· Partial Differential Equations
· Harmonic Analysis
· Geometric Analysis
Most of my work is in the area of PDE's and applications to
geometry. Please, see my CV for more
details.
· Heather Griffin-Rowe, Garrett Rea (current)
· Emily Shores (PhD 2005)
· Erin Haller (PhD 2008)
· Samantha Wempe (current, Undergraduate Honors thesis, 2009)
· April Gentry (Undergraduate honor thesis advisor, 2005)
· Diana Dunn (Undergraduate honor thesis advisor, 2004)
"Harmonic measure from a geometric and an analytic point of view", with C. Kenig and L. Lanzani, University Lectures Series 35, 2005, ISBN 0-8218-2728-6.
"An introduction to the Heisenberg Group and to the sub-Riemannian Isoperimetric problem", with D. Danielli, S. Pauls and J. Tyson. Birkhäuser , "Progress in Mathematics". 2007, Approx. 230 p., Hardcover, ISBN-10: 3-7643-8132-9
ISBN-13: 978-3-7643-8132-5
- A sub-Riemannian analogue of the mean curvature flow, Oberwolfach Abstract 2008.
· " Smoothness of Lipschitz intrinsic minimal graphs in the Heisenberg group H^n, n>1." , To appear in Crelle's Journal (with Giovanna Citti and Maria Manfredini).
· Generalized mean curvature flow in Carnot groups, To appear in Communications in PDE, arXiv:0808.3467 (with Giovanna Citti)
·Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$, To appear in Indiana University Mathematical Journal, arXiv:0804.3406 , (with Giovanna Citti and Maria Manfredini).
· Mutual Absolute Continuity of Harmonic and Surface Measures for Hormander Type Operators, To appear in Proceedings of Symposia in Pure Mathematics (2008), arXiv:0803.0784, (with Nicola Garofalo and Duy-Minh Nhieu)
· " Isoperimetric inequalities in the Heisenberg group and in the plane." , Lecture Notes of “Seminario Interdisciplinare di Matematica”, Vol. 6 (2007), 93-106
· "On the Mixed boundary value problem for Laplace's operator on Lipschitz planar domains." Math. Annalen, 2008 (with R. Brown and L. Lanzani)
· "Ill-posedness of the Cauchy problem for Heisenberg Wave maps" Proc. Amer. Math. Soc. 2008 (with J. Shatah)
· "Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups'' , to appear in Trans. Amer. Math. Soc. (with S. D. Pauls and J. Tyson)
· "A note on the engulfing property and the $\Gamma^{1+ \alpha}$-regularity of convex functions in Carnot Groups", Proc. Amer. Math. Soc. 134 (2006) 3191-3199. (with D. Maldonado)
· "The Aronsson-Euler equation for absolutely minimizing lipschitz extensions in Carnot-Caratheodory spaces." Trans. Amer. Math. Soc. 357 (2005), no. 2, 795--823. (with Thomas Bieske).
· "$1-$Quasiconformal maps in Carnot groups" Duke Math. Journal Vol. 135, No. 3 (2006) (with Michael Cowling).
· "Ahlfors type estimates for the perimeter measure in Carnot-Caratheodory spaces". Journal of Geometric Analysis, Vol. 16 (3) (2006) 455-497 . (with Nicola Garofalo).
· Pointwise Schauder estimates for subelliptic operators in Carnot groups , Proceedings for the AMS-SIAM Harmonic Analysis conference in Mt. Holyhoke, 2001. (with Qing Han).
· "Examples of Uniform and NTA domains in Carnot groups. " , Proceedings on Analysis and Geometry (Novosibirsk Akademgorodok, 1999) (Russian), 103--121, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000. (with D.M. Nhieu and N. Garofalo).
·
· "Properties of harmonic measures in the Dirichlet problem for nilpotent Lie groups of Heisenberg type" , American Journal of Mathematics 124, vol 2, 273--306. (2002) (with D.M. Nhieu and N. Garofalo).
· "The Dirichlet problem for sub-Laplacians" , unpublished paper (2001) (with D. M. Nhieu and N. Garofalo).
· A version of a theorem of Dahlberg for the subelliptic Dirichlet problem. Math. Res. Lett. 5 (1998), no. 4, 541--549 (DM Nhieu and N. Garofalo).
· Legendrian
energy minimizers. ,
(with Fang-Hua Lin) Calc. Var. Partial
Differential Equations 12 (2001), no. 2, 145--171.
· Regularity
for quasilinear equations and $1-$quasiconformal maps in Carnot
groups,
Math. Annalen 313 (1999), no. 2, 263--295.
· "Regularity of
minimizers of the calculus of variations in Carnot groups via
hypoellipticity of systems of H\"ormander type " ,
Journal
of the European Mathematical Society, 2 (2003). (with N. Garofalo).
· "
Boundary behavior of nonnegative solutions of subelliptic equations
in NTA domains for Carnot-Carathéodory metrics.", ,
,
J. Fourier Anal. Appl. 4 (1998), no. 4-5, 403--432. (with N.
Garofalo).
· Regularity
for quasilinear equations in the Heisenberg group,
Communications
in Pure and Applied Mathematics vol. L, (1997), pages 867--889.
· Capacitary
estimates and the local behavior of solutions of nonlinear
subelliptic equations,
American Journal of Mathematics,
vol. 118, no. 6 (1996), pages 1153-1196 (with D. Danielli and N.
Garofalo).
· Uniform
domains and quasiconformal mappings on the Heisenberg group ,
Man.
Math., 86 (1995), pages 267-281 (with P. Tang).
· Subelliptic
mollifiers and a basic pointwise estimate of Poincar\'e
type,
Math. Z. (1997) (with D. Danielli and N. Garofalo).
· The
geometric Sobolev embedding for vector fields and the isoperimetric
inequality,
Comm. in Analysis and Geometry 2(2) (1994),
pages 203--215 (with D. Danielli and N. Garofalo).
· Subelliptic
mollifiers and a characterization of Rellich and Poincar\'e
domains,
Rend. Sem. Mat. Univ. Pol. Torino, 51 (4) (1993)
pages 361-386 (with D. Danielli and N. Garofalo).
· An
embedding theorem and the Harnack inequality for nonlinear
subelliptic equations,
Comm. Partial Diff. Eq., 18 (1993),
pages 1765-1794 (with D. Danielli and N. Garofalo).