We resolve four questions concerning Karpukhin's Dirichlet-to-Neumann operator on differential forms. First, for $0\leq p\leq n-2$, we prove that the operator is classical pseudodifferential of order one and compute its principal symbol, which is elliptic on the coclosed symbol space. This answers Open Question 7.19 in...
We prove sharp two-eigenvalue isoperimetric inequalities for Neumann and Aharonov--Bohm Neumann spectra on surfaces. If $\Omega\subset\mathbb S^2$ is smooth, simply connected and proper, then the harmonic mean of $\mu_2(\Omega)$ and $\mu_3(\Omega)$ is bounded above by the first positive Neumann eigenvalue of the equal-...
Let $\mathbb{M}_\kappa^n$ be the space form of sectional curvature $\kappa\in\{-1,0,1\}$, so that $\mathbb{M}_{-1}^n=\mathbb{H}^n, \mathbb{M}_{0}^n=\mathbb{R}^n, \mathbb{M}_{1}^n=\mathbb{S}^n$. Let $\Omega\subset\mathbb{M}_\kappa^n$ be a nonempty bounded open set with Lipschitz boundary, and assume that $0<|\Omega|<|\m...
Da-Guang Chen, Cheng-Xi Yang· 1 citation· ⚡1
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