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Proceedings of the American Mathematical Society
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
Published electronically: October 27, 2006
MathSciNet review: 2276651
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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.


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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: http://dx.doi.org/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
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.