Appears in collection : Harmonic analysis techniques for elliptic operators / Techniques d'analyse harmonique pour des opérateurs elliptiques

Given $T>0$ and Hilbert spaces $V$ and $H$ where $V \hookrightarrow H$, a mapping $a:[0, T] \times V \times V \rightarrow \mathbb{K}$ is called non-autonomous form if $a(\cdot, v, w):[0, T] \rightarrow \mathbb{K}$ is measurable for all $v, w \in V$ and$$|a(t, v, w)| \leq M|v|_{V}|w|_{V} \quad \text { for all } t \in[0, T] \text { and } v, w \in V$$for some $M \geq 0$. The form is said to be coercive if there exists $\alpha>0$ with$$\operatorname{Re} a(t, v, v) \geq \alpha|v|_{V}^{2} \quad \text { for all } t \in[0, T] \text { and } v \in V$$An elegant result of Lions shows well-posedness of the problem$$\begin{cases}u^{\prime}(t)+\mathscr{A}(t) u(t) & =f(t) \tag{P}\\ u(0) & =u_{0}\end{cases}$$where $f \in L^{2}\left(0, T ; V^{\prime}\right)$ and $u_0 \in H$. Here, we consider the usual embedding $H \hookrightarrow V^{\prime}$ and the family of operators $\mathscr{A}(t) \in$ $\mathscr{L}\left(V, V^{\prime}\right)$ given by$$\langle\mathscr{A}(t) u, v\rangle=a(t, u, v) \quad \text { for all } t \in[0, T] \text { and } v, w \in V .$$In fact, one has maximal regularity in $V^{\prime}$, i.e.$$u \in H^1\left(0, T ; V^{\prime}\right) \cap L^2(0, T ; V) .$$Particularly, all the terms $u^{\prime}, \mathscr{A}(\cdot) u(\cdot)$ and $f$ belong to $L^{2}\left(0, T ; V^{\prime}\right)$. Frequently however, the part $A(t)$ of $\mathscr{A}(t)$ in $H$ given by$$D(A(t))=\{v \in V: \mathscr{A}(t) v \in H}, \quad A(t) v=\mathscr{A}(t) v$$is more important because this operator incorporates the boundary conditions. Thus, an important problem is the following Lions' Problem (1961). If $f \in L^{2}(0, T ; H)$ and $u_{0} \in V$, does this imply $u \in H^{1}(0, T ; H)$ ? The answer is "No", even if $u_{0}=0$. A first counterexample has been given by Dominik Dier (2014). It is based on the counterexample of McIntosh showing that $V \neq D\left(A^{\frac{1}{2}}\right)$ is possible. On the other hand, if the form $a$ is sufficiently regular in time, then positive results hold by results of D. Dier, S. Fackler, E.M. Ouhabaz, C. Spina and others.

Organization of the project: The project is organized in the following parts.

1. Consider the Gelfand triple $V \hookrightarrow H \hookrightarrow V^{\prime}$ and let $\mathscr{A}: V \rightarrow V^{\prime}$ be the operator associated to an autonomous, coercive form $a$ on $V$ and let $A$ be the part of $\mathscr{A}$ in $H$. Moreover, denote by $(T(t))_{t}$ the contractive, holomorphic $C_{0}$-semigroup on $H$ generated by $-A$, cf. [AVV19, Theorem 5.8]. The goal is then to prove that $$\begin{equation²} T(\cdot) x \in H^{1}(0, T ; H) \quad \text { if and only if } \quad x \in D\left(A^{\frac{1}{2}}\right) \tag{2.5} \end{equation²}$$ The main steps in the proof are outlined in [ADF17, Section 4]. One of the main ingredients and the main focus of this talk lies in understanding that $D\left(A^{\frac{1}{2}}\right)=[H, D(A)]_{\frac{1}{2}}$. That is, the domain of the square root is an interpolation space! The proof of this fact is a special case of [Haa06, Theorem 6.6.9].

2. Lions' theorem on maximal regularity in $V^{\prime}$, cf. [AVV19, Theorem 17.15] and the above introduction. A key argument in the proof involves Lions' representation theorem, cf. [AVV19, Theorem 17.11].

3. Dier's counterexample, cf. [ADF17, Example 5.1] and [Die14].

4. A positive result: maximal regularity in $H$ for Lipschitz continuous forms. More precisely, we suppose that the nonautonomous form $a:[0, \tau] \times V \times V \rightarrow \mathbb{K}$ can be written as $a=a_{1}+b$ where $a_{1}$ and $b$ are bounded non-autonomous forms on $V$ with the following requirements: (i) $a_{1}$ is symmetric, i.e. $a_{1}(t, x, y)=\overline{a_{1}(t, y, x)}$ for $x, y \in V$ and $0 \leq t \leq \tau$; (ii) $a_{1}$ is coercive, i.e. there exists $\alpha>0$ with $a_{1}(t, x, x) \geq \alpha|x|_{V}^{2}$ for all $x \in V, 0 \leq t \leq \tau$; (iii) $a_{1}$ is Lipschitz continuous, i.e. there exist $M_{1}^{\prime} \geq 0$ with $$\left|a_{1}(t, x, y)-a_{1}(s, x, y)\right| \leq M_{1}^{\prime}|t-x||x|_{V}|y|_{V}$$ for all $0 \leq t \leq \tau$ and all $x, y \in V$; (iv) There exists $M_{b} \geq 0$ with $|b(t, x, y)| \leq M_{b}|x|_{V}|y|_{H}$ for all $0 \leq t \leq \tau$ and $x, y \in V$. Then the statement of Lions' Problem as above holds true. For reference, cf. [ADLO14] and [AVV19, Theorem 18.2]. This talk's goal is giving a proof, possibly under somewhat stronger regularity assumptions on the form. For instance, if one even assumes $C^{1}$ regularity instead of Lipschitz regularity in time, then many technicalities become easier to handle.