Proceedings of the American Mathematical Society

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

Author(s): Frances Hammock; Peter Luthy; Alexander M. Meadows; Phillip Whitman
Journal: Proc. Amer. Math. Soc. 135 (2007), 1419-1430.
MSC (2000): Primary 35J60, 26B05
Posted: October 27, 2006
Retrieve article in: PDF DVI PostScript

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.

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 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.: 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.: Gilbarg, David and Trudinger, Neil S.,
Elliptic Partial Differential Equations of Second Order,
Springer, New York, revised 2nd editon, 2001. MR 1814364 (2001k:35004)
7.: Meadows, Alexander,
``Stable and singular solutions of the equation $\Delta u =\frac{1}{u}$ ,''
Indiana Univ. Math. J., 2005. MR 2106341 (2005i:35094)
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.

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

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: 10.1090/S0002-9939-06-08618-7
PII: S 0002-9939(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
Posted: 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
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS