Strictly elliptic operators with Dirichlet boundary conditions on spaces of continuous functions on manifolds

Abstract

We study strictly elliptic differential operators with Dirichlet boundary conditions on the space $\mathrm {C}(\overline{M})$ of continuous functions on a compact Riemannian manifold $\overline{M}$ with boundary and prove sectoriality with optimal angle $\frac{\pi }{2}$.

Appendix A. Bessel functions

The solutions of the ordinary differential equation

$$\begin{aligned} z^2 \frac{d^2}{dz^2} f(z) + z \frac{d}{dz} f(z) = (z^2 + \alpha ^2) f(z) \end{aligned}$$

(3.4)

for $z \in \mathbb {C}$ are called modified Bessel functions of order $\alpha \in \mathbb {R}$. In particular, we have the following.

Proposition A.1

The modified Bessel functions of first kind of order $\alpha \in \mathbb {R}$ are given by

$$\begin{aligned} I_\alpha (z) = \sum _{k = 0}^\infty \frac{ \left( \frac{z}{2} \right) ^{2k + \alpha } }{\Gamma ( k + \alpha + 1 )k!} \end{aligned}$$

for $z \in \mathbb {C}{\setminus } \mathbb {R}_-$, where $\Gamma$ denotes the Gamma function. Moreover, we obtain the modified Bessel function of second kind of order $\alpha \in \mathbb {R}{\setminus } \mathbb {Z}$ by

$$\begin{aligned} K_\alpha (z) = \frac{\pi }{2} \cdot \frac{I_{-\alpha }(z) - I_\alpha (z) }{\sin ( \pi \alpha )} \end{aligned}$$

for $z \in \mathbb {C}{\setminus } \mathbb {R}_-$. If $\alpha \in \mathbb {Z}$, there exists a sequence $(\alpha _n)_{n \in \mathbb {N}} \subset \mathbb {R}{\setminus } \mathbb {Z}$ such that $\alpha _n \rightarrow \alpha$ and $K_\alpha$ is the limit

$$\begin{aligned} K_\alpha (z) := \lim \limits _{n \rightarrow \infty } K_{\alpha _n} (z) \end{aligned}$$

for $z \in \mathbb {C}{\setminus } \mathbb {R}_-$.

First, we prove an estimate for the modified Bessel function of second kind.

Lemma A.2

Let $\alpha \in \mathbb {R}$ and $\eta > 0$. Then, there exists a constant $C(\eta ) > 0$ such that

$$\begin{aligned} |K_\alpha (z)| \le K_\alpha (C(\eta )|z|) \end{aligned}$$

for all $z \in \Sigma _{\frac{\pi }{2}-\eta }$.

Proof

Since ${{\,\mathrm{Re}\,}}(z) > 0$ for all $z \in \Sigma _{\frac{\pi }{2}-\eta }$ and $\alpha \in \mathbb {R}$ it follows by [23, p. 181] that

$$\begin{aligned} |K_\alpha (z)|&= \left| \int _0^\infty e^{-z \cosh (t)} \cosh (\alpha t) \, \mathrm{d}t \right| \le \int _0^\infty e^{-{{\,\mathrm{Re}\,}}(z) \cosh (t)} \cosh (\alpha t) \, \mathrm{d}t . \end{aligned}$$

Note that $z = |z|e^{i\varphi }$ with $|\varphi | \in [0,\nicefrac {\pi }{2}-\eta )$. The monotony of the cosinus implies

$$\begin{aligned} \frac{{{\,\mathrm {Re}\,}}(z)}{|z|} = \cos (\varphi ) \ge \cos \left( {\pi }/{2}-\eta \right) = \sin (\eta ) =: C(\eta ) > 0 . \end{aligned}$$

Using the monotony of the exponential function and the positivity of $\cosh$, we conclude

$$\begin{aligned} \int _0^\infty e^{-{{\,\mathrm{Re}\,}}(z) \cosh (t)} \cosh (\alpha t) \, \mathrm{d}t \le \int _0^\infty e^{- C(\varepsilon )|z|\cosh (t) } \cosh (\alpha t) \, \mathrm{d}t = K_\alpha (C(\eta ) |z|) \end{aligned}$$

for all $z \in \Sigma _{\frac{\pi }{2}-\eta }$. $\square$

Therefore, we obtain an estimate for the kernel.

Lemma A.3

Let $\alpha \in \mathbb {R}$, $k \in [0,\infty )$ and $\lambda \in \Sigma _{\pi -\eta }$ for $\eta > 0$. If $k + \alpha < n$, we obtain

$$\begin{aligned} \sup _{x \in M} \int _M \frac{K_{\alpha }(C(\eta ) \sqrt{|\lambda |} \rho (x,y))}{\rho (x,y)^k} \, \mathrm{d}y \le C(\eta ) \sqrt{|\lambda |}^{ k - n} \end{aligned}$$

for $|\lambda |\ge 1$.

Proof

Remark that

$$\begin{aligned} \int _{M} \frac{K_{\alpha }(C(\eta ) \sqrt{|\lambda |} {\rho }(x,y))}{{\rho }(x,y)^{k}} \, \mathrm{d}y =&\int _{B_{R}(x)} \frac{K_{\alpha }(C(\eta ) \sqrt{|\lambda |} {\rho }(x,y))}{{\rho }(x,y)^{k}} \, \mathrm{d}y \\&+ \int _{M {\setminus } B_{R}(x)} \frac{K_{\alpha }(C(\eta ) \sqrt{|\lambda |} {\rho }(x,y))}{{\rho }(x,y)^{k}} \, \mathrm{d}y . \end{aligned}$$

For the first term, one obtains

$$\begin{aligned} \int _{B_{R}(x)} \frac{K_{\alpha }(C(\eta ) \sqrt{|\lambda |} {\rho }(x,y))}{{\rho }(x,y)^{k}} \, \mathrm{d}y&\le \tilde{C} \int _{{\mathbb {R}}^n} \frac{K_{\alpha }(C(\eta ) \sqrt{|\lambda |} |y| )}{|y|^{k}} \, \mathrm{d}y \\&= \hat{C}(\eta ) \sqrt{|\lambda |}^{k} \frac{1}{\sqrt{|\lambda |}^{n}} \int _{{\mathbb {R}}^n} \frac{K_{\alpha }(|z| )}{|z|^{k}} \, dz \\&= \hat{C}(\eta ) \sqrt{|\lambda |}^{ k - n} \int _0^\infty \int _{\mathbb {S}^{n-1}_r} \frac{K_{\alpha }(r)}{r^{k}} \, \text {dvol}_{\mathbb {S}^{n-1}_r} \, dr \\&= \check{C}(\eta ) \sqrt{|\lambda |}^{ k - n} \int _0^\infty K_{\alpha }(r) r^{n-1-k} \, dr . \end{aligned}$$

Since

$$\begin{aligned} K_{\alpha }(r) = \mathcal {O}(r^{-\alpha }) \end{aligned}$$

for small $r \in \mathbb {R}_+$ and

$$\begin{aligned} K_{\alpha }(r) = \mathcal {O}\left( \frac{e^{-r}}{\sqrt{r}}\right) \end{aligned}$$

for large $r \in \mathbb {R}_+$, we have

$$\begin{aligned} r^{n-1-k} K_{\alpha }(r) = \mathcal {O}(r^{n-1-k-\alpha }) \end{aligned}$$

for small $r \in \mathbb {R}_+$ and

$$\begin{aligned} r^{n-1-k} K_{\alpha }(r) = \mathcal {O}(r^{n-\frac{3}{2}-k} e^{-r}) \end{aligned}$$

for large $r \in \mathbb {R}_+$. Hence, there exists a constant $\bar{C} < \infty$ such that

$$\begin{aligned} \int _0^\infty K_{\alpha }(r) r^{n-1-k} \, dr < \bar{C} \end{aligned}$$

and we conclude that

$$\begin{aligned} \int _{B_{R}(x)} \frac{K_{\alpha }(C(\eta ) \sqrt{|\lambda |} {\rho }(x,y))}{{\rho }(x,y)^{k}} \, \mathrm{d}y \le C(\eta ) \sqrt{|\lambda |}^{k-n} . \end{aligned}$$

If $y \in \overline{M} {\setminus } B_R(x)$, we have $\rho (x,y) \ge R$ and therefore

$$\begin{aligned} \int _{M {\setminus } B_{R}(x)} \frac{K_{\alpha }(C(\eta ) \sqrt{|\lambda |} {\rho }(x,y))}{{\rho }(x,y)^{k}} \, \mathrm{d}y&\le \frac{K_{\alpha }(C(\eta )R \sqrt{|\lambda |})}{R^k} \text {vol}_g(M {\setminus } B_{R}(x)) \\&\le \hat{C}(\eta ) e^{-\tilde{C}(\eta )\sqrt{|\lambda |}} \\&\le \bar{C}(\eta ) \sqrt{|\lambda |}^{k-n} \end{aligned}$$

for $|\lambda |$ since

$$\begin{aligned} K_{\alpha }(r) = \mathcal {O}\left( \frac{e^{-r}}{\sqrt{r}}\right) \end{aligned}$$

for large $r \in \mathbb {R}_+$. $\square$

Replacing x by $x^*$ this yields an estimate for the reflected kernel.

Corollary A.4

Let $\alpha \in \mathbb {R}$, $k \in [0,\infty )$ and $\lambda \in \Sigma _{\pi -\eta }$ for $\eta > 0$. Moreover, let $x \in S_{2\varepsilon }$. If $k + \alpha < n$, we obtain

$$\begin{aligned} \sup _{x \in S_{2\varepsilon }} \int _M \frac{K_{\alpha }(C(\eta )\sqrt{\lambda } \overline{\rho }(x^*,y))}{\overline{\rho }(x^*,y)^k} \, \mathrm{d}y \le C \sqrt{|\lambda |}^{ k - n} \end{aligned}$$

for $|\lambda |\ge 1$.