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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.

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A Theorem of Radó-type in \(R^n\), with possible extensions to Carnot Groups, Part III

Zachary Forrest

2:00pm-3:15pm

CMC 108

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A Theorem of Radó-type in \(R^n\), with possible extensions to Carnot Groups, Part II

Zachary Forrest

2:00pm-3:15pm

CMC 108

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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.

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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.

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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

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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 \(2\)).

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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 \(p=2\).