On a mean value property related to the p-laplacian and p-harmonious functions

Transcription

1 On a mean value property related to the p-laplacian and p-harmonious functions Ángel Arroyo García Universitat Autònoma de Barcelona Workshop on Complex Analysis and Operator Theory Málaga, June 23, 2016

2 The Mean Value Property for Harmonic Functions Let Ω R N be a domain. Definition A function u C 2 (Ω) is said to be harmonic in Ω if it is a solution to the Laplace equation u(x) = N 2 u x 2 i=1 i = 0 x Ω. Definition A function u C(Ω) satisfies the Mean Value Property (MVP) in Ω if 1 u(x) = u = u B(x, r) Ω. B(x,r) B(x, r) B(x,r)

3 The Mean Value Property for Harmonic Functions Theorem (1840, Gauss) u harmonic in Ω = u satisfies the MVP in Ω. Definition Let r : Ω R + be a function such that 0 < r(x) dist(x, Ω) for each x Ω. We say that a function r satisfying this condition is an admissible radius function on Ω. Definition We say that a function u C(Ω) satisfies the (restricted) Mean Value Property with respect to r if u(x) = u x Ω. B x

4 The Mean Value Property for Harmonic Functions The Converse Mean Value Property Under which conditions can we say that a function u satisfying the restricted MVP is harmonic? One Radius Theorem (1934, Kellogg) Ω R N bounded domain r admissible radius function in Ω u C(Ω) u satisfies MVP (with respect to r) = u is harmonic in Ω.

5 Games and Mean Value Properties G (x, y) E G Lets consider a game in which a particle moves randomly jumping from one point to another contiguous point. Then: - Player I wins if it reaches G on E. - Player II wins if it reaches G on G \ E. We want to know the probability u(x, y) that Player I wins if the particle starts from a point (x, y) G.

6 Games and Mean Value Properties G u(x, y) E G 1 The function u should satisfy: 1 { 1 if (x, y) E u(x, y) = 1 0 if (x, y) G \ E 0 1 u(x, y) = 1 4 u(x + r, y) + 1 u(x r, y) This is a discrete version of the formula u(x) = u(x, y + r) + 1 u(x, y r) 4 1 u B(x, r) B(x,r)

7 Games and Mean Value Properties G E G Lets consider a game in which each step is decided by the players, who try to maximize their respective probabilities (x, y) of winning. By tossing a coin at each step, the winner can choose where the particle moves: - Player I would choose the point in which the probability of reaching E is greater. - Player II would choose the point in which the probability of reaching E is smaller. As in the random game, Player I wins if the particle reaches E, and Player II wins otherwise.

8 Games and Mean Value Properties The Tug-of-War game Lake Albert Farmers Tug-of-War team, From left to right: James Angel, W. Hennessy, C. Annison, H. Holder, J. Monks, Henry Angel, J. Brooker and J.J. Wild. Ángel Arroyo García Universitat Autònoma de Barcelona On a mean value property related to the p-laplacian and p-

9 Games and Mean Value Properties G u(x, y) E G 1 1 The function u should satisfy: { 1 if (x, y) E u(x, y) = 0 if (x, y) G \ E 0 1 u(x, y) = 1 2 min V (x,y) u max V (x,y) u This is a discrete version of the formula u(x) = 1 2 inf u + 1 B(x,r) 2 sup u. B(x,r)

10 Games and Mean Value Properties Let 0 < α < 1. We can also consider a combination of the two previous games. At each step it can be decided which game is going to be played: with probability α, the players can participate actively in the game by tossing a coin and maximizing or minimizing the function u, with probability 1 α, the particle moves randomly to a contiguous point. Then, the function u should satisfy [ 1 u(x, y) = α 2 min u + 1 ] V (x,y) 2 max u + (1 α) 1 V (x,y) 4 v V (x,y) The analogous in the continuous case is [ ] 1 u(x) = α 2 inf u + 1 B(x,r) 2 sup 1 u + (1 α) B(x,r) B(x, r) u( v). B(x,r) u.

11 The Asymptotic MVP Let Ω R N be a domain and u C(Ω). We define the following averages over B(x, r) Ω: Mu(x, r) = u, Su(x, r) = 1 B(x,r) 2 sup u + 1 B(x,r) 2 inf u, B(x,r) T α u(x, r) = αsu(x, r) + (1 α)mu(x, r) (where 0 < α < 1).

12 The Asymptotic MVP Let x Ω such that u(x) 0. We compute each of these mean values in the second order Taylor s expansion of u C 2 (Ω): u(x + rζ) = u(x) + r u(x), ζ + r 2 D 2 u(x) ζ, ζ + o(r 2 ) 2 where r > 0 and ζ 1. r 2 Mu(x, r) = u(x) + 2(N + 2) u(x) + o(r 2 ) Su(x, r) = u(x) + r 2 D 2 u(x) u(x) 2 u(x), u(x) +o(r 2 ) u(x) }{{} u(x)

13 The Asymptotic MVP u stands for the -Laplace operator defined for C 2 functions by u(x) = D 2 u(x) u(x) u(x), u(x) u(x) = 1 N 2 u u u u(x) 2 x i x j x i x j i,j=1 Why the -Laplace operator is "infinity"? We can obtain as a limit of p as p, where p is the p-laplace operator defined by p u = div( u p 2 u) It turns out that p can be written as a combination of and : p u = u p 2 [ u + (p 2) u].

14 The Asymptotic MVP Combining Su(x, r) and Mu(x, r) with α = p 2 N+p expansion: in the Taylor s T α u(x, r) = u(x) + r 2 2(N + p) [ u(x) + (p 2) u(x)] + o(r 2 ) p u(x) T α u(x, r) = u(x) + 2(N + p) u(x) p 2 + o(r 2 ) r 2

15 The Asymptotic MVP Let u C 2 (Ω). It turns out that, the p-laplacian can be asymptotically characterized by: u(x) = 2(N + 2) lim r 0 Mu(x, r) u(x) r 2 u(x) = 2 lim r 0 Su(x, r) u(x) r 2 p u(x) = 2(N + p) u(x) p 2 lim r 0 T α u(x, r) u(x) r 2 These equalites allows us to define the p-laplace operator and talk about p-harmonicity without using derivatives.

16 The Asymptotic MVP Definition A function u C(Ω) W 1,p loc (Ω) is said to be p-harmonic if it is a (weak) solution of the p-laplace equation Definition p u = 0. A function u C(Ω) satisfies the p-asymptotic mean value property (p-amvp) if the expansion u(x) = T α u(x, r) + o(r 2 ) (r 0) holds at each x Ω in a viscosity sense. The viscosity sense means that we need to replace the function u by C 2 test functions that touch u at x from above and from below.

17 The Asymptotic MVP Theorem (2011, Manfredi-Parviainen-Rossi) u is p-harmonic in Ω R N u(x) = T α u(x, r) + o(r 2 ) in a viscosity sense, x Ω, with α = p 2 N+p. Theorem u is p-harmonic in Ω R 2 u(x) = T αu(x, r) + o(r 2 ) x Ω, with α = p 2 p+2. for 1 < p < (2014, Manfredi-Lindqvist) for 1 < p < (2015, A.-Llorente)

18 The Asymptotic MVP However, the viscosity sense is needed for p =, for instance, consider the -harmonic function u(x, y) = x 4/3 y 4/3 at the point (x, y) = (1, 0). This function does not satisfy the corresponding -AMVP in a classical sense since and therefore lim r 0 2 r 2 [Su((1, 0), r) u(1, 0)] = Su((1, 0), r) = u(1, 0) + r o(r 2 ) In dimension N 3 it is not known if the asymptotic mean value property can be interpreted in the classical sense.

19 The Restricted MVP: Harmonious Functions Definition We say that a function r : Ω R such that 0 < r(x) < dist(x, Ω) for each x Ω is an admissible radius function, and we write B(x) = B(x, r(x)). Definition Given an admissible radius function r, we define the following operators C(Ω) C(Ω): Su(x) = Su(x, r(x)) = 1 2 sup u + 1 B(x) 2 inf Mu(x) = Mu(x, r(x)) = u B(x) B(x) u T α u(x) = T α u(x, r(x)) = αsu(x) + (1 α)mu(x)

20 The Restricted MVP: Harmonious Functions Definition A function u C(Ω) satisfies the (Restricted) Mean Value Property if T α u = u. Functions satisfying the restricted MVP have been called p-harmonious by some authors. Note that, by Kellogg s Theorem, 2-harmonious functions are harmonic, and vice versa.

21 The Dirichlet problem for harmonious functions Let Ω R N be a bounded domain. Given a function f C( Ω), we want to show that there exists a (unique) solution to the Dirichlet problem { Tα u = u in Ω u = f on Ω Define K f as the set of all continuous extensions of f to Ω: K f = { u C(Ω) : u Ω f }. Then, to solve this Dirichlet problem is equivalent to the problem of finding the fixed points of T α in K f.

22 The Dirichlet problem for harmonious functions Theorem (1998, Le Gruyer-Archer) Ω R N bounded and convex domain r C(Ω) admissible radius function such that r(x) r(y) x y Then the Dirichlet problem { Su = u in Ω u = f on Ω has a unique solution u C(Ω), where f C( Ω) is any given boundary data. Moreover, given any u 0 K f, S k u 0 u.

23 The Dirichlet problem for harmonious functions Definition Let u C(Ω). We define ω u,ω : R + R + the concave modulus of continuity of a function u C(Ω) in Ω as the lower bound of all concave functions ω : R + R + such that u(x) u(y) ω( x y ) x, y Ω. Lemma (Case α = 1) (1998, Le Gruyer-Archer) Let u C(Ω). Then, for each x, y Ω, Su(x) Su(y) ω Su,Ω ( x y ) ω u,ω ( x y ).

24 The Dirichlet problem for harmonious functions Idea of the Proof (Case α = 1, existence of fixed points) Lets consider the set } C = {u K f : u,ω f,ω and ω u,ω ω f, Ω. C C(Ω) is equicontinuous. By Arzéla-Ascoli theorem, C is a compact subset of C(Ω). Since S maps C into C, the Schauder s Fixed Point Theorem shows that there exists a fixed point u = Su in C K f.

25 The Dirichlet problem for harmonious functions Theorem (2014, A.-Llorente) Ω R N bounded and strictly convex domain 0 α < 1 and 0 < λ < ε < 1 α r C(Ω) admissible radius function such that r(x) r(y) x y λ dist(x, Ω) r(x) ε dist(x, Ω) Then the Dirichlet problem { Tα u = u in Ω u = f on Ω has a unique solution u C(Ω), where f C( Ω) is any given boundary data.

26 The Dirichlet problem for harmonious functions Idea of the Proof In order to show the Theorem, we will use an auxiliary operator C(Ω) C(Ω) defined by H α = 1 2 (Id +T α). Assume that the sequences { T k α u } k and { H k αu } k are equicontinuous at each point x Ω. By the Arzéla-Ascoli { } theorem, there exists a convergent subsequence H k j α u j.

27 The Dirichlet problem for harmonious functions Idea of the Proof By a technical theorem due to Ishikawa, it can be shown that the sequence { Hαu k } is asymptotically regular, i.e., k lim H k+1 u Hαu k = 0. k α It turns out that the sequence { Hαu k } is in fact convergent k to a fixed point. Since the fixed points of H α and T α are the same, then, given any u 0 K f, H k αu 0 = H α H k 1 α u 0 u = H α u = T α u.

28 The Dirichlet problem for harmonious functions Theorem (1976, Ishikawa) Let (X, ) be a Banach space, K X a bounded, closed and convex subset of X and let T : K X be a nonexpansive self-mapping of K: Tx Ty x y x, y K. Define H = 1 2 (I + T ). Then, for each x K, lim H k+1 x H k x = 0. k Applying this result with X = C(Ω), K = K f and T = T α we obtain that lim H k+1 u Hαu k = 0. k α

29 Estimates and interior equicontinuity Lemma (Case α = 0) Let u C(Ω). There exists a constant C = C(N) > 0 such that, if x, y Ω, then Lemma (Case 0 < α < 1) Mu(x) Mu(y) C u x y r(x). Let G Ω a proper subdomain of Ω. Then, for each x, y G, combining the cases α = 1 and α = 0 we get: T α u(x) T α u(y) αω u,ω ( x y ) + (1 α)c u x y inf G r

30 Estimates and interior equicontinuity We define the sequence of domains Ω 1 Ω 2 Ω by Ω n = {x Ω : dist(x, Ω) > (1 ε) n } (n N) Since r(x) ε dist(x, Ω), then, for each x Ω n, B(x) Ω n+1.

31 Estimates and interior equicontinuity Since r(x) λ dist(x, Ω), then inf Ωn r λ(1 ε) n, and since for every G Ω there is a n N such that G Ω n, then, Lemma Let G Ω be an open subdomain, and u C(Ω), then ω T k α u,g (t) ω u,ω (t) + C u t ω H k α u,g (t) ω u,ω (t) + C u t for each k N, where C = C(N, Ω, G, α, ε, λ) > 0. Corollary The sequences { Tα k u } k and { Hαu k } k interior point x Ω. are equicontinuous at each

32 THANKS! Ángel Arroyo García Universitat Autònoma de Barcelona On a mean value property related to the p-laplacian and p-