Gradient walks and 𝑝-harmonic functions

Luiro, Hannes; Parviainen, Mikko

doi:10.1090/proc/13540

Gradient walks and $p$-harmonic functions
HTML articles powered by AMS MathViewer

by Hannes Luiro and Mikko Parviainen

Proc. Amer. Math. Soc. 145 (2017), 4313-4324

DOI: https://doi.org/10.1090/proc/13540

Published electronically: June 9, 2017

PDF | Request permission

Abstract:

We consider a class of stochastic processes having a connection to $p$-harmonic functions. In particular, we obtain stochastic approximations that converge uniformly to a $p$-harmonic function, and also with an explicit convergence rate. The main difficulty is how to deal with the zero set of the gradient of the underlying function.

References

B. Bojarski and T. Iwaniec, $p$-harmonic equation and quasiregular mappings, Partial differential equations (Warsaw, 1984) Banach Center Publ., vol. 19, PWN, Warsaw, 1987, pp. 25–38. MR 1055157
E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827–850. MR 709038, DOI 10.1016/0362-546X(83)90061-5
Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333–363. MR 649348, DOI 10.1002/cpa.3160350303
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
Vesa Julin and Petri Juutinen, A new proof for the equivalence of weak and viscosity solutions for the $p$-Laplace equation, Comm. Partial Differential Equations 37 (2012), no. 5, 934–946. MR 2915869, DOI 10.1080/03605302.2011.615878
Petri Juutinen, Peter Lindqvist, and Juan J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal. 33 (2001), no. 3, 699–717. MR 1871417, DOI 10.1137/S0036141000372179
Bernd Kawohl, Juan Manfredi, and Mikko Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl. (9) 97 (2012), no. 2, 173–188 (English, with English and French summaries). MR 2875296, DOI 10.1016/j.matpur.2011.07.001
Hannes Luiro, Mikko Parviainen, and Eero Saksman, Harnack’s inequality for $p$-harmonic functions via stochastic games, Comm. Partial Differential Equations 38 (2013), no. 11, 1985–2003. MR 3169768, DOI 10.1080/03605302.2013.814068
Hannes Luiro, Mikko Parviainen, and Eero Saksman, On the existence and uniqueness of $p$-harmonious functions, Differential Integral Equations 27 (2014), no. 3-4, 201–216. MR 3161602
Juan J. Manfredi, Mikko Parviainen, and Julio D. Rossi, An asymptotic mean value characterization for $p$-harmonic functions, Proc. Amer. Math. Soc. 138 (2010), no. 3, 881–889. MR 2566554, DOI 10.1090/S0002-9939-09-10183-1
Juan J. Manfredi, Mikko Parviainen, and Julio D. Rossi, On the definition and properties of $p$-harmonious functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 2, 215–241. MR 3011990
Sean Meyn and Richard L. Tweedie, Markov chains and stochastic stability, 2nd ed., Cambridge University Press, Cambridge, 2009. With a prologue by Peter W. Glynn. MR 2509253, DOI 10.1017/CBO9780511626630
Yuval Peres and Scott Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian, Duke Math. J. 145 (2008), no. 1, 91–120. MR 2451291, DOI 10.1215/00127094-2008-048
Yuval Peres, Oded Schramm, Scott Sheffield, and David B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 167–210. MR 2449057, DOI 10.1090/S0894-0347-08-00606-1
Peter Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150. MR 727034, DOI 10.1016/0022-0396(84)90105-0
K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), no. 3-4, 219–240. MR 474389, DOI 10.1007/BF02392316
N. N. Ural′ceva, Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184–222 (Russian). MR 0244628