Donatella Danielli

 

Research Interests 

 

My research focuses on the study of analytic and geometric properties of partial differential equations and variational inequalities. More specifically, recently I have been interested in lower dimensional obstacle problems and free boundary problems arising in the theory of flame propagation.

 

 

Curriculum Vitae

 

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 LQ 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, Memoirs AMS., 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: C1,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. 

  29. (with N. Garofalo and D. M. Nhieu)  Sub-Riemannian Calculus on Hypersurfaces in Carnot Groups, Adv. Math., 215 (1), 292-378, 2007. 

  30. (with N. Garofalo and D. M. Nhieu)  A partial solution of the isoperimetric problem for the Heisenberg group, Forum Mathematicum, 20(1), 99-143, 2008.

  31. (with N. Garofalo and D. M. Nhieu) A notable family of entire intrinsic minimal graphs in the Heisenberg group which are not perimeter minimizing, Amer. Jour. Math., 130 (2), 317-339, 2008. 

  32. (with N. Garofalo) Interior Cauchy-Schauder estimates for the heat flow in Carnot- Carathéodory spaces, Methods Appl. Anal., 15 (1) (Special issue dedicated to N. Trudinger), 121-136, 2008. 

  33. (with N. Garofalo, D. M. Nhieu, and S. Pauls), Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group, Jour. Diff. Geom., 81 (2), 251-296, 2009. 

  34. (with N. Garofalo and P. C. Nguyen) Inequalities of Hardy-Sobolev Type in Carnot- Carathéodory Spaces, Sobolev Spaces in Mathematics I (V. Maz’ya ed.), International Mathematical Series, Vol. 8, Springer, 117-152, 2009. 

  35. (with N. Garofalo and N. Marola) Local behavior of p-harmonic Green's functions in metric spaces, Potential Anal., 32 (4), 343-362, 2010. 

  36. (with N. Garofalo and D. M. Nhieu) Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips, Math. Zeit., 265 (3), 617-637, 2010. 

  37. (with N. Garofalo, D. M. Nhieu, and S. Pauls) The Bernstein problem for embedded surfaces in the Heisenberg group $\Bbb H^1$H1, Indiana Univ. Math. Jour., 59 (2), 563-594, 2010. 

  38. (with N. Garofalo and P.C. Nguyen) Sharp Hardy-Sobolev type inequalities in Carnot- Carathéodory spaces. Potential Analysis, 34 (3), 223-242, 2011. 

  39. (with N. Garofalo and D.M. Nhieu) Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg group. Proc. AMS., 140, 811-821, 2012. 

  40. (with N. Garofalo, A. Petrosyan, and T. To) Optimal regularity and the free boundary in the parabolic Signorini problem. Memoirs AMS, 249 (1181), 103 p., 2017.    

  41. (with S. Salsa) Obstacle problems involving the fractional Laplacian. Recent Developments in the Nonlocal Theory (T. Kuusi and G. Palatucci, Eds.), Book Series on Measure Theory, De Gruyter, Berlin, 81-164, 2018. 

  42. (with A. Petrosyan and C. Pop) Obstacle problems for nonlocal operators. New developments in the analysis of nonlocal operators, 191–214, Contemp. Math., vol. 723, Amer. Math. Soc., 2019.

  43. (with A. Petrosyan, and C. Pop) Obstacle problems for nonlocal operators: A brief overview, To appear in ISNPS 2018 Proceedings. arXiv:1807.10910

  44. (with A. Banerjee, N. Garofalo, and A. Petrosyan) The structure of the singular set in the thin obstacle problem for degenerate parabolic equations. arXiv:1902.07457 

  45. (with A. Banerjee, N. Garofalo, and A. Petrosyan) The regular free boundary in the thin obstacle problem for degenerate parabolic equations. arXiv:1906.06885 

  46. A singular perturbation approach to a two phase parabolic free boundary problem arising in combustion theory, preprint.  

  47.  (with T. Backing and R. Jain) Regularity results for a penalized boundary obstacle problem, in preparation.