Journal of the Mathematical Society of Japan

Diffusion with nonlocal Robin boundary conditions

Wolfgang ARENDT, Stefan KUNKEL, and Markus KUNZE

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

Abstract

We investigate a second order elliptic differential operator $A_{\beta, \mu}$ on a bounded, open set $\Omega\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $0\leq \beta\in L^{\infty}(\partial\Omega)$ and $\mu\colon\partial\Omega\to{\mathscr{M}}(\overline{\Omega})$, and boundary conditions of the form $$ \partial_{\nu}^{{\mathscr{A}}}u(z)+\beta(z)u(z)=\int_{\overline{\Omega}}u(x)\mu(z)(\mathrm{d}x), \quad z\in\partial\Omega, $$ where $\partial_{\nu}^{{\mathscr{A}}}$ denotes the weak conormal derivative with respect to our differential operator. Under suitable conditions on the coefficients of the differential operator and the function $\mu$ we show that $A_{\beta, \mu}$ generates a holomorphic semigroup $T_{\beta,\mu}$ on $L^{\infty}(\Omega)$ which enjoys the strong Feller property. In particular, it takes values in $C(\overline{\Omega})$. Its restriction to $C(\overline{\Omega})$ is strongly continuous and holomorphic. We also establish positivity and contractivity of the semigroup under additional assumptions and study the asymptotic behavior of the semigroup.

Article information

Source
J. Math. Soc. Japan, Volume 70, Number 4 (2018), 1523-1556.

Dates
Received: 24 October 2016
Revised: 23 April 2017
First available in Project Euclid: 3 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1538553644

Digital Object Identifier
doi:10.2969/jmsj/76427642

Mathematical Reviews number (MathSciNet)
MR3868216

Zentralblatt MATH identifier
07009711

Subjects
Primary: 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 35B35: Stability

Keywords
diffusion process non-local boundary condition stability

Citation

ARENDT, Wolfgang; KUNKEL, Stefan; KUNZE, Markus. Diffusion with nonlocal Robin boundary conditions. J. Math. Soc. Japan 70 (2018), no. 4, 1523--1556. doi:10.2969/jmsj/76427642. https://projecteuclid.org/euclid.jmsj/1538553644


Export citation

References

  • M. S. Agranovich, Sobolev spaces, their generalizations and elliptic problems in smooth and Lipschitz domains, Revised translation of the 2013 Russian original, Springer Monographs in Math., Springer, Cham, 2015.
  • W. Arendt, Gaussian estimates and interpolation of the spectrum in $L^p$, Differential Integral Equations, 7 (1994), 1153–1168.
  • W. Arendt, Semigroups and evolution equations: Functional calculus, regularity and kernel estimates, vol. 1 of Handbook of Differential Equations: Evolutionary Equations, North-Holland, 2002, 1–85.
  • W. Arendt, Heat kernels, lecture notes of the 9th internet seminar, 2005.
  • W. Arendt and C. J. K. Batty, Domination and ergodicity for positive semigroups, Proc. Amer. Math. Soc., 114 (1992), 743–747.
  • W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, 96, Birkhäuser/Springer Basel AG, Basel, second ed., 2011.
  • W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-parameter semigroups of positive operators, Lecture Notes in Math., 1184, Springer-Verlag, Berlin, 1986.
  • W. Arendt, S. Kunkel and M. Kunze, Diffusion with nonlocal boundary conditions, J. Funct. Anal., 270 (2016), 2483–2507.
  • W. Arendt and R. Mazzeo, Spectral properties of the dirichlet-to-neumann operator, Ulmer Seminare, 12 (2007), 23–37.
  • W. Arendt and A. F. M. ter Elst, Gaussian estimates for second order elliptic operators with boundary conditions, J. Operator Theory, 38 (1997), 87–130.
  • W. Arendt and A. F. M. ter Elst, The Dirichlet-to-Neumann operator on $C(\partial\Omega)$, preprint, 2017.
  • V. I. Bogachev, Measure theory, Vol. I, II, Springer-Verlag, Berlin, 2007.
  • D. M. Bošković, M. Krstić and W. Liu, Boundary control of an unstable heat equation via measurement of domain-averaged temperature, IEEE Trans. Automat. Control, 46 (2001), 2022–2028.
  • D. Daners, Heat kernel estimates for operators with boundary conditions, Math. Nachr., 217 (2000), 13–41.
  • D. Daners, Inverse positivity for general Robin problems on Lipschitz domains, Arch. Math. (Basel), 92 (2009), 57–69.
  • B. de Pagter, Irreducible compact operators, Math. Z., 192 (1986), 149–153.
  • D. Dier, Non-autonomous forms and invariance, preprint, available at arXiv:1609.03857, 2016.
  • K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Math., 194, Springer-Verlag, New York, 2000.
  • W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2), 55 (1952), 468–519.
  • W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1–31.
  • W. Feller, Generalized second order differential operators and their lateral conditions, Illinois J. Math., 1 (1957), 459–504.
  • E. I. Galakhov and A. L. Skubachevskiĭ, On Feller semigroups generated by elliptic operators with integro-differential boundary conditions, J. Differential Equations, 176 (2001), 315–355.
  • G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213–229.
  • P. L. Gurevich, W. Jäger and A. L. Skubachevskiĭ, On the existence of periodic solutions of some nonlinear problems of thermal control, Dokl. Akad. Nauk, 418 (2008), 151–154.
  • M. Kunze, A Pettis-type integral and applications to transition semigroups, Czechoslovak Math. J., 61 (2011), 437–459.
  • M. Kunze, Perturbation of strong Feller semigroups and well-posedness of semilinear stochastic equations on Banach spaces, Stochastics, 85 (2013), 960–986.
  • H. P. Lotz, Uniform convergence of operators on $L^\infty$ and similar spaces, Math. Z., 190 (1985), 207–220.
  • A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995.
  • A. Manavi, H. Vogt and J. Voigt, Domination of semigroups associated with sectorial forms, J. Operator Theory, 54 (2005), 9–25.
  • R. Nittka, Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains, J. Differential Equations, 251 (2011), 860–880.
  • E. M. Ouhabaz, Analysis of heat equations on domains, London Math. Soc. Monographs Series, 31, Princeton University Press, Princeton, NJ, 2005.
  • K. Sato and T. Ueno Multi-dimensional diffusion and the Markov process on the boundary, J. Math. Kyoto Univ., 4 (1964/1965), 529–605.
  • A. L. Skubachevskiĭ, Some problems for multidimensional diffusion processes, Dokl. Akad. Nauk SSSR, 307 (1989), 287–291. translation in Soviet Math. Dokl., 40 (1990), 75–79.
  • A. L. Skubachevskiĭ, Nonlocal elliptic problems and multidimensional diffusion processes, Russian J. Math. Phys., 3 (1995), 327–360.
  • K. Taira, Semigroups, boundary value problems and Markov processes, Springer Monographs in Math., Springer, Heidelberg, second ed., 2014.
  • K. Taira, Analytic semigroups and semilinear initial boundary value problems, London Math. Soc. Lecture Note Series, 434, Cambridge University Press, Cambridge, second ed., 2016.
  • A. D. Venttsel', On boundary conditions for multi-dimensional diffusion processes, Theor. Probability Appl., 4 (1959), 164–177.