Articles of sobolev spaces

Sobolev spaces on boundaries

Is the following a good definition for a Sobolev space on a boundary: Can anyone show me another source where such a space is defined? In the definition, $v \in W^{s,p}(\partial\Omega)$ if $v \circ g_i \in W^{s,p}(D_i)$. Now, does this only need to hold for one such representation $\{g_i\}$ (as the author suggests) or all […]

$H^s(\mathbb{T})$ space is a Banach algebra with pointwise product

I have ran across the following theorem but the given proof does not convince me. Theorem Let $u, v \in H^s(\mathbb{T})$ with $s>1/2$. Then the pointwise product $uv$ is in $H^s(\mathbb{T})$ and $\lVert uv\rVert_s \le C\lVert u\rVert_s\lVert v \rVert_s$ for a constant $C$ not depending on $u$ and $v$. In functional terms, this theorem is […]

Scale invariant definition of the Sobolev norm $\|\|_{m,\Omega}$ for $H^m(\Omega)$

I learned the following from Constantin and Foias’s Navier-Stokes Equations (Chapter 4): We say that a function of a bounded open set $\Omega\subset\mathbb{R}^n$, $c(\Omega)$, is scale invariant if $c(\Omega)=c(\Omega’)$ for all $\Omega’$ obtained from $\Omega$ by a rigid transformation and a dilation $x\mapsto\delta x$. Denote by $T_\delta$ the operation mapping functions defined on $\Omega$ to […]

Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the localized Sobolev space $H_{s}^{loc}$ is the set of all distributions $f\in \mathcal{D’}(U)$ such that for every precompact open set $V$ with $\overline{V}\subset U$ there exists $g\in H_{s}$ such that $g=f$ on $V.$ Fact. A […]

Example that $u\in W^{1,2}$, but $u \notin W^{1,3}$

I’m doing the calculations about the following assertion Let $\Omega$ be $\{(x,y):0<y<x^2, 0<x<1\}$. The function $u(x,y)=\log (x^2+y^2)$ belongs to $W^{1,2}(\Omega)$, which you can check by integrating $|\nabla u|^2\approx 1/x^2$ within $\Omega$. We have $\Delta u=0$, which is the nicest equation one could ask for. However, $u$ does not belong to $W^{1,3}(\Omega)$, which you also can […]

Dual space of a closed subspace of a Hilbert space

I’m reading Girault and Raviart’s book concerning Finite Element Methods for Navier-Stokes equations, and they use in the proof of one result, the following argument: As $V=\{v\in H_0^1(\Omega)^N; {\rm div}\;v=0\; \text{in}\;\Omega\}$ is closed in $H_0^1(\Omega)^N$, then V* (dual space of V) can be identified with a subspace of $H^{-1}(\Omega)^N$, where $\Omega$ is an open set […]

Can we conclude that $\Delta (\Phi\circ u)$ is a measure, given that $\Phi$ is a particular smooth function and $u$ is in some Sobolev space?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$ is such that $\Delta u$ is a measure. Let $\Phi$ be a smooth function in $\mathbb{R}$, such that $\Phi'(u)\in W_0^{1,2}(\Omega)$. Is it true that $\Delta (\Phi\circ u)$ is a measure and $$\Delta (\Phi\circ u)=(\Phi”\circ u)|\nabla u|^2+(\Phi’\circ u)\Delta u, \tag{1}$$ If the equality $$\int_\Omega \phi […]

Understanding of the characterization of $H^{-1}$ in Evans's PDE book

The following is the characterization theorem for $H^{-1}(U):=(H_0^1(U))^*$ in Evans’s Partial Differential Equations: Here is my question: The proof says that (iii) directly follows from (i). Would anybody elaborate why it is so? $U$ is an open subset of $\mathbb{R}^{n}$ $H^{1}(U)$ is the Sobolev space of $L^{2}(U)$ functions with weak derivatives in $L^{2}(U)$ $H_{0}^{1}(U)$ is […]

One-sided smooth approximation of Sobolev functions

I’m currently trying to specialise a rather general variational inequality to known simple examples to check if my assumptions on the problem are plausible. While doing this, I stepped over the following question: Given a function $u$ in some Sobolev space. (say, for instance, $u\in W^{1,2}_0(\Omega)$, $\Omega \subset \mathbb R^n$ bounded with smooth boundary) Is […]

antiderivative of $\psi'(u)$ for $u\in W^{1,2}_0((a,b))$

Let $u\in W^{1,2}_0((a,b))$ and $\psi’$ the derivative of a convex function $\Psi\in C^1(\mathbb{R})$. If I want to consider the antiderivative of $\psi'(u)$, what happens with the $u$ inside $\Psi(u)$? Is it possible to say how does the antiderivative looks like? I need this for an other proof. Regards