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# Discrete Mathematics (Leader: Dr. Diego Ricciotti <ricciotti (at) usf.edu>document.write('<a href="mai' + 'lto:' + 'ricciotti' + '&#64;' + 'usf.edu' + '">Dr. Diego Ricciotti</a>');)

## Wednesday, October 17, 2018

Title
Speaker
Time
Place

A $$\mathrm{p}(\cdot)$$-Poincaré Type Inequality and $$\mathrm{p}(\cdot)$$-Capacity in Carnot Groups
Robert Freeman
2:00pm-3:15pm
CMC 108

Abstract

We discuss the $$\mathrm{p}(\cdot)$$-Capacity and quasi-continuity in the sub-Riemannian setting of Carnot groups. We will use the $$\mathrm{p}(\cdot)$$-capacity and quasi-continuity to show a $$\mathrm{p}(\cdot)$$-Poincaré-type inequality. We will then use this inequality to establish the existence of a minimizer to the $$\mathrm{p}(\cdot)$$-Dirichlet energy integral. Moreover, we use the $$\mathrm{p}(\cdot)$$-Poincaré-type inequality to show the uniqueness of the minimizer, up to a set of zero $$\mathrm{p}(\cdot)$$-capacity, in the Sobolev sense.

## Wednesday, October 10, 2018

Title
Speaker
Time
Place

A Theorem of Radó-type in $$R^n$$, with possible extensions to Carnot Groups, Part III
Zachary Forrest
2:00pm-3:15pm
CMC 108

## Wednesday, October 2, 2018

Title
Speaker
Time
Place

A Theorem of Radó-type in $$R^n$$, with possible extensions to Carnot Groups, Part II
Zachary Forrest
2:00pm-3:15pm
CMC 108

## Wednesday, September 26, 2018

Title
Speaker
Time
Place

A Theorem of Radó-type in $$R^n$$, with possible extensions to Carnot Groups
Zachary Forrest
2:00pm-2:50pm
CMC 108

Abstract

We explore a Theorem for Euclidean spaces in the style of Radó for $$p$$-Harmonic functions as presented in a paper of Juutinen & Lindqvist, utilizing techniques of Viscosity Solutions. We consider the connection between the underlying Geometry and the proof of an essential Lemma in the paper, and speculate regarding the adaptability of this proof to the Heisenberg and (more generally) Carnot spaces.

## Wednesday, September 19, 2018

Topic
Speaker
Time
Place

Regularity of weak solutions of the $$p$$-Laplace equation in the Heisenberg group — Part IV
Diego Ricciotti
2:00pm-2:50pm
CMC 108

Abstract

We discuss De Giorgi classes and Holder continuity of the horizontal derivatives of p-harmonic functions in the Heisenberg group.

## Wednesday, September 12, 2018

Topic
Speaker
Time
Place

Regularity of weak solutions of the $$p$$-Laplace equation in the Heisenberg group — Part III
Diego Ricciotti
2:00pm-2:50pm
CMC 108

Abstract

We present a proof of Lipschitz regularity for $$p$$-harmonic functions in the Heisenberg group for $$1 ## Wednesday, September 5, 2018 Topic Speaker Time Place Regularity of weak solutions of the \(p$$-Laplace equation in the Heisenberg group — Part II
Diego Ricciotti
2:00pm-2:50pm
CMC 108

Abstract

Continuing from the regularity results presented last time, we will start discussing the nonlinear case ($$p$$ different than $$2$$).

## Wednesday, August 29, 2018

Topic
Speaker
Time
Place

Regularity of weak solutions of the $$p$$-Laplace equation in the Heisenberg group — Part I
Diego Ricciotti
2:00pm-2:50pm
CMC 108

Abstract

I will present some regularity results for weak solutions of the $$p$$-Laplace equation in the Heisenberg group based on difference quotients techniques, starting with the case $$p=2$$.