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Suppose that $\Omega \subset \Bbb R^d$ is a sufficiently nice domain. From the examples of orthogonal bases in Hilbert space cases (or looking at a wavelets basis), it seems natural to me that one may …

Let $\Omega\subset \Bbb R^n$ be a $C^{2}$ domain (open and bounded) and let $p\in(1,\infty)$. Suppose $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ is a weak solution to the fourth-order elliptic system …

Recently, I asked a somewhat related question here. In the comment section, I found the formula $$ \lim_{r\to 0}\frac{1}{r}\int_{\Omega_r} f(x)\,dx = \int_{\partial \Omega}f(\sigma)\,d\mathcal{H}^{n-1 …

Let $\Omega$ be an open domain with nice boundary and $u\in W^{1,p}(\Omega)$. I believe that $|u|^p\in W^{1,1}$ with $$ \nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u $$ but couldn't find a good r …

Let $\Omega$ be a domain with smooth boundary. Let $S\subset W^{1,1}(\Omega)$ be a relatively weakly compact set. Is it true that $(S,w)$ is metrizable? Since $S$ is relatively weakly compact, i …

In the book Sobolev Spaces with Application of Maz'ya, $\mathring {L^k_p}(\Omega)$ is defined to be the completion of $\mathcal D(\Bbb R^n)$ under the norm $||\nabla^ku||_{L_p(\Omega)}$. For nice dom …

Following Aubin's book "Some nonlinear problems in Riemannian geometry", we use the notation $$ |\nabla^r \psi|^2 = \nabla_{\alpha_1}\cdots \nabla_{\alpha_r}\psi \nabla^{\alpha_1}\cdots \nabla^{\alpha …

For concreteness let's assume that $u\in W^{1,2}(\Bbb R^2).$ It is well known that $$ \|u\|_4\le C \|u\|_2^{\frac 12} \|\nabla u\|_2^{\frac 12}. $$ This is also true if $u\in W^{1,2}_0(\Omega)$ for a …

Let the space $L_{p,q}(\Bbb R^n_T)$ be defined as the set of all measurable $f:\Bbb R^{n}\times(0,T)\to\Bbb R$ such that $||f||_{p,q}<\infty$, where $$ ||f||_{p,q}:=\left(\int_0^T\left( \int_{\Bbb R^n …