Generalized Morrey-Campanato estimates for elliptic equations with coefficients of integrable oscillation

Abstract: This work concerns regularity properties of weak solutions to elliptic equations in divergence form -div(a$\nabla$u) = div F , under low regularity assumptions on both the coefficient a and the source term F . We introduce generalized Morrey and Campanato spaces extending the classical definitions by replacing uniform boundedness requirements with suitable integrability conditions. Within this framework, we establish regularity estimates for the gradient of weak solutions in these generalized spaces. As applications, we recover classical H{ö}lder and Lebesgue estimates and derive fractional Sobolev regularity results. In particular, the proposed approach yields fractional Sobolev estimates in situations where the coefficient may be discontinuous and the gradient of the solution is not expected to be locally bounded.

• The paper introduces generalized Morrey-Campanato spaces, extending classical elliptic regularity to systems with integrable oscillation coefficients.
• It establishes a two-step transfer mechanism where gradient regularity advances to fractional Sobolev continuity even with discontinuous coefficients.
• The results offer new embedding rules and parameter regimes, providing sharp estimates that extend classical Schauder and Calderón-Zygmund frameworks.

Generalized Morrey-Campanato Regularity Theory for Elliptic Equations with Integrable Oscillation Coefficients

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Introduction and Context

The paper develops a robust extension of classical regularity theory for weak solutions to elliptic equations in divergence form $$-\operatorname{div}(a \nabla u) = \operatorname{div} F$$, under substantially weakened regularity assumptions on both the coefficient field $a$ and source term $F$. By introducing generalized Morrey and Campanato spaces—where uniform bound conditions are replaced with integrability requirements—the author formulates and proves gradient regularity results that include, but go well beyond, traditional Hölder and Lebesgue estimates. The work leverages integrable oscillation properties of coefficients and achieves fractional Sobolev continuity for $u$ even in cases with discontinuous $a$ or non-locally bounded $\nabla u$. This expands the toolkit available for elliptic regularity, especially in settings that defy classical Schauder or Calderón-Zygmund frameworks.

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Generalized Morrey and Campanato Spaces

Traditional Morrey spaces $L^{p,\lambda}$ require uniform control over the concentration function $r^{-\lambda/p} \|u\|_{L^p(B(x,r))}$. The paper generalizes this by substituting the uniformity in $x$ with an $L^q$-integrability on the domain, leading to spaces denoted $a$0 with norm $a$1. Similarly, Campanato spaces are extended using oscillation functions $a$2 integrated in $a$3, denoted $a$4. These spaces interpolate between Morrey-type, Lebesgue-type, and Hölder-type regularity, and critical parameter values ($a$5) allow equivalence with Lebesgue spaces.

Comprehensive injection and embedding rules are established, e.g., $a$6; for $a$7, $a$8; and for $a$9 on $F$0-smooth domains, there is equivalence between $F$1 and $F$2. The construction aligns well with fractional Sobolev scales and provides a flexible foundation for regularity analysis.

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Elliptic Coefficient Oscillation and Hypotheses

Elliptic coefficients are required to fulfill integrability of their oscillation, specified as:

• (H1): $F$3 and $F$4 a.e.
• (H2): There exists $F$5 and $F$6 so that $F$7 for a.e. $F$8.

This (H2) is stricter than some prior fractional Hajłasz conditions but allows powerful translation into oscillation-based integrability estimates. Crucially, $F$9 (the $u$0-oscillation function) is equivalent to having $u$1 satisfy (H2) with exponents $u$2, and $u$3 almost everywhere.

The parameter range $u$4 ensures Hölder regularity; $u$5 may permit discontinuities in $u$6, and regularity arguments are forced to exploit local integrability and oscillation estimates.

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Main Gradient Regularity Results

The essence of the paper is the two-step transport mechanism for regularity:

Step 1: Generalized Morrey Regularity Transfer

Theorem 4.4 demonstrates that, under (H1)-(H2) and with $u$7, Morrey-type regularity transfers to $u$8, i.e.,

$u$9

for appropriate $a$0. This holds even if $a$1 is not uniformly Hölder, as classical Schauder theory demands.

Step 2: Generalized Campanato Regularity Upgrade

Theorem 4.5 proves that, given $a$2 and $a$3, further Campanato-type regularity is transferred:

$a$4

The Campanato exponent increases by $a$5, echoing the elliptic regularizing effect, while integrability may decrease if exponents are not compatible.

These theorems allow recovery of classical estimates as corollaries, but also produce fractional Sobolev regularity in low-regularity regimes $a$6, for which no local boundedness of $a$7 is expected.

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Notable Regularity Consequences

A succession of implications is derived:

• Schauder Regularity: When $a$8 (i.e., $a$9), $\nabla u$0 inherits $\nabla u$1 regularity.
• $\nabla u$2 Regularity (for $\nabla u$3): With $\nabla u$4 in appropriate Morrey-Campanato regime and $\nabla u$5, $\nabla u$6; this matches classical $\nabla u$7 theory but is recovered via the generalized transfer framework.
• Fractional Sobolev in Low Regularity: For $\nabla u$8, $\nabla u$9 lies in a fractional Sobolev space $L^{p,\lambda}$0 for $L^{p,\lambda}$1 and $L^{p,\lambda}$2 dependent on $L^{p,\lambda}$3. This holds even with discontinuous $L^{p,\lambda}$4 and constitutes a strong result beyond prior theory.

Explicit parameter regimes are mapped, and sharp constraints on exponents are outlined, e.g., for $L^{p,\lambda}$5, boundedness of regularity transfer is lost but positive fractional differentiability for $L^{p,\lambda}$6 remains.

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Implications and Future Directions

Practically, the extension of Morrey-Campanato machinery to integrable oscillation coefficients enables resolvent regularity analysis for PDEs with rough coefficients, particularly in stochastic, composite, or discontinuous media where uniform controls are unattainable. The theory suggests new avenues for regularity in elliptic systems and variational settings, and the embedding with fractional Sobolev spaces may be leveraged in nonlinear and nonlocal contexts as well.

Theoretically, the results clarify the transmission of regularity from source terms to gradients in settings well outside the reach of classical frameworks. The identification of explicit parameter boundaries and direct embedding rules for generalized spaces furnishes a modular approach adaptable to broader families of PDEs.

Future directions include exploring further extensions to systems, nonlinear equations, and the interplay with singular integrals and potential theory, as well as possible refinement of integrability requirements for sharper regularity bounds.

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Conclusion

The paper provides a rigorous extension of Morrey-Campanato regularity theory for elliptic equations with coefficients exhibiting integrable oscillations. The generalized spaces and transfer theorems encapsulate and extend classical results, offering new fractional Sobolev regularity estimates in challenging discontinuous regimes. These advancements furnish both a unified theoretical framework and practical sharp tools for PDE analysis with non-smooth coefficients (2606.20237).