques, with key advances over the last decade significantly extending classical Calderón–Zygmund and Schauder frameworks.

1. Model Operators and Definitions

The canonical formulation for operators with singular-degenerate coefficients involves a weight, often of power type, in both divergence and non-divergence structure:

• Divergence-form: $$\operatorname{div}\left(\mu(x) A(x, \nabla u)\right) = \operatorname{div}\left(\mu(x) F\right)$$ in $\Omega$
• Weighted coefficients: $\mu(x)=\mathrm{dist}(x,\partial\Omega)^\alpha$, $\alpha\in(-1,\infty)$, or more generally, $\mu(x) = |x|^\beta$, $x\in\mathbb{R}^n$
• Degeneracy at the boundary: $\alpha>0$, $\mu(x)\to0$
• Singularity at the boundary: $\alpha<0$, $\mu(x)\to\infty$

The weight $\mu(x)$ appears both in the principal symbol and in the natural (weighted) function spaces for solutions (e.g., $L^p(\Omega, \mu)$, $W^{1,p}(\Omega, \mu)$). In parabolic settings, similar structures with weights in time or space yield models such as $u_t - \mathrm{div}(\mu(x)A(x,t)\nabla u) = f$, see (Dong et al., 2023, Dong et al., 2020, Dong et al., 2018).

In interior or mixed-dimensional models, singularities/degeneracies may also occur along submanifolds–for example, degeneracy along ${x_j = 0}$ for $j=1,2,3$ in multidimensional Helmholtz-type equations or fast-diffusion SPDEs with multivalued monotone operators (Ergashev, 2018, Gess et al., 2015).

2. Weighted Norms, Sobolev Spaces, and Hypotheses

The functional analytic framework is fundamentally shaped by the singular-degenerate weight:

• Weighted Lebesgue and Sobolev spaces: $L^p(\Omega, \mu)$; $W^{1,p}(\Omega, \mu)$; their mixed-norm and parabolic analogues.
• Admissible weights: Muckenhoupt $A_p$-class ($\mu$ satisfies the $A_p$ doubling and reverse Hölder conditions)
• Parameter ranges: $|\alpha|<1$ for power weights $\mu(x)=x_n^\alpha$ to guarantee that $\mu\in A_2$ and so that standard weighted estimates hold, cf. (Fujishima et al., 2017, Dong et al., 2018, Cao et al., 2016).

In cases where $\mu$ fails the $A_2$ condition (e.g., $\alpha\ge1$ or $\alpha\le-1$), advanced perturbative and localization methods allow extension of regularity theory beyond the classical scope (Dong et al., 2023).

3. Main Regularity and A Priori Estimates

3.1 Weighted Calderón–Zygmund Theory

For weak solutions $u$ of the weighted elliptic or parabolic equation in divergence form, global and local $W^{1,p}$ or $W^{1,q}_\mu$-type estimates of the form

$$\|\nabla u\|_{L^p(\Omega, \mu)} \leq C \|F\|_{L^p(\Omega, \mu)}$$

hold under small weighted mean oscillation of the coefficients (in a weighted BMO or DMO sense), uniform (weighted) ellipticity, and boundary regularity (e.g., Reifenberg flatness) (Mengesha et al., 2016, Cao et al., 2016, Dong et al., 2023, Dong et al., 2020, Dong et al., 2018).

These estimates rely on weighted reverse Hölder inequalities, energy methods, and real-variable (level-set or maximal function) arguments, extending Caffarelli–Peral's approach to the weighted setting.

3.2 Weighted Schauder and Boundary Regularity

Given coefficients with Dini mean oscillation (DMO), solutions possess weighted $C^{k,\omega}$ or $C^{k+α}$ regularity up to the boundary for both degenerate ($\alpha>0$) and singular ($\alpha<0$) cases:

$$\sum_{|\beta|\leq k}\|D^\beta u\|_{L^\infty(B^+_{1/2})} + \sum_{|\beta|=k}[D^\beta u]_{C^{0,\omega}(B^+_{1/2})} \le C (\|\cdot\|_{L^2(\cdot)} + ... )$$

with the correct weighted norms and modulus (Dong et al., 2023, Dong et al., 13 Feb 2025, Dong et al., 30 Jul 2025).

Boundary and interior regularity techniques crucially use freezing of coefficients in tangential or time variables, Dini control, and new Lipschitz estimates in weighted settings.

3.3 Beyond Classical $A_2$ Regime

Weights $\mu(x)=\mathrm{dist}(x,\partial\Omega)^\alpha$ fail $A_2$ for $\alpha\geq1$ or $\alpha\leq-1$. Perturbative regimes combine small weighted mean oscillation and explicit control of the weight's behavior near the boundary, leading to weighted Sobolev regularity for all $\alpha\in(-1,\infty)$ (Dong et al., 2023).

4. Methodological Innovations

Technique	Key Feature	Reference
Weighted perturbation/small BMO/DMO	Localized oscillation controls, beyond global A2	(Dong et al., 2023, Cao et al., 2016)
Level-set/Calderón–Zygmund/maximal function	$L^2$ integrability bootstrapped to $L^p$ via density and covering lemmas	(Cao et al., 2016, Fang et al., 22 Oct 2025)
Lipschitz and Campanato-type boundary estimates	Sharp pointwise regularity near singular/degenerate boundary	(Dong et al., 2023, Dong et al., 2018, Mengesha et al., 2016)
Weighted Caccioppoli/reverse Hölder	Self-improvement of integrability in weighted setting	(Dong et al., 2018, Cao et al., 2016)
Higher-order boundary Harnack principles	Weighted Schauder inputs for ratio estimates of positive solutions	(Dong et al., 2023, Dong et al., 13 Feb 2025, Dong et al., 30 Jul 2025)

Advanced results also involve regularized/approximating weights (with $A_{1+1/n}$-BMO control), weighted Aubin–Lions compactness theorems, and energy methods for backward SPDEs with degeneracy and singularity (Fang et al., 7 Jan 2026, Fang et al., 22 Oct 2025, Horst et al., 2014).

5. Paradigmatic Examples and Applications

• Spectral Fractional Laplacian: The Caffarelli–Silvestre extension formula for $L^s$ and $\mathrm{div}(y^{1-2s}A(x)\nabla U)=0$ links singular-degenerate weights to nonlocal operators (Mengesha et al., 2016, Dong et al., 2018).
• Nonlinear Fast Diffusion and Fujita Exponents: Thresholds for blow-up and global existence are weighted by the singular-degenerate behavior (e.g., $w(x)=|x|^\beta$ or $w(x)=|x_1|^\alpha$ with Muckenhoupt $A_2$ structure) (Fujishima et al., 2017, Gess et al., 2015).
• Stochastic/BSPDEs: Degenerate backward SPDEs with singular terminal values and non-uniform diffusions in optimal control of liquidation and self-organized criticality problems (Horst et al., 2014, Gess et al., 2015).
• Multidimensional Equations with Several Singular Coefficients: Construction of fundamental solutions in terms of multivariable hypergeometric functions for elliptic and Helmholtz equations with multiple singularities (Ergashev, 2018, Ergashev, 2018).

6. Analytical and Structural Challenges

6.1 Smallness Conditions and Sharpness

The necessity of small oscillation (weighted BMO or DMO) in the coefficients is critical: without this, higher integrability or regularity fails even if the coefficient is Hölder or VMO in the classical sense but not with respect to the underlying weight (Cao et al., 2016). Counterexamples illustrate failure of Calderón–Zygmund estimates in the absence of such smallness.

6.2 Boundary and Interior Singular/ Degenerate Sets

Analysis near the sets where the diffusion vanishes or blows up (e.g., $x_n=0$) demands fine perturbation techniques, weighted Sobolev inequalities, and Hardy-type inequalities. In evolutionary problems, anisotropic parabolic cylinders weighted by $\mu$ capture the intrinsic geometry and scaling (Fang et al., 22 Oct 2025, Fang et al., 7 Jan 2026).

6.3 Extensions Beyond Scalar and Linear Theory

While the scalar theory is now well-understood, further extension to systems and fully nonlinear (or fully degenerate/singular) operators remains open, with partial results for SPDEs and stochastic variational inequalities (Gess et al., 2015). Parabolic Carleman estimates and control theory for interior degeneracy/singularity are in active development (Fragnelli et al., 2015).

7. Connections to Fractional, Nonlocal, and Geometric Problems

Singular-degenerate coefficients appear naturally in nonlocal and fractional diffusion via extension problems, in geometry (singular Yamabe, fast-diffusion, and stratified spaces), and in mathematical finance (Heston/SABR models). The regularity and analysis of solutions to such equations require the above singular-degenerate frameworks in both divergence and non-divergence settings (Dong et al., 2018, Mengesha et al., 2016).

These analytic and geometric perspectives are currently being synthesized with advances in fully nonlinear equations, stochastic analysis, optimal transport, and geometric measure theory.