TY - JOUR

T1 - Global gradient estimates for general nonlinear parabolic equations in nonsmooth domains

AU - Byun, Sun Sig

AU - Ok, Jihoon

AU - Ryu, Seungjin

PY - 2013/6/1

Y1 - 2013/6/1

N2 - We establish the natural Calderón-Zygmund theory for a nonlinear parabolic equation of p-Laplacian type in divergence form,. (0.1)ut-diva(Du,x,t)=div(|F|p-2F)in ΩT, by essentially proving that. (0.2)|F|p∈Lq(ΩT)⇒|Du|p∈Lq(ΩT), for every q∈. [1, ∞). The equation under consideration is of general type and not necessarily of variation form, the involved nonlinearity a= a(ξ, x, t) is assumed to have a small BMO semi-norm with respect to (x, t)-variables and the lateral boundary ∂. Ω of the domain is assumed to be δ-Reifenberg flat. As a consequence, we are able to not only relax the known regularity requirements on the nonlinearity for such a regularity theory, but also extend local results to a global one in a nonsmooth domain whose boundary has a fractal property. We also find an optimal regularity estimate in Orlicz-Sobolev spaces for such nonlinear parabolic problems.

AB - We establish the natural Calderón-Zygmund theory for a nonlinear parabolic equation of p-Laplacian type in divergence form,. (0.1)ut-diva(Du,x,t)=div(|F|p-2F)in ΩT, by essentially proving that. (0.2)|F|p∈Lq(ΩT)⇒|Du|p∈Lq(ΩT), for every q∈. [1, ∞). The equation under consideration is of general type and not necessarily of variation form, the involved nonlinearity a= a(ξ, x, t) is assumed to have a small BMO semi-norm with respect to (x, t)-variables and the lateral boundary ∂. Ω of the domain is assumed to be δ-Reifenberg flat. As a consequence, we are able to not only relax the known regularity requirements on the nonlinearity for such a regularity theory, but also extend local results to a global one in a nonsmooth domain whose boundary has a fractal property. We also find an optimal regularity estimate in Orlicz-Sobolev spaces for such nonlinear parabolic problems.

KW - BMO nonlinearity

KW - Calderón-Zygmund theory

KW - Global estimate

KW - Reifenberg domain

UR - http://www.scopus.com/inward/record.url?scp=84875621810&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2013.03.004

DO - 10.1016/j.jde.2013.03.004

M3 - Article

AN - SCOPUS:84875621810

SN - 0022-0396

VL - 254

SP - 4290

EP - 4326

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 11

ER -