MAnET Metric Analysis for Emergent Technologies

Shirsho Mukherjee

Ph.D. student @ Jyväskylän Yliopisto (JYU)

since 1/11/2014 for 3 years.

under the supervision of Prof. Xiao Zhong

Ph.D. Thesis

“Regularity of Quasilinear Elliptic Equations on the Heisenberg Group” on

Research

My main goal is to prove the Holder regularity of the horizontal gradient for the p-laplace type equations in the Heisenberg Group for all p> 1. We, also plan to generalize the result for more general elliptic operator in this setting. There are even scopes of obtaining results of boundary regularity for these problems, and perhaps improve the settings to general Carnot groups as well. I sincerely hope that my research would yield to substantial contributory improvement in the areas related to analysis on Carnot Groups and Subriemannian manifolds, amongst research works that all MANET fellows have undertaken.

Complementary training activities:

• [Course] Language courses - a.a. 2014-2015 Jyväskylä, Finland

Local training activities:

• [Course] Analysis on Manifolds (lecturer M. Salo) - September 2014 to December 2014 Jyväskylä, Finland
• [Course] Optimal Mass Transportation (lecturer T. Rajala) - September 2014 to December 2015 Jyväskylä, Finland
• [Course] Sobolev Spaces (lecturer P. Koskela) - 1° semester 2014-2015 Jyväskylä, Finland
• [Course] Topological Group Theory (lecturer E. le Donne) - 1° semester 2014-2015 Jyväskylä, Finland
• [Course] Viscosity Theory (lecturer M. Parvaine) - 1° semester 2014-2015 Jyväskylä, Finland
• [Course] Bernoulli Convolution (lecturer P. Shmerkin) - August 10-14, 2015 Jyväskylä, Finland
• [Course] Dimension Theory of Smooth Dynamical System (lecturer J. Schmeling) - August 17-21, 2015 Jyväskylä, Finland

Network-wide training activities:

• [Workshop] Sub-Riemannian Analysis, PDE and Applications - January 26-30, 2015 Berna, Swiss
• [Workshop] Geometric mesaure theory, optimal mass transportation and PDEs - June 15-19, 2015 Sant Feliu de Guíxols, Spain
• [Workshop] A mathematical tribute to Ennio De Giorgi - September 19-23, 2016 Pisa, Italy
• [Conference] Lecce Conference in Calculus of Variations and Partial Differential Equation - CDV2016 - October 4-7, 2016 Lecce, Italy
• [Workshop] Around Analysis, domains and mappings, conference on the occasion of the 75th birthday of Olli Martio - December 15-17, 2016 Jyväskylä, Finland
• [Workshop] OT meets DFT - May 31 to June 7, 2017 Jyvaskyla, Finland
• [Conference] 23rd Rolf Nevanlinna Colloquium - June 12-16, 2017 Zurich, Switzerland

Talks and Poster Presentations:

• [Talk] Regularity of Variational Problems in the Heisenberg Group at Mid-term Review Meeting - December 9, 2015 Helsinki, Finland
• [Talk] Regularity of p-laplace equations in the Heisenberg Group at Analysis Seminar - May 25, 2016 Jyvaskyla, Finland
• [Talk] Existence of minimizers for spectral optimization at Analysis Seminar - May 10, 2017 Jyvaskyla
• [Talk] C^1,α-regularity of p-laplace equation in the Heisenberg Group at Mathematical Days 2018 - January 5, 2018 Jyvaskyla
• [Talk] C^1,α-regularity of p-laplace equation in the Heisenberg Group at Sub-Riemannian Geometry Harmonic Analysis, PDE and Applications - January 24-27, 2018 Bologna, Italy

Secondment:

• Unihe - August 1 to October 30, 2017 Helsinki, Finland

Publications and Preprints

• S. Mukherjee, X. Zhong, C^1,α -Regularity for variational problems in the Heisenberg group preprint
• S. Mukherjee, Regularity of Variational Problems in the Heisenberg Group report
• S. Mukherjee, Regularity of Quasilinear Equations in the Heisenberg Groupg report
• S. Mukherjee, C^1,α -Regularity of p-Laplace Equation in the Heisenberg Group report