Let p>2, let Omega⊂R^n×R be a space-time domain, and let z_0∈partial Omega. For the p-parabolic equation partial_tu-operatornamediv(|nabla u|^p-2nabla u)=0, does the existence of a single traditional barrier at z_0 imply that z_0 is regular for the Dirichlet problem? Equivalently, does a traditional barrier characterize boundary regularity in the degenerate range p>2?

One-barrier characterization of boundary regularity for degenerate parabolic equations

The problem asks whether existence of a single traditional barrier characterizes boundary regularity for degenerate parabolic equations. Avelin and Parviainen’s recent soda-can classification claims a negative answer in specified $p$ -parabolic ranges.

Known results

Avelin and Parviainen (2015): for $1<p<2$ , soda-can domains can be irregular while admitting a traditional barrier.
Avelin and Parviainen (2015): for $p>2$ , a positive lower bound on the boundary radius implies irregularity and absence of a traditional barrier.
The earlier work left open whether the one-barrier failure also occurs in the degenerate range $p>2$ .

2026 soda-can classification

Avelin and Parviainen claim that, for specified $p$ and $l$ ranges, irregularity coexists with a traditional barrier, thereby disproving the one-barrier characterization in those regimes. The result is presented in an unrefereed preprint and has not been independently assessed in the retrieved sources.

Current status (as of October 2026): A preprint claims the characterization fails in specified degenerate $p$ -parabolic ranges, but that claim is unverified and cases outside those ranges remain open.

One-barrier characterization of boundary regularity for degenerate parabolic equations

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