Strictly elliptic operators with Dirichlet boundary conditions on spaces of continuous functions on manifolds
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Journal of Evolution Equations
Strictly elliptic operators with Dirichlet boundary conditions on spaces of continuous functions on manifolds Tim Binz
Abstract. We study strictly elliptic differential operators with Dirichlet boundary conditions on the space C(M) of continuous functions on a compact Riemannian manifold M with boundary and prove sectoriality with optimal angle π2 .
1. Introduction Our starting point is a smooth compact Riemannian manifold M of dimension n with smooth boundary ∂ M and Riemannian metric g and the initial value-boundary problem ⎧ √ g g d ⎪ √1 a∇ u(t) + b, ∇ u(t) + cu(t) for t > 0, u(t) = |a|div ⎪ g M M ⎨ dt |a| u(t)|∂ M = 0 for t > 0, ⎪ ⎪ ⎩u(0) =u . 0
(IBP) Here, a is a smooth (1, 1)-tensorfield, b ∈ C(M, Rn ) and c ∈ C(M, R). We are interested in existence, uniqueness and qualitative behaviour of the solution of this initial value-boundary problem. To study these properties systematically, the theory of operator semigroups (cf. [4,11,13,18]) can be used. We choose the Banach space C(M) and define the differential operator with Dirichlet boundary condition
1 g g A0 f := |a|divg √ a∇ M f + b, ∇ M u(t) + c f |a| with domain ⎧ ⎫ ⎨ ⎬ D(A0 ) := f ∈ W 2, p (M) ∩ C0 (M) : A0 f ∈ C(M) . ⎩ ⎭ p≥1
Mathematics Subject Classification: 47D06, 34G10, 47E05, 47F05 Keywords: Dirichlet boundary conditions, Analytic semigroup, Riemmanian manifolds.
T. Binz
J. Evol. Equ.
Then, the initial value-boundary problem (IBP) is equivalent to the abstract Cauchy problem d dt u(t) = A0 u(t) for t > 0, (ACP) u(0) = u 0 in C(M). In this paper, we show that the solution u of the above problems can be extended analytically in the time variable t to the open complex right half-plane. In operator theoretic terms this corresponds to the fact that A0 is sectorial of angle π2 . Here is our main theorem. Theorem 1.1. The operator A0 is sectorial of angle C(M).
π 2
and has compact resolvent on
For domains ⊂ Rn , the generation of analytic semigroups by elliptic operators with Dirichlet boundary conditions on different spaces is well known. It was first shown by Browder in [8] for L 2 (), by Agmon in [3] for L p () (see also [18, Chap. 3.1.1]) and by Stewart in [22] for C() (see also [18, Chap. 3.1.5]). By Stewart’s method, one even gets the angle of analyticity. Later Arendt proved in [5] (see also [1, Chap. III. 6]), using the Poisson operator, that the angle of the analytic semigroup generated by the Laplacian on the space C() is π2 . However, this method does not work on manifolds with boundary. The angle π2 of analyticity of A0 plays an important role in the generation of analytic semigroups by elliptic differential operators with Wentzell boundary conditions on spaces of continuous functions. Many authors are interested in this topic, and we refer, e.g. to [9,10,12,14,15]. In this context, one starts from the “maximal” operator Am : D(Am ) ⊆ C(M) → C(M) in divergence form, given by
1 g g Am f := |a|divg √ a∇ M f + b, ∇ M f + c f |a| with domain ⎧ ⎫ ⎨ ⎬ D(Am ) := f ∈ W 2, p (M) : A
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