BASIC ESTIMATES FOR SOLUTIONS OF A CLASS OF NONLOCAL ELLIPTIC AND PARABOLIC EQUATIONS

摘要

In this work we consider the problems {ℓu = f in Ω, u = 0 inR~NΩ, and {u_t+ℓu = f in Q_τ ≡ Ω x (0,T), u(x, t) = 0 in (R~n Ω) x (0, T), u(x,0)= 0 in Ω, where ℓ is a nonlocal differential operator and Ω is a bounded domain in R~N, with Lipschitz boundary. The main goal of this work is to study existence, uniqueness and summa-bility of the solution u with respect to the summability of the datum /. In the process we establish an L~p-theory, for p ≥ 1, associated to these problems and we prove some useful inequalities for the applications.