Table Of ContentSPRINGER BRIEFS IN MATHEMATICS
Diego Ricciotti
p-Laplace
Equation in
the Heisenberg
Group
Regularity of
Solutions
123
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Diego Ricciotti
p
-Laplace Equation
in the Heisenberg Group
Regularity of Solutions
123
DiegoRicciotti
Department ofMathematics
University of Pittsburgh
Pittsburgh, PA
USA
ISSN 2191-8198 ISSN 2191-8201 (electronic)
SpringerBriefs inMathematics
ISBN978-3-319-23789-3 ISBN978-3-319-23790-9 (eBook)
DOI 10.1007/978-3-319-23790-9
LibraryofCongressControlNumber:2015958335
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Preface
This work is based on my master’s thesis from the University of Bologna, written
under the supervision of my advisors Juan Manfredi and Bruno Franchi, and is
intended to present a self-contained introduction to the p-Laplace equation and
related regularity theory in the Heisenberg group. We also obtain new regularity
results in the nondegenerate case 1 < p < 2. I thank Drs. Franchi and Manfredi for
encouragingmetopreparethismanuscriptandtosubmitittotheBCAMSpringer
Briefs.
The regularity theory of nonlinear elliptic equations is quite well understood in
the Euclidean case, but in many problems in science and engineering the most
naturalsetupissubelliptic.Thisisthecaseinnon-holonomicmechanics,nonlinear
elasticity,roboticcontrol theory, andcertainmodelsintheneurobiology of vision,
tonamebutafewexamples.Inthesubellipticcasethevelocityfieldsarerestricted
toreflectnon-holonomicconstraints,andthisleadstothestudyofnon-commuting
vector fields generating nilpotent Lie algebras. The simplest, yet very important,
exampleistheHeisenberggroup,whichwedescribeinChap.2.Thecomplications
resulting from the lack of commutativity of the primary vector fields generate new
challengesinregularitytheory,especiallyinthenonlinearcasebutalsointhelinear
case, where the key result was proved by Hormander.
When we minimize non-quadratic energy functionals, the resulting Euler
equations are quasilinear, of p-Laplacian type, as described in Chap. 3. The
expectedregularity would be that solutionshave Holder continuous derivatives.In
Chap.5wepresentZhong’sLipschitzcontinuityresultsforp-harmonicfunctionsin
thefullrange1<p<∞.Inthenondegenerate case,wherevanishing gradientsdo
not present a difficulty, we would expect C∞ regularity. This was known for the
casep≥2.Ourmainnewresultpresentedinthismanuscriptisaproofofthisfact
valid for the full range 1 < p < ∞.
Pittsburgh Diego Ricciotti
May 2015
vii
Acknowledgments
I would like to thank my thesis advisors Bruno Franchi, for all his support during
my years in Bologna, and Juan Manfredi, for all the time he has dedicated to me
since before my arrival in Pittsburgh.
IwishtoexpressmyappreciationtoProf.AndrasDomokos,whoseanswersand
comments have been a great help in the development of this work.
I am also particularly grateful to Prof. Giovanna Citti, coordinator of the
CAP EXCEL exchange program, for her help and guidance.
ix
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 The Heisenberg Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Carnot Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Heisenberg Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Difference Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Morrey and Campanato Spaces . . . . . . . . . . . . . . . . . . . . . . . . 23
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 The p-Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Hilbert–Haar Existence Theory . . . . . . . . . . . . . . . . . . . . . . . . 38
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 C1 Regularity for the Non-degenerate Equation . . . . . . . . . . . . . . 43
4.1 Summability of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5 Lipschitz Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1 Caccioppoli Type Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Lipschitz Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
xi
Acronyms
r Euclidean gradient
r2 Euclidean Hessian
k (cid:2) k Euclidean norm
E
h (cid:2) i Euclidean scalar product
BEðxÞ Euclidean ball of radius r and center x
r
H Heisenberg group
g Lie algebra associated to H
C1ðΩÞ Smooth functions with compact support in Ω
0
rH Horizontal gradient
divH Horizontal divergence
r2 Horizontal Hessian
H
HW1;pðΩÞ Horizontal Sobolev space
HW1;pðΩÞ Local horizontal Sobolev space
loc
HW1;pðΩÞ Closure of C1ðΩÞ in the horizontal Sobolev norm
0 0
dcc, d Carnot–Carathéodory distance
BrðxÞ Carnot–Carathéodory ball of radius r and center x
ΓαðΩÞ Hölder functions with respect to dcc
k (cid:2) k Korányi norm
K
supp Support of a function
Ω ðxÞ ¼Ω\B ðxÞ
r r
Mp;λðΩÞ Morrey space
Lp;λðΩÞ Campanato space
DpðuÞ p-Dirichlet functional
⨏Ω Integral average over Ω
LieðXÞ Lie algebra generated by the set of vector fields X
xiii
