The Dirichlet-to-Neumann operator on C(∂ Ω)

2017/07/18 by Wolfgang Arendt, Arendt, W., A. F. M. ter Elst +1 · 1 citation
Computer Science · Mathematics · #35J15 #35J57 #35K08 #47D06 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1707.05556

openalex publication_date 2017/07/18 · openalex created_date 2017/07/31 · openalex updated_date 2026/07/28

Abstract

Let Ω⊂ \bf R^d be an open bounded set with Lipschitz boundary Γ. Let D_V be the Dirichlet-to-Neumann operator with respect to a purely second-order symmetric divergence form operator with real Lipschitz continuous coefficients and a positive potential V. We show that the semigroup generated by -D_V leaves C(Γ) invariant and that the restriction of this semigroup to C(Γ) is a C₀-semigroup. We investigate positivity and spectral properties of this semigroup. We also present results where V is allowed to be negative. Of independent interest is a new criterium for semigroups to have a continuous kernel.

Cited by

• Boundary behavior of Robin problems in non-smooth domains

Related

• Analyticity of the Dirichlet-to-Neumann semigroup on continuous functions
• Semigroups Generated by Elliptic Operators in Non-Divergence Form on C₀(/omega)
• Dirichlet-to-Neumann semigroup with respect to a general second order\n eigenvalue problem
• Operators with Wentzell boundary conditions and the Dirichlet-to-Neumann operator
• The Dirichlet-to-Neumann operator on rough domains