Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Tornado solutions for semilinear elliptic equations in $ \mathbb{R}^2$: regularity


Author: Alexander M. Meadows
Journal: Proc. Amer. Math. Soc. 135 (2007), 1411-1417
MSC (2000): Primary 35J60, 26B05
DOI: https://doi.org/10.1090/S0002-9939-06-08617-5
Published electronically: October 27, 2006
MathSciNet review: 2276650
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give conditions under which bounded solutions to semilinear elliptic equations $ \Delta u = f(u)$ on domains of $ \mathbb{R}^2$ are continuous despite a possible infinite singularity of $ f(u)$. The conditions do not require a minimization or variational stability property for the solutions. The results are used in a second paper to show regularity for a familiar class of equations.


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

  • 1. Brauner, C.M. and Nicolaenko, B.,
    ``On nonlinear eigenvalue problems which extend into free boundaries problems,''
    Bifurcation and nonlinear eigenvalue problems (Proc., Session, Univ. Paris XIII, Villetaneuse, 1978), pp. 61-100, Lecture Notes in Math., 782, Springer, Berlin, New York, 1980. MR 0572251 (83e:35127)
  • 2. Caffarelli, Luis and Salsa, Sandro,
    A geometric approach to free boundary problems, Graduate Studies in Mathematics, 68, Amer. Math. Soc., 2005.MR 2145284
  • 3. Dávila, Juan,
    ``Global regularity for a singular equation and local $ H^1$ minimizers of a nondifferentiable functional,''
    Commun. Contemp. Math., vol. 6, no. 1, pp. 165-193, 2004. MR 2048779 (2005e:35061)
  • 4. Dávila, Juan and Montenegro, Marcelo,
    ``Positive versus free boundary solutions to a singular elliptic equation,''
    J. Anal. Math., vol. 90, pp. 303-335, 2003. MR 2001074 (2005d:35278)
  • 5. Hammock, Frances; Luthy, Peter; Meadows, Alexander and Whitman, Phillip,
    ``Tornado solutions for semilinear elliptic equations in $ {\mathbb{R}}^2$: Applications,''
    Proc. Amer. Math. Soc., this issue.
  • 6. Meadows, Alexander,
    ``Stable and singular solutions of the equation $ \Delta u =\frac{1}{u}$,''
    Indiana Univ. Math. J., vol. 53, no. 6, pp. 1681-1703, 2004. MR 2106341 (2005i:35094)
  • 7. Phillips, Daniel,
    ``A minimization problem and the regularity of solutions in the presence of a free boundary,''
    Indiana Univ. Math. J., vol. 32, no. 1, pp. 1-17, 1983. MR 0684751 (84e:49012)
  • 8. Simon, Leon,
    ``The minimal surface equation,''
    Geometry V, Springer, Berlin, pp. 239-272, 1997. MR 1490041 (99b:53014)
  • 9. Struwe, Michael,
    Plateau's Problem and the Calculus of Variations, Math. Notes 35, Princeton University Press, 1989.MR 0992402 (90h:58016)
  • 10. Taylor, Michael,
    Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, 81, Amer. Math. Soc., 2000.MR 1766415 (2001g:35004)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35J60, 26B05

Retrieve articles in all journals with MSC (2000): 35J60, 26B05


Additional Information

Alexander M. Meadows
Affiliation: Department of Mathematics and Computer Science, St. Mary’s College of Maryland, St. Mary’s City, Maryland 20686
Email: ammeadows@smcm.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08617-5
Keywords: Semilinear elliptic equations, regularity theory, singular solutions
Received by editor(s): September 11, 2005
Received by editor(s) in revised form: December 5, 2005
Published electronically: October 27, 2006
Additional Notes: This work was partially supported by NSF grants DMS-9983660 and DMS-0306495 at Cornell University
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society