Degenerate Sobolev spaces and regularity of subelliptic equations

Authors:
Eric T. Sawyer and Richard L. Wheeden

Journal:
Trans. Amer. Math. Soc. **362** (2010), 1869-1906

MSC (2000):
Primary 35B65, 35D10, 35H20, 46E35

Published electronically:
October 30, 2009

MathSciNet review:
2574880

Full-text PDF Free Access

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****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**, 10.1080/03605308208820218**3.**Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano,*Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields*, Houston J. Math.**22**(1996), no. 4, 859–890. MR**1437714****4.**Nicola Garofalo and Duy-Minh Nhieu,*Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces*, Comm. Pure Appl. Math.**49**(1996), no. 10, 1081–1144. MR**1404326**, 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A**5.**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****6.**Norman G. Meyers and James Serrin,*𝐻=𝑊*, Proc. Nat. Acad. Sci. U.S.A.**51**(1964), 1055–1056. MR**0164252****7.**Cristian Rios, Eric T. Sawyer, and Richard L. Wheeden,*A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations*, Adv. Math.**193**(2005), no. 2, 373–415. MR**2137289**, 10.1016/j.aim.2004.05.009**8.**Cristian Rios, Eric T. Sawyer, and Richard L. Wheeden,*Regularity of subelliptic Monge-Ampère equations*, Adv. Math.**217**(2008), no. 3, 967–1026. MR**2383892**, 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**, 10.1090/memo/0847

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:
https://doi.org/10.1090/S0002-9947-09-04756-4

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.