Pointwise differentiability of weak solutions of parabolic equations with measurable coefficients
Paweł Strzelecki · Annales Academiae Scientiarum Fennicae Series A I Mathematica · 1992
We prove that weak solutions of a parabolic equation with measurable coefficients u1-div[a(a,t).r,]= (b(r,t),u,) arc totally differentiable (in the classical sense) almost everywhere with respect to the Lebesgue measure if u1 € ,L[" We assume that the equation (1,) is defined in an open domain Q : G x (0, ?)C R'*1 , where G is an open domain in R" and 7 ) 0, and that the coefficients qH: akl bt are bounded measurable functions of (r,t) fulfilling the following conditions with some K ) 1:(2) I{-r < Do*,(r, t)VkV S K &,1 for all (r,t) e Q and all unit vectors V € R", and k for all (x,t) e Q.Let Wr'z(Q) denote the Sobolev space of square integrable functions on Q with first order distributional partial derivatives in Lz(Q).Then u e W1'2(Q) is called a weak solution of (1) if and only if the integral identity (4) [ax.Ex""#,-\*,#)drdt:0 a 1991