Towards the $C^{p^\prime}$-Regularity Conjecture in Higher Dimensions

Sharp regularity estimates for quasilinear evolution equations

“…which is the expected optimal regularity in certain scenarios of elliptic equations (see [5], [6] and [8]).…”

“…We then iterate this estimate in a systematic manner, properly adjusted to the intrinsic scaling of equation. Inspired by the recent results from [5,6,8] and [39], we obtain an estimate (Theorem 3.2), which provides a precise control of the oscillation of weak solution of (1.1) in terms of the magnitude of its gradient.…”

“…[10] and [25] for complete essays on regularity of evolution equations with degenerate diffusion). Although weak solutions of (1.1) under the compatibility assumptions (C) are known to be locally of the class C 1+α (in the parabolic sense) for some α ∈ (0, 1), the sharp exponent is known only for some specific cases (see [5,6,26,28,33]). This type of quantitative information plays an essential role in the study of blow-up analysis, related geometric and free boundary problems and for proving Liouville type results (see [3,4,12,16,18,40] for some enlightening examples).…”

Regularity properties for p−dead core problems and their asymptotic limit as p→∞

“…However, for such an example, fixed , one observes that , where Here stands for the integer part. Note that when , In fact, regularity with is the optimal regularity expected (and proved in some cases) for weak solutions of the Laplacian operator with bounded right‐hand side, see for a systematic treatment of this subject (called regularity conjecture) and compare with in the context of obstacle problems. Furthermore, note that provided that , which me...…”

“…In fact, C α loc regularity with α := p p−1 = 1 + 1 p−1 is the optimal regularity expected (and proved in some cases) for weak solutions of the p−Laplacian operator with bounded right-hand side, see [3] for a systematic treatment of this subject (called C p regularity conjecture) and compare with [2] in the context of p−obstacle problems. Furthermore, note that α(p, q) > 2 provided that q > max{0, p−2 2 }, which means that it is a classical solution, even across the free boundary.…”