U.U.D.M. Project Report 2018:36 The p-Laplace equation – general properties and boundary behaviour Frida Fejne Examensarbete i matematik, 30 hp Handledare: Kaj Nyström Ämnesgranskare: Wolfgang Staubach Examinator: Denis Gaidashev Augusti 2018 Department of Mathematics Uppsala University The p-Laplace equation – general properties and boundary behaviour. Frida Fejne August 27, 2018 Abstract In this thesis we investigate the properties of the solutions to the p-Laplace equation, which is the Euler- Lagrange equation of the p-Dirichlet integral, a generalization of the well known Dirichlet integral. It turns out that many of the properties of the harmonic functions also hold for the so called p-harmonic functions. After giving a comprehensive introduction to the subject, where we establish the existence of weak solutions on bounded domains, we discuss general properties such as the Harnack inequality, Hölder continuity, differentiability, and Perron’s method. In the last part of the thesis we study boundary behaviour and in particular the behaviour of the ratio of two p-harmonic functions near a portion of the boundary where they both vanish. 2 Acknowledgements I would like to thank Kaj Nyström for introducing me to this fascinating subject and for supervising this thesis. I also thank Lukas Schoug for showing me how to do the pictures and for letting me explain some of the concepts in order to gain deeper understanding of the theory. 3 Contents 1 Introduction 5 1.1 Preliminaries . 5 1.1.1 Notation . 5 1.1.2 Weighted Sobolev spaces . 6 1.2 Variational integrals and the p-Dirichlet integral . 8 1.3 Structure of Thesis . 10 2 Basic definitions and existence of weak solutions 10 3 Regularity of weak solutions 17 3.0.1 The case 1 < p < n ................................... 19 3.0.2 The case p = n ..................................... 25 3.0.3 The case n<p<∞ .................................... 27 4 Differentiability 27 5 The p-superharmonic functions and their properties 34 6 Perron’s method 41 7 Boundary behaviour 45 7.1 Boundary estimates . 50 7.2 Halfspace . 51 7.3 The fundamental inequality for the gradient of a p-harmonic function . 55 7.4 Estimates for degenerate elliptic equations in weighted Sobolev spaces . 62 7.5 Reduction to linear equation and final proof . 68 A Proofs of basic properties 73 B Some useful inequalities 75 4 1 Introduction One of the central problems of modern analysis has been the Dirichlet problem, that is, given an open n connected set Ω Ï R , and a real-valued function f, continuous on the boundary )Ω, to find a function 2 ̄ u Ë C (Ω) ã C.Ω), such that T Δu = 0 in Ω; u = f on )Ω: The literature on the problem is extensive and goes deep into the realms of many mathematical subjects, such as complex analysis (see e.g. [5]) and probability theory (see e.g. [24]). Another approach to the Dirichlet problem is via the Dirichlet energy integral. The Dirichlet energy integral of a function u :Ω → R is defined as 0 12 0 12 2 )u )u E.u/ = ð(uð dx = + § + dx; (1) ÊΩ ÊΩ )x1 )xn 1 E.u/ (sometimes written as multiplied by 2 ). The Euler-Lagrange equation for is easily shown to be the Laplace equation Δu = 0, that is, solving the Dirichlet problem with boundary conditions f is the same as minimizing the Dirichlet integral over functions with boundary data f. In this thesis, we consider a generalized version of (1), namely the p-Dirichlet integral p I.u/ = ð(uð dx; ÊΩ of which the Euler-Lagrange equation is the p-Laplace equation p*2 Δpu ≡ div.ð(uð (u/ = 0: 1.1 Preliminaries 1.1.1 Notation n In the following we let Ω be a domain in R , which is not necessarily bounded. G and D will always denote n n open sets in R . We let ê⋅; ⋅ë and dx denote the inner product and the Lebesgue measure on R , respectively. We will express the Euclidean distance between the two points x1 and x2 by d.x1; x2/. c will always be a positive constant, depending on at most p and n unless otherwise stated, such that c g 1. The value of c may vary from line to line and between occurrences. Furthermore, we will use the notation c.a1; a2; § ; as/ when n c also depends on the the additional constants a1; a2; § as. For x Ë R and r > 0 we define the open ball n B.x; r/ = ^y Ë R : ðx * yð < r`. When the center of the ball is arbitrary or clear from the context we will denote the ball by Br, and balls with different radii are always assumed to be concentric unless otherwise stated. The average of a function U over the set E is defined as 1 .u/E = u.x/ dx = u.x/ dx: ÕE ðEð ÊE The reader is assumed to be familiar with basic facts of Sobolev spaces, roughly corresponding to the content in chapter 7 in [6] or chapter 5 in [1]. We now, however, repeat the most basic definitions and properties. 5 1 u u Ë L (Ω) = . 1; § ; n/ Let be a locally summable function, i.e., loc , and let be a multiindex of order 1 = 1 + § + n v Ë L (Ω) th u ð ð . Then loc is the weak derivative of if the equation uD ' dx = .*1/ð ð v' dx; ÊΩ ÊΩ where )ð ðu D u = )x 1 ; )x n 1 § n ' C∞ D u v holds for all Ë 0 (Ω), i.e., all smooth functions in Ω with compact support. We use the notation = . k;p The Sobolev space W (Ω) is defined as the normed space that consists of equivalence classes of locally p summable functions u :Ω → R such that D u exists in the weak sense and D u Ë L (Ω) for all ð ð f k: k;p The norm of u Ë W (Ω) is defined as 1_p h ³ p n k ∫Ω ðD uð dx 1 f p < ∞; u k;p ð ðf ñ ñW (Ω) = l ³ D u p = ∞: n fk ð ð j ð ð ess supΩ (u = .u ; § ; u / Unless otherwise stated, x1 xn will always denote the distributional gradient. Furthermore, W k;p C∞ W k;p W k;p W k;p 0 (Ω) is the closure of 0 (Ω) in (Ω) and we note that both (Ω) and 0 (Ω) are Banach spaces. W k;p(Ω) Lp (Ω) u Ë W k;p(Ω) u Ë W k;p.D/ The space loc is defined analogously to loc : loc if and only if for each open set D ÏÏ Ω. In the following we will almost exclusively deal with the case k = 1 and 1 < p < ∞. For u W 1;p Ë 0 (Ω) and Ω bounded we recall the Poincaré inequality 0 11_n ðΩð ñuñLp(Ω) f ñ(uñLp(Ω); !n n where !n is the volume of a unit ball in R . We will sometimes use the notation A ù B which means that the ratio of A and B is bounded from above and below by constants. The dependence of the constants will be specified at each occurrence. 1.1.2 Weighted Sobolev spaces We will now give a very brief introduction to weighted Sobolev spaces. These function spaces will only occur in Section 7.4 but it actually turns out that most of the results we discuss in this thesis are also valid for functions that belong to the weighted Sobolev spaces. For a better and deeper introduction to the subject, see Ë L1 .Rn/ [4] or [9]. We consider a non-negative real-valued function loc and define the Radon measure by .E/ = .x/ dx ÊE n p ̃ p p whenever EÏ R . We will denote the L -space corresponding to the measure by L (Ω) or L (Ω; /. We have the following definition: Definition 1.1. A weight is called p-admissible if the following conditions hold: n (i) .x/ Ë .0; ∞) a.e. in R and the corresponding measure is a doubling measure, i.e. .B2r/ f c.Br/ n for all Br Ï R . 6 ̃ p ∞ p (ii) If v Ë L .G/ is a vector-valued function and ^'i` ÏC .G/ such that ∫G ð'ið d → 0 and ∫G ð('i * p vð d → 0 as i → ∞, then v = 0. n (iii) The weighted Sobolev inequality holds, i.e., there exists > 1 such that for all Br Ï R it holds that 0 11_p 0 11_p 1 1 ' p d cr (' p d ; .B / ð ð f .B / ð ð r ÊBr r ÊBr ' C∞ B where Ë 0 . r/. (iv) The weighted Poincaré inequality holds, i.e., ' *.'/ p d crp (' p d ð Br ð f ð ð ÊBr ÊBr ' Ë C∞.B / .'/ B for all bounded r . Here Br is the average over the ball r using the weighted measure . We note from condition (i) that the Lebesgue measure dx is absolutely continuous with respect to . In the 1;p 1;p following we assume that .x/ is a p-admissible weight. The weighted Sobolev space W (Ω) or W (Ω; / is defined as the closure of smooth functions in Ω, with respect to the weighted Sobolev norm 0 11_p 0 11_p ̃ p p ñ'ñ = ð'ð d + ð('ð d ÊΩ ÊΩ 1;p p ̃ p Thus, a function u is in W (Ω) if and only if u Ë L (Ω) and there exists a vector-valued function v Ë L (Ω) such that for some sequence of smooth functions ^'i` such that the following conditions are satisfied: p lim ð'i * uð d = 0 i→∞ ÊΩ p lim ð('i * vð d = 0: i→∞ ÊΩ v u W1;p(Ω) v = (u W1;p(Ω) W1;p(Ω) We say that is the gradient of in and use the notation .