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Do you know any reference which says something about the: Smoothness of the Sobolev space $W^{1,p}(\Omega)$ i.e. if the duality mapping $J\colon W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ is a singleton. …

When I study parabolic pde's I often came across the following type of Bochner spaces $L^p([a,b];L^{q}(\Omega),\ W^{1,p}([a,b];L^{q}(\Omega))$ and $L^{q}([a,b];W^{1,p}(\Omega))$ where $p,q\geq 1$ and …

Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$. My question: knowing that $f,g\in W^{1,p}(\Omega)$ for what $p$ can we conclude that $f\cdot g\in …

Let $w_1,w_2\in W^{1,p}(\Omega)$ be two functions with $w_1,w_2>0$ and $\dfrac{w_2}{w_1},\dfrac{w_1}{w_2}\in L^{\infty}(\Omega)$, where $\Omega\subset\mathbb{R}^N$ is a bounded domain (i.e. open and c …

I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on semigroups of linear operators, I found on many places properties of the Neumann Laplacian. In …

In the book of S.Osher & R.Fedkiw - Level set methods and dynamic implicit surfaces, at page 15, is stated without proof, a formula like that: $$\text{Per}_{\Omega}(\omega)=\lim_{\varepsilon\to 0} \in …

Is there a way to show in a relatively simple manner that for a Lipschitz bounded, connected, open domain $\Omega\subset\mathbb{R}^N$ for any $f\in L^2(\Omega)$ the solution of the problem: $$\begin{c …

Consider that $u_1,u_2:\Omega\to (0,\infty)$ where $\Omega\subset\mathbb{R}^N$ is an open set. We know that $u_1,u_2\in W^{1,p}(\Omega)$ for some $p>1$ and $\dfrac{u_1}{u_2},\ \dfrac{u_2}{u_1}\in L^{\ …

Let $\Omega\subseteq\mathbb{R}^N$ be a bounded and smooth domain. If $z:[0,T]\to L^2(\Omega)$ is a function in $H^1([0,T],L^2(\Omega))$ with the property that $z'(t)(x):=z'(t,x)>0$ a.e. on $\Omega$ fo …

The question I have in mind is the following: how can we prove that for any $v\in H^1(\Omega)$ and for any $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ the Gauss-Green identity takes place $$\int …

Let $\Omega$ be an open set from $\mathbb{R}^N$. How can we prove that if $u,v\in H^1(0,T,L^2(\Omega))$ (in Bochner sense) then $(u\cdot v)'\in L^2(0,T,L^1(\Omega))$ with $(uv)'=u'v+v'u$ and the integ …

I have a general question. Suppose that we have the following simple evolution problem $\begin{cases} \dfrac{\partial u}{\partial t}-\Delta u=f(u), & (t,x)\in (0,\infty)\times\Omega\\ \dfrac{\partial …

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, and $u\in L^2(\Omega)$ with $0\leq u(x)\leq 1$ a.e. on $\Omega$. It is well known that $C^{\infty}_c(\Omega)$ is dense in $L^2(\Omega)$. Because $C …

I try to understand the proof the lemma given at page 54 in Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations. Here it is a screenshot: Here is what I did: $$-u(x)=u(y)-u(x)=\int …

I am interested in proving the existence and regularity of solution to the following problem: $$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-\Delta y(t,x)+c(t,x)y(t,x)=f(t,x), & (t,x)\in (0,T)\ti …