Strict Faber-Krahn type inequality for the mixed local-nonlocal operator under polarization

2023/09/14 by Kumar, K Ashok, Biswas, Nirjan · 1 citation
#47J10 #49Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35R11 #Secondary 35B51

paper · doi:10.48550/arxiv.2309.07520

Abstract

Let Ω⊂ ℝ^d with d≥ 2 be a bounded domain of class C^1,β for some β∈ (0,1). For p∈ (1, ∞ ) and s∈ (0,1), let Λ^s_p(Ω) be the first eigenvalue of the mixed local-nonlocal operator -Δ_p+(-Δ_p)^s in Ω with the homogeneous nonlocal Dirichlet boundary condition. We establish a strict Faber-Krahn type inequality for Λ_p^s(⋅ ) under polarization. As an application of this strict inequality, we obtain the strict monotonicity of Λ_p^s(⋅ ) over annular domains and characterize the rigidity property of the balls in the classical Faber-Krahn inequality for -Δ_p+(-Δ_p)^s.

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