Calderón–Zygmund estimates for a parabolic Schrodinger system on Reifenberg domains

摘要

Let N∈{1,2,3,…},n∈{2,3,4,…}$$ Nin left{1,2,3,dots right},nin left{2,3,4,dots right} $$ and Ω⊂ℝn$$ Omega subset {mathbb{R}}#x0005E;n $$ be a Reifenberg flat domain. Consider the parabolic Schrodinger system ut−div(a(x,t)∇u)+V(x)u=divFinΩT:=Ω×(0,T),u=0on∂Ω×0,T,u=u0onΩ×{0}.$$ left{begin{array}{rll}{u}_t-operatorname{div}left(aleft(x,tright)nabla uright)#x0002B;V(x)kern0.1em uamp; #x0003D;operatorname{div}Famp; mathrm{in}kern0.4em {Omega}_T:#x0003D; Omega times left(0,Tright), {}uamp; #x0003D;0amp; mathrm{on}kern0.4em mathrm{partial Omega}times left0,Tright, {}uamp; #x0003D;{u}_0amp; mathrm{on}kern0.4em Omega times left{0right}.end{array}right. $$ Within an appropriate framework, we show that F∈Lq(ΩT,ℝNn)∇u0∈Lq(Ω,ℝN)⟹∇u,V12u∈Lq(ΩT,ℝN)$$ left{begin{array}{c}mid Fmid in {L}#x0005E;qleft({Omega}_T,{mathbb{R}}#x0005E;{Nn}right) {}mid nabla {u}_0mid in {L}#x0005E;qleft(Omega, {mathbb{R}}#x0005E;Nright)end{array}right.kern0.30em Longrightarrow kern0.30em mid nabla umid, {V}#x0005E;{frac{1}{2}}kern0.1em mid umid in {L}#x0005E;qleft({Omega}_T,{mathbb{R}}#x0005E;Nright) $$ for a certain range of q$$ q $$.