Global gradient estimates for general nonlinear parabolic equations in nonsmooth domains

Global estimates for quasilinear parabolic equations on Reifenberg flat domains and its applications to Riccati type parabolic equations with distributional data

“…We remark that the global gradient estimates of solutions of (1.1) obtained in Theorem 1.2 extend results in [2,3,4] to more general nonlinear structure and in the setting of weighted Lorentz spaces. Notice that Theorem 1.1 and 1.2 in the quasilinear elliptic framework are obtained in [14].…”

“…(x, t) ∈ R N × R, where Λ 1 and Λ 2 are positive constants. In addtion, we also assume that the derivatives of A with respect to ζ are bounded, that is, 4) for any ζ ∈ R N and (x, t) ∈ R N . We remark that the condition (1.4) is needed in order to ensure that the reference problems (2.5) and (2.17) in the next section have C 0,1 regularity solutions (see [11,12]), which will be used in the sequel.…”

Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

“…This result extends those in [8]. Note that in [9], the assumptions only require the nonlinearity a to have a small BMO norms with respect to both x and t, and the domain Ω is flat in Reifenberg's sense. It is worth noticing that although the estimates for the gradient of the weak solutions to the parabolic problems on Lebesgue spaces L p have been well-known, the global estimates for the gradient of the weak solutions on the Lorentz spaces for the case p = 2 are less well-known, and even have not been established so far.…”

“…Moreover, the Calderón-Zygmund estimates with measurable dependence with respect to time was obtained in [13]. (c) In [9], the global Calderón-Zygmund theory for the weak solution to the problem (1) was investigated. It was proved that if |F | p ∈ L q (Ω T ) for q > 1 then |Du| p ∈ L q (Ω T ).…”

Global Gradient Estimates for Nonlinear Elliptic Equations

“…It should be point out that one may obtain the same results based on the so-called "Harmonicanalysis-free" technique in [1]. This approach is quite effective for L p regularity estimates for nonlinear parabolic problems with no invariance property under normalization, see [2,4,14]. In addition, note that the so-called "sharp maximal function method", first introduced in [18] and later modified in [11,12,19], is useful when differential operators related to problems are bounded and linear.…”