Maximal $$L_{p}$$ L p -regularity of non-local boundary value problems

Abstract

We investigate the R-boundedness of operator families belonging to the Boutet de Monvel calculus. In particular, we show that weakly and strongly parameter-dependent Green operators of nonpositive order are Rbounded. Such operators appear as resolvents of non-local (pseudodifferential) boundary value problems. As a consequence, we obtain maximal Lp-regularity for such boundary value problems. An example is given by the reduced Stokes equation in waveguides.

Fourier methods for fractional-order operators

“…e.g. Denk and Seiler [DS15]. We expect that perturbation methods would allow x-dependent symbols to some extent; this remains to be investigated.…”

Generation of Semigroups for Vector-Valued Pseudodifferential Operators on the Torus

“…Then u ε ∈ C ∞ (T n , E), and Theorem 3. 16 ≤ C 4 u − u ε B r 1 ∞,1 (T n ,E) → 0 (ε ց 0). Thus, (λ + A a,k ) −1 u ε → (λ + A a,k )u (ε ց 0) in W k ∞ (T n , E).…”

“…This approach was used, e.g., by Denk-Nau [15], Favini-Giudetti-Yakubov [17], Nau-Saal [23], and Rabinovich [24]. For an application to the Stokes system, see Denk-Seiler [16].…”

“…= lim εց0 T n K ε (η, λ)(∂ α u)(x − η)dη (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) with K ε from Lemma 4.8. From (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16), Lemma 4.8 and dominated convergence, we get…”

“…) is called a classical solution of (4-18) if u(t) ∈ D(A a,k (t)) for all t ∈ (0, T ], u(0) = u 0 , and if (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) holds for all t ∈ (0, T ]. Theorem 4.12.…”