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$: Applications


Authors: Frances Hammock, Peter Luthy, Alexander M. Meadows and Phillip Whitman
Journal: Proc. Amer. Math. Soc. 135 (2007), 1419-1430
MSC (2000): Primary 35J60, 26B05
DOI: https://doi.org/10.1090/S0002-9939-06-08618-7
Published electronically: October 27, 2006
MathSciNet review: 2276651
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show partial regularity of bounded positive solutions of some semilinear elliptic equations $ \Delta u = f(u)$ in domains of $ \mathbb{R}^2$. As a consequence, there exists a large variety of nonnegative singular solutions to these equations. These equations have previously been studied from the point of view of free boundary problems, where solutions additionally are stable for a variational problem, which we do not assume.


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. Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, Graduate Studies in Mathematics, vol. 68, American Mathematical Society, Providence, RI, 2005. MR 2145284
  • 3. Juan Dávila, Global regularity for a singular equation and local 𝐻¹ minimizers of a nondifferentiable functional, Commun. Contemp. Math. 6 (2004), no. 1, 165–193. MR 2048779, https://doi.org/10.1142/S0219199704001240
  • 4. Juan Dávila and Marcelo Montenegro, Positive versus free boundary solutions to a singular elliptic equation, J. Anal. Math. 90 (2003), 303–335. MR 2001074, https://doi.org/10.1007/BF02786560
  • 5. Diaz, J.I., Morel, J.M. and Oswald, L.,
    ``An elliptic equation with singular nonlinearity,''
    Comm. P.D.E., vol. 12, pp. 1333-1344, 1987. MR 0912208 (89m:35077)
  • 6. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
  • 7. Alexander M. Meadows, Stable and singular solutions of the equation Δ𝑢=1/𝑢, Indiana Univ. Math. J. 53 (2004), no. 6, 1681–1703. MR 2106341, https://doi.org/10.1512/iumj.2004.53.2560
  • 8. Meadows, Alexander,
    ``Tornado solutions for semilinear elliptic equations in $ \mathbb{R}^2$: regularity,''
    Proc. Amer. Math. Soc., this issue.
  • 9. 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)
  • 10. Phillips, Daniel,
    ``Hausdorff measure estimates of a free boundary for a minimum problem,''
    Comm. P.D.E., vol. 8, pp. 1409-1454, 1983. MR 0714047 (85b:35068)
  • 11. Simon, Leon,
    ``Some examples of singular minimal hypersurfaces,''
    in preparation, 2001.

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

Frances Hammock
Affiliation: Department of Mathematics, UCLA, Los Angeles, California 90095-1555
Email: hammockf@math.ucla.edu

Peter Luthy
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14850
Email: pmlut@math.cornell.edu

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

Phillip Whitman
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08540
Email: pwhitman@math.princeton.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08618-7
Keywords: Semilinear elliptic equations, regularity theory, singular solutions, free boundary problems
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 REU grant DMS-0139229 at Cornell University
The third author 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