Abstract
This paper is devoted to a functional analytic approach to the study of the hypoelliptic Robin problem for a second-order, uniformly elliptic differential operator with a complex parameter \(\lambda \), under the probabilistic condition that either the absorption phenomenon or the reflection phenomenon occurs at each point of the boundary. We solve the long-standing open problem of the asymptotic eigenvalue distribution for the homogeneous Robin problem when \(\vert \lambda \vert \) tends to \(\infty \). More precisely, we prove the spectral properties of the closed realization of the uniformly elliptic differential operator, similar to the elliptic (non-degenerate) case. However, in the degenerate case we cannot use Green’s formula to characterize the adjoint operator of the closed realization. Hence, we shift our attention to its resolvent. In the proof, we make use of the Boutet de Monvel calculus in order to study the resolvent and its adjoint in the framework of \(L^{2}\) Sobolev spaces.
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References
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces (Pure and Applied Mathematics), 2nd edn. Elsevier, Amsterdam (2003)
Agmon, S.: Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton (1965)
Bourdaud, G.: \(L^{p}\)-estimates for certain non-regular pseudo-differential operators. Comm. Partial Differ. Equ. 7, 1023–1033 (1982)
Boutet de Monvel, L.: Boundary problems for pseudo-differential operators. Acta Math. 126, 11–51 (1971)
Chazarain, J., Piriou, A.: Introduction à la Théorie des équations aux Dérivées Partielles Linéaires. Gauthier-Villars, Paris (1981)
Fujiwara, D., Uchiyama, K.: On some dissipative boundary value problems for the Laplacian. J. Math. Soc. Jpn. 23, 625–635 (1971)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order (Classics in Mathematics), reprint of the 1998th edn. Springer, New York (2001)
Greiner, P.: An asymptotic expansion for the heat equation. Arch. Ration. Mech. Anal. 41, 163–218 (1971)
Grubb, G.: Functional Calculus of Pseudodifferential Boundary Problems. Progress in Mathematics, vol. 65. Birkhäuser Inc, Boston (1986)
Grubb, G., Kokholm, N.J.: A global calculus of parameter-dependent pseudodifferential boundary problems in \(L_{p}\) Sobolev spaces. Acta Math. 171, 165–229 (1993)
Hörmander, L.: Pseudo-differential operators and hypoelliptic equations. In: Proc. Sym. Pure Math., X, Singular integrals, A.P. Calderón (ed.), 138–183. American Mathematical Society, Providence, Rhode Island (1967)
Hörmander, L.: The analysis of linear partial differential operators III: pseudo-differential operators, reprint of the 1994th edn. Classics in Mathematics. Springer, Berlin, Heidelberg, New York, Tokyo (2007)
Iwasaki, C.: The asymptotic expansion of the fundamental solution for parabolic initial-boundary value problems and its application. Osaka J. Math. 31, 663–728 (1994)
Kannai, Y.: Hypoellipticity of certain degenerate elliptic boundary value problems. Trans. Am. Math. Soc. 217, 311–328 (1976)
Krietenstein, T., Schrohe, E.: Bounded \(H^{\infty }\)-calculus for a degenerate elliptic boundary value problem arXiv:1711.00286 (under review)
Kumano-go, H.: Pseudodifferential Operators. MIT Press, Cambridge (1981)
Lions, J.-L., Magenes. E.: Problèmes aux limites non-homogènes et applications 1, 2. Dunod, Paris (1968)
Mizohata, S.: The Theory of Partial Differential Equations. Cambridge University Press, London (1973)
Munkres, J.R.: Elementary Differential Topology. Annals of Mathematics Studies, No. 54. Princeton University Press, Princeton (1966)
Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations, Corrected Second Printing. Springer, New York (1984)
Rempel, S., Schulze, B.-W.: Index Theory of Elliptic Boundary Problems. Akademie, Berlin (1982)
Schrohe, E.: A short introduction to Boutet de Monvel’s calculus. In: J. Gil, D. Grieser and M. Lesch (eds.) Approaches to Singular Analysis, pp. 85–116, Oper. Theory Adv. Appl., vol. 125, Birkhäuser, Basel (2001)
Seeley, R.T.: Singular integrals and boundary value problems. Am. J. Math. 88, 781–809 (1966)
Seeley, R.T.: Topics in pseudo-differential operators. In: L. Nirenberg (ed.) Pseudo-Differential Operators (C.I.M.E., Stresa, 1968), pp. 167–305. Edizioni Cremonese, Roma (1969). Reprint of the first edition, Springer, Berlin (2010)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, no. 30, Princeton University Press, Princeton (1970)
Taira, K.: On some degenerate oblique derivative problems. J. Fac. Sci. Univ. Tokyo Sect. 23, 259–287 (1976)
Taira, K.: Semigroups, Boundary Value Problems and Markov Processes. Springer Monographs in Mathematics, 2nd edn. Springer, Berlin (2014)
Taira, K.: Analytic Semigroups and Semilinear Initial-boundary Value Problems, 2nd edn. London Mathematical Society Lecture Note Series, vol. 434. Cambridge University Press, Cambridge (2016)
Taira, K.: Spectral analysis of the subelliptic oblique derivative problem. Ark. Mat. 55, 243–270 (2017)
Taylor, M.E.: Pseudodifferential Operators. Princeton Mathematical Series, Vol. 34, Princeton University Press, Princeton (1981)
Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987)
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Dedicated to the memory of Professor Kyuya Masuda (1937–2018).
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Taira, K. Spectral analysis of the hypoelliptic Robin problem. Ann Univ Ferrara 65, 171–199 (2019). https://doi.org/10.1007/s11565-018-0308-4
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DOI: https://doi.org/10.1007/s11565-018-0308-4
Keywords
- Robin problem
- Hypoelliptic operator
- Non self-adjoint eigenvalue problem
- Asymptotic eigenvalue distribution
- Boutet de Monvel calculus