All Questions

Let $H$ be the space of measurable functions on $(0,1)$ such that $$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$ Let $C>0$ be a constant. Suppose that $W \...

In my work I stumbled upon a proposition (without proof, alas), which I can't really prove. Suppose we have a family of functions $\left\{\phi_\epsilon (t,x,v)\right\}_{\epsilon\in(0,1]}$, and $M(v)$ ...