The paper is concerned with studying strictly elliptic operators on Riemannian manifolds with smooth boundary where the operator is acting on the space of continuous functions. Let \(A_m\) denote such an operator and let \((\overline{M},g)\) be the Riemannian manifold with the boundary \(\partial M\). Given this, the author considers \(A_m\) under various boundary conditions like Dirichlet-to-Neumann and Wentzell. The connection between the two boundary conditions was explored in the work of [K.-J. Engel and G. Fragnelli, Adv. Differ. Equ. 10, No. 11, 1301–1320 (2005; Zbl 1108.47039)] where an abstract framework was developed to deal with the generalized Wentzell boundary conditions. The (generalized) Wentzell boundary conditions are intimately related to (abstract) Cauchy problem which provides a good motivation to study the problem in the paper.
The main result of the paper is that \(A_m\) under the Wentzell boundary condition generates an analytic semigroup of angle \(\frac{\pi}{2}\) over \(C(\overline{M})\). As shown by the author and Engel previously [T. Binz and K.-J. Engel, Math. Nachr. 292, No. 4, 733–746 (2019; Zbl 1422.35014)], this is closely related to the problem of generation of analytic semigroups of Dirichlet-to-Neumann operators on the boundary \(C(\partial M)\). Specifically, as per Theorem 3.1 in [K.-J. Engel and G. Fragnelli, Adv. Differ. Equ. 10, No. 11, 1301–1320 (2005; Zbl 1108.47039)], if the condition that Dirichlet-to-Neumann operator generates an analytic semigroup of angle \(\alpha\) on the boundary \(C(\partial M)\) is satisfied(along with other boundedness conditions on feedback operators, Dirichlet operator, etc.), then \(A_m\) generates an analytic semigroup on \(\overline M\) of angle \(\alpha\). The first half of the paper deals with the special case of Laplace-Beltrami operator, \(A_m := \triangle_g\) where the existence of analytic semigroup with angle \(\frac{\pi}{2}\) is proved for Dirichlet-to-Neumann and Wentzel boundary conditions. For the case of Dirichet-to-Neumann operator, the author employs a clever use of perturbation by considering \(W := \sqrt{-\triangle_{\partial M}}\). This is a pseudo-differential operator of order 1 and it follows from [W. Arendt et al., Vector-valued Laplace transforms and Cauchy problems. 2nd ed. Basel: Birkhäuser (2011; Zbl 1226.34002)] that it generates analytics semigroup of angle \(\frac{\pi}{2}\) as its true for \(\triangle_{\partial M}\), which was proved by author previously [J. Evol. Equ. 20, No. 3, 1005–1028 (2020; Zbl 1516.58007)]. The result then follows by showing that \(W-P\), where \(P := N - W\), generates an analytic semigroup. The existence of analytic semigroup of angle \(\frac{\pi}{2}\) follows from Theorem 3.1 in [K.-J. Engel and G. Fragnelli, Adv. Differ. Equ. 10, No. 11, 1301–1320 (2005; Zbl 1108.47039)]. The second half of the paper extends the results for \(\triangle_g\) to strictly elliptic operators on \(\overline M\). To that, the strategy is to reduce the problem to the case of Laplace-Beltrami operator (for a different metric) and express the original operator as its perturbation.

The paper is very well organized and the results proved for Laplacian are nicely leveraged to prove the main theorem dealing with the strictly elliptic operators. This interesting paper provides important generalizations of previously known results and offers a lot to gain from a careful reading to even the non-experts in the area.