p-Laplace Equation in the Heisenberg Group

p-Laplace Equation in the Heisenberg Group PDF

Author: Diego Ricciotti

Publisher: Springer

Published: 2015-12-28

Total Pages: 96

ISBN-13: 331923790X

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This works focuses on regularity theory for solutions to the p-Laplace equation in the Heisenberg group. In particular, it presents detailed proofs of smoothness for solutions to the non-degenerate equation and of Lipschitz regularity for solutions to the degenerate one. An introductory chapter presents the basic properties of the Heisenberg group, making the coverage self-contained. The setting is the first Heisenberg group, helping to keep the notation simple and allow the reader to focus on the core of the theory and techniques in the field. Further, detailed proofs make the work accessible to students at the graduate level.

Generalizations of a Laplacian-Type Equation in the Heisenberg Group and a Class of Grushin-Type Spaces

Generalizations of a Laplacian-Type Equation in the Heisenberg Group and a Class of Grushin-Type Spaces PDF

Author: Kristen Snyder Childers

Publisher:

Published: 2011

Total Pages:

ISBN-13:

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In [2], Beals, Gaveau and Greiner find the fundamental solution to a 2-Laplace-type equation in a class of sub-Riemannian spaces. This fundamental solution is based on the well-known fundamental solution to the p-Laplace equation in Grushin-type spaces [4] and the Heisenberg group [6]. In this thesis, we look to generalize the work in [2] for a p-Laplace-type equation. After discovering that the "natural" generalization fails, we find two generalizations whose solutions are based on the fundamental solution to the p-Laplace equation.

On the P(x)-Laplace Equation in Carnot Groups

On the P(x)-Laplace Equation in Carnot Groups PDF

Author: Robert D. Freeman

Publisher:

Published: 2020

Total Pages: 83

ISBN-13:

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In this thesis, we examine the p(x)-Laplace equation in the context of Carnot groups. The p(x)-Laplace equation is the prototype equation for a class of nonlinear elliptic partial differential equations having so-called nonstandard growth conditions. An important and useful tool in studying these types of equations is viscosity theory. We prove a p()-Poincar ́e-type inequality and use it to prove the equivalence of potential theoretic weak solutions and viscosity solutions to the p(x)-Laplace equation. We exploit this equivalence to prove a Rad ́o-type removability result for solutions to the p-Laplace equation in the Heisenberg group. Then we extend this result to the p(x)-Laplace equation in the Heisenberg group.

Analysis of the Hodge Laplacian on the Heisenberg Group

Analysis of the Hodge Laplacian on the Heisenberg Group PDF

Author: Detlef Muller

Publisher: American Mathematical Soc.

Published: 2014-12-20

Total Pages: 104

ISBN-13: 1470409399

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The authors consider the Hodge Laplacian \Delta on the Heisenberg group H_n, endowed with a left-invariant and U(n)-invariant Riemannian metric. For 0\le k\le 2n+1, let \Delta_k denote the Hodge Laplacian restricted to k-forms. In this paper they address three main, related questions: (1) whether the L^2 and L^p-Hodge decompositions, 1

Maximal Subellipticity

Maximal Subellipticity PDF

Author: Brian Street

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2023-07-03

Total Pages: 768

ISBN-13: 3111085643

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Maximally subelliptic partial differential equations (PDEs) are a far-reaching generalization of elliptic PDEs. Elliptic PDEs hold a special place: sharp results are known for general linear and even fully nonlinear elliptic PDEs. Over the past half-century, important results for elliptic PDEs have been generalized to maximally subelliptic PDEs. This text presents this theory and generalizes the sharp, interior regularity theory for general linear and fully nonlinear elliptic PDEs to the maximally subelliptic setting.

Notes on the Stationary p-Laplace Equation

Notes on the Stationary p-Laplace Equation PDF

Author: Peter Lindqvist

Publisher: Springer

Published: 2019-04-26

Total Pages: 104

ISBN-13: 3030145018

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This book in the BCAM SpringerBriefs series is a treatise on the p-Laplace equation. It is based on lectures by the author that were originally delivered at the Summer School in Jyväskylä, Finland, in August 2005 and have since been updated and extended to cover various new topics, including viscosity solutions and asymptotic mean values. The p-Laplace equation is a far-reaching generalization of the ordinary Laplace equation, but it is non-linear and degenerate (p>2) or singular (p2). Thus it requires advanced methods. Many fascinating properties of the Laplace equation are, in some modified version, extended to the p-Laplace equation. Nowadays the theory is almost complete, although some challenging problems remain open./pbrp

Function Spaces and Potential Theory

Function Spaces and Potential Theory PDF

Author: David R. Adams

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 372

ISBN-13: 3662032821

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"..carefully and thoughtfully written and prepared with, in my opinion, just the right amount of detail included...will certainly be a primary source that I shall turn to." Proceedings of the Edinburgh Mathematical Society

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