## On the regularity theory for mixed local and nonlocal quasilinear elliptic equations

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- by Prashanta Garain and Juha Kinnunen PDF
- Trans. Amer. Math. Soc.
**375**(2022), 5393-5423 Request permission

## Abstract:

We consider a combination of local and nonlocal $p$-Laplace equations and discuss several regularity properties of weak solutions. More precisely, we establish local boundedness of weak subsolutions, local Hölder continuity of weak solutions, Harnack inequality for weak solutions and weak Harnack inequality for weak supersolutions. We also discuss lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. Our approach is purely analytic and it is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. The main results apply to sign changing solutions and capture both local and nonlocal features of the equation.## References

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## Additional Information

**Prashanta Garain**- Affiliation: Department of Mathematics, Ben-Gurian University of the Negev, P.O.B.-653, Be’er Sheva-8410501, Israel
- MR Author ID: 1239252
- Email: pgarain92@gmail.com
**Juha Kinnunen**- Affiliation: Department of Mathematics, P.O. Box 11100, FI-00076 Aalto University, Finland
- MR Author ID: 349676
- Email: juha.k.kinnunen@aalto.fi
- Received by editor(s): February 26, 2021
- Received by editor(s) in revised form: October 22, 2021
- Published electronically: March 17, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 5393-5423 - MSC (2020): Primary 35B45, 35B65, 35D30, 35J92, 35R11
- DOI: https://doi.org/10.1090/tran/8621
- MathSciNet review: 4469224