Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

Abstract

Abstract. Consider the nonlinear parabolic equation in the formwhere T > 0 and Ω is a Reifenberg domain. We suppose that the nonlinearity a(ξ, x, t) has a small BMO norm with respect to x and is merely measurable and bounded with respect to the time variable t. In this paper, we prove the global Calderón-Zygmund estimates for the weak solution to this parabolic problem in the setting of Lorentz spaces which includes the estimates in Lebesgue spaces. Our global Calderón-Zygmund estimates extend certain previous results to equations with less regularity assumptions on the nonlinearity a(ξ, x, t) and to more general setting of Lorentz spaces.

Global Calderón--Zygmund theory for parabolic $p$-Laplacian system: the case $1

“…Considering the nonlinearity to have a small BM O norm, and Ω is flat in Reifenberg's sense, they obtain that if |F | p ∈ L q (Ω T ) for q > 1, then |∇u| p ∈ L q (Ω T ). Authors in [4,5] extended the results to more general setting of Lorentz spaces. We also infer interested readers to [21,25,26,27,28,29,36] for potential estimates for the spatial gradient of solutions.…”

Global Gradient Estimates for Dirichlet Problems of Elliptic Operators with a BMO Anti-Symmetric Part

“…Moreover, some estimates similar to (6.3) and (6.5) for the Dirichlet problem of some nonlinear elliptic or parabolic equations on Reifenberg flat domains were obtained in [2,10,11,44,45].…”

“…A typical example of Reifenberg flat domains is the well-known Van Koch snowflake (see, for instance, [57]). In recent years, boundary value problems of elliptic or parabolic equations on Reifenberg flat domains have been widely concerned and studied (see, for instance, [8,10,11,13,16,17,25,44,64,65]).…”

Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains

“…We point out that, for the Dirichlet problem (1.9) with F ≡ 0, the estimate (2.3) was established in[1, Theorem 2.3] under the assumptions that A is symmetric and satisfies the (δ, R)-BMO condition for some small δ ∈ (0, ∞) and some R ∈ (0, ∞), and that Ω is a bounded Lipschitz domain with a small Lipschitz constant. Moreover, estimates similar to (2.3) with F ≡ 0 for the Dirichlet problem of some nonlinear elliptic or parabolic equations on Reifenberg flat domains were obtained in[2,3,11,12,51,52]. Furthermore, for the Neumann problem (N) p in bounded (semi-)convex domains, the estimate (2.3), with F M θ q (Ω) replaced by F M θ p (Ω) , was obtained in [74, Corollary 2.5].…”

“…The Reifenberg flat domain was introduced by Reifenberg [62], which naturally appears in the theory of minimal surfaces and free boundary problems. In recent years, boundary value problems of elliptic or parabolic equations on Reifenberg flat domains have been widely concerned and studied (see, for instance, [9,10,11,12,14,16,17,18,51,52] (i) Then there exists a positive constant δ 0 ∈ (0, ∞), depending only on n, p and Ω, such that, if A satisfies the (δ, R)-BMO condition for some δ ∈ (0, δ 0 ) and R ∈ (0, ∞) or A ∈ VMO(R n ), then the Robin problem (R) p with f ∈ L p (Ω; R n ), F ∈ L p * (Ω) and g ∈ W −1/p,p (∂Ω) is uniquely solvable and there exists a positive constant C, depending only on n, p and Ω, such that, for any weak solution u of the Robin problem (R) p , u ∈ W 1,p (Ω) and…”