Two alternative proofs of weak Harnack inequality for mixed local and nonlocal $p$-Laplace equations with a nonhomogeneity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914075904049152 |
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| author | Garain, Prashanta |
| author_facet | Garain, Prashanta |
| contents | We study a class of mixed local and nonlocal $p$-Laplace equations with prototype \[ -Δ_p u + (-Δ_p)^s u = f \quad \text{in } Ω, \] where $Ω\subset \mathbb{R}^n$ is bounded and open. We provide sufficient condition on $f$ to ensure weak Harnack inequality with a tail term for sign-changing supersolutions. Two different proofs are presented, avoiding the Krylov--Safonov covering lemma and expansion of positivity: one via the John--Nirenberg lemma, the other via the Bombieri--Giusti lemma. To our knowledge, these approaches are new, even for $p = 2$ with $f \equiv 0$, and include a new proof of the reverse Hölder inequality for supersolutions. Further, we establish Harnack inequality for solutions by first deriving a local boundedness result, together with a tail estimate and an initial weak Harnack inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04065 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two alternative proofs of weak Harnack inequality for mixed local and nonlocal $p$-Laplace equations with a nonhomogeneity Garain, Prashanta Analysis of PDEs 35B45, 35B65, 35D30, 35J92, 35R11 We study a class of mixed local and nonlocal $p$-Laplace equations with prototype \[ -Δ_p u + (-Δ_p)^s u = f \quad \text{in } Ω, \] where $Ω\subset \mathbb{R}^n$ is bounded and open. We provide sufficient condition on $f$ to ensure weak Harnack inequality with a tail term for sign-changing supersolutions. Two different proofs are presented, avoiding the Krylov--Safonov covering lemma and expansion of positivity: one via the John--Nirenberg lemma, the other via the Bombieri--Giusti lemma. To our knowledge, these approaches are new, even for $p = 2$ with $f \equiv 0$, and include a new proof of the reverse Hölder inequality for supersolutions. Further, we establish Harnack inequality for solutions by first deriving a local boundedness result, together with a tail estimate and an initial weak Harnack inequality. |
| title | Two alternative proofs of weak Harnack inequality for mixed local and nonlocal $p$-Laplace equations with a nonhomogeneity |
| topic | Analysis of PDEs 35B45, 35B65, 35D30, 35J92, 35R11 |
| url | https://arxiv.org/abs/2510.04065 |