Tornado solutions for semilinear elliptic equations in : 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 on domains of are continuous despite a possible infinite singularity of . 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.

**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.**Hammock, Frances; Luthy, Peter; Meadows, Alexander and Whitman, Phillip,

``Tornado solutions for semilinear elliptic equations in : Applications,''*Proc. Amer. Math. Soc.*, this issue.**6.**Meadows, Alexander,

``Stable and singular solutions of the equation ,''*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)**

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.