The Robin heat kernel and its expansion via Robin eigenfunctions

2025/05/21 by Meng Yu, Kui Wang, Meng, Yifeng +1
Mathematics · #Numerical methods in inverse problems #advanced mathematical theories #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2505.15092

Abstract

We prove the existence and uniqueness of the Robin heat kernel on compact Riemannian manifolds with smooth boundary for Robin parameter α∈ℝ, expressed as a spectral expansion in terms of Robin eigenvalues and eigenfunctions. For the non-negative parameter regime (α≥ 0), we present a direct proof based on trace Sobolev inequalities and eigenfunction estimates. The case of negative parameters (α<0) requires novel analytical techniques to handle L^∞ estimates of Robin eigenfunctions, addressing challenges not present in the non-negative case. Our result extends the the classical Dirichlet and Neumann cases to the less-studied negative parameter regime.

Citations

• Robin heat kernel comparison on manifolds
• Witten Deformation on Non-compact Manifold: Heat Kernel Expansion and Local Index Theorem

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