Abstract
We prove that strictly elliptic operators with generalized Wentzell boundary conditions generate analytic semigroups of angle \(\frac{\pi }{2}\) on the space of continuous functions on a compact manifold with boundary.
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1 Introduction
We start from a strictly elliptic differential operator \(A_m\) with domain \(D(A_m)\) on the space \(C({\overline{M}})\) of continuous functions on a smooth, compact, orientable Riemannian manifold \(({\overline{M}},g)\) with smooth boundary \(\partial M\). Moreover, let C be a strictly elliptic differential operator on the boundary, take \(\frac{\partial ^a}{\partial \nu ^g}:D(\frac{\partial ^a}{\partial \nu ^g})\subset C({\overline{M}}) \rightarrow C(\partial {M})\) to be the outer conormal derivative, and functions \(\eta , \gamma \in C(\partial M)\) with \(\eta \) strictly positive and a constant \(q > 0\). In this setting, we define the operator \(A^B \subset A_m\) with generalized Wentzell boundary conditions by requiring
On a bounded domain \(\Omega \subset {\mathbb {R}}^n\) with sufficiently smooth boundary \(\partial \Omega \), Favini, Goldstein, Goldstein, Obrecht, and Romanelli [8] showed that for \(A_m = \Delta _\Omega \) and \(C = \Delta _{\partial \Omega }\) the operator \(A^B\) generates an analytic semigroup of angle \(\frac{\pi }{2}\) on \(C(\overline{\Omega })\). In a preprint Goldstein, Goldstein, and Pierre [9] generalized this statement to arbitrary elliptic differential operators of the form \(A_m f := \sum _{l,k = 1}^n \partial _l (a^{kl} \partial _k f)\) and \(C \varphi := \sum _{l,k = 1}^n \partial _l (\alpha ^{kl} \partial _k \varphi )\).
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 \(B_1^+(0) := \{ x \in {\mathbb {R}}^n :x_n \ge 0, | x | \le 1 \} \subset {\mathbb {R}}^n\). With the restriction g of the metric of \({\mathbb {R}}^n\) 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 \({\mathbb {R}}^n\) since the boundary is only \(\partial B_1^+(0) = \{ x \in {\mathbb {R}}^n :x_n = 0, | x | \le 1 \}\).
The situation \(q = 0\) on bounded, smooth domains in \({\mathbb {R}}^n\) 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 \(\sum _{i=1}^{n}x_i y^i\). Moreover, we denote by \(\hookrightarrow \) a continuous and by \({\mathop {\hookrightarrow }\limits ^{c}}\) 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 \({\partial X}\), called state and boundary space, respectively;
-
(ii)
a densely defined maximal operator \(A_m :D(A_m) \subset X \rightarrow X\);
-
(iii)
a boundary (or trace) operator \(L \in {\mathcal {L}}(X,{\partial X})\);
-
(iv)
a feedback operator \(B :D(B) \subseteq X \rightarrow {\partial X}\).
Using these spaces and operators, we define the operator \(A^B:D(A^B)\subset X\rightarrow X\) 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 \(A_0\subset A_m\) given by
The abstract Dirichlet operator associated with \(A_m\) is, if it exists,
i.e. \(L^{A_m}_0 \varphi = f\) is the unique solution of the abstract Dirichlet problem
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 \(N=N^{A_m,B}\) and call it the (abstract) Dirichlet-to-Neumann operator.
3 Laplace–Beltrami operator with generalized Wentzell boundary conditions
Take now as maximal operator \(A_m :D({A}_m) \subset \mathrm {C}({\overline{M}}) \rightarrow \mathrm {C}({\overline{M}})\) the Laplace-Beltrami operator \(\Delta _M^g\) with domain \(D(A_m) := \left\{ f \in \bigcap _{p > 1} \mathrm {W}^{2,p}_{{loc}}(M) \cap \mathrm {C}({\overline{M}}) \right. :\left. A_m f \in \mathrm {C}({\overline{M}}) \right\} \). Moreover, consider another strictly elliptic differential operator \(C :D(C) \subset \mathrm {C}(\partial M) \rightarrow \mathrm {C}(\partial M)\) in divergence form on the boundary space. To this end, take real valued functions
such that \(\alpha _j^k\) are strictly elliptic, i.e.
for all co-vectorfields \(X_k,X_l\) on \(\partial M\) with \((X_1(q),\dots ,X_n(q)) \not = (0,\dots ,0)\). Let \(\alpha = (\alpha _j^k)_{j,k = 1,\dots ,n}\) denote the 1-1-tensorfield and \(\beta = (\beta _j)_{j = 1,\dots ,n}\). Moreover, we denote by \(|\alpha |\) the determinate of \(\alpha \) and define \(C :D(C) \subset \mathrm {C}(\partial M) \rightarrow \mathrm {C}(\partial M)\) by
In order to define the feedback operator, we first consider \(B_0 :D(B_0) \subset \mathrm {C}({\overline{M}}) \rightarrow \mathrm {C}(\partial M)\) given by
This leads to the feedback operator \(B :D(B) \subset \mathrm {C}({\overline{M}}) \rightarrow \mathrm {C}(\partial M)\) given by
where \(L :\mathrm {C}({\overline{M}}) \rightarrow \mathrm {C}(\partial M) , f\mapsto f|_{\partial M}\) denotes the trace operator and \(q > 0\) and \(\eta \in \mathrm {C}({\overline{M}})\) is positive. Using these operators \(A_m\) and B, we define the operator \(A^B\) with Wentzell boundary conditions on \(\mathrm {C}({\overline{M}})\) as in (2.1).
Note that the feedback operator B can be splitted into
The following proof is inspired by [7] and similar to [2, Ex. 5.3].
Lemma 3.1
The operator B is relatively \(A_0\)-bounded of bound 0.
Proof
Since \(D(A_0) \subset \ker (L)\), the operators B and \(\eta \cdot B_0\) coincide on \(D(A_0)\). Hence it remains to prove the statement for the operator \(B_0\). By [13, Chap. 5., Thm. 1.3] and the closed graph theorem, we obtain
Rellich’s embedding (see [1, Thm. §3 2.10, Part III.]) implies
for \(p > \frac{n-1}{1-\alpha }\), where n denotes the dimension of \({\overline{M}}\). So we obtain
Therefore, by Ehrling’s lemma (cf. [12, Thm. 6.99]), for every \(\varepsilon >0\), there exists a constant \(C_\varepsilon >0\) such that
for every \(f \in D(A_0)\). Since \(B_0 \in {\mathcal {L}}(\mathrm {C}^1({\overline{M}}),{\partial X})\), this implies the claim. \(\square \)
Lemma 3.2
The operator \(N^{\Delta _m,B_0}\) is relatively C-bounded of bound 0.
Proof
Let \(W := -(\Delta _{\partial M}^g)^{\frac{1}{2}}\) and remark that by the proof of [3, Thm. 3.8], there exists a relatively W-bounded perturbation P of bound 0 such that
Therefore [11, Thm. 3.8] implies that \(N^{\Delta _m,B_0}\) is relatively \(\Delta _{\partial M}^g\)-bounded of bound 0. Using the (uniform) ellipticity of C, there exists a constant \(\Lambda > 0\) such that
for \(\varphi \in D(C) = D(\Delta _{\partial M}^g)\). Hence \(N^{\Delta _m,B_0}\) is relatively C-bounded of bound 0.
\(\square \)
Now the abstract results of [2] lead to the desired result.
Theorem 3.3
The operator \(A^B\) with Wentzell boundary conditions associated to the Laplace-Beltrami operator \(\Delta _m=\Delta _M^g\) generates a compact and analytic semigroup of angle \(\frac{\pi }{2}\) on \(\mathrm {C}({\overline{M}})\).
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 \(L_0 \in {\mathcal {L}}(\mathrm {C}(\partial M),\mathrm {C}({\overline{M}}))\) 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 \(\frac{\pi }{2}\) on \(\mathrm {C}({\overline{M}})\) and has compact resolvent. Moreover, by [4, Cor. 3.6], the operator C generates a compact and analytic semigroup of angle \(\frac{\pi }{2}\) on \(\mathrm {C}(\partial M)\). Finally, the claim follows by [2, Thm. 4.3]. \(\square \)
4 Elliptic operators with generalized Wentzell boundary conditions
Consider a strictly elliptic differential operator \(A_m :D(A_m) \subset \mathrm {C}({\overline{M}}) \rightarrow \mathrm {C}({\overline{M}})\) in divergence form on the boundary space. To this end, let
be real-valued functions, such that \(a_j^k\) are strictly elliptic, i.e.
for all co-vectorfields \(X_k,X_l\) on \({\overline{M}}\) with \((X_1(q),\dots ,X_n(q)) \not = (0,\dots ,0)\). Let \(a = (a_j^k)_{j,k = 1,\dots ,n}\) be the 1-1-tensorfield and \(b = (b_j)_{j = 1,\dots ,n}\). Then we define \(A_m :D(A_m) \subset \mathrm {C}({\overline{M}}) \rightarrow \mathrm {C}({\overline{M}})\) by
Note that, since \({\overline{M}}\) is compact, every strictly elliptic operator is uniformly elliptic (and of course vice versa).
We consider a (2, 0)-tensorfield on \({\overline{M}}\) given by
Its inverse \({\tilde{g}}\) is a (0, 2)-tensorfield on \({\overline{M}}\), which is a Riemannian metric since \(a^k_j g^{jl}\) is strictly elliptic on \({\overline{M}}\). We denote \({\overline{M}}\) with the old metric by \({\overline{M}}^g\) and with the new metric by \({\overline{M}}^{{\tilde{g}}}\) and remark that \({\overline{M}}^{{\tilde{g}}}\) is a smooth, compact, orientable Riemannian manifold with smooth boundary \(\partial M\). Since the differentiable structures of \({\overline{M}}^g\) and \({\overline{M}}^{{\tilde{g}}}\) coincide, the identity
is a \(\mathrm {C}^\infty \)-diffeomorphism. Hence the spaces
coincide. Moreover, [10, Prop. 2.2] implies that the following spaces coincide
for all \(p > 1\) and \(k \in {\mathbb {N}}\). Denote by \({\hat{A}}_m\) the maximal operator defined in (4.1) with \(b_j = c = 0\) and by \({\hat{C}}\) the operator given in (3.1) for \(\beta _j = \gamma = 0\). Moreover, denote the corresponding feedback operator by \({\hat{B}}\).
Next, we look at the operators \(A_m\), \(B_0\), and C with respect to the new metric \({\tilde{g}}\).
Lemma 4.1
The operator \({\hat{A}}_m\) and the Laplace-Beltrami operator \(\Delta _M^{{\tilde{g}}}\) coincide on \(\mathrm {C}({\overline{M}})\).
Proof
Using local coordinates, we obtain
for \(f \in D({\hat{A}}_m)= D(\Delta ^{{\tilde{g}}}_m)\) since \(|g| = |a| \cdot |{\tilde{g}}|\). \(\square \)
Now we compare the maximal operators \(A_m\) and \({\hat{A}}_m\).
Lemma 4.2
The operators \(A_m\) and \({\hat{A}}_m\) differ only by a relatively bounded perturbation of bound 0.
Proof
Using (4.2), we define
for \(f \in D(A_m) \cap D({\hat{A}}_m)\). Since \(b_l \in \mathrm {C}_c(M)\), there exist compact sets \(K_l := \text {supp}(b_l)\). Let \(K := \bigcup _{l = 1}^n K_l\) and note that it is a compact set and every \(b_l\) and hence \(P_1 f\) vanishes outside of K. We define
Morreys embedding ([1, Thm. §3 2.10, Part III.]) implies
Moreover, we obtain
and therefore \(P_1 \in {\mathcal {L}}(\mathrm {C}^1(K), \mathrm {C}({\overline{M}}))\). Hence \(D({\hat{A}}_m)=D({\tilde{A}}_m)\). By (4.3), we conclude from Ehrling’s lemma (see [12, Thm. 6.99]) that
for \(f \in D({\hat{A}}_m)\) and all \(\varepsilon > 0\). Hence \(P_1\) is relatively \(A_m\)-bounded of bound 0. Finally remark that
is bounded and that
for \(f \in D({\hat{A}}_m)\). \(\square \)
Lemma 4.3
The operators \(B_0\) and the negative conormal derivative \(-\frac{\partial }{\partial \nu ^{{\tilde{g}}} }\) coincide.
Proof
Since the Sobolev spaces coincide, we compute in local coordinates
for \(f \in D(B)=D(\frac{\partial ^{{\tilde{g}}}}{\partial \nu })\). \(\square \)
Define \({\tilde{C}} :D({\tilde{C}}) \subset \mathrm {C}(\partial M) \rightarrow \mathrm {C}(\partial M)\) by
where \(\tilde{\alpha }(q) := a(q)^{-1} \cdot \alpha (q)\).
Lemma 4.4
The operators \({\hat{C}}\) and \({\tilde{C}}\) coincide on \(\mathrm {C}(\partial M)\).
Proof
An easy calculation shows
Hence we obtain in local coordinates
for \(\varphi \in D({\hat{C}})=D({\tilde{C}})\). \(\square \)
Next we compare the operators C and \({\hat{C}}\).
Lemma 4.5
The operators C and \({\hat{C}}\) differ only by a relatively bounded perturbation of bound 0.
Proof
Denote by
and note that \(P \in {\mathcal {L}}(\mathrm {C}^1({\partial M}), \mathrm {C}(\partial M))\). The Sobolev embeddings and the closed graph theorem imply
Finally, the claim follows by Ehrling’s lemma (cf. [12, Thm. 6.99]). \(\square \)
Now we are prepared to prove our main theorem.
Theorem 4.6
The operator \(A^B\) with Wentzell boundary conditions generates a compact and analytic semigroup of angle \(\frac{\pi }{2}\) on \(\mathrm {C}({\overline{M}})\).
Proof
Since \({\tilde{C}}\) is a strictly elliptic differential operator in divergence form on \(\mathrm {C}(\partial M)\), we obtain by Theorem 3.3 that the Laplace-Beltrami operator with Wentzell boundary conditions given by
generates a compact and analytic semigroup of angle \(\frac{\pi }{2}\) on \(\mathrm {C}({\overline{M}})\). Now Lemma 4.1, Lemma 4.3, and Lemma 4.4 imply that the operator \({\hat{A}}^{{\hat{B}}}\) generates a compact and analytic semigroup of angle \(\frac{\pi }{2}\) on \(\mathrm {C}({\overline{M}})\). Note that \(A_m\) and \({\hat{A}}_m\) differ only by a relatively \(A_m\)-bounded perturbation of bound 0 by Lemma 4.2. By Lemma 4.5, one obtains that the perturbation on the boundary is relatively \({\hat{C}}\)-bounded. Now the claim follows from [2, Thm. 4.2]. \(\square \)
Remark 4.7
Theorem 4.6 generalizes the main theorem in [9] for the case \(p = \infty \).
Corollary 4.8
The initial-value boundary problem
on \(\mathrm {C}({\overline{M}})\) is well-posed. Moreover, the solution \(\left( \begin{matrix} u(t) \\ \varphi (t) \end{matrix}\right) \in \mathrm {C}^\infty (M) \times \mathrm {C}^\infty (\partial M)\) for \(t > 0\) depends analytically on the initial value \(\left( \begin{matrix} u_0 \\ u_0|_{\partial M} \end{matrix}\right) \) and is governed by a compact and analytic semigroup, which can be extended to the right half plane.
Change history
07 July 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00013-021-01627-8
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Binz, T. Strictly elliptic operators with generalized Wentzell boundary conditions on continuous functions on manifolds with boundary. Arch. Math. 115, 111–120 (2020). https://doi.org/10.1007/s00013-020-01457-0
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DOI: https://doi.org/10.1007/s00013-020-01457-0