Abstract
We obtain weighted Poincaré inequalities in bounded domains, where the weight is given by a symmetric nonnegative definite matrix, which can degenerate on submanifolds. Furthermore, we investigate uniqueness and nonuniqueness of solutions to degenerate elliptic and parabolic problems, where the diffusion matrix can degenerate on subsets of the boundary of the domain. Both the results are obtained by means of the distance function from the degeneracy set, which is used to construct suitable local sub– and supersolutions.
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Acknowledgements
The authors would like to thank Prof. U. Gianazza for bringing to their attention the global regularity result in [1].
The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Funding
Open access funding provided by Politecnico di Milano within the CRUI-CARE Agreement. D.D.M. and F.P. are partially supported by 2020 GNAMPA project “Equazioni Ellittiche e Paraboliche ed Analisi Geometrica”. F.P. is supported by the PRIN-201758MTR2 project “Direct and inverse problems for partial differential equations: theoretical aspects and applications.”
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D.D.M. and F.P. wrote the entire manuscript text.
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Monticelli, D.D., Punzo, F. Weighted Poincaré Inequalities and Degenerate Elliptic and Parabolic Problems: An Approach via the Distance Function. Potential Anal 60, 1421–1444 (2024). https://doi.org/10.1007/s11118-023-10094-5
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DOI: https://doi.org/10.1007/s11118-023-10094-5
Keywords
- Weighted Poincaré inequality
- Uniqueness of solutions
- Degenerate elliptic equations
- Degenerate parabolic equations
- Sub– supersolutions