Hölder regularity for the gradient of the inhomogeneous parabolic normalized p-Laplacian

Abstract

In this paper we study an evolution equation involving the normalized p-Laplacian and a bounded continuous source term. The normalized p-Laplacian is in non divergence form and arises for example from stochastic tug-of-war games with noise. We prove local C α, α 2 regularity for the spatial gradient of the viscosity solutions. The proof is based on an improvement of flatness and proceeds by iteration.Recently, a connection between the theory of stochastic tug-of-war games and non-linear equations of p-Laplacian type has been investigated. In the elliptic case, this connection started with the seminal

Remarks on regularity for p-Laplacian type equations in non-divergence form

“…Let us mention two special cases. The case γ = 0 corresponds to the normalized p-Laplacian ∆ N p u := ∆u + (p − 2) D 2 u Du |Du| , Du |Du| , and the regularity of the gradient was studied in [3,22] using viscosity theory methods. The case γ = p − 2 corresponds to the usual parabolic p-Laplace equations, and it was shown in [23] that bounded weak solutions and viscosity solutions are equivalent.…”

“…The method of improvement of flatness was already used in the elliptic case [4,8,20] and in the uniformly parabolic case [3]. In these works, one ends up working with equations with ellipticity constants not depending on the slope, making the improvement of flatness working for all k ∈ N. The method of alternatives is classical when studying the regularity for p-Laplacian type equations [14,15,28].…”

Gradient regularity for a singular parabolic equation in non-divergence form

“…In the case γ = 0 and f ≡ 0, Jin and Silvestre [24] showed the Hölder regularity of the gradient for solutions of (1.1), and the result was generalized by Imbert, Jin, and Silvestre [22] to the whole range −1 < γ < ∞. In the non-homogeneous case, Attouchi and Parviainen [3] treated C 1,α -regularity in the uniformly parabolic case γ = 0, and later the same result was proved by Attouchi [2] in the degenerate case γ ∈ (0, ∞). For related regularity results in the elliptic setting, we refer to [35,10,23,11,4,5,7].…”

“…Another special case is γ = 0, when the equation reads u t − ∆ N p u = f. The motivation to study parabolic equations involving the normalized p-Laplacian stems partially from connections to time-dependent tug-of-war games [29,31,20] 5956 AMAL ATTOUCHI AND EERO RUOSTEENOJA and image processing [18]. For regularity results concerning this equation, we refer to [6,24,3,8,21,19].…”

The tusk condition and Petrovskiĭ criterion for the normalized p‐parabolic equation

“…Boundary regularity for the normalized ‐parabolic equation was first studied by Banerjee–Garofalo (see also their earlier paper ). More recently, Jin–Silvestre established the interior ‐regularity for solutions of (see also Imbert–Jin–Silvestre , Attouchi–Parviainen and Parviainen–Ruosteenoja for related regularity results).…”