Maximal regularity for non-autonomous evolution equations governed by forms having less regularity

2014/11/01 by Ouhabaz, El Maati · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1411.0139

Abstract

We consider the maximal regularity problem for non-autonomous evolution equations \ u'(t) + A(t) u(t) · amp;= · amp; f(t), t ∈ (0, τ] u(0) · amp;= · amp;u₀. . Each operator A(t) is associated with a sesquilinear form \mathfraka(t) on a Hilbert space H. We assume that these forms all have the same domain V. It is proved in \citeHO14 that if the forms have some regularity with respect to t (e.g., piecewise α-Hölder continuous for some α> 1/2) then the above problem has maximal L_p--regularity for all u₀ in the real-interpolation space (H, D(A(0)))_1-1/p,p. In this paper we prove that the regularity required there can be improved for a class of sesquilinear forms. The forms considered here are such that the difference \mathfraka(t;⋅,⋅) - \mathfraka(s; ⋅,⋅) is continuous on a larger space than the common domain V. We give three examples which illustrate our results.

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