Domokos, Andras
(2004)
On the regularity of p-harmonic functions in the Heisenberg group.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
In this thesis we first implement iteration methods for fractional difference quotients of weak solutions to the p-Laplace equation in the Heisenberg group. We obtain that $Tu in L^p_{loc} (Omega )$ for $1<p<4$, where $u$ is a p-harmonic function.Then we give detailed proofs for $HW^{2,2}$-regularity for $p$ in the range $2 leq p <4$ and $HW^{2,p}$-regularity in the case $frac{sqrt{17}-1}{2} leq p leq 2$ for $ep$-approximate p-harmonic functions in the Heisenberg group. These last estimates however are not uniform in $ep$. The method to prove uniform estimates is based on Cordes type estimates for subellipticlinear partial differential operators in non-divergence form with measurable coefficients in the Heisenberg group. In this way we establish interior $HW^{2,2}$-regularity for p-harmonic functions in the Heisenberg group ${mathbb H}^n$ for $p$ in an interval containing $2$. We will also show that the $C^{1,alpha}$ regularity is true for $p$ in a neighborhood of $2$. In the last chapter we extend our results to the more general case of Carnot groups.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| Date: |
23 September 2004 |
| Date Type: |
Completion |
| Defense Date: |
22 March 2004 |
| Approval Date: |
23 September 2004 |
| Submission Date: |
27 May 2004 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
Heisenberg group; p-Laplacian; regularity; subelliptic PDE; weak solutions |
| Other ID: |
http://etd.library.pitt.edu/ETD/available/etd-05272004-124513/, etd-05272004-124513 |
| Date Deposited: |
10 Nov 2011 19:45 |
| Last Modified: |
15 Nov 2016 13:44 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/7962 |
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