SOLUTIONS TO A CLASS OF PARABOLIC INHOMOGENEOUS NORMALIZED p-LAPLACE EQUATIONS

doi:10.1016/S0252-9602(15)60016-9

Abstract

Abstract:

In this article, we prove that viscosity solutions of the parabolic inhomogeneous equations
(n+p)/pu_t-Δ_p^Nu=f(x,t)
can be characterized using asymptotic mean value properties for all p≥1, including p = 1 and p = ∞. We also obtain a comparison principle for the non-negative or non-positive inhomogeneous term f for the corresponding initial-boundary value problem and this in turn implies the uniqueness of solutions to such a proble

Key words: Parabolic normalized p-Laplace equation, viscosity solution, asymptotic mean value property, comparison principle, uniqueness theorem, infinity Laplacian

CLC Number:

35C15

Fang LIU. SOLUTIONS TO A CLASS OF PARABOLIC INHOMOGENEOUS NORMALIZED p-LAPLACE EQUATIONS[J].Acta mathematica scientia,Series B, 2015, 35(2): 477-494.

Trendmd

References

[1] Manfredi J J, Parviainen M, Rossi J D. An asymptotic mean value characterization of p-harmonic functions. Proc Amer Math Soc, 2010, 138: 881-889

[2] Giorgi T, Smits R. Mean value property for p-harmonic functions. Proc Amer Math Soc, 2012, 140(7): 2453-2463

[3] Kawohl B, Manfredi J, Parviainen M. Solutions of nonlinear PDEs in the sense of averages. J Math Pures Appl, 2012, 97: 173-188

[4] Manfredi J J, Parviainen M, Rossi J D. An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games. SIAM J Math Anal, 2010, 42(5): 2058-2081

[5] Barron E N, Evans L C, Jensen R. The infinity Laplacian, Aronsson's equation and their generalizations. Trans Amer Math Soc, 2008, 360: 77-101

[6] Le Gruyer E, Archer J C. Harmonious extensions. SIAM J Math Anal, 1998, 29: 279-292

[7] Le Gruyer E. On absolutely minimizing Lipschitz extensions and PDE Δ_∞u = 0. Nonlinear Differential Equations Appl, 2007, 14: 29-55

[8] Peres Y, Schramm O, Sheffield S, Wilson D. Tug of war and the infinity Laplacian. J Amer Math Soc, 2009, 22(1): 167-210

[9] Lu G, Wang P. Infinity Laplace equation with non-trivial right-hand side. Electr J Diff Equ, 2010, 77: 1-12

[10] Lu G, Wang P. Inhomogeneous infinity Laplace equation. Advances in Mathematics, 2008, 217(4): 1838- 1868

[11] Lu G, Wang P. A PDE perspective of the normalized infinity Laplacian. Communications in Partial Dif- ferential Equations, 2008, 33: 1788-1817

[12] Lu G, Wang P. A uniqueness theorem for degenerate elliptic equations//Lecture Notes of Seminario Inter- disciplinare di Matematica, Conference on Geometric Methods in PDE's. On the Occasion of 65th Birthday of Ermanno Lanconelli Bologna, 2008: 207-222

[13] Liu F, Yang X. Solutions to an inhomogeneous equation involving infinity-Laplacian. Nonlinear Analysis: Theory, Methods and Applications, 2012, 75(14): 5693-5701

[14] Peres Y, Sheffield S. Tug-of-war with noise: A game theoretic view of the p-Laplacian. Duke Math J, 2008, 145(1): 91-120

[15] Caselles V, Morel J M, Sbert C. An axiomatic approach to image interpolation. IEEE Trans Image Process, 1998, 7(3): 376-386

[16] Portilheiro M, Vázquez J L. Degenerate homogeneous parabolic equations associated with the infinity- Laplacian. Calc Var and Partial Differential Equations, 2012, 31: 457-471

[17] Portilheiro M, Vázquez J L. A porous medium equation involving the infinity-Laplacian. Viscosity solutions and asymptotic behaviour. Communications in Partial Differential Equations, 2012, 37: 753-793

[18] Crandall M G, Wang P. Another way to say caloric. J Evol Equ, 2004, 3: 653-672

[19] Akagi G, Suzuki K. On a certain degenerate parabolic equation associated with the infinity-Laplacian. Discrete and Continuous Dynamical Systems (Supplement), 2007: 18-27

[20] Akagi G, Suzuki K. Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian. Calc Var and Partial Differential Equations, 2008, 31: 457-471

[21] Akagi G, Juutinen P, Kajikiya R. Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian. Math Ann, 2009, 343: 921-953

[22] Laurencot P, Stinner C. Refined asymptotics for the infinite heat equation with homogeneous Dirichlet boundary conditions. Communications in Partial Differential Equations, 2010, 36(3): 532-546

[23] Does K. An evolution equation involving the normalized p-Laplacian. Comm Pure Appl Anal, 2011, 10: 361-396; University of Cologne: Dissertation under the same title, 2009

[24] Kawohl B. Variations on the p-Laplacian//Nonlinear Elliptic Partial Differential Equations. Contemporary Mathematics, 2011, 540: 35-46

[25] Manfredi J J, Parviainen M, Rossi J D. On the definition and properties of p-harmonious functions. Ann Sc Norm Super Pisa CI Sci, 2012, 11(2): 215-241

[26] Manfredi J J, Parviainen M, Rossi J D. Dynamic programming principle for tug-of-war games with noise. ESAIM Control Optim Cal Var, 2012, 18(1): 81-90

[27] Juutinen P, Kawohl B. On the evolution governed by the infinity Laplacian. Math Ann, 2006, 335: 819-851

[28] Evans L C, Spruck J. Motion of level sets by mean curvature I. J Differential Geom, 1991, 33: 635-681

[29] Crandall M G, Ishii H, Lions P L. User's guide to viscosity solutions of second-order partial differential equations. Bull A M S, 1992, 27: 1-67

[30] Ohnuma S, Sato K. Singular degenerate parabolic equations with applications to the p-Laplace diffusion equation. Communications in Partial Differential Equations, 1997, 22: 381-411

[31] Falcome M, Finzi Vita S, Giorgi T, Smits R. A semi-Lagrangian scheme for the game p-Laplacian via p-averaging. Applied Numerical Mathematics, 2013, 73: 63-80