Journal of the Mathematical Society of Japan

Analytic semigroups for the subelliptic oblique derivative problem

Kazuaki TAIRA

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper is devoted to a functional analytic approach to the subelliptic oblique derivative problem for second-order, elliptic differential operators with a complex parameter $\lambda$. We prove an existence and uniqueness theorem of the homogeneous oblique derivative problem in the framework of $L^{p}$ Sobolev spaces when $\vert\lambda\vert$ tends to $\infty$. As an application of the main theorem, we prove generation theorems of analytic semigroups for this subelliptic oblique derivative problem in the $L^{p}$ topology and in the topology of uniform convergence. Moreover, we solve the long-standing open problem of the asymptotic eigenvalue distribution for the subelliptic oblique derivative problem. In this paper we make use of Agmon's technique of treating a spectral parameter $\lambda$ as a second-order elliptic differential operator of an extra variable on the unit circle and relating the old problem to a new one with the additional variable.

Article information

J. Math. Soc. Japan, Volume 69, Number 3 (2017), 1281-1330.

First available in Project Euclid: 12 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 35S05: Pseudodifferential operators 47D03: Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20} 35P20: Asymptotic distribution of eigenvalues and eigenfunctions

oblique derivative problem subelliptic operator analytic semigroup asymptotic eigenvalue distribution Agmon's method


TAIRA, Kazuaki. Analytic semigroups for the subelliptic oblique derivative problem. J. Math. Soc. Japan 69 (2017), no. 3, 1281--1330. doi:10.2969/jmsj/06931281.

Export citation


  • R. A. Adams and J. J. F. Fournier, Sobolev spaces, Second edition, Pure and Applied Math., Elsevier/Academic Press, Amsterdam, 2003.
  • S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand, Princeton, New Jersey, 1965.
  • S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math., 12 (1959), 623–727.
  • M. S. Agranovich and M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Uspehi Mat. Nauk, 19(117), 1964, 53–161 (in Russian); English translation: Russian Math. Surveys, 19 (1964), 53–157.
  • S. Axler, P. Bourdon and W. Ramey, Harmonic function theory, Second edition, Graduate Texts in Math., No. 137, Springer-Verlag, New York, 2001.
  • J. Bergh and J. Löfström, Interpolation spaces, An introduction, Springer-Verlag, Berlin Heidelberg New York, 1976.
  • A. Bjerhammar and L. Svensson, On the geodetic boundary-value problem for a fixed boundary surface – satellite approach, Bull. Géod., 57 (1983), 382–393.
  • L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math., 126 (1971), 11–51.
  • A. P. Calderón, Boundary value problems for elliptic equations, In: Outlines Joint Sympos. Partial Differential Equations (Novosibirsk, 1963), 303–304. Moscow: Acad. Sci. USSR Siberian Branch.
  • J. Chazarain and A. Piriou, Introduction à la théorie des équations aux dérivées partielles linéaires, Gauthier-Villars, Paris, 1981.
  • R. Čunderlík, K. Mikula and M. Mojzeš, Numerical solution of the linearized fixed gravimetric boundary-value problem, J. Geod., 82 (2008), 15–29.
  • Ju. V. Egorov, Subelliptic operators, Uspekhi Mat. Nauk 30:2 (182) (1975), 57–114, 30:3 (183) (1975), 57–104 (in Russian); English translation: Russian Math. Surveys, 30:2 (1975), 59–118, 30:3 (1975), 55–105.
  • Ju. V. Egorov and V. A. Kondratev, The oblique derivative problem, Math. USSR Sb., 7 (1969), 139–169.
  • A. Friedman, Partial differential equations, Holt, Rinehart and Winston Inc., New York, New York, 1969.
  • D. Fujiwara, On some homogeneous boundary value problems bounded below, J. Fac. Sci. Univ. Tokyo Sec. IA, 17 (1970), 123–152.
  • D. Fujiwara and K. Uchiyama, On some dissipative boundary value problems for the Laplacian, J. Math. Soc. Japan, 23 (1971), 625–635.
  • D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Reprint of the 1998 edition, Classics in Math., Springer-Verlag, Berlin, 2001.
  • P. Guan and E. Sawyer, Regularity estimates for the oblique derivative problem, Ann. of Math., 137 (1993), 1–70.
  • P. Holota, Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation, J. Geod., 71 (1997), 640–651.
  • L. Hörmander, Pseudodifferential operators and non-elliptic boundary problems, Ann. of Math., 839 (1966), 129–209.
  • L. Hörmander, Pseudo-differential operators and hypoelliptic equations. In: Proc. Sympos. Pure Math., X, Singular integrals, (ed. A. P. Calderón), 138–183. Amer. Math. Soc., Providence, Rhode Island, 1967.
  • L. Hörmander, Subelliptic operators. In: Seminar on singularities of solutions of linear partial differential equations, Ann. of Math. Stud., 91 (1979), 127–208.
  • K. R. Koch and A. J. Pope, Uniqueness and existence for the geodetic boundary-value problem using the known surface of the Earth, Bull. Géod., 46 (1972), 467–476.
  • H. Kumano-go, Pseudodifferential operators, MIT Press, Cambridge, Massachusetts, 1981.
  • J.-L. Lions and E. Magenes, Problèmes aux limites non-homogènes et applications, 1, 2, Dunod, Paris, 1968; English translation: Non-homogeneous boundary value problems and applications, 1, 2, Springer-Verlag, Berlin Heidelberg New York, 1972.
  • K. Masuda, Evolution equations (in Japanese), Kinokuniya-Shoten, Tokyo, 1975.
  • A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
  • S. Rempel and B.-W. Schulze, Index theory of elliptic boundary problems, Akademie-Verlag, Berlin, 1982.
  • R. T. Seeley, Extension of $C^{\infty}$ functions defined in a half-space, Proc. Amer. Math. Soc., 15 (1964), 625–626.
  • R. T. Seeley, Singular integrals and boundary value problems, Amer. J. Math., 88 (1966), 781–809.
  • H. Smith, The subelliptic oblique derivative problem, Comm. Partial Differential Equations, 15 (1990), 97–137.
  • E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Ser. Princeton University Press, Princeton, New Jersey, 1970.
  • H. B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc., 259 (1980), 299–310.
  • K. Taira, On some degenerate oblique derivative problems, J. Fac. Sci. Univ. Tokyo Sec. IA, 23 (1976), 259–287.
  • K. Taira, Un théorème d'existence et d'unicité des solutions pour des problèmes aux limites non-elliptiques, J. Funct. Anal., 43 (1981), 166–192.
  • K. Taira, Diffusion processes and partial differential equations, Academic Press, San Diego New York London Tokyo, 1988.
  • K. Taira, Boundary value problems and Markov processes, Second edition, Lecture Notes in Math., 1499, Springer-Verlag, Berlin, 2009.
  • K. Taira, Semigroups, boundary value problems and Markov processes, Second edition, Springer Monographs in Math., Springer-Verlag, Heidelberg, 2014.
  • K. Taira, Analytic semigroups and semilinear initial-boundary value problems, Second edition, London Math. Soc. Lecture Note Series, 434, Cambridge University Press, Cambridge, 2016.
  • K. Taira, Spectral analysis of the subelliptic oblique derivative problem, to appear in Arkiv för Matematik.
  • M. Taylor, Pseudodifferential operators, Princeton Mathematical Series, 34, Princeton University Press, Princeton, New Jersey, 1981.
  • F. Treves, A new method of the subelliptic estimates, Comm. Pure Appl. Math., 24 (1971), 71–115.
  • H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, 1978.
  • J. Wloka, Partial differential equations, Cambridge University Press, Cambridge, 1987.
  • K. Yosida, Functional analysis, reprint of the sixth (1980) edition, Classics in Math., Springer-Verlag, Berlin, 1995.