For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

learn more… | top users | synonyms

2
votes
0answers
50 views
2
votes
0answers
34 views

Poincaré-like inequality

Let $\Omega\subset\mathbb{R}^3$ be an open bounded set. Let $\partial\Omega=\Gamma^1\cup\Gamma^2$, with $\Gamma^1\cap\Gamma^2=\emptyset$. We denote as $\Gamma^1_j$, $j=1,\dots,p_{\Gamma^1}+1$, the ...
2
votes
0answers
81 views

$\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$

Let $\Omega\subset \mathbb R^d$ ($d=2,3$) is a bounded Lipschitz domain. Question: Is it true that for each function $g(x)\in L^2(\Omega)$ one can find a sequence $\{g_n\}_1^\infty$ of $H_0^1(\...
2
votes
0answers
24 views

Lebesgue Space/Bochner Space interpolation Theorem

I need the embedding, for $I\subset\mathbb{R}$ is a bounded intervall and $\Omega\subset\mathbb{R}^n$ is a bounded domain, $$L^{q_1}(I;L^{p_1}(\Omega))\cap L^{p_2}(I;L^{p_2}(\Omega))\hookrightarrow L^...
2
votes
0answers
24 views

Hilbert space and traces

Let $\Omega$ be the open unit ball in $\mathbb{R}^n$, and $\Gamma := \Omega \cap \{x_n=0\}$. Let $\Omega_1 = \{ x \in \Omega: x_n > 0 \}$ and $\Omega_2 = \{ x \in \Omega: x_n < 0 \}$. Define $...
2
votes
0answers
38 views

Hahn Banach Theorem extending distribution

For any given distribution $T\in D'(\Omega)$, could $T$ has a coutinuous extension $$\widetilde{T}:C_0(\Omega)\rightarrow R,\ \ \ \widetilde{T}\in(C_0(\Omega))'\ \ ?$$ Could you state a general ...
2
votes
0answers
20 views

Poisson equation, $L^2$ bounds

Consider a bounded domain $\Omega\subset\mathbb{R}^d$ and $u\in H^1_0(\Omega)$. I know that $$ \|u\|_{H^m}\leq C\|\Delta u\|_{H^{m-2}} $$ for $m\geq 1$. Is the same true for $m=0$, i.e. for the $L^2$ ...
2
votes
0answers
55 views

Does continuity imply weak differentiability?

I have recently been reading about weak derivatives. I have found few examples of only weakly differentiable functions and they were all continuous. Is there an example of a continuous function which ...
2
votes
0answers
42 views

The convergence rate of the derivative of a sequence of function

Let $v_\delta$ be a sequence of continuously differentiable functions on $(-1,1)$ and $0\leq v_\delta\leq 1$. For each $\delta>0$, assume that $v_\delta(\delta)=v_\delta(-\delta)=1$ and $v_\delta(0)...
2
votes
0answers
27 views

Proving that $W^{1,2,0}(U)$ is dense in $H^1(U)$ for an open set $U\in\mathbb{R}$

I am trying to show that $W^{1,2,0}(U)$ is dense in $H^1(U)$ for an open, bounded set $U\subset\mathbb{R}.$ Where $W^{1,2,0}(U)$ is defined to be the space of smooth functions $f:U\to\mathbb{C}$ such ...
2
votes
0answers
34 views

$H^1_0(M)$ for non-compact $M$

Consider a complete non-compact Riemannian manifold $M$. My question is, is it possible for a non-zero constant function $c$ to be in $H^1_0(M)$? My guess is, this should be possible when $M$ has ...
2
votes
0answers
55 views

Strong convergence

I have a sequence $(u_n)$ such that for a functional $I:W^{1,p}_0(\mathbb{R}^N)\rightarrow \mathbb{R}$ of $C^1-$classe we have $$I'(u_n)u_n=0, \forall n\in \mathbb{N}$$ and $$ \nabla u_n (x) \to \...
2
votes
0answers
24 views

seeking for a general strategy to identify the right space for the domain of semigroup generator

I wish to show the domain of a strongly continuous semigroup $S(t)$ is some sobolev space, for instance, for the heat semigroup, it is known that $$D(A)=W^{2,p}$$ I believe a general strategy to get ...
2
votes
0answers
30 views

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 $\Psi(u)$...
2
votes
0answers
44 views

Dense subset of Nikol'skii spaces?

I'm looking for a dense subset of the Nikol'skii space $N^{s,p}(\mathbb{R}^N)=B_{p,\infty}^s(\mathbb{R}^N)$, $s\in(0,1)$, $p\in(1,\infty)$, the definition of which is recalled below: $N^{s,p}(\mathbb{...
2
votes
0answers
37 views

$W^{1,p}$ is separable for $1\leq p<\infty$

I've been asked to prove that the Sobolev spaces $W^{1,p}(\Omega)$, $\Omega$ open in $\mathbb R^n$, are separable for $1\leq p <\infty$ using the map $$i\colon W^{1,p}(\Omega)\to L^p(\Omega)\times ...
2
votes
0answers
52 views

Multi-index notation confusion

While trying to understand a proof of equivalence of norms for $H^k(\mathbb{R}^n)$ (Fourier Transforms) I came across a possible inconsistency in the multi-index notation. Can somebody please clarify ...
2
votes
0answers
60 views

Embedding of $W^{d, 1}(\overline{\Omega)}$ into $C(\overline{\Omega})$

I've been trying to prove the following assertion: Assume that $\Omega\in C^{0,1}(\mathbb{R}^d)$. Prove that $W^{d,1}(\Omega)\hookrightarrow C(\overline{\Omega}).$ My approach: I have proven that $...
2
votes
0answers
56 views

Is the image of gradient map from Sobolev space to Lebesgue space weakly closed?

Suppose f is a map defined between $W_0^{1,p}(\Omega)$ and $L^{p'}(\Omega)$ as follows - $u \mapsto |\nabla u|^{p-1}$. Is the range of this map weakly closed in $L^{p'}$?.
2
votes
0answers
81 views

Compactness of the trace operator

Is it true that for a set $\Omega$ with Lipschitz boundary the trace operator $T : H^1(\Omega) \to L^2(\partial \Omega)$ is compact? Can you please give a reference? I found a theorem in Necas' ...
2
votes
0answers
45 views

Minimizing sequences and topology (direct method)

To show the importance of the choice of the topology for the direct method we have been assigned the following exercise which I've not been able to solve due to my lack of understanding on how strong ...
2
votes
0answers
38 views

A quick question in PDE and evaluation of norms, need verification

Given $I\subset \mathbb R^N$ open bounded. Then we define two quantities $$ Q_1=\sup\left\{ \int_\Omega u\,\text{div}\phi\,dx, \,\phi\in C_c^\infty(\Omega;\,\mathbb R^2), \,\|\phi\|_{L^\infty}\leq 1\...
2
votes
0answers
54 views

How to “uniformly” mollify a sequence without knowing the higher derivative.

The motivation and previous discussion of some related ideas can be seen here. My question: given a sequence $u_n\in u_0+H^1_0(\Omega)$ where $u_0\in H^1(\Omega)$ and $\Omega\subset \mathbb R^N$ is ...
2
votes
0answers
30 views

Is one dimesional cubic NLS is globally wellposed in $H^{s}(\mathbb R), (0<s<1)$?

We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS): $$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$ $$\phi (x,0) = \...
2
votes
0answers
29 views

Weak convergence and norm convergnce along a subsequnece in $H^1(\Omega)$

Let $\Omega$ be an open subset of $\mathbb{R}^d$. Let $(f_n)_n$ be a sequence in $H^2(\Omega)$. Let $f\in H^2(\Omega)$. Assume that $f_n\rightarrow f$ weakly in $H^1(\Omega)$ and that $D^{\alpha}f_n\...
2
votes
0answers
80 views

Limit under the integral sign and partition of unity

Let $U \subset \mathbb{R}^N$ be a bounded open set and let $\{ U_j \}_{j=1}^\infty$ be an open covering of $U$ such that $U = \bigcup\limits_{j=1}^\infty U_j$. Suppose $\{ \psi_j \}_{j=1}^\infty$ is ...
2
votes
0answers
50 views

Property implying weak differentiability

What property does imply that a function $f \in L^1_{loc}(\Omega)$ ($\Omega \subset \mathbb{R}^n$) is weakly differentiable, namely there exists $g \in L^1_{loc}(\Omega)$ such that $\int_{\Omega} f\...
2
votes
0answers
65 views

question about density of Sobolev spaces

I have a short question about density of spaces. Consider: $C_c^{\infty}(0,1)=\{f\in C^{\infty}(0,1); supp(f)\subset (0,1)\;\text{compact}\}, $ $H_0^1(0,1)=\overline{C_c^{\infty}(0,1)}^{\|\cdot\|_{1,...
2
votes
0answers
47 views

Extension of Sobolev function

Let $D$ be a convex bounded domain in $\mathbb{R}^{n-1}$. Let $A:D\to\mathbb{R}^{+}$ be a Lipschitz continuous function. Let $\,\Omega\,$ be a bounded domain in $\mathbb{R}^{n}$ of the form $\,\Omega=\...
2
votes
0answers
40 views

Why this function is in $C^\infty((0,\infty);H^1(\Omega))$?

Let $\lambda_j$ be the eigenvalues of the Neumann Laplacian on a bounded domain $\Omega$ with eigenvectors $\varphi_j$, which we know are smooth. Given a function $u \in H^{\frac 12}(\Omega)$ with $\...
2
votes
0answers
102 views

Poincare inequality on a domain for functions with mean value zero

Let $C=\Omega \times (0,\infty)$ for a bounded $C^1$ domain $\Omega$. Consider a function $u \in H^1(C)$. Write $u=u(x,y)$ where $x \in \Omega$ and $y \in (0,\infty).$ Is it true that if $u \in H^1(C)...
2
votes
0answers
153 views

Leibnitz rule for fractional derivatives

I need to estimate the following norm $$\Vert fg\Vert_{H^{\frac{1}{2}}(\mathbb{R}^3)}$$ Is there some product rule for the fractional derivative?
2
votes
0answers
43 views

Inequality involving $H^s$ and $L^2$.

I have this inequality which I don't see how to prove it. We have $F \in C^s$, and $u\in H^s$. I want to show that: $$\| F\circ u \|_{H^s} \leq C(\| F \circ u \|_{L^2}+\sum_{r=1}^s \sum_{j=1}^r \...
2
votes
0answers
292 views

The Co-area formula for $BV$ function V.S. the co-area formula for $C^\infty$ functions

I am working on the proof of the co-area formula for $BV$ functions. Suppose $u\in BV(\Omega)$ then the co-area formula states that $$ \|Du\|(\Omega)=\int_{-\infty}^\infty \|\partial E_t(u)\|(\Omega)\...
2
votes
0answers
43 views

Spaces of the derivative in a direction

I have two question regarding the spaces where the first, and second, directional derivatives of a functional are. Let $\Omega \in \mathbb{R}^2$ an open subset. Let the functional: $$\phi =L^p (\...
2
votes
0answers
66 views

use of Strichartz estimates

I have been learning(understanding) the Strichartz estimates. My naive Questions: (1) Can you give some ideas, how Strichartz estimates plays role in solving nonlinear dispersive equations(e.g. ...
2
votes
0answers
65 views

counterexample to strauss inequality

I am looking for a counterexample to Strauss inequality in dimension 1, where it supposedly fails. How can one construct an $H^1(\mathbb{R})$ function which does not decay at infinity, for which for ...
2
votes
0answers
27 views

A $L^1$-bounded sequence from a $H^m$-bounded sequence

I am trying to show the following: for any $m > 0$ and $\alpha \in \mathbb{N}^n$, assume $(f_j)$ is a sequence of functions which is bounded in $H^m(\mathbb{R}^n).$ Assume moreover that all the $...
2
votes
0answers
86 views

Weak convergence of determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that $u^n\...
2
votes
0answers
53 views

Proving a PDE has a particular weak form (check my proof please!)

Let $u_t - \Delta u = f$ hold in $L^2(0,T;H^{-1})$ for a solution $u \in L^2(0,T;H^1_0)$ with $u_t \in L^2(0,T;H^{-1})$. This means $$\int_0^T \left(\langle u_t(t), v(t)\rangle + \int_\Omega \nabla u(...
2
votes
0answers
120 views

Traces of Sobolev functions in an unbounded domain

I have a doubt concerning the trace of Sobolev functions. Let $C=\Omega\times(0,\infty)$ an infinite cylinder of basis a smooth domain $\Omega$ of $\mathbb{R}^{N}$ and consider the classical Sobolev ...
2
votes
0answers
89 views

Alternative derivation of Poincaré inequality

I've been trying to prove the Poincaré inequality via a representation formula for Sobolev functions $u\in W^{1,p}(\mathbb{R}^{n})$, $1\leq p < n$, wlog with compact support. The setting is ...
2
votes
0answers
495 views

Can we characterize homogeneous Sobolev spaces by means of Sobolev embedding?

For $\alpha>0^{[1]}$, let $D^\alpha=(-\Delta)^{\frac{\alpha}{2}}$ be the operator defined on all Schwartz class functions $f$ as follows: $$ \widehat{D^\alpha f}(\xi)=\lvert \xi\rvert^\alpha \hat{...
2
votes
0answers
89 views

Gagliardo-Nirenberg-Sobolev inequality

There is one step of the textbook's proof that I wish could be clarified. Pages 277-278 of PDE Evans, 2nd edition, says: Integrate this inequality with respect to $x_1$: \begin{align} \int_{-\...
2
votes
0answers
68 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
2
votes
0answers
61 views

Mollification of functions in $L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$

Let $u \in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain. I know that (eg. from Wloka or Hunter's PDE notes) that there is a mollification ...
2
votes
0answers
116 views

Pointwise (in time) convergence in $H^{-1}$ implies pointwise weak convergence in $L^q$, why?

Let $u_n \to u$ in $C^0([0,T];H^{-1}(\Omega))$ and suppose $\lVert u_n \rVert_{L^\infty(0,T;L^\infty(\Omega))} \leq C$ for all $n$. It follows that for almost all $t$, $u_n(t)$ is bounded in $L^\...
2
votes
0answers
50 views

Local Mollification of a $L^1$ function

Let $f$ be a $L^1(\Omega)$ function, where $\Omega\subseteq \mathbb{R}^2$ is a smooth and bounded domain. Then can we find a fuction $\phi\in W_0^{2,2}$ suct that $\phi>0$ in $\Omega$ and $\int_\...
2
votes
0answers
80 views

Why is this bilinear form elliptic on $H^1_0$?

I have the following bilinear form: $$ a(v,w) = \int_\Omega \nabla v(x) \cdot \nabla w(x)dx + \int_\Omega \left( \sum_{i=1}^n \beta_i v_{x_i}(x) \right) w(x) dx $$ where $\Omega \subset \mathbb{R}^...
2
votes
0answers
220 views

Functions of fractional-order Sobolev spaces

In fracture mechanics one might end up dealing with functions such as $$w(r,\theta) = \sqrt{r} \sin \frac \theta 2,$$ which is defined, for example, on a cracked unit circle, $\Omega = B(0,1)\setminus\...