Tornado solutions for semilinear elliptic equations in : 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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show partial regularity of bounded positive solutions of some semilinear elliptic equations in domains of . 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.**C.-M. Brauner and B. Nicolaenko,*On nonlinear eigenvalue problems which extend into free boundaries problems*, Bifurcation and nonlinear eigenvalue problems (Proc., Session, Univ. Paris XIII, Villetaneuse, 1978) Lecture Notes in Math., vol. 782, Springer, Berlin-New York, 1980, pp. 61–100. MR**572251****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**, 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**, 10.1007/BF02786560**5.**J. I. Diaz, J.-M. Morel, and L. Oswald,*An elliptic equation with singular nonlinearity*, Comm. Partial Differential Equations**12**(1987), no. 12, 1333–1344. MR**912208**, 10.1080/03605308708820531**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**, 10.1512/iumj.2004.53.2560**8.**Meadows, Alexander,

``Tornado solutions for semilinear elliptic equations in : regularity,''*Proc. Amer. Math. Soc.*, this issue.**9.**Daniel Phillips,*A minimization problem and the regularity of solutions in the presence of a free boundary*, Indiana Univ. Math. J.**32**(1983), no. 1, 1–17. MR**684751**, 10.1512/iumj.1983.32.32001**10.**Daniel Phillips,*Hausdorff measure estimates of a free boundary for a minimum problem*, Comm. Partial Differential Equations**8**(1983), no. 13, 1409–1454. MR**714047**, 10.1080/03605308308820309**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

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.