Strictly elliptic operators with generalized Wentzell boundary conditions on continuous functions on manifolds with boundary

We prove that strictly elliptic operators with generalized Wentzell boundary conditions generate analytic semigroups of angle π2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\pi }{2}$$\end{document} on the space of continuous functions on a compact manifold with boundary.


Introduction.
We start from a strictly elliptic differential operator A m with domain D(A m ) on the space C(M ) of continuous functions on a smooth, compact, orientable Riemannian manifold (M, g) with smooth boundary ∂M . Moreover, let C be a strictly elliptic differential operator on the boundary, take ∂ a ∂ν g : D( ∂ a ∂ν g ) ⊂ C(M ) → C(∂M ) to be the outer conormal derivative, and functions η, γ ∈ C(∂M ) with η strictly positive and a constant q > 0. In this setting, we define the operator A B ⊂ A m with generalized Wentzell boundary conditions by requiring On a bounded domain Ω ⊂ R n with sufficiently smooth boundary ∂Ω, Favini, Goldstein, Goldstein, Obrecht, and Romanelli [8] showed that for A m = Δ Ω and C = Δ ∂Ω the operator A B generates an analytic semigroup of angle π 2 on C(Ω). In a preprint Goldstein, Goldstein, and Pierre [9] generalized this statement to arbitrary elliptic differential operators of the form A m f := n l,k=1 ∂ l (a kl ∂ k f ) and Cϕ := n l,k=1 ∂ l (α kl ∂ k ϕ). Our main theorem (Theorem 4.6) generalizes these results to arbitrary strictly elliptic operators A m and C on smooth, compact, orientable Riemannian manifolds with smooth boundary.
Consider a half-ball B + 1 (0) := {x ∈ R n : x n ≥ 0, |x| ≤ 1} ⊂ R n . With the restriction g of the metric of R n to (B + 1 (0), g), we obtain a smooth, compact, orientable Riemannian manifold B + 1 (0) with smooth boundary. It is not the closure of a domain in R n since the boundary is only ∂B + 1 (0) = {x ∈ R n : x n = 0, |x| ≤ 1}.
The situation q = 0 on bounded, smooth domains in R n was studied by Engel and Fragnelli [5] and on smooth, compact, orientable Riemannian manifolds in [3].
For q = 0, the boundary condition is a partial differential equation of first order whereas for q > 0 it is a partial differential equation of second order. Using the theory developed in [5] and [2], this yields two different abstract Dirichlet-to-Neumann operators: In the case q = 0, it is a pseudo differential operator of first order, in the case q > 0, it is an elliptic differential operator of second order perturbed by a pseudo differential operator of first order.
The paper is organized as follows. In the second section, we introduce the abstract setting from [5] and [2] for our problem. In the third section, we study the special case that A m is the Laplace-Beltrami operator and B is the normal derivative. In the last section, we generalize to arbitrary strictly elliptic operators and their conormal derivatives.
Throughout the whole paper, we use the Einstein notation for sums and write x i y i shortly for n i=1 x i y i . Moreover, we denote by → a continuous and by c → a compact embedding.
2. The abstract setting. As in [5,Sect. 2], the basis of our investigation is the following.

Abstract setting 2.1. Consider
(i) two Banach spaces X and ∂X, called state and boundary space, respectively; (ii) a densely defined maximal operator Using these spaces and operators, we define the operator A B : D(A B ) ⊂ X → X with abstract generalized Wentzell boundary conditions as For an interpretation of Wentzell boundary conditions as "dynamic boundary conditions", we refer to [5,Sect. 2].
In the sequel, we need the following operators.
Notation 2.2. The kernel of L is a closed subspace and we consider the restriction A 0 ⊂ A m given by Vol. 115 (2020)Elliptic operators with Wentzell boundary conditions 113 The abstract Dirichlet operator associated with A m is, if it exists, If it is clear which operator A m is meant, we simply write L 0 . Finally, we introduce the abstract Dirichlet-to-Neumann operator associated with (A m , B), defined by . If it is clear which operators A m and B are meant, we write N = N Am,B and call it the (abstract) Dirichlet-to-Neumann operator.
Now the abstract results of [2] lead to the desired result. Proof. We verify the assumptions of [2,Thm. 4.3]. Remark that by [3,Lem. 3.6] and Lemma 3.1 above, the Dirichlet operator L 0 ∈ L(C(∂M ), C(M )) exists and B is relatively A 0 -bounded of bound 0. By multiplicative perturbation, we assume without loss of generality that q = 1. Now [4, Thm. 1.1] implies that A 0 is sectorial of angle π 2 on C(M ) and has compact resolvent. Moreover, by [4,Cor. 3.6], the operator C generates a compact and analytic semigroup of angle π 2 on C(∂M ). Finally, the claim follows by [2,Thm. 4.3].

Elliptic operators with generalized
Wentzell boundary conditions. Consider a strictly elliptic differential operator A m : in divergence form on the boundary space. To this end, let be real-valued functions, such that a k j are strictly elliptic, i.e.
for all co-vectorfields X k , X l on M with (X 1 (q), . . . , X n (q)) = (0, . . . , 0). Let a = (a k j ) j,k=1,...,n be the 1-1-tensorfield and b = (b j ) j=1,...,n . Then we define Note that, since M is compact, every strictly elliptic operator is uniformly elliptic (and of course vice versa). We consider a (2, 0)-tensorfield on M given bỹ Its inverseg is a (0, 2)-tensorfield on M , which is a Riemannian metric since a k j g jl is strictly elliptic on M . We denote M with the old metric by M g and with the new metric by Mg and remark that Mg is a smooth, compact, orientable Riemannian manifold with smooth boundary ∂M . Since the differentiable structures of M g and Mg coincide, the identity