All Questions

The question is the following: Suppose $f : [0,\infty) \rightarrow \mathbb{R}$ is a continuous function. Can I find $\delta \in (0,\infty)$ and a function $g : [0,\delta] \rightarrow \mathbb{R}$ such ...

Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, how i can prove that the following problem: $$\text{div}(t^a\nabla u)=0,\quad\text{in }\mathbb{R}^n\times(0,\infty),$$ $$ u(x,0)=f(x),\quad\forall x\...

Given two $n$-tuple vectors $\vec{\alpha}=(\alpha_1,\cdots,\alpha_n)$ and $\vec{h}=(h_1,\cdots,h_n)$, where $h_i\ge0$, $\sum_{i=1}^nh_i=1$, and $\alpha_i\in(0,1)$, we consider a random variable $S$ on ...

Can anyone please suggest how to justify widely used formula for interchange of integral and infimum: $ \inf_{u(t)\in U}\int_{t_0}^{t_1}g(t,u(t))dt=\int_{t_0}^{t_1}\inf_{u\in U}g(t,u)dt, $ where $ U\...