The Laplacian on Cylindrical Domains

$L^{p}-L^{q}$-Maximal regularity of the Van Wijngaarden–Eringen equation in a cylindrical domain

“…This proves to be a strong tool for the treatment of related elliptic and parabolic, as well as hyperbolic problems. Operators in cylindrical domains with a similar splitting property as in the present paper were, in the case of an infinite cylinder, also considered by Nau et al in [9,[19][20][21][22].…”

“…has L p -L q -maximal regularity. Then, using the results of [20], we establish our main findings concerning (1.2), namely: for any given f ∈ L p (T, L q ( )) and under the condition…”

“…where denotes a cylindrical decomposition of the Dirichlet Laplacian operator on L q ( ) with respect to the two cross-sections, i.e., = 1 + 2 , where i acts on the according component of . Following [20] we introduce L q -realizations q,i = i as follows:…”

“…It is well known that many situations in applied sciences naturally lead to problems in cylindrical domains . We refer, e.g., to the textbooks [5] and [6] and the references [9,20,22] for a demonstration of the significance of problems on such .…”

“…In this paper, we directly apply general results of [3] and [20] to our case of the VWE equation and obtain a L p -L q -maximal regularity result. The main difficulty relies in the verification of the so-called R-boundedness property that must be satisfied by certain sets of operators.…”