Degenerate Sobolev spaces and regularity of subelliptic equations
Authors:
Eric T. Sawyer and Richard L. Wheeden
Journal:
Trans. Amer. Math. Soc. 362 (2010), 18691906
MSC (2000):
Primary 35B65, 35D10, 35H20, 46E35
Published electronically:
October 30, 2009
MathSciNet review:
2574880
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We develop a notion of degenerate Sobolev spaces naturally associated with nonnegative quadratic forms that arise from a large class of linear subelliptic equations with rough coefficients. These Sobolev spaces allow us to make the widest possible definition of a weak solution that leads to local Hölder continuity of solutions, extending our results in an earlier work, where we studied regularity of classical weak solutions. In cases when the quadratic forms arise from collections of rough vector fields, we study containment relations between the degenerate Sobolev spaces and the corresponding spaces defined in terms of weak derivatives relative to the vector fields.
 1.
Richard
Bouldin, The norm continuity properties of square roots, SIAM
J. Math. Anal. 3 (1972), 206–210. MR 0310673
(46 #9771)
 2.
Eugene
B. Fabes, Carlos
E. Kenig, and Raul
P. Serapioni, The local regularity of solutions of degenerate
elliptic equations, Comm. Partial Differential Equations
7 (1982), no. 1, 77–116. MR 643158
(84i:35070), http://dx.doi.org/10.1080/03605308208820218
 3.
Bruno
Franchi, Raul
Serapioni, and Francesco
Serra Cassano, MeyersSerrin type theorems and relaxation of
variational integrals depending on vector fields, Houston J. Math.
22 (1996), no. 4, 859–890. MR 1437714
(98c:49037)
 4.
Nicola
Garofalo and DuyMinh
Nhieu, Isoperimetric and Sobolev inequalities for
CarnotCarathéodory spaces and the existence of minimal
surfaces, Comm. Pure Appl. Math. 49 (1996),
no. 10, 1081–1144. MR 1404326
(97i:58032), http://dx.doi.org/10.1002/(SICI)10970312(199610)49:10<1081::AIDCPA3>3.0.CO;2A
 5.
David
Gilbarg and Neil
S. Trudinger, Elliptic partial differential equations of second
order, Classics in Mathematics, SpringerVerlag, Berlin, 2001. Reprint
of the 1998 edition. MR 1814364
(2001k:35004)
 6.
Norman
G. Meyers and James
Serrin, 𝐻=𝑊, Proc. Nat. Acad. Sci. U.S.A.
51 (1964), 1055–1056. MR 0164252
(29 #1551)
 7.
Cristian
Rios, Eric
T. Sawyer, and Richard
L. Wheeden, A higherdimensional partial Legendre transform, and
regularity of degenerate MongeAmpère equations, Adv. Math.
193 (2005), no. 2, 373–415. MR 2137289
(2006f:35099), http://dx.doi.org/10.1016/j.aim.2004.05.009
 8.
Cristian
Rios, Eric
T. Sawyer, and Richard
L. Wheeden, Regularity of subelliptic MongeAmpère
equations, Adv. Math. 217 (2008), no. 3,
967–1026. MR 2383892
(2009k:35086), http://dx.doi.org/10.1016/j.aim.2007.07.004
 9.
C. RIOS, E. SAWYER AND R. L. WHEEDEN, Hypoellipticity for infinitely degenerate quasilinear equations and the Dirichlet problem, in preparation.
 10.
S. RODNEY, Existence of weak solutions to subelliptic partial differential equations in divergence form and the necessity of the Sobolev and Poincaré inequalities, thesis.
 11.
Eric
T. Sawyer and Richard
L. Wheeden, Hölder continuity of weak solutions to subelliptic
equations with rough coefficients, Mem. Amer. Math. Soc.
180 (2006), no. 847, x+157. MR 2204824
(2007f:35037), http://dx.doi.org/10.1090/memo/0847
 1.
 R. BOULDIN, The norm continuity properties of square roots, SIAM J. Math. Anal. 4 (1972). MR 0310673 (46:9771)
 2.
 E. FABES, C. KENIG AND R. SERAPIONI, The local regularity of solutions of degenerate elliptic equations, Comm. P.D.E. 7 (1982), 77116. MR 643158 (84i:35070)
 3.
 B. FRANCHI, R. SERAPIONI AND F. SERRA CASSANO, MeyersSerrin type theorems and relaxation of variational integrals depending on vector fields, Houston J. Math. 22 (1996), 859889. MR 1437714 (98c:49037)
 4.
 N. GAROFALO AND D. M. NHIEU, Isoperimetric and Sobolev inequalities for CarnotCarathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996),10811144. MR 1404326 (97i:58032)
 5.
 D. GILBARG AND N. TRUDINGER, Elliptic Partial Differential Equations of Second Order,SpringerVerlag, Berlin, 2001. MR 1814364 (2001k:35004)
 6.
 N. MEYERS AND J. SERRIN, , Proc. Nat. Acad. Sci. U.S.A. 51 (1964),10551056. MR 0164252 (29:1551)
 7.
 C. RIOS, E. SAWYER AND R. L. WHEEDEN, A higherdimensional partial Legendre transform, and regularity of degenerate MongeAmpère equations, Advances in Mathematics 193 (2005), 373415. MR 2137289 (2006f:35099)
 8.
 C. RIOS, E. SAWYER AND R. L. WHEEDEN, Regularity of subelliptic MongeAmpère equations, Advances in Mathematics 217 (2008), 9671026. MR 2383892
 9.
 C. RIOS, E. SAWYER AND R. L. WHEEDEN, Hypoellipticity for infinitely degenerate quasilinear equations and the Dirichlet problem, in preparation.
 10.
 S. RODNEY, Existence of weak solutions to subelliptic partial differential equations in divergence form and the necessity of the Sobolev and Poincaré inequalities, thesis.
 11.
 E. SAWYER AND R. L. WHEEDEN, Hölder continuity of subelliptic equations with rough coefficients, Memoirs Amer. Math. Soc. 847 (2006). MR 2204824 (2007f:35037)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
35B65,
35D10,
35H20,
46E35
Retrieve articles in all journals
with MSC (2000):
35B65,
35D10,
35H20,
46E35
Additional Information
Eric T. Sawyer
Affiliation:
Department of Mathematics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Richard L. Wheeden
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
DOI:
http://dx.doi.org/10.1090/S0002994709047564
PII:
S 00029947(09)047564
Received by editor(s):
September 6, 2007
Published electronically:
October 30, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
