Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Robin boundary value problems on arbitrary domains


Author: Daniel Daners
Journal: Trans. Amer. Math. Soc. 352 (2000), 4207-4236
MSC (2000): Primary 35J25; Secondary 35D10, 35B45
DOI: https://doi.org/10.1090/S0002-9947-00-02444-2
Published electronically: March 21, 2000
MathSciNet review: 1650081
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

We develop a theory of generalised solutions for elliptic boundary value problems subject to Robin boundary conditions on arbitrary domains, which resembles in many ways that of the Dirichlet problem. In particular, we establish $L_p$-$L_q$-estimates which turn out to be the best possible in that framework. We also discuss consequences to the spectrum of Robin boundary value problems. Finally, we apply the theory to parabolic equations.


References [Enhancements On Off] (What's this?)

  • 1. Robert A. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 56:9247
  • 2. W. Arendt, Gaussian estimates and interpolation of the spectrum in $L_p$, Differential and Integral Equations 7 (1994), 1153-1168. MR 95e:47066
  • 3. J. Thomas Beale, Scattering frequencies of resonators, Comm. Pure Appl. Math. 26 (1973), 549-563. MR 50:5217
  • 4. M-H. Bossel, Membranes élastiquement liées inhomogènes ou sur une surface: une nouvelle extension du théorème isopérimérique de Raleigh-Faber-Krahn, Z. Angew. Math. Phys. 39 (1988), 733-742. MR 90a:73103
  • 5. E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations 74 (1988), 120-156. MR 89h:35256
  • 6. E. N. Dancer and D. Daners, A priori bounds for solutions and spectra of Robin boundary value problems, Tech. Report No. 94-33, University of Sydney, 1994.
  • 7. -, Domain perturbation for elliptic equations subject to Robin boundary conditions, J. Differential Equations 138 (1997), 86-132. MR 98e:35017
  • 8. Daniel Daners, Heat kernel estimates for operators with boundary conditions, Math. Nachr., to appear.
  • 9. R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5, Springer, Berlin, 1992. MR 92k:00006
  • 10. E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. MR 90e:35123
  • 11. K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, Chichester, 1990. MR 92j:28008
  • 12. Herbert Federer, Geometric Measure Theory, Springer, Berlin, 1969. MR 41:1976
  • 13. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1983. MR 86c:35035
  • 14. P. Grisvard, Elliptic Equations in Nonsmooth Domains, Pitman, London, 1985. MR 86m:35044
  • 15. M. A. Krasnosel'ski{\u{\i}}\kern.15em, On a theorem of M. Riesz, Soviet Math. Dokl. 1 (1960), 229-231. MR 22:9852
  • 16. Peter D. Lax and Ralph S. Phillips, On the scattering frequencies of the Laplace operator for exterior domains, Comm. Pure Appl. Math. 25 (1972), 85-101. MR 45:5531
  • 17. M. Marcus and V. J. Mizel, Every superposition operator mapping one Sobolev Space into another is continuous, J. Funct. Anal. 33 (1979), 217-229. MR 80h:47039
  • 18. V. G. Maz'ja, Classes of regions and embedding theorems for function spaces, Soviet Math. Dokl. 1 (1960), 882-885. MR 23A:3448
  • 19. -, Zur Theorie Sobolewscher Räume, Teubner-Texte zur Mathematik, Teubner, Leipzig, 1981.
  • 20. -, Sobolev Spaces, Springer, Berlin, 1985. MR 87g:46056
  • 21. R. Nagel et al., One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, Springer, Berlin, 1986. MR 88i:47022
  • 22. Jindrich Necas, Les méthodes directes en théorie des équations elliptiques, Academia, Prague, 1967. MR 37:3168
  • 23. L. E. Payne and H. F. Weinberger, Lower bounds for vibration frequencies of elastically supported membranes and plates, SIAM J. 5 (1957), 171-182. MR 19:1110c
  • 24. A. Pazy, Semigroups of lieear operators and applications to partial differential equations, Springer, New York, 1983. MR 85g:47061
  • 25. Derek W. Robinson, Elliptic Operators on Lie Groups, Oxford University Press, Oxford, 1991. MR 92m:58133
  • 26. H. H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin, 1974. MR 54:11023
  • 27. René P. Sperb, Untere und obere Schranken für den tiefsten Eigenwert der elastisch gestützten Membran, Z. Angew. Math. Phys. 23 (1972), 231-244. MR 47:1355
  • 28. -, Maximum Principles and Their Applications, Academic Press, New York, 1981. MR 84a:35033
  • 29. N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa 27 (1973), 265-308. MR 51:6113
  • 30. N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992. MR 95f:43008
  • 31. William P. Ziemer, Weakly Differentiable Functions, Springer, New York, 1989. MR 91e:46046

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J25, 35D10, 35B45

Retrieve articles in all journals with MSC (2000): 35J25, 35D10, 35B45


Additional Information

Daniel Daners
Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
Email: D.Daners@maths.usyd.edu.au

DOI: https://doi.org/10.1090/S0002-9947-00-02444-2
Received by editor(s): April 5, 1996
Received by editor(s) in revised form: May 19, 1998
Published electronically: March 21, 2000
Additional Notes: Supported by a grant of the Australian Research Council
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society