Quasilinear Divergence Form Elliptic Equations

• Quasilinear divergence form elliptic equations are nonlinear PDEs with the principal term in divergence form and natural growth conditions, generalizing the p-Laplace operator.
• They leverage advanced methods such as reverse Hölder techniques, nonlinear potential theory, and capacity estimates to establish existence and regularity.
• The framework addresses anisotropic, degenerate, and weighted problems, extending inverse problem methodologies to recover nonlinear coefficients from boundary data.

A quasilinear divergence form elliptic equation is a nonlinear partial differential equation of the form

$$-\operatorname{div} \mathbf{A}(x,u,\nabla u) + B(x,u,\nabla u) = f(x)$$

posed in a domain $\Omega \subset \mathbb{R}^n$, subject to either Dirichlet, Neumann, or more general boundary conditions. The leading term is in divergence form and quasilinear, that is, the principal part $\mathbf{A}$ depends nonlinearly (but typically polynomially or analytically) on the gradient $\nabla u$, while possibly depending on $u$ and $x$ as well. Such equations encompass a broad class of nonlinear elliptic PDEs including the $p$-Laplace equation, anisotropic and degenerate variants, and systems with natural growth, measure data, singular drift, or anisotropic higher-order nonlinearities.

1. Structural Prototypes and Analytical Setting

The canonical representative is the divergence-form equation

$$-\operatorname{div} \mathbf{A}(x,u,\nabla u) + B(x,u,\nabla u) = f(x) \quad \text{in }\Omega$$

where:

• $$\mathbf{A} : \Omega \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$$ is a Carathéodory function:
  • Measurable in $x$, continuous in $(u,\xi)$.
  • Satisfies coercivity/monotonicity: $$(\mathbf{A}(x,u,\xi)-\mathbf{A}(x,u,\eta))\cdot (\xi-\eta) \geq \mu |\xi-\eta|^2$$ (for $p=2$; higher $p$ need more general $p$-growth structure).
  • Growth: $|\mathbf{A}(x,u,\xi)| \leq C(1 + |\xi|^{p-1})$.
  • Examples: $\mathbf{A}(x,u,\xi) = a(x,u)|\xi|^{p-2}\xi$, $p$-Laplace operator, or analytic polynomial in $\xi$, or anisotropic/homogeneous nonlinearities (Cârstea et al., 2020).
• $B$ is a lower-order Carathéodory term (often controlled growth in $u$, $\nabla u$).

Associated function spaces are Sobolev or Orlicz–Sobolev, depending on the growth structure. Ellipticity and monotonicity conditions can be uniform, degenerate, or even degenerate on a set $K$ (allowing, e.g., dead core or plateau phenomena) (Ciraolo, 2012).

2. Existence, Regularity, and Maximum/Comparison Principles

Existence theory relies on monotonicity and coercivity, natural-growth conditions, and often uses sub/supersolution methods, degree theory for variational inequalities, or minimization for convex functionals. For operators of $m$-Laplace type with data or coefficients in Morrey spaces, global $L^\infty$ bounds are proved through refined De Giorgi-type techniques and capacity/morrey-trace inequalities (Palagachev et al., 21 Dec 2025, Degiovanni et al., 2018). Existence is robust to lower order terms of natural or singular growth, as in models with drift or measure data (Degiovanni et al., 2018, Bidaut-Véron et al., 2018).

Regularity theories provide:

• Local and global $W^{1,p}$ and higher-integrability (via reverse Hölder, Gehring lemma, and bootstrapping, see (Dong et al., 2010)).
• Sharp boundary regularity, including $C^{\alpha}$ or $C^{1,\alpha}$ up to minimal geometric constraints (e.g., $p$-capacity thickness of the boundary, Morrey or BMO smallness of coefficients) (Tran et al., 2019, Nguyen et al., 2015).
• Second-order regularity: For model equations $-\operatorname{div}(a(|\nabla u|)\nabla u)=f$ (e.g., $p$-Laplace), higher regularity for nonlinear "stress fields" is established, i.e., $a(|\nabla u|)\nabla u \in W^{1,2}$, through nonlinear Calderón–Zygmund theory (Cianchi et al., 2017, Spadaro et al., 22 Jul 2025, Guarnotta et al., 2021). In convex domains, minimal boundary regularity is needed for full $W^{1,2}$ estimates (Spadaro et al., 22 Jul 2025).
• Boundedness and Harnack inequalities by weighted Moser iteration and Stummel–Kato or Morrey class control of lower-order coefficients, even in degenerate weighted settings (Fazio et al., 2010).

Comparison principles hold for very degenerate equations as long as monotonicity is preserved outside the degeneracy set and the forcing is nontrivial almost everywhere (Ciraolo, 2012).

3. Regularity Paradigms: Boundary Effects and Anisotropy

Boundary regularity is dictated by the geometric and measure-theoretic properties of $\partial\Omega$. Novel techniques include:

• Fine analysis on Lipschitz domains with curvature in weak Lorentz–Zygmund spaces or convex boundaries (no regularity required) for second-order regularity (Spadaro et al., 22 Jul 2025, Cianchi et al., 2017).
• Barrier techniques relying on (weighted) $p$-capacity conditions further extend solvability to very rough domains and enable quantitative estimates for Poisson equations with singular measure data or boundary singularities (Hara, 2022).

For anisotropic and higher-order analytic nonlinearities (i.e., those polynomial or analytic in $\nabla u$), advanced linearization and polarization techniques, employing Gaussian quasi-modes, establish unique recoverability of nonlinear coefficients from boundary data—significantly generalizing the Calderón inverse problem to this quasilinear regime (Cârstea et al., 2020).

4. Key Analytical Techniques and Gradient Policy

Main methodological advances include:

• Reverse Hölder, Gehring bootstrapping, and maximal function arguments (calibrated for both classical and fractional/maximal function settings) give interior $W^{1,q}$ estimates and sharp Lorentz-space gradient bounds (Dong et al., 2010, Tran et al., 2019, Nguyen et al., 2015).
• Integral Bernstein methods and Bochner identities adapted to nonlinear/quasilinear settings yield endpoint gradient estimates in strong and superlinear regimes (Cirant et al., 2022).
• Nonlinear potential theory and Wolff/Riesz potentials dominate the existence theory for measure-driven quasilinear equations (Bidaut-Véron et al., 2018): well-posedness is characterized in terms of capacitary smallness and nonlinear potential invariance.
• Use of Adams trace inequalities and Morrey-space embeddings is central for obtaining $L^\infty$ bounds in the presence of Morrey data (Palagachev et al., 21 Dec 2025).

5. Degeneracy, Weights, and Extension to Metric Spaces

Degenerate quasilinear elliptic equations—where ellipticity is compromised or vanishes on nontrivial sets or according to a weight—are controlled by:

• Strong $A_\infty$ weights, which support the analogs of Sobolev, Poincaré, and Harnack-type inequalities (Fazio et al., 2010).
• Weighted regularity and capacity methods, encompassing equations with weights (Muckenhoupt or general doubling) and providing robust barrier constructions for potential-theoretic applications and geometric Hardy inequalities (Hara, 2022).
• Weak comparison principles and uniqueness are retained provided strict monotonicity outside the degenerate region and nontrivial data (Ciraolo, 2012).

These frameworks extend to metric measure spaces with sufficiently strong doubling and Poincaré structures (Hara, 2022).

6. Recent Directions: Functional Frameworks and Application Scope

Contemporary developments target:

• Generalization of regularity from classical uniform ellipticity to the context where the gradient map is quasiconformal (quasiuniform convexity), permitting sharp regularity for the stress field and encompassing many physically relevant constitutive laws (Guarnotta et al., 2021).
• Analysis in Orlicz and Lorentz–Zygmund scales, optimizing the description of solutions’ integrability and regularity (Spadaro et al., 22 Jul 2025, Nguyen et al., 2015).
• Treatment of measure-valued sources and singular or source reaction nonlinearities via capacity and nonlinear potential methods (Bidaut-Véron et al., 2018, Degiovanni et al., 2018).
• Conormal boundary value problems with Morrey data, generalizing classical $L^p$ results to lower integrability scales and more singular inhomogeneities (Palagachev et al., 21 Dec 2025).

7. Inverse and Uniqueness Problems for Nonlinear Media

A significant recent advance is the unique identifiability in inverse boundary value problems for certain analytic, anisotropic quasilinear divergence-form equations. By exploiting higher-order linearization and Gaussian mode constructions, it is shown that the full family of analytic nonlinearities in the current density can be uniquely recovered from Dirichlet-to-Neumann data, extending the classical Calderón paradigm to a notably broader nonlinear, anisotropic context (Cârstea et al., 2020).

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Selected Table: Model Equations and Regularity Frameworks

Equation Class	Main Regularity/Existence Results	Reference
$p$-Laplace, analytic nonlinearities	$C^{\alpha}$, $C^{1,\alpha}$, gradient Lorentz/Orlicz, $W^{2,q}$	(Dong et al., 2010, Nguyen et al., 2015, Spadaro et al., 22 Jul 2025)
Natural/singular growth	Existence via variational inequality, a priori $L^{\infty}$	(Degiovanni et al., 2018, Palagachev et al., 21 Dec 2025)
Anisotropic analytic divergence-form	Uniqueness in inverse problem, recovery of nonlinear coefficient	(Cârstea et al., 2020)
Degenerate weighted equations	Harnack, local boundedness, weighted $C^{1,\alpha}$	(Fazio et al., 2010, Hara, 2022)
Stress field $DF(Du)$ in $W^{1,2}$	Quasiconformal map condition as optimal ellipticity	(Guarnotta et al., 2021, Cianchi et al., 2017)

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Quasilinear divergence form elliptic equations constitute a highly developed theoretical framework with active research extending from regularity and boundedness to inverse problems and equations with degenerate or weighted structures, relying on advanced harmonic, potential, and functional analytic methodologies as well as deep geometric-measure-theoretic underpinnings.