Table Of ContentSPRINGER BRIEFS IN MATHEMATICS
Peter Lindqvist
Notes on the
Stationary
p-Laplace
Equation
123
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Peter Lindqvist
Notes on the Stationary
p-Laplace Equation
123
PeterLindqvist
Department ofMathematical Sciences
NorwegianUniversity of Science
andTechnology
Trondheim, Norway
ISSN 2191-8198 ISSN 2191-8201 (electronic)
SpringerBriefs inMathematics
ISBN978-3-030-14500-2 ISBN978-3-030-14501-9 (eBook)
https://doi.org/10.1007/978-3-030-14501-9
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©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2019
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Preface
Themostimportantpartialdifferentialequationofthesecondorderisthecelebrated
Laplace equation. This is the prototype for linear elliptic equations. It is less well
known that it also has a non-linear counterpart, the so-called p-Laplace equation,
dependingonaparameterp.Thep-Laplaceequationhasbeenmuchstudiedduring
the last 50 years and its theory is by now well developed. Some challenging open
problems remain, however. The p-Laplace equation is a degenerate or singular
elliptic equation in divergence form. It deserves a treatise of its own, without any
extracomplicationsandgeneralizations.Thisismyhumbleattempttowritesucha
treatise. The interested reader may wish to consult the monograph Nonlinear
Potential Theory ofDegenerate Elliptic Equations by J.Heinonen,T. Kilpeläinen,
and O. Martio when it comes to more advanced and general questions.
ThesenoteswereoriginallywrittenupaftermylecturesattheSummerSchoolin
JyväskyläinAugust2005.Gradually,thetexthasbeenupdatedandextended,with
new chapters about viscosity solutions and asymptotic mean values, among other
topics. I am grateful to Juan Manfredi, who read the entire original text and con-
tributed valuable comments and improvements. I thank Fredrik Hoeg, Marta
Lewicka, Erik Lindgren, and Eero Ruosteenoja for help with proofreading.
Trondheim, Norway Peter Lindqvist
January 2019
v
About This Book
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
SummerSchoolinJyväskylä,Finland,inAugust2005andhavesincebeenupdated
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 (p < 2). 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.
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 The Dirichlet Problem and Weak Solutions . . . . . . . . . . . . . . . . . . 7
3 Regularity Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 The Case p[n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 The Case p¼n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 The Case 1\p\n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 On p-Superharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.1 Definition and Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 The Obstacle Problem and Approximation . . . . . . . . . . . . . . . . . 40
5.3 Infimal Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4 The Poisson Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.5 Summability of Unbounded p-Superharmonic Functions . . . . . . . 48
5.6 About Pointwise Behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.7 Summability of the Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6 Perron’s Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7 Some Remarks in the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . 65
8 The Infinity Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9 Viscosity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
10 Asymptotic Mean Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
11 Some Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
12 Inequalities for Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Literature .... .... .... .... ..... .... .... .... .... .... ..... .... 101
ix
About the Author
Peter Lindqvist is Professor of Mathematics in the Department of Mathematical
Sciences, Norwegian University of Science and Technology, Trondheim, Norway.
His research focus is analysis, including in particular partial differential equations
and “non-linear potential theory”.
xi
Chapter 1
Introduction
TheLaplaceequation(cid:2)u =0or
∂2u ∂2u ∂2u
+ +···+ =0
∂x2 ∂x2 ∂x2
1 2 n
istheEuler-LagrangeequationfortheDirichletintegral
(cid:2) (cid:2) (cid:2) (cid:3)(cid:4) (cid:5) (cid:4) (cid:5) (cid:6)
∂u 2 ∂u 2
D(u)= |∇u|2dx = ··· +···+ dx ...dx .
∂x ∂x 1 n
1 n
(cid:3) (cid:3)
Ifwechangethesquaretoapthpower,wehavetheintegral
(cid:2) (cid:2) (cid:2) (cid:3)(cid:4)∂u (cid:5)2 (cid:4) ∂u (cid:5)2(cid:6)2p
I(u)= |∇u|pdx = ··· +···+ dx ...dx .
∂x ∂x 1 n
1 n
(cid:3) (cid:3)
ThecorrespondingEuler-Lagrangeequationis
div(|∇u|p−2∇u)=0.
Thisisthe p-Laplaceequationandthe p-Laplaceoperatorisdefinedas
(cid:2) u =div(|∇u|p−2∇u)
p
(cid:7) (cid:8)n ∂u ∂u ∂2u (cid:9)
=|∇u|p−4 |∇u|2(cid:2)u+(p−2) .
∂x ∂x ∂x ∂x
i,j=1 i j i j
Usually p ≥1.Atthecriticalpoints(∇u =0)theequationisdegeneratefor p >2
andsingularfor p <2.Thesolutionsarecalled p-harmonicfunctions.
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2019 1
P.Lindqvist,NotesontheStationaryp-LaplaceEquation,
SpringerBriefsinMathematics,https://doi.org/10.1007/978-3-030-14501-9_1
