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


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 \(\frac{\pi }{2}\).

This is a preview of subscription content, access via your institution.


  1. 1.

    W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander. Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, vol. 96. Birkhäuser (2001).

  2. 2.

    R. A. Adams. Sobolev Spaces, Academic Press, New York-London (1975).

    Google Scholar 

  3. 3.

    S. Agmon. On th eigenfunctions and the eigenvalues of general boundary value problems. Comm. Pure Appl. Math. 25 (1962).

  4. 4.

    H. Amann. Linear and Quasilinear Parabolic Problems, vol. 1. Birkhäuser (2001).

  5. 5.

    W. Arendt. Resolvent positive operators and inhomogeneous boundary value problems. Ann. Scuola Norm. Sup. Pisa 24.70 (2000), 639–670.

    MATH  Google Scholar 

  6. 6.

    T. Binz and K. Engel Operators with Wentzell boundary conditions and the Dirichlet-to-Neumann Operator. Math. Nachr. (to appear 2018).

  7. 7.

    T. Binz Strictly elliptic operators with Wentzell boundary conditions on spaces of continuous functions on manifolds. (preprint 2018).

  8. 8.

    F. Browder. On the spectral theory of elliptic differential operators I. Math. Ann. 142.1 (1961), 22–130.

    MathSciNet  Article  Google Scholar 

  9. 9.

    M. Campiti and G. Metafune. Ventcel’s boundary conditions and analytic semigroups. Arch. Math. 70 (1998), 377–390.

    MathSciNet  Article  Google Scholar 

  10. 10.

    K.-J. Engel and G. Fragnelli. Analyticity of semigroups generated by operators with generalized Wentzell boundary conditions. Adv. Differential Equations 10 (2005), 1301–1320.

    MathSciNet  MATH  Google Scholar 

  11. 11.

    K.-J. Engel and R. Nagel. One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., vol. 194. Springer (2000).

  12. 12.

    K.-J. Engel. The Laplacian on\(C(\overline{\Omega })\)with generalized Wentzell boundary conditions. Arch. Math. 81 (2003), 548–558.

    MathSciNet  Article  Google Scholar 

  13. 13.

    L. C. Evans. Partial Differential Equations, Graduate Studies in Mathematics., vol. 19. Amer. Math. Soc. (1998).

  14. 14.

    A. Favini, G. R. Goldstein, J. A. Goldstein, E. Obrecht, and S. Romanelli. Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem. Math. Nachr. 283 (2010), 504–521.

    MathSciNet  Article  Google Scholar 

  15. 15.

    A. Favini, G. R. Goldstein, J. A. Goldstein, and S. Romanelli. The heat equation with generalized Wentzell boundary condition. J. Evol. Equ. 2 (2002), 1–19.

    MathSciNet  Article  Google Scholar 

  16. 16.

    Gilbarg, D. and Trudinger, N. S. Elliptic partial differential equations of second order, Classics in Mathematics. Springer (2001).

  17. 17.

    E. Hebey. Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes. Amer. Math. Soc. (2000).

  18. 18.

    A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser (1995).

  19. 19.

    M. Rudin. Real and Complex Analysis, Higher Mathematics Series, vol 3. McGraw-Hill (1986).

  20. 20.

    M. Renardy and R. C. Rogers. An Introduction to Partial Differential Equations, Texts in Appl. Math., vol 13. Springer (1993).

  21. 21.

    R. T. Seeley. Extenstion of\({\rm C}^\infty \)functions defined in a half space. Proc. Amer. Math. Soc. 15 (1964), 625–626.

    MathSciNet  MATH  Google Scholar 

  22. 22.

    B. Stewart. Generation of analytic semigroups by strongly elliptic operators. Trans. Amer. Math. Soc. 199 (1974), 141–161.

    MathSciNet  Article  Google Scholar 

  23. 23.

    G. N. Watson A Treatise on the Theory of Bessel Functions, Cambridge University Press (1995).

Download references


The author wishes to thank Professor Simon Brendle and Professor Klaus Engel for important suggestions and fruitful discussions. Moreover the author thanks the referee for his many helpful comments.

Author information



Corresponding author

Correspondence to Tim Binz.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A. Bessel functions

Appendix A. Bessel functions

The solutions of the ordinary differential equation

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

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 \(z \in \Sigma _{\frac{\pi }{2}-\eta }\).


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 \(|\lambda |\ge 1\).


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

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


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

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 \(|\lambda |\ge 1\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Binz, T. Strictly elliptic operators with Dirichlet boundary conditions on spaces of continuous functions on manifolds. J. Evol. Equ. 20, 1005–1028 (2020).

Download citation


  • Dirichlet boundary conditions
  • Analytic semigroup
  • Riemmanian manifolds

Mathematics Subject Classification

  • 47D06
  • 34G10
  • 47E05
  • 47F05