MoRight: Morrey-Space Regularity Theory

• MoRight is a Morrey-space based C¹,α-regularity theory for weak solutions of degenerate elliptic equations of p-Laplacian type.
• The framework employs homogeneous comparison, the Fefferman–Phong inequality, and Campanato–Meyers estimates to quantify local gradient regularity.
• It bridges degenerate and singular regimes by optimizing regularity thresholds under minimal data assumptions in Morrey spaces.

Searching arXiv for the specified paper and closely related context. MoRight denotes the Morrey-space-based $C^{1,\alpha}$-regularity theory for degenerate elliptic equations of $p$-Laplacian type developed in "Higher Hölder regularity for degenerate elliptic PDEs with data in Morrey spaces" (Fazio et al., 11 Oct 2025). It concerns weak solutions of nonlinear elliptic equations with low-integrability right-hand sides controlled in Morrey spaces and establishes sharp local $C^{1,\alpha}$-regularity by combining a homogeneous comparison argument, the Fefferman–Phong inequality, and the Campanato–Meyers characterization of Hölder continuity. In the model case, the framework treats

$$-\Delta_p u=f \quad \text{in } \Omega, \qquad \Delta_p u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u),$$

for $1$f \in L^{1,\lambda}(\Omega)$ and $n-1<\lambda<n$, and it extends to general quasilinear operators with $p$-growth and degenerate or singular ellipticity.

1. PDE class and structural setting

The model equation underlying MoRight is the $p$-Poisson problem

$-\Delta_p u=f \quad \text{in } \Omega,$

equivalently written as

$p$0

The general equation treated is

$p$1

where $p$2 is differentiable and satisfies $p$3-growth, degenerate ellipticity, and controlled $p$4-dependence (Fazio et al., 11 Oct 2025).

The structural assumptions are stated in terms of the $p$5-Jacobian of $p$6. For all $p$7,

$p$8

which is the uniform monotonicity condition in degenerate form. The $p$9-derivatives obey

$C^{1,\alpha}$0

and the $C^{1,\alpha}$1-dependence is controlled by

$C^{1,\alpha}$2

with $C^{1,\alpha}$3 and $C^{1,\alpha}$4 depending only on structural data.

The lower-order term is assumed to satisfy Morrey-controlled growth,

$C^{1,\alpha}$5

with

$C^{1,\alpha}$6

This places the entire right-hand side within a Morrey-scale regime compatible with interior gradient Hölder estimates.

2. Morrey-space framework and Fefferman–Phong control

Morrey spaces are central to the MoRight formulation. For $C^{1,\alpha}$7 and $C^{1,\alpha}$8, the space $C^{1,\alpha}$9 consists of all $$-\Delta_p u=f \quad \text{in } \Omega, \qquad \Delta_p u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u),$$0 such that

$$-\Delta_p u=f \quad \text{in } \Omega, \qquad \Delta_p u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u),$$1

The paper uses the basic facts $$-\Delta_p u=f \quad \text{in } \Omega, \qquad \Delta_p u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u),$$2, $$-\Delta_p u=f \quad \text{in } \Omega, \qquad \Delta_p u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u),$$3, and $$-\Delta_p u=f \quad \text{in } \Omega, \qquad \Delta_p u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u),$$4 if $$-\Delta_p u=f \quad \text{in } \Omega, \qquad \Delta_p u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u),$$5, as well as the embedding criterion: if $$-\Delta_p u=f \quad \text{in } \Omega, \qquad \Delta_p u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u),$$6 and $$-\Delta_p u=f \quad \text{in } \Omega, \qquad \Delta_p u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u),$$7, then $$-\Delta_p u=f \quad \text{in } \Omega, \qquad \Delta_p u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u),$$8 (Fazio et al., 11 Oct 2025).

The route from Morrey data to elliptic regularity proceeds through Stummel–Kato classes and the Fefferman–Phong inequality. For $$-\Delta_p u=f \quad \text{in } \Omega, \qquad \Delta_p u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u),$$9 and $1

$1

If $1

$1

This yields the Fefferman–Phong estimate in the form used in the paper: for any ball $1

$1

and, for Morrey data,

$1

In the model problem, a weak solution is $f \in L^{1,\lambda}(\Omega)$0 such that

$f \in L^{1,\lambda}(\Omega)$1

The Fefferman–Phong mechanism is used to show that the right-hand side is finite even at this low integrability level. The paper also notes that the same mechanism applies to measure data of Morrey type, namely measures $f \in L^{1,\lambda}(\Omega)$2 satisfying $f \in L^{1,\lambda}(\Omega)$3.

3. Main local $f \in L^{1,\lambda}(\Omega)$4-regularity theorem

The principal result of MoRight is a sharp interior $f \in L^{1,\lambda}(\Omega)$5-estimate for weak solutions of the $f \in L^{1,\lambda}(\Omega)$6-Poisson equation with Morrey data (Fazio et al., 11 Oct 2025). Let $f \in L^{1,\lambda}(\Omega)$7 be bounded Lipschitz, $f \in L^{1,\lambda}(\Omega)$8, $f \in L^{1,\lambda}(\Omega)$9, and $n-1<\lambda<n$0. If $n-1<\lambda<n$1 is a weak solution of

$n-1<\lambda<n$2

then

$n-1<\lambda<n$3

with

$n-1<\lambda<n$4

where $n-1<\lambda<n$5 is the interior gradient Hölder exponent for the corresponding homogeneous equation.

The positivity constraints on $n-1<\lambda<n$6 are explicit. In the branch $n-1<\lambda<n$7, the quantity $n-1<\lambda<n$8 lies in $n-1<\lambda<n$9 because $p$0. In the branch $p$1, one additionally requires

$p$2

which ensures $p$3. Thus the singular regime admits local $p$4-regularity only above a dimensionally determined threshold.

The theorem is quantified through Campanato decay of the gradient oscillation. For balls $p$5, the argument yields an estimate of the form

$p$6

and the Campanato–Meyers characterization then gives $p$7, hence $p$8. The exponent $p$9 is therefore determined by the competition between the homogeneous gradient exponent $p$0 and the Morrey-scaling contribution of the forcing term.

4. Comparison scheme and proof mechanism

The proof is organized around homogeneous comparison. Fix a ball $p$1, enlarge it slightly to $p$2, and let $p$3 solve the homogeneous comparison problem

$p$4

Classical theory due to Ladyzhenskaya–Uraltseva, Evans, and Tolksdorf gives

$p$5

for some $p$6 (Fazio et al., 11 Oct 2025).

The oscillation of $p$7 is decomposed as

$p$8

The term involving $p$9 is estimated by the homogeneous $-\Delta_p u=f \quad \text{in } \Omega,$0-regularity: $-\Delta_p u=f \quad \text{in } \Omega,$1 The mean-difference term is controlled by

$-\Delta_p u=f \quad \text{in } \Omega,$2

Accordingly, the core issue is to estimate

$-\Delta_p u=f \quad \text{in } \Omega,$3

At this point the proof bifurcates according to the degenerate and singular regimes. For $-\Delta_p u=f \quad \text{in } \Omega,$4, strict $-\Delta_p u=f \quad \text{in } \Omega,$5-monotonicity of the $-\Delta_p u=f \quad \text{in } \Omega,$6-Laplacian yields

$-\Delta_p u=f \quad \text{in } \Omega,$7

The term on the right is handled by Hölder combined with Fefferman–Phong: $-\Delta_p u=f \quad \text{in } \Omega,$8 This leads to the Morrey-driven decay exponent that enters the $-\Delta_p u=f \quad \text{in } \Omega,$9 formula for $p$00.

For $p$01, the operator is singular where $p$02 is small, and the argument uses the refined monotonicity inequality

$p$03

After Hölder with exponents $p$04 and $p$05, together with the energy comparison and the same Fefferman–Phong step, one obtains

$p$06

which after averaging over $p$07 yields the singular-branch decay. The common endpoint of both branches is a Campanato estimate

$p$08

and therefore $p$09.

5. Extension to general quasilinear equations

MoRight extends the model $p$10-Poisson theorem to the quasilinear equation

$p$11

under the structural assumptions on $p$12 and the Morrey-growth condition on $p$13. The conclusion is unchanged: any weak solution satisfies

$p$14

with the same exponent

$p$15

The comparison function $p$16 is now taken as the solution of

$p$17

and the same scheme applies (Fazio et al., 11 Oct 2025).

The coercivity estimates are adapted to the nonlinear operator $p$18. For $p$19,

$p$20

while for $p$21,

$p$22

The $p$23-term is controlled by the Morrey assumptions

$p$24

in conjunction with Fefferman–Phong and standard Young-type absorptions. In this sense, MoRight is not restricted to the prototype $p$25-Laplacian but rather provides a full interior regularity scheme for degenerate and singular $p$26-growth operators with mild coefficient dependence.

The paper makes a precise distinction between the degenerate and singular regimes. In the degenerate case $p$27, the nonlinearity degenerates where $p$28 is small, and strict $p$29-monotonicity is sufficient to control the perturbation term. In the singular case $p$30, the operator is singular where $p$31 is small, and the weighted quadratic inequality replaces strict $p$32-monotonicity. The additional restriction $p$33 is therefore intrinsic to the singular-branch exponent.

6. Sharpness, thresholds, and position within the literature

A central feature of MoRight is the optimality of the Morrey threshold $p$34. The paper invokes Serrin’s radial examples

$p$35

which solve

$p$36

with

$p$37

For these examples,

$p$38

In that range, solutions are at most Lipschitz, which shows that the condition $p$39 cannot be relaxed (Fazio et al., 11 Oct 2025).

The exponent $p$40 is likewise sharp in the sense stated by the paper: it is the minimum of the homogeneous gradient exponent $p$41 and the Morrey-driven scaling exponents. Without improving either the data integrability or the homogeneous exponent, $p$42 cannot be increased. This identifies the regularity threshold as a genuinely two-parameter phenomenon, depending simultaneously on the nonlinear homogeneous theory and the low-integrability scaling of the forcing term.

The paper also situates the result relative to classical linear theory. For $p$43 and smooth coefficients, Di Fazio (1992) proved $p$44 under $p$45, $p$46, using Green’s function methods; MoRight recovers this linear case in the nonlinear degenerate and singular setting without Green’s function tools. More broadly, the paper states that prior Morrey-data results mainly yielded $p$47 regularity or differentiability under stronger assumptions, whereas the present framework upgrades the conclusion to local $p$48 in the full $p$49-Laplacian-type regime.

The scope remains explicitly interior. Although $p$50 is assumed bounded Lipschitz, no boundary regularity is claimed. The examples emphasized in the paper reflect the low-integrability reach of the theory: power-type singular data $p$51 in $p$52 belong to $p$53 if $p$54, so when $p$55, one may take $p$56 just below $p$57 and still obtain

$p$58

This suggests that MoRight provides a near-endpoint interior regularity theory for degenerate elliptic equations with Morrey-controlled forcing.