AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS

Transcription

1 AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Abstract. We characterize p-harmonic functions in terms of an asymptotic mean value property. A p-harmonic function u is a viscosity solution to pu = div u p u) = 0 with < p in a domain Ω if and only if the expansion ux) = α max u + min u B x) B x) ) + Z β u dy + o ) B x) B x) holds as 0 for x Ω holds in a weak sense, which we call viscosity sense. Here the coefficients α, β are determined by α + β = and α/β = p )/N + ). To the memory of our friend and colleague Fuensanta Andreu. Introduction and statement of the main result A well known fact that one can find in any elementary PDE textbook states that u is harmonic in a domain Ω R N that is u verifies u = 0 in Ω) if and only if it verifies the mean value property ux) = B x) B x) uy) dy, whenever B x) Ω. In fact, we can relax this condition by requiring that it holds asymptotically ux) = uy) dy + o ), B x) B x) as 0. This follows easily for C functions by using the Taylor expansion and for continuous functions by using the theory of viscosity solutions. Interestingly, a weak asymptotic mean value formula holds in some nonlinear cases as well. Our goal in this paper is to characterize p-harmonic functions, < p, by means of this type of asymptotic mean value properties. Key words and phrases. p Laplacian, infinity Laplacian, mean value property, viscosity solutions. 000 Mathematics Subject Classification. 35J0, 35J60, 35J70.

2 JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI We begin by stating what we mean by weak asymptotic expansions and why is it reasonable to say that our asymptotic expansions hold in a viscosity sense. As is the case in the theory of viscosity solutions, we test the expansions of a function u against test functions φ that touch u from below or above at a particular point. Select α and β determined by the conditions α + β = and α/β = p )/N + ). That is, we have.) α = p p + N, and β = + N p + N. Observe that if p = above, then α = 0 and β =, and if p =, then α = and β = 0. We follow the usual convention to denote the mean value of a function fy) dy = fy) dy. B B Definition. A continuous function u verifies ux) = α max u + min u + β uy) dy + o ), as 0, B x) B x) in the viscosity sense if B B x) ) for every φ C such that u φ has a strict minimum at the point x Ω with ux) = φx), we have 0 φx) + α max φ + min φ + β φy) dy + o ). B x) B x) B x) ) for every φ C such that u φ has a strict maximum at the point x Ω with ux) = φx), we have 0 φx) + α max φ + min φ + β ψy) dy + o ). B x) B x) B x) This condition is actually weaker than asking for the asymptotic expansion to hold in the classical sense as the following example suggested in Le Gruyer [0] shows. Example: Set p = and consider Aronsson s function ux, y) = x 4/3 y 4/3 near the point x, y) =, 0). Aronsson s function is -harmonic in the viscosity sense but it is not of class C, see Aronsson [, ]. It will follow from Theorem below that u satisfies ux) = max u + min u B x) B x) + o ) as 0,

3 ASYMPTOTIC MEAN VALUE CHARACTERIZATION 3 in the viscosity sense of Definition. However, let us verify that the expansion does not hold in the classical sense. Clearly, we have max u = u +, 0) = + ) 4/3. B,0) To find the minimum, we set x = cosθ), y = sinθ) and solve the equation d dθ u + cosθ), sinθ)) = cosθ))/3 sinθ) 4 3 sinθ))/3 cosθ) = 0. By symmetry, we can focus our attention on the solution ) 4 + θ = arccos. Hence, we obtain min u = u + cosθ ), sinθ )) B,0) = + ) ) 4/ We are ready to compute 4 + ) ) 4/3. max u + min u u, 0) B,0) B,0) lim 0+ = 8. But if an asymptotic expansion held in the classical sense, this limit would have to be zero. The following theorem states our main result and provides a characterization to the p-harmonic functions. Theorem. Let < p and let u be a continuous function in a domain Ω R N. The asymptotic expansion ux) = α max u + min u + β uy) dy + o ), as 0, B x) B x) B x) holds for all x Ω in the viscosity sense if and only if p ux) = 0 in the viscosity sense. Here α and β are determined by.). We use the notation div u p u) = p u for the regular p-laplacian and u = u D u u, u for the -homogeneous infinity Laplacian. The infinity Laplacian appears naturally when one considers limits as p of p harmonic functions in the viscosity sense and has applications to best

4 4 JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Lipschitz extensions, image processing and mass transport problems, see [], [3], [4], [7], [8], [9]. We observe that the notions of a viscosity solution and a Sobolev weak solution for the p-laplace equation agree for < p <, see Juutinen- Lindqvist-Manfredi []. Therefore, Theorem characterizes weak solutions when < p <. Finally, we note that Wang [4] has also used Taylor series to give sufficient conditions for p-subharmonicity in terms of asymptotic mean values of ux) u0)) p.. Proof of Theorem To gain some intuition on why such asymptotic mean value formula might be true, let us formally expand the p-laplacian as follows.) p u = p ) u p 4 D u u, u + u p u. This formal expansion was used by Peres and Sheffield in [3] see also Peres et. al. []) to find p harmonic functions as limits of values of Tug-of-War games. Suppose that u is a smooth function with u 0. We see from.), that u is a solution to p u = 0 if and only if.) p ) u + u = 0. It follows from the classical Taylor expansion that.3) ux) u dy = ux) N z dz + o ) and.4) ux) B x) max u + min B x) B x) ux) u u x + ux) ux) = ux) + o ). B0,) ) + u x ux) ) ux) The volume of the unit ball in R N will be denoted by ω N and the N dimensional area of the unit sphere will be denoted by σ N. Observe that since σ N /ω N = N we have N B0,) z dz = N +. Multiply.3) and.4) by suitable constants and add up the formulas so that we have the operator from.) on the right hand side. This process gives us the choices of the constants α and β in.) needed to obtain the asymptotic expansion of Theorem.

5 ASYMPTOTIC MEAN VALUE CHARACTERIZATION 5 The main idea of the proof of Theorem is just to work in the viscosity setting and use the expansions.3) and.4). The derivation of.4) also needs some care. We start by recalling the viscosity characterization of p-harmonic functions for p <, see []. Definition 3. For < p < consider the equation div u p u ) = 0. ) A lower semi-continuous function u is a viscosity supersolution if for every φ C such that u φ has a strict minimum at the point x Ω with φx) 0 we have p ) φx) φx) 0. ) An upper semi-continuous function u is a subsolution if for every φ C such that u φ has a strict maximum at the point x Ω with φx) 0 we have p ) φx) φx) 0. 3) Finally, u is a viscosity solution if it is both a supersolution and a subsolution. For the case p = we must restrict the class of test functions as in []. Let Sx) denote the class of C functions φ such that either φx) 0 or φx) = 0 and the limit exists. φy) φx)) lim y x y x = φx) Definition 4. Consider the equation u = 0. ) A lower semi-continuous function u is a viscosity supersolution if for every φ Sx) such that u φ has a strict minimum at the point x Ω we have φx) 0. ) An upper semi-continuous function u is a subsolution if for every φ Sx) such that u φ has a strict maximum at the point x Ω we have φx) 0. 3) Finally, u is a viscosity solution if it is both a supersolution and a subsolution. Proof of Theorem. We first consider asymptotic expansions for smooth functions that involve the infinity Laplacian p = ) and the regular Laplacian p = ).

6 6 JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Choose a point x Ω and a C -function φ defined in a neighborhood of x. Let x and x be the point at which φ attains its minimum and maximum in B x) respectively; that is, φx ) = min φy) and φx ) = max φy). y B x) y B x) Next, we use some ideas from [5]. Consider the Taylor expansion of the second order of φ φy) = φx) + φx) y x) + D φx)y x), y x) + o y x ) as y x 0. Evaluating this Taylor expansion of φ at the point x at x, we get φx ) = φx) + φx)x x) + D φx)x x), x x) + o ), as 0. Evaluating the Taylor expansion at the antipodal point of x with respect to x, given by x = x x, we have φ x ) = φx) φx)x x) + D φx)x x), x x) + o ). Adding the expressions, we obtain φ x ) + φx ) φx) = D φx)x x), x x) + o ). Since x is the point where the minimum of φ is attained, it follows that and then.5) φ x ) + φx ) φx) max y B x) φy) + min y B x) max y B x) φy) φy) + φx) min y B x) φy) φx), D φx)x x), x x) + o ). Repeating the same process at the point x we get instead φy) + min φy) φx).6) max y B x) y B x) D φx)x x), x x) + o ). Next we derive a counterpart for the expansion with the usual Laplacian p = ). Averaging both sides of the classical Taylor expansion of φ at x we

7 ASYMPTOTIC MEAN VALUE CHARACTERIZATION 7 get B x) φy) dy = φx) + N i,j= φ x x) i B 0) z iz j dz + o ). The values of the integrals in the sum above are zero when i j. Using symmetry, we compute zi dz = z dz B 0) N B 0) = Nω N N ρ σ N ds dρ = = 0 B ρ NN + )ω N N + ), with the notation introduced after.4). We end up with.7) φy) dy φx) = N + ) φx) + o ). B x) Assume for the moment that p so that α 0. Multiply.5) by α and.7) by β and add. We arrive at the expansion valid for any smooth function φ:.8) α max y B x) β N + ) + o ). φy) + min y B x) p ) φy) + β x D φx) x B x) ), φy) dy φx) ) x x + φx)) We remark that x B x) for > 0 small enough whenever φx) 0. In fact, suppose, on the contrary, that there exists a subsequence x j B j x) of minimum points of φ. Then, φx j ) = 0 and, since x j x as j 0, we have by continuity that φx) = 0. A simple argument based on Lagrange multipliers then shows that x.9) lim x = φ 0 φ x). We are ready to prove that if the asymptotic mean value formula holds for u, then u is a viscosity solution. Suppose that function u satisfies the asymptotic expansion in the viscosity sense according to Definition. Consider a smooth φ such that u φ has a strict minimum at x and φ Sx) if p =. We obtain 0 φx) + α max φ + min φ B x) B x) + β φy) dy + o ), B x)

8 8 JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI and thus, by.8), 0 β N + ) p ) x D φx) x ), x x ) + φx)) + o ). If φx) 0 we take limits as 0. Taking into consideration.9) we get 0 β N + ) p ) φx) + φx)). Suppose now that p = and that the limit φy) φx) lim y x y x = L exists. We need to deduce that L 0 from 0 max φ + min φ B x) B x) φx). But the limit condition implies the given δ > 0 for small we have and thus L δ) φx) φy) L + δ), 0 max φ φx)) + B x) min φ φx)) B x) L δ). Dividing by we get L δ 0. Since δ > 0 is arbitrary we conclude that L 0. This proves that u is a viscosity supersolution. To prove that u is a viscosity subsolution, we first derive a reverse inequality to.8) by considering the maximum point of the test function, that is, using.6) and.7), and then choose a function φ that touches u from above. We omit the details. To prove the converse implication, assume that u is a viscosity solution. In particular u is a supersolution. Let φ be a smooth test function such that u φ has a strict local maximum at x Ω. If p =, we also assume φ Sx). If φx) 0, we get.0) p ) φx) φx) 0. The statement to be proven is lim sup 0+ φx) + α max φ + min φ B x) B x) ) + β φy) dy 0. B x) This again follows from.8). Indeed, divide.8) by, use.9), and deduce from.0) that the limit on the right hand side is bounded from below by zero.

9 ASYMPTOTIC MEAN VALUE CHARACTERIZATION 9 For the case p = with φx) = 0 we assume the existence of the limit and observe that lim sup 0+ φy) φx) lim y x y x = L 0 ) φx) + max φ + min φ 0. B x) B x) The argument for the case of supersolutions is analogous. Finally, we need to address the case < p <. Since α 0 we use.6) instead of.5) to get a version of.8) with x in place of x. The argument then continues in the same way as before. Acknowledgements. Part of this work was done during a visit of JDR to University of Pittsburgh. He wants to thank for the friendly and stimulating atmosphere found there. JDR partially supported by project MTM004-03, MEC, Spain, by UBA X066 and by CONICET, Argentina. MP supported by The Emil Aaltonen Foundation, The Fulbright Center, and The Magnus Ehrnrooth Foundation. References [] G. Aronsson. Extensions of functions satisfying Lipschitz conditions. Ark. Mat., 6, 967), [] G. Aronsson, M.G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions. Bull. Amer. Math. Soc., 4, 004), [3] E.N. Barron, L.C. Evans and R. Jensen, The infinity laplacian, Aronsson s equation and their generalizations. Trans. Amer. Math. Soc. 360, 008), [4] T. Bhattacharya, E. Di Benedetto and J. Manfredi. Limits as p of pu p = f and related extremal problems. Rend. Sem. Mat. Univ. Politec. Torino, 99), [5] F. Charro, J. Garcia Azorero and J. D. Rossi. A mixed problem for the infinity laplacian via Tug-of-War games. To appear in Calc. Var. Partial Differential Equations. [6] M.G. Crandall, H. Ishii and P.L. Lions. User s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc., 7, 99), 67. [7] L.C. Evans and W. Gangbo, Differential equations methods for the Monge- Kantorovich mass transfer problem. Mem. Amer. Math. Soc., 37, 999), no [8] J. García-Azorero, J.J. Manfredi, I. Peral and J.D. Rossi, The Neumann problem for the -Laplacian and the Monge-Kantorovich mass transfer problem. Nonlinear Analysis TM&A., 66), 007), [9] R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Rational Mech. Anal ), [0] E. Le Gruyer, On absolutely minimizing lipschitz extensions and PDE u = 0, NoDEA Nonlinear Differ. Equ. Appl. 4, 007), [] P. Juutinen, P. Lindqvist and J.J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation, SIAM J. Math. Anal., 33, 00),

10 0 JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI [] Y. Peres, O. Schramm, S. Sheffield and D. Wilson; Tug-of-war and the infinity Laplacian. J. Amer. Math. Soc.,, 009), [3] Y. Peres, S. Sheffield; Tug-of-war with noise: a game theoretic view of the p- Laplacian. Duke Math. J. 45), 008), 9 0. [4] P. Wang, A formula for smooth -Harmonic Functions, PanAmerican Mathematical Journal, Number, Juan J. Manfredi Department of Mathematics University of Pittsburgh Pittsburgh, PA 560. manfredi@pitt.edu Mikko Parviainen Institute of Mathematics Helsinki University of Technology P.O. Box 00, 005 TKK, Helsinki, Finland Mikko.Parviainen@tkk.fi Julio D. Rossi IMDEA Matemáticas, C-IX, Campus UAM, Madrid, Spain On leave from Departamento de Matemática, FCEyN UBA 48) Buenos Aires, Argentina. jrossi@dm.uba.ar