Holder regularity and Liouville properties for nonlinear elliptic inequalities with power-growth gradient terms

Alessandro Goffi

Research output: Contribution to journalArticlepeer-review


This note studies local integral gradient bounds for distributional solutions of a large class of partial differential inequalities with diffusion in divergence form and power-like first-order terms. The applications of these estimates are two-fold. First, we show the (sharp) global Hölder regularity of distributional semi-solutions to this class of diffusive PDEs with first-order terms having supernatural growth and right-hand side in a suitable Morrey class posed on a bounded and regular open set Ω. Second, we provide a new proof of entire Liouville properties for inequalities with superlinear first-order terms without assuming any one-side bound on the solution for the corresponding homogeneous partial differential inequalities. We also discuss some extensions of the previous properties to problems arising in sub-Riemannian geometry and also to partial differential inequalities posed on noncompact complete Riemannian manifolds under appropriate area-growth conditions of the geodesic spheres, providing new results in both these directions. The methods rely on integral arguments and do not exploit maximum and comparison principles.
Original languageEnglish (US)
Pages (from-to)1-25
Number of pages25
StatePublished - Nov 10 2022
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2022-12-07
Acknowledged KAUST grant number(s): CRG2021-4674
Acknowledgements: The author is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The author has been partially supported by the INdAM-GNAMPA Project 2022 ‘Proprietà quantitative e qualitative per EDP non lineari con termini di gradiente’ and by the King Abdullah University of Science and Technology (KAUST) project CRG2021-4674 ‘Mean-Field Games: models, theory and computational aspects’.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

  • Mathematics(all)


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