All Questions

Let $\Omega \subset \mathbb R^n$ be a bounded domain with a Lipschitz boundary and $n > p \ge 1$. It is well known that up to the critical exponent $p^* = pn/(n − p)$, i.e. $q < p^*$, the ...

NOTE: I also asked this question here in MSE. In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...

I was reading one preprint and stumbled upon a part in the proof where one particular embedding was used. Namely: Let $\Omega\subset{\bf R}^2$ be open and bounded and let $p\in\langle 1,2\rangle$. ...

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 \...

From the Rellich-Kondrachov theorem we know that $H^m(\Omega)\hookrightarrow_c L^2(\Omega)$ when $\Omega$ is bounded of class $C^1$ and $m\geq 1$ is an integer. Also this is not true if $\Omega:=\...

Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...