Sharp regularity estimates for second order fully nonlinear parabolic equations

Abstract

ABSTRACT. We prove sharp regularity estimates for viscosity solutions of fully nonlinear parabolic equations of the formwhere F is elliptic with respect to the Hessian argument and f ∈ L p,q (Q 1 ). The quantity κ(n, p,q) := n p + 2 q determines to which regularity regime a solution of (Eq) belongs. We prove that when 1 < κ(n, p,q) < 2 − ε F , solutions are parabolic-Hölder continuous for a sharp, quantitative exponent 0 < α(n, p,q) < 1. Precisely at the critical borderline case, κ(n, p,q) = 1, we obtain sharp Log-Lipschitz regularity estimates. When 0 < κ(n, p,q) < 1, solutions are locally of class C 1+σ, 1+σ 2 and in the limiting case κ(n, p,q) = 0, we show C 1,Log-Lip regularity estimates provided F has "better" a priori estimates.

Sharp regularity estimates for quasilinear evolution equations

“…where {L k } k∈N is sequence of affine functions (compare with [4], [14,Theorem 2] and [19,Section 5 and 6] in the fully nonlinear setting). Nevertheless, it provides the following information on the oscillation of u inQ λ .…”

“…v t − div(A(x, t)∇v) = f, where A(x, t) is (Hölder) continuous and 0 < a < A(x, t) < b < ∞, for some constants a and b. Thus, v ∈ C 1+β * locally (see [19,28]), where the sharp exponent is given by…”

“…weak solutions of (1.1) are of class C α for some α ∈ (0, 1). Using compactness and geometric tangential methods (see [3,4,11,12,14,19,36,37]) and intrinsic scaling techniques (see [16,20,24,41] In this work we will solve it in the second scenario. More precisely, our main result reveals that bounded week solutions of (1.1) are locally of the class C 1+α (in the parabolic sense) in the critical zone (i.e.…”

“…along the set where gradient vanishes), where the diffusivity of the equation collapses (see, for example, [15,16,17,18,37] and [38], where sharp and improved regularity estimates are obtained along the set of certain degenerate points of solutions). Furthermore, unlike [16], [19], [39], we also treat the singular case, i.e., when max 1, 2n n+2 < p < 2, which is a non-trivial task, because in this setting the degeneracy degree of p-Laplacian blows-up along critical points.…”

“…Heuristically, in order to obtain the desired C 1+α regularity estimate, one should approach solutions by suitable affine functions. However, in a specific iterative scheme, these functions "are not in the kernel of operator", which provides an accumulative error in each step of approximation (see section 3 for details, also compare with [4], [19,Section 4 and 5] and [37]). To overcome this obstacle, we use a technique based on the notion of geometric tangential analysis.…”