Given:
$\phi(x,\lambda) : \Omega$X$\mathbb R \to \mathbb R^{n}$ be a Caratheodory vector such that for each $M \gt 0 $, $\alpha_{M}(x) = max_{|u| \leq M} |\phi(x,u)| \in L^{2}_{loc} (\Omega)$ .
Also given:
for each $p \in \mathbb R$ ; $div_{x}[\int_{p}^{\infty}( \phi(x,\lambda) - \phi(x,p))d\nu_{x}^{k}(\lambda)]$ is pre-compact in $H^{-1}_{loc}(\Omega)$ ; where $\nu_{x}^{k}$ (for $k \in \mathbb N)$ is a bounded sequence of measure valued functions.
Now,first question arises for me:
What is the meaning of "for each $p \in \mathbb R$ ; $div_{x}[\int_{p}^{\infty}( \phi(x,\lambda) - \phi(x,p))d\nu_{x}^{k}(\lambda)]$ is pre-compact in $H^{-1}_{loc}(\Omega)$ " here??
Does that mean $|| div_{x}[\int_{p}^{\infty}( \phi(x,\lambda) - \phi(x,p))d\nu_{x}^{k}(\lambda)]||_{H^{-1}_{loc}(\Omega)} \to 0$ as $k \to \infty$ ??
i.e. For any $\psi \in H^{1}_{0}(\Omega)$; $|\int \int div_{x}[\int_{p}^{\infty}( \phi(x,\lambda) - \phi(x,p))d\nu_{x}^{k}(\lambda)]\psi(x,\lambda)| \leq C k^{-p}||\psi||_{H^{1}_{0}(\Omega)}$ for some $p \gt 0$ ??
Secondly:
If additionally it is given that:
$\nu^{k_{r}}_{x} \rightharpoonup \nu_{x}^{0}$ i.e. [$\forall f(\lambda) \in C(\mathbb R)$;$ \int_{\Omega} [\int_{\mathbb R} f(\lambda) d\nu^{k_{r}}_{x}]g(x)dx \to \int_{\Omega} [\int_{\mathbb R} f(\lambda) d\nu^{0}_{x}]g(x)dx$; $ \forall g(x) \in L^{1}(\Omega)$ ] as $r \to \infty$ ;
then combining with the pre compactness criterion how to show that $div_{y}[\int_{p}^{\infty}(\phi(y,\lambda)-\phi(y,p))d\gamma_{y}^{k_{r}}(\lambda)] \to 0$ ; as $r \to \infty$ in $H^{-1}_{loc}(\Omega)$ (Where $\gamma_{x}^{k_{r}} = \nu_{x}^{k_{r}} - \nu_{x}^{0}$)
Someone please let me know !! Thank You!!
