We consider viscosity solutions to non-homogeneous degenerate and singular parabolic equations of the p-Laplacian type and in non-divergence form. We provide local Hölder and Lipschitz estimates for the solutions. In the degenerate case, we prove the Hölder regularity of the gradient. Our study is based on a combination of the method of alternatives and the improvement of flatness estimates.
We study local regularity properties of value functions of time-dependent tug-of-war games. For games with constant probabilities we get local Lipschitz continuity. For more general games with probabilities depending on space and time we obtain Hölder and Harnack estimates. The games have a connection to the normalized p(x, t)parabolic equation (n + p(x, t))ut = ∆u + (p(x, t) − 2)∆ N ∞ u.
Abstract. We prove local Lipschitz continuity and Harnack's inequality for value functions of the stochastic game tug-of-war with noise and running payoff. As a consequence, we obtain game-theoretic proofs for the same regularity properties for viscosity solutions of the inhomogeneous p-Laplace equation when p > 2.
We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in Ω ⊂ R n . The method of the proof is based on a game-theoretic idea to estimate the value of a related game defined in Ω × Ω via couplings.Date: June 29, 2018. 2010 Mathematics Subject Classification. 91A05, 91A15, 91A50, 35B65, 35J60, 35J92.
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