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 \(\frac{\pi }{2}\).
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References
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).
R. A. Adams. Sobolev Spaces, Academic Press, New York-London (1975).
S. Agmon. On th eigenfunctions and the eigenvalues of general boundary value problems. Comm. Pure Appl. Math. 25 (1962).
H. Amann. Linear and Quasilinear Parabolic Problems, vol. 1. Birkhäuser (2001).
W. Arendt. Resolvent positive operators and inhomogeneous boundary value problems. Ann. Scuola Norm. Sup. Pisa 24.70 (2000), 639–670.
T. Binz and K. Engel Operators with Wentzell boundary conditions and the Dirichlet-to-Neumann Operator. Math. Nachr. (to appear 2018).
T. Binz Strictly elliptic operators with Wentzell boundary conditions on spaces of continuous functions on manifolds. (preprint 2018).
F. Browder. On the spectral theory of elliptic differential operators I. Math. Ann. 142.1 (1961), 22–130.
M. Campiti and G. Metafune. Ventcel’s boundary conditions and analytic semigroups. Arch. Math. 70 (1998), 377–390.
K.-J. Engel and G. Fragnelli. Analyticity of semigroups generated by operators with generalized Wentzell boundary conditions. Adv. Differential Equations 10 (2005), 1301–1320.
K.-J. Engel and R. Nagel. One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., vol. 194. Springer (2000).
K.-J. Engel. The Laplacian on\(C(\overline{\Omega })\)with generalized Wentzell boundary conditions. Arch. Math. 81 (2003), 548–558.
L. C. Evans. Partial Differential Equations, Graduate Studies in Mathematics., vol. 19. Amer. Math. Soc. (1998).
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.
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.
Gilbarg, D. and Trudinger, N. S. Elliptic partial differential equations of second order, Classics in Mathematics. Springer (2001).
E. Hebey. Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes. Amer. Math. Soc. (2000).
A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser (1995).
M. Rudin. Real and Complex Analysis, Higher Mathematics Series, vol 3. McGraw-Hill (1986).
M. Renardy and R. C. Rogers. An Introduction to Partial Differential Equations, Texts in Appl. Math., vol 13. Springer (1993).
R. T. Seeley. Extenstion of\({\rm C}^\infty \)functions defined in a half space. Proc. Amer. Math. Soc. 15 (1964), 625–626.
B. Stewart. Generation of analytic semigroups by strongly elliptic operators. Trans. Amer. Math. Soc. 199 (1974), 141–161.
G. N. Watson A Treatise on the Theory of Bessel Functions, Cambridge University Press (1995).
Acknowledgements
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.
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Appendix A. Bessel functions
Appendix A. Bessel functions
The solutions of the ordinary differential equation
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
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
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
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
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
Note that \(z = |z|e^{i\varphi }\) with \(|\varphi | \in [0,\nicefrac {\pi }{2}-\eta )\). The monotony of the cosinus implies
Using the monotony of the exponential function and the positivity of \(\cosh \), we conclude
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
for \(|\lambda |\ge 1\).
Proof
Remark that
For the first term, one obtains
Since
for small \(r \in \mathbb {R}_+\) and
for large \(r \in \mathbb {R}_+\), we have
for small \(r \in \mathbb {R}_+\) and
for large \(r \in \mathbb {R}_+\). Hence, there exists a constant \(\bar{C} < \infty \) such that
and we conclude that
If \(y \in \overline{M} {\setminus } B_R(x)\), we have \(\rho (x,y) \ge R\) and therefore
for \(|\lambda |\) since
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
for \(|\lambda |\ge 1\).
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Binz, T. Strictly elliptic operators with Dirichlet boundary conditions on spaces of continuous functions on manifolds. J. Evol. Equ. 20, 1005–1028 (2020). https://doi.org/10.1007/s00028-019-00548-y
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DOI: https://doi.org/10.1007/s00028-019-00548-y