Problem: (1) Narrate the resonance theorem. (2) Let $X$ be a Banach space, and denote by $C_w([0,T];X)$ be all the maps $$\bex \ba{cccc} u:&[0,T]&\to& X\\ &t&\mapsto &u(t) \ea \eex$$ such that for any functional $\phi\in X‘$, the function $[0,T]\ni t\mapsto \sef{\phi,u(t)}$ is continuous. Utilize (1) to show $C_w([0,T];X)\subset L^\infty([0,T];X)$.

Proof: (1) The resonance theorem reads as follows. Let $X$ be a Banach space, $Y$ be a normed linear space. Suppose that $\sed{A_\lm}_{\lm\in \vLm}$ is a family of bounded linear map from $X$ to $Y$, and satisfies $$\bex \sup_{\lm\in \vLm} \sen{A_\lm x}<\infty,\quad\forall\ x\in X. \eex$$ Then there exists a positive constant $M$ such that for any $\lm\in \vLm$, $\sen{A_\lm}\leq M$.

(2) As is well-known, $X$ can be viewed as a bounded linear functional on $X‘$; that is, $X\subset X‘‘$. Let $u\in C_w([0,T];X)$. We will use (1) to show $u\in L^\infty(0,T;X)$. Indeed, for $\forall\ \phi\in X‘$, $\sef{\phi,u(t)}$ is a continuous function on $[0,T]$, and thus is bounded, i.e., $$\bex \sup_{t\in [0,T]}|\sef{\phi,u(t)}|<\infty,\quad\forall\phi\in X‘. \eex$$ Consequently, $\sed{u(t)}_{t\in [0,T]}$ viewed as a family of bounded linear functional on $X‘$, satisfies the hypothesis of (1), and hence there exists an $M>0$ such that $$\bex \sup_{t\in [0,T]}\sen{u(t)}_{X} =\sup_{t\in [0,T]}\sen{u(t)}_{X‘‘} \leq M. \eex$$

2017-2018-2偏微分方程復習題解析8