"Newsweek, a fat-wallet book that criticizes other people for rehashing the news, is represented here [in Rio] by a virtually unpaid British stringer whose stories are published with all the tack-sharp regularity of total eclipses of the sun," Thompson wrote.
We obtain sharp geometric regularity estimates for solutions along the boundary of plateaus sets.
We establish sharp interior and boundary regularity estimates for solutions to ∂tu−Lu= f t,x) in I×Ω, with I⊂R and Ω⊂Rn.
Under some regularity conditions, we obtain sharp bounds for the minimal errors for approximation and upper bounds for the minimal errors for integration.
For example, in [19], they studied the existence of nonuniform exponential dichotomy, and obtained sharp lower and upper bound for the regularity coefficient.
Now, we give the time regularity of solution (3.2) and sharp upper and lower bounds for the (L^{2} -norm of increments.
Moreover, we address a borderline problem in the regularity theory and give a sharp criterion for continuity of the gradient of a solution.
This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations with a nonlocal operator P Pu+∂tu= f x,t), for x∈Ω⊂Rn,t∈I⊂R.
