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Two alternative proofs of weak Harnack inequality for mixed local and nonlocal $p$-Laplace equations with a nonhomogeneity

Prashanta Garain

TL;DR

This work establishes weak Harnack and Harnack inequalities for a class of mixed local/nonlocal $p$-Laplace equations with a nonhomogeneous term, under mild regularity on the source $f$. It develops two independent analytic routes to the weak Harnack inequality with a tail term: (i) a Moser–type iteration built on the John–Nirenberg lemma, and (ii) a Bombieri–Giusti variant that bypasses Positivity expansion. Central to both approaches are robust energy estimates, a reverse Hölder inequality for supersolutions, and careful tail control linking interior behavior to exterior data. The results extend the weak Harnack theory to the mixed operator setting, yield local boundedness and a Harnack principle for solutions, and provide new proofs even for the homogeneous linear case $f\equiv0$. The framework accommodates nonzero $f$ in $L^{q/p}_{\mathrm{loc}}(\Omega)$ with $q>n$ and yields quantitative tail terms that capture the influence of the nonlocal component.

Abstract

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.

Two alternative proofs of weak Harnack inequality for mixed local and nonlocal $p$-Laplace equations with a nonhomogeneity

TL;DR

This work establishes weak Harnack and Harnack inequalities for a class of mixed local/nonlocal -Laplace equations with a nonhomogeneous term, under mild regularity on the source . It develops two independent analytic routes to the weak Harnack inequality with a tail term: (i) a Moser–type iteration built on the John–Nirenberg lemma, and (ii) a Bombieri–Giusti variant that bypasses Positivity expansion. Central to both approaches are robust energy estimates, a reverse Hölder inequality for supersolutions, and careful tail control linking interior behavior to exterior data. The results extend the weak Harnack theory to the mixed operator setting, yield local boundedness and a Harnack principle for solutions, and provide new proofs even for the homogeneous linear case . The framework accommodates nonzero in with and yields quantitative tail terms that capture the influence of the nonlocal component.

Abstract

We study a class of mixed local and nonlocal -Laplace equations with prototype where is bounded and open. We provide sufficient condition on 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 with , 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.

Paper Structure

This paper contains 17 sections, 19 theorems, 125 equations.

Table of Contents

  1. Introduction and main results
  2. Functional setting and auxiliary results
  3. Functional setting
  4. Auxiliary results
  5. Energy estimates
  6. Energy estimate for supersolutions
  7. Energy estimate for subsolutions
  8. Supersolution and tail estimates
  9. Inverse estimate for supersolutions
  10. Reverse Hölder inequality for supersolutions
  11. Logarithmic estimate for supersolutions
  12. Tail estimate
  13. Proof of the main results
  14. Proof of Theorem \ref{['mthm']} (Using John-Nirenberg lemma)
  15. Proof of Theorem \ref{['mthm']} (Using Bombieri--Giusti lemma)
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $x_0 \in \mathbb{R}^n$ and $0 < r \leq 1$ satisfy $r < \frac{R}{2}$. Assume $0 < s < 1 < p < n$ and $f \in L^{\frac{q}{p}}(B_R(x_0))$ with $q > n$. Suppose $u$ is a weak supersolution of meqn with $u \geq 0$ in $B_R(x_0) \subset \Omega$. Then, for every $0 < \eta < \kappa(p-1)$, there exists a c where the tail term $\mathrm{Tail}(u_-; x_0, R)$ is defined in tail.

Theorems & Definitions (28)

  • Theorem 1.1: Weak Harnack Inequality
  • Theorem 1.2: Local boundedness
  • Theorem 1.3: Harnack Inequality
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 2.7
  • ...and 18 more