Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maximal $L_p$-regularity for fractional problem driven by non-autonomous forms

Jia Wei He, Shi Long Li, Yong Zhou

Abstract

We investigate the maximal $L_p$-regularity in J.L. Lions' problem involving a time-fractional derivative and a non-autonomous form $a(t;\cdot,\cdot)$ on a Hilbert space $H$. This problem says whether the maximal $L_p$-regularity in $H$ hold when $t \mapsto a(t ; u, v)$ is merely continuous or even merely measurable. We prove the maximal $L_p$-regularity results when the coefficients satisfy general Dini-type continuity conditions. In particular, we construct a counterexample to negatively answer this problem, indicating the minimal Hölder-scale regularity required for positive results.

Maximal $L_p$-regularity for fractional problem driven by non-autonomous forms

Abstract

We investigate the maximal -regularity in J.L. Lions' problem involving a time-fractional derivative and a non-autonomous form on a Hilbert space . This problem says whether the maximal -regularity in hold when is merely continuous or even merely measurable. We prove the maximal -regularity results when the coefficients satisfy general Dini-type continuity conditions. In particular, we construct a counterexample to negatively answer this problem, indicating the minimal Hölder-scale regularity required for positive results.

Paper Structure

This paper contains 6 sections, 17 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The proofs of maximal $L_p$-regularity
  4. A non-symmetric counterexample
  5. Badly behaved function and extending forms
  6. The Holder exponent of the form

Key Result

Theorem 1.1

Assume that the non-autonomous forms $\{a(t , \cdot, \cdot)\}_{0 \leq t \leq \tau}$ satisfy the hypothesis (A0) and the regularity condition where $\omega:[0, \tau] \rightarrow[0, \infty)$ is a non-decreasing continuous function such that Then the problem (FP) with $u_0\in Tr^{p}_{\alpha}$ has maximal $L_p$-regularity in $H$ for all $p \in(1, \infty)$. When $\alpha>\frac{1}{p}$, If $\omega$ addi

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • ...and 27 more