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arxiv: 2604.20597 · v1 · submitted 2026-04-22 · 🧮 math.AP

Widely degenerate anisotropic diffusion: local boundedness and semicontinuity

Pasquale Ambrosio , Simone Ciani , Giovanni Cupini This is my paper

Pith reviewed 2026-05-09 23:28 UTC · model grok-4.3

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Local weak solutions to fully anisotropic widely degenerate parabolic PDEs with measurable (x,t)-dependent coefficients are bounded if in a non-homogeneous De Giorgi class and possess semicontinuous representatives under suitable assumptions on the exponents p_i.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The equations studied describe diffusion processes where the speed and behavior can differ sharply in each coordinate direction, and diffusion can effectively stop when the gradient in that direction falls below a threshold delta_i. The coefficients multiplying each term are allowed to be merely measurable and bounded, and can change with both position and time. Different directions are governed by their own growth exponents p_i greater than 1. The authors first show that any weak solution sitting inside a certain generalized parabolic De Giorgi class must be locally bounded. They then prove that sub-solutions and super-solutions possess representatives that are upper or lower semicontinuous. This work broadens earlier results that handled either isotropic cases, single exponents, or smoother coefficients. The proofs rely on adapting De Giorgi iteration techniques to the anisotropic and widely degenerate setting while carefully tracking the different p_i and the degeneracy thresholds.

Core claim

We first show that the local boundedness of weak solutions follows from their membership in an appropriate non-homogeneous parabolic De Giorgi class. We then establish the existence of semicontinuous representatives for local weak sub(super)-solutions.

Load-bearing premise

Under suitable assumptions on the exponents p_i (the precise restrictions on the p_i are not detailed in the abstract but are required for the De Giorgi-class membership to imply boundedness).

read the original abstract

We investigate the regularity of local weak solutions to evolution equations of the form \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\,\partial_{x_{i}}\left[a_{i}(x,t)\,(\vert\partial_{x_{i}}u\vert-\delta_{i})_{+}^{p_{i}-1}\,\frac{\partial_{x_{i}}u}{\vert\partial_{x_{i}}u\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,\Omega_{T}\,=\,\Omega\times(0,T)\,, \] where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $n\geq2$, the coefficients $a_{i}$ are measurable and bounded, $p_{i}>1$ and $\delta_{i}\geq0$ are fixed parameters. Under suitable assumptions on the exponents $p_{i}$, we first show that the local boundedness of weak solutions follows from their membership in an appropriate non-homogeneous parabolic De Giorgi class. We then establish the existence of semicontinuous representatives for local weak sub(super)-solutions. Our analysis extends analogous results available for less degenerate operators and generalizes the local boundedness results obtained in [7] to fully anisotropic, widely degenerate parabolic PDEs with non-smooth coefficients depending additionally on the space-time variables $(x,t)$, whose growth is governed by a family of exponents $p_{i}$ rather than by a single exponent.

Editorial analysis

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Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The central claims rest on deriving Caccioppoli-type inequalities from the given divergence-form PDE with measurable coefficients, placing weak solutions into a non-homogeneous parabolic De Giorgi class, and then applying standard iteration for boundedness; semicontinuity follows from a separate argument on sub- and super-solutions. These steps use the explicit structure of the operator and stated restrictions on the p_i exponents; no step reduces a result to a fitted parameter, self-definition, or load-bearing self-citation whose content is merely renamed. The extension of De Giorgi techniques is independent of the target conclusions and remains falsifiable against the PDE itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background from parabolic PDE theory and De Giorgi classes together with the stated assumptions on coefficients and exponents; no new entities are postulated.

axioms (2)
  • domain assumption Coefficients a_i are measurable and bounded
    Explicitly stated in the abstract as part of the equation setup.
  • ad hoc to paper Suitable assumptions on the exponents p_i
    Invoked in the abstract to ensure the De Giorgi-class implication holds, but the precise restrictions are not given.

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