Donatella Danielli

Education

Professional Experience

Honors

Research Interests

Partial Differential Equations

Applied Mathematics

Harmonic Analysis

Publications

  1.       A compact embedding theorem for a class of degenerate Sobolev spaces.  Rendiconti Sem. Mat. Univ. Polit. Torino, 49(3); 399-420, 1991.

  2.       Formules de representation et theoremes d’inclusion pour des operateurs sous-elliptiques.  Comptes Rendus Acad. Sci. Paris, t. 314, Serie I; 987-990, 1992.

  3.       (with L. Capogna and N. Garofalo)  Embedding theorems and the Harnack inequality for solutions of nonlinear subelliptic equations.  Comptes Rendus Acad. Sci. Paris, t. 316, Serie I; 809-814, 1993.

  4.       (with L. Capogna and N. Garofalo)  An embedding theorem and the Harnack inequality for nonlinear subelliptic equations.  Comm. in PDE, 18; 1765-1794, 1993.

  5.       (with L. Capogna and N. Garofalo)  Subelliptic mollifiers and a characterization of Rellich and Poincaré domains. Rend Sem. Mat. Univ. Polit. Torino, 51(4); 361-386, 1993.

  6.       (with L. Capogna and N. Garofalo)  An isoperimetric inequality and the geometric Sobolev embedding for vector fields.  Mathematical Research Letters, 1; 263-268, 1994.

  7.       (with L. Capogna and N. Garofalo)  The geometric Sobolev embedding for  vector fields and the isoperimetric inequality.  Comm. in Analysis and Geometry, 2(2); 203-215, 1994.

  8.       Regularity at the boundary for solutions of nonlinear subelliptic equations. Indiana Univ. Math. J., 44(1); 269-286, 1995.

9.              (with L. Capogna and N. Garofalo)  Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations. Amer. Jour. Mathematics, 118(6); 1153-1196, 1996.

10.        (with L. Capogna and N. Garofalo)  Subelliptic mollifiers and a basic pointwise estimate of Poincaré type. Math. Zeitschrift, 226; 147-154, 1997.

11.              (with N. Garofalo) Geometric properties of solutions to subelliptic equations in nilpotent Lie groups. Lecture Notes in Pure and Applied Mathematics,   vol.194,  Reaction Diffusion Systems,  Marcel Dekker Inc., 89-105, 1998.

12.       (with N. Garofalo and D. M. Nhieu) Isoperimetric and trace inequalities with respect to Carnot-Carathéodory metrics.  Geometry Seminars, 1996-97,  Univ. of Bologna, Bologna, 51-62, 1998.

13.              (with N. Garofalo and D. M. Nhieu)  Trace inequalities for Carnot-Carathéodory  spaces and applications. Annali Scuola Normale Sup. Pisa  (4), 27; 195-252, 1998.

14.              A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations. Potential Analysis, 11 (4); 387-413, 1999.

15.             (with  N. Garofalo and D. M. Nhieu) Sub-elliptic Besov spaces and the characterization of traces on lower dimensional manifolds. Contemporary  Mathematics, vol. 277, Amer. Math. Soc., 19-37, 2001.

16.       (with N. Garofalo) Properties of entire solutions of non-uniformly elliptic equations arising in geometry and in phase transitions. Calculus of Variations, 15 (4); 451-491, 2002.

17.        (with  N. Garofalo and S. Salsa) Variational inequalities with lack of ellipticity. Part I: interior regularity and non-degeneracy of the free boundary, Indiana  Univ. Math. Jour., 52; 361-398, 2003.

18.              (with  A. Petrosyan and H. Shahgholian) A singular perturbation problem for the p-Laplace operator, Indiana Univ. Math. Jour., 52; 457-476, 2003.

19.             (with N. Garofalo and D. M. Nhieu) On the best possible character of the $L^Q$ norm in some a priori estimates for non-divergence form equations in Carnot groups, Proc. Amer. Math. Soc., 131; 3487-3498, 2003.

20.        (with  N. Garofalo and D. M. Nhieu) Notions of convexity in Carnot groups, Comm. Anal. Geom., 11 (2); 263-341, 2003.

21.        (with  N. Garofalo, D. M. Nhieu, and F. Tournier) The Theorem of Busemann-Feller-Alexandrov in Carnot Groups, Comm. Anal. Geom., 12 (4); 853-886, 2004.

22.        (with  A. Petrosyan) A minimum problem with free boundary for a degenerate quasilinear operator, Calculus of Variations, 23 (1); 97-124, 2005.

23.        (with M. Korten) On the pointwise jump condition of the free boundary in the 1-phase Stefan problem, Comm. Pure Appl. Anal., 4 (2), 357-366, 2005.

24.        (with N. Garofalo and D. M. Nhieu) Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces, Mem. Amer. Math. Soc., 182 (857), 2006.

25.        (with A. Petrosyan) Full regularity of the free boundary in a Bernoulli-type problem in two dimensions,  Math. Res. Lett., 13 (4), 667-681, 2006.

26.        (with N. Garofalo and A. Petrosyan)  The sub-elliptic obstacle problem: C^1,α regularity of the free boundary in Carnot groups of step two, Adv. Math., 211, 485-516, 2007.

27.        (with L. Capogna, S. Pauls, and J. Tyson) An Introduction to the Heisenberg Group and the sub-Riemannian Isoperimetric Inequality, Progress in Mathematics, 259. Birkhäuser Verlag, Basel, 2007.

28.        (Editor) Recent Developments in Nonlinear Partial Differential Equations, Contemporary Mathematics. Amer. Math. Soc., 2007.

Colloquia and Invited Lectures at Meetings

  1.       “Boundary regularity in the Dirichlet problem for nonlinear subelliptic equations,” meeting on Partial Differential Equations,  organizers E. Lanconelli and G. Talenti, Bologna, Italy, June 1994.

  2.       “Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations,” meeting “Harmonic Analysis and its Applications,”

             I.S.I. Guccia, Palermo, Italy, December 1994.

3.              “Interior and boundary regularity for solutions of nonlinear subelliptic equations and capacitary estimates”,   meeting on Partial Differential Equations of Elliptic Type, Cortona, Italy, May 1996.

4.              “A Fefferman-Phong type inequality and applications”, meeting on Partial Differential Equations of Elliptic Type, Cortona, Italy, May 1996.

5.              “A singular perturbation approach to a two phase parabolic free boundary problem arising in flame propagation”, Purdue University, January 1999. 

6.        “Singular perturbations of a parabolic free boundary problem arising in combustion theory”, meeting “Geometry and regularity of free boundaries”,

           Royal Institute of Technology, Stockholm, Sweden, February 2000.

7.              “A singular perturbation problem for the p-Laplace operator: Uniform estimates and limits", Kansas State University, October 2000.
 

8.              “The subelliptic obstacle problem”, AMS Sectional Meeting, Lawrence, Kansas, March 2001.

9.       “Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Caratheodory spaces”, 49^th Midwest PDE Seminar, (Conference in honor of D. Adams, J. Lewis, and R. Gariepy), Lexington, Kentucky, March 2002.

10.      “The Theorem of Busemann-Feller-Alexandrov on Carnot Groups”, Kansas State University, October 2003.

11.      “Variational inequalities and free boundaries for the p-Laplace operator", University of Missouri at Columbia, April 2004.

12.     “The Theorem of Busemann-Feller-Alexandrov on Carnot Groups”, The 2nd Chicago area PDE seminar, University of Chicago, May 2004.

13.     “Variational inequalities and free boundaries for the p-Laplace operator", International Conference on Harmonic Analysis and Partial Differential Equations in honor of Carlos E. Kenig, University of Chicago, Illinois, September 2004.

14.    “Hypersurfaces of minimal type in sub-Riemannian geometry”, Seventh New Mexico Analysis Seminar, University of New Mexico, October 2004.

15.    “Variational inequalities and free boundaries for the p-Laplace operator”, AMS Sectional Meeting, Albuquerque, New Mexico, October 2004.

16.      “A Partial Solution of the Isoperimetric Problem for the Heisenberg Group”, AMS Sectional Meeting, Notre Dame, Indiana, April 2006.
 

17.     “Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H¹”, AIMS Sixth International Conference on Dynamical Systems, Differential Equations and Applications, University of Poitiers, France, June 2006.

18.     “The sub-elliptic obstacle problem: C^1,α regularity of the free boundary in Carnot groups of step two”, 6th International Congress on Industrial and Applied Mathematics, ETH, Zurich, Switzerland, July 2007.

 Individual Invited Lectures 

1.  “The local behavior of solutions of nonlinear subelliptic equations", University  of Bologna, Italy, July 1994.

2.   “A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations”, University  of Bologna, Italy, June 1996.

3.  “Trace inequalities of Fefferman-Phong type and applications”, Royal Institute of  Technology, Stockholm, Sweden, June 1997.

4.  “A two phase parabolic free boundary problem arising in flame propagation: Uniform estimates and limits”, Purdue University, September 1997.

5.  “A singular perturbation approach to a two phase parabolic free boundary problem arising in flame propagation”, Brown University, February 1999.

6.  “A singular perturbation approach to a two phase parabolic free boundary problem arising in flame propagation”, University of Pittsburgh, February 1999.

7.  “A singular perturbation approach to a two phase parabolic free boundary problem arising in  flame propagation”, Texas A&M University, February 1999.

8. “A singular perturbation approach to a two phase parabolic free boundary problem arising in flame propagation,” Johns Hopkins University, February 1999.

9.  “A singular perturbation approach to a two phase parabolic free boundary problem arising in  flame propagation”, University of Arkansas, February 1999.

10.  “A singular perturbation approach to a two phase parabolic free boundary problem arising in  flame propagation”, Georgia Institute of Technology, February 1999.

11.  “A singular perturbation approach to a two phase parabolic free boundary problem arising in flame propagation”, University of Texas at Austin, April 1999.

12.  “A singular perturbation approach to a two phase parabolic free boundary problem arising in  flame propagation”, Ohio State University, June 1999.

13.  “Singular perturbations of a two phase parabolic free boundary problem: Uniform estimates and limits, Polytechnic of Milan, July 1999.

14.  “Singular perturbations of a two phase parabolic free boundary problem: Uniform estimates and limits, University of Bari, July 1999.

15.  “A singular perturbation approach to a two phase parabolic free boundary problem arising in  flame propagation”, University of Pennsylvania, October 1999.

16.  “Singular perturbations of a parabolic free boundary problem arising in flame propagation”, University of Maryland, December 1999.

17.  “Singular perturbations of a parabolic free boundary problem arising in combustion theory”,  Linköping University, Sweden, March 2000.

18.  “Optimal regularity in the subelliptic obstacle problem”, Institut Mittag-Leffler, Stockholm,  Sweden, April 2000.

19.  “Singular perturbation problems arising in combustion theory”, Notre Dame University, October 2000.   

20.  “Variational inequalities with lack of ellipticity”, Purdue University, August 2001.

21.   “The method of moving spheres for elliptic Weingarten surfaces”, Purdue University, September 2002.

22.  “The method of moving spheres for elliptic Weingarten surfaces”, The University of Texas at Austin, April 2003.

23.   "The isoperimetric inequality", Dean's Freshman Honors Seminar, Purdue University, April 2003.

24.  "An introduction to partial differential equation", Liftoff Seminar, Purdue University, October 2003.

25.  “A Partial Solution of the Isoperimetric Problem for the Heisenberg Group”, University of Virginia, February 2006.

26.   “A Partial Solution of the Isoperimetric Problem for the Heisenberg Group”, University of California at Los Angeles, March 2006.

27. “Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group,” Johns Hopkins University, March 2007.

28. “Minimal Surfaces in the Heisenberg Group”, Purdue University, October 2007.

Professional Service

Undergraduate Advisor Coordinator, Johns Hopkins University, 2000-01.

AMS – UMI First Joint International Meeting, Special Session Organizer, June 2002.

Exciting Science @ Purdue (School of Science recruitment event), Organizer, April 2003 and April 2004.

Symposium on Analysis and PDEs, Purdue University, Organizer, May 2003 and June 2004.

Symposium on Analysis and PDEs, Purdue University, co-Organizer, May 2007.

Referee for: NSF, Proc. AMS, Jour. Diff. Eq., Comm. PDE, Jour. Math. Anal. Appl., SIAM Jour. Math. Anal., Jour. London Math. Soc., Math. Ann., Pot. Anal., Trans. AMS, Adv. Math., Annals of Math,  Memoirs AMS., Interfaces and Free Boundaries, Math. Nachr.