Elliptic operators and maximal regularity on periodic little-Hölder spaces

Abstract

We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is satisfied in the setting of periodic little-Hölder spaces, provided the coefficients of the differential operator satisfy minimal regularity assumptions. We address parameter-dependent elliptic equations, deriving invertibility and resolvent bounds which lead to results on generation of analytic semigroups. We also demonstrate that the techniques and results of the paper hold for elliptic differential operators with operator-valued coefficients, in the setting of vector-valued functions. IntroductionIn this paper we consider the following abstract periodic inhomogeneous equationwhere A(·, D) := 2m k=0 b k (·)D k is a differential operator of order 2m, with variable coefficients b k : R → C. Further, we enforce periodic boundary conditions on the problem by assuming that the given functions u 0 , b k , f (t, ·), for t ≥ 0, k = 0, . . . , 2m, are all 2π -periodic in x ∈ R. Hence, we will be looking for solutions u(t, ·) which also exhibit 2π -periodicity on R, for t > 0. We will also consider the more general setting of vector-valued functions u 0 , f (t, ·), u(t, ·) : R → E, and operator-valued coefficients b k : R → L(E), for an arbitrary Banach space E over C. This more general setting is discussed in Sect. 6.Understanding the nature of solutions (i.e., existence, uniqueness and regularity) to inhomogeneous equations of this form is integral to the study of abstract quasilinear equations. In the quasilinear setting, we see that (0.1) takes the form ∂ t u + A(u, D)u = F(u), (2000): Primary 35G16, 35J30; Secondary 35K59, 42A45. Mathematics Subject Classification

Stability and bifurcation of equilibria for the axisymmetric averaged mean curvature flow

“…where {e L−t : t ≥ 0} is the analytic semigroup in P − (E 0 ) generated by L − and {e L+t : t ∈ R} is the group in P + (E 0 ) generated by the bounded operator L + . From [22,Theorem 5.2] one sees that E 0 (J), E 1 (J) is a pair of maximal regularity for −L and it is easy to see that −L − inherits the property of maximal regularity. In particular, the pair P − (E 0 (J)), P − (E 1 (J)) is a pair of maximal regularity for −L − .…”

On Well-Posedness, Stability, and Bifurcation for the Axisymmetric Surface Diffusion Flow

“…Additionally, we state conditions for global existence of the semiflow induced by (1.3). We rely on the theory developed in [46] and the well-posedness results for quasilinear equations with maximal regularity provided in [12]. We include comments on how we prove these well-posedness results in an appendix.…”

“…There is a natural equivalence between functions defined on T and 2π-periodic functions on R which preserves properties of (Hölder) continuity and differentiability. In particular, we will be working with the so-called periodic little-Hölder spaces h σ (T), for σ ∈ R + \ Z. Definitions and basic properties of periodic little-Hölder spaces, as well as details on the connection between spaces of functions on T and 2π-periodic functions on R can be found in [46] and the references therein. For the readers convenience, we provide a brief definition of h σ (T) below.…”

“…for k ∈ N 0 and α ∈ (0, 1) which is a Banach algebra with pointwise multiplication of functions and equipped with the norm · h k+α := · C k+α (T) inherited from C k+α (T). For equivalent definitions and more properties of the periodic little-Hölder spaces, see [46,Section 1]. In order to make explicit the quasilinear structure of (1.3), we reformulate the problem.…”

UMD Banach spaces and the maximal regularity for the square root of several operators

“…We consider the Cauchy problem for a certain C > 0 which does not depend on f . This property of maximal regularity is important, for instance, in nonlinear problems and a lot of authors have studied it in the last years (see [3], [5], [6], [7], [8], [9], [24], [27] and [41], amongst others).…”