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On the regularity of p-harmonic functions in the Heisenberg group

Domokos, Andras (2004) On the regularity of p-harmonic functions in the Heisenberg group. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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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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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Domokos, Andrasand36@pitt.eduAND36
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairManfredi, Juan J.manfredi@pitt.eduMANFREDI
Committee MemberSaunders, Davidsaunders@pitt.eduSAUNDERS
Committee MemberBeatrous, Frank H.beatrous@pitt.eduBEATROUS
Committee MemberChaparro, Luis F.chaparro@ee.pitt.eduLFCH
Committee MemberMetzger, Thomas Ametzger@pitt.eduMETZGER
Committee MemberPan, Yibiaoyibiao@pitt.eduYIBIAO
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:, etd-05272004-124513
Date Deposited: 10 Nov 2011 19:45
Last Modified: 15 Nov 2016 13:44


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