Gradient Higher Integrability

Updated 11 January 2026

Gradient higher integrability results refer to the property where gradients of weak solutions to nonlinear PDEs improve their integrability beyond the natural L^p class using reverse Hölder inequalities.
These techniques leverage intrinsic scaling, covering arguments, and parabolic Sobolev–Poincaré inequalities to unify analysis across singular, degenerate, multi-phase, and variable exponent models.
The results are crucial for regularity theory, enabling optimal convergence estimates and refined defect set measures in complex variational and parabolic systems.

Gradient higher integrability results refer to quantitative self-improvement phenomena whereby the spatial gradient of a minimizer or weak solution to a nonlinear PDE or variational problem—initially known to belong to some natural $L^p$ (or Orlicz) class—actually enjoys additional integrability: for instance, $|\nabla u|^p \in L^{1+\varepsilon}_\text{loc}$ for some $\varepsilon > 0$ . These results originated with classical work on Gehring’s lemma, but recent advances provide sharp and unified frameworks for a wide array of singular, degenerate, variable-exponent, multi-phase, doubly nonlinear, and non-standard growth problems. Intrinsic geometry, covering and stopping-time arguments, and reverse Hölder inequalities are universal themes. The field underpins advances in regularity theory, fine measure estimates on defect sets, and optimal convergence results for complex variational models.

1. Foundational Principles and Historical Context

The self-improving (higher integrability) property is rooted in Gehring’s lemma, which states that if a non-negative function $f$ satisfies a certain reverse Hölder inequality, then $f$ belongs to a better Lebesgue space. This principle was first systematically exploited in the context of elliptic PDE and functionals with $p$ -growth. For the parabolic $p$ -Laplacian, Kinnunen–Lewis established that weak solutions possess spatial gradients in $L^{p+\varepsilon}$ for some $\varepsilon>0$ . Subsequent developments extended these methods to non-standard growth, Orlicz, double-phase, porous medium, variable exponent, and multi-phase PDEs, unifying the approach via intrinsic scaling, space-time adapted cylinders, and general covering arguments (Hästö et al., 2019, Hästö et al., 24 Nov 2025, Kim et al., 2022, Sen, 2024).

Intrinsic geometry—where the space-time cylinder’s time scale adapts to the local energy—became essential for unifying singular ( $p<2$ ) and degenerate ( $p>2$ ) regimes, for example in doubly nonlinear, fast-diffusion, or double-/multi-phase systems (Bögelein et al., 2018, Moring et al., 2023, Bögelein et al., 2018, Sen, 2024).

2. Structural Conditions and Model Problems

Higher integrability results apply to a variety of underlying equations and energy densities. Prototype structures include:

Model	Energy/Operator Structure	Nonlinear Regime
Parabolic $p$ -Laplacian	$\partial_tu-\operatorname{div}(\|\nabla u\|^{p-2}\nabla u)=0$	$p$ -growth, degenerate/singular
Double-phase problems	$u_t-\operatorname{div} (\|\nabla u\|^{p-2}\nabla u + a(z)\|\nabla u\|^{q-2}\nabla u)=0$	Transition between $p$ - and $q$ -growth, $a(z)\ge0$
Multi-phase	$u_t-\div\big(\|\nabla u\|^{p-2}\nabla u + a(z)\|\nabla u\|^{q-2}\nabla u + b(z)\|\nabla u\|^{s-2}\nabla u\big)=0$	Competition of several growth modes
Variable exponent	$-\div(\|\nabla u\|^{p(x)-2}\nabla u) + b(x)\cdot\nabla u = \div h(x)$	$p(x)$ -Laplacian
Doubly nonlinear systems	$\partial_t(\|u\|^{q-1}u) - \div(\|\nabla u\|^{p-2}\nabla u) = \div(\|F\|^{p-2}F)$	Time- and space-nonlinearity, $q>1$ , $p>1$
Porous medium/fast diffusion	$\partial_tu - \div(A(x,t,u,\nabla(\|u\|^{m-1}u))) = \div F$	$m$ -homogeneous, $0

Properly handling these settings requires structural assumptions, notably uniform ellipticity and growth balance, and (for double/multi-phase) suitable regularity of the modulating coefficients (typically $C^{\alpha,\alpha/2}$ -Hölder continuity in space-time) with sharp relations on exponents and regularity (Kim et al., 2023, Sen, 2024, Oh et al., 22 Apr 2025).

3. Methodologies: Intrinsic Geometry, Reverse Hölder, and Covering

The key steps in establishing higher integrability are:

A. Intrinsic Scaling:

Define parabolic cylinders whose spatial and temporal scales depend–in a self-consistent way–on the local size of the energy density $H(z,|\nabla u|)$ . For the multi-phase case (Sen, 2024), cylinders are adapted as follows:

p-phase: $Q_{\rho}^{\lambda,p}$ , time scale $\sim \lambda^{2\mu}\rho^2/\lambda^p$
(p,q)-phase: $Q_{\rho}^{\lambda,(p,q)}$ , time scale $\sim \lambda^{2\mu}\rho^2/(\lambda^p + a(z)\lambda^q)$
(p,s)-phase: ... (analogous adjustment with $b(z), s$ )
(p,q,s)-phase: full combination.

The exponent $\mu$ captures the rate at which the scaling adapts, and is tailored to bridge degenerate/singular regimes (Sen, 2024).

B. Parabolic Sobolev–Poincaré Inequalities:

On each intrinsic cylinder, prove a version of the parabolic Sobolev–Poincaré inequality, matching the scaling and growth rate dictated by the phase (p, (p,q), etc.). These allow bounding the normalized averaged oscillation in terms of the local gradient energy (Sen, 2024).

C. Reverse Hölder Inequalities:

Demonstrate that the energy density $H(z,|\nabla u|)$ (or appropriate Orlicz/variable exponent versions) satisfies a reverse Hölder inequality in the intrinsic geometry, i.e.,

$\iint_Q H(z,|\nabla u|)\leq c \left(\iint_{2Q} H(z,|\nabla u|)^\theta \right)^{1/\theta}$

for some $\theta<1$ and controlled $c$ , within each phase (Kim et al., 2023, Hästö et al., 24 Nov 2025, Sen, 2024).

D. Vitali Covering and Gehring’s Lemma:

Cover the superlevel set $\{z : H(z,|\nabla u|)>\Lambda\}$ by a maximal disjoint family of intrinsic cylinders, apply the reverse Hölder in each, and sum. This produces an integral inequality of the form

$\iint_{E_{\Lambda}} H(z,|\nabla u|) \leq C \Lambda^{1-\theta} \iint_{E_{\Lambda/c}} H(z,|\nabla u|)^\theta + \cdots$

Bootstrapping via a one-dimensional Gehring lemma yields $H(z,|\nabla u|)\in L^{1+\varepsilon}$ with explicit exponent (Sen, 2024, Hästö et al., 24 Nov 2025).

4. Unified Interior Higher Integrability Results

The interior higher integrability theorem for parabolic multi-phase systems, as established in (Sen, 2024), states: For $u$ a weak solution to

$u_t - \operatorname{div}\left( |\nabla u|^{p-2}\nabla u + a(z)|\nabla u|^{q-2}\nabla u + b(z)|\nabla u|^{s-2}\nabla u\right) = 0$

with H\"older continuous coefficients $a(z),b(z)\geq0$ and exponents satisfying sharp relations,

$q \leq p+\min\left\{ \alpha\left(\frac p2 - \frac n{n+2}\right),\,\frac{2\alpha}{n+2} \right\},\quad s \leq p+\min\left\{ \beta\left(\frac p2 - \frac n{n+2}\right),\,\frac{2\beta}{n+2} \right\}$

there holds

$\iint_{Q_r} H(z,|\nabla u|)^{1+\varepsilon}\leq c\left(\iint_{Q_{2r}}H(z,|\nabla u|)+1\right)^{1 + \frac{\varepsilon s}{(n+2)\mu-n}}$

for some $\varepsilon>0$ and with $H(z,|\xi|)=|\xi|^p+a(z)|\xi|^q+b(z)|\xi|^s$ (Sen, 2024).

This result is robust: it covers both degenerate ( $p\ge 2$ ) and singular ( $p<2$ ) regimes without case-splitting, and extends, via multi-phase intrinsic geometry, to any finite number of phases.

a) Generalization to Double- and Multi-Phase:

Gradient higher integrability now holds for systems involving arbitrary finite sums

$H(z,|\xi|) = |\xi|^{p} + \sum_{i=1}^m a_i(z)|\xi|^{p_i}$

with each $a_i$ H\"older continuous and suitable bounds on $p_i$ . The same intrinsic-cylinder/stopping-time/Gehring methodology applies (Sen, 2024).

b) Orlicz, Variable Exponent, and Weighted Settings:

Analogous higher integrability results have been established for Orlicz-type and variable exponent flows, even in the presence of weights (via Muckenhoupt $A_p$ theory), subject to suitable structural conditions on the modular or $\varphi$ -function and modulus of continuity (Hästö et al., 24 Nov 2025, Karppinen, 2019, Hietanen et al., 25 Nov 2025, Chen et al., 2023, Adimurthi et al., 2018).

c) Optimality and Sharpness:

The scaling restrictions (e.g., on $q$ and $s$ in relation to $p$ and the modulus of H\"older continuity of $a(\cdot)$ and $b(\cdot)$ ) are known to be sharp, matching counter-examples based on "frozen" phase geometries or failings of reverse Hölder at thresholds (Kim et al., 2023, Sen, 2024).

d) Connections to Regularity Theory:

Gradient higher integrability is fundamental for developing $C^{1,\alpha}$ regularity, partial regularity, and dimension estimates for singular sets in variational and free boundary problems (Focardi, 2016, Labourie et al., 2021). Enhanced integrability directly reduces the Hausdorff dimension of defect sets (e.g., for Mumford–Shah minimizers or for the free boundary in the thermal insulation problem).

e) Unified Treatment Across Regimes:

Unified frameworks are now available for:

Doubly nonlinear parabolic systems (in both $u$ - and $Du$ -phases) (Moring et al., 2023, Bögelein et al., 2018)
Singular porous medium and fast diffusion flows, including critical and subcritical regimes (Bögelein et al., 2018, Bögelein et al., 16 Jan 2025)
Systems with obstacles (i.e., variational inequalities) (Karppinen, 2019, Cho et al., 2020, Li, 2020)

6. Technical Innovations and Future Directions

The field continues to advance via:

Development of universal, phase-insensitive intrinsic scaling dependent on a tunable parameter (e.g., $\mu$ ), unifying all ranges of exponents (Sen, 2024)
Parabolic covering lemmas adapted to the geometry of the problem, allowing efficient control of overlapping regions without dichotomy into discrete regimes (Sen, 2024)
Reverse Hölder in general Orlicz and Musielak–Orlicz frameworks, with modular continuity conditions (e.g., intrinsic (A1)-condition), relaxing previous strict log-H\"older or Dini continuity requirements (Hästö et al., 24 Nov 2025, Hietanen et al., 25 Nov 2025)
Weighted gradient higher integrability under $A_p$ weights, variable exponent, and double phase ( $t^p+a(x)t^q$ ) for minimizers under nonstandard growth (Hietanen et al., 25 Nov 2025)
Quantitative versions with explicit dependence of the higher exponent $\varepsilon$ on the data, and precise energy scaling in the gain

Open directions include optimality at "borderline" exponents, fine dependence of $\varepsilon$ on constants, further extension to time-measurable coefficients with minimal space regularity, and gradient estimates up to the boundary under minimal geometric hypotheses on the domain.

References