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

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40 views

The weak convergence between $L^2$ and $H^1$

If the sequence $u_n\in H^1(R^n)$, $n\geq 3$ and $u_n$ weak converges to $u$ in $H^1(R^n)$, if we can get $u_n$ weak converges to $u$ in $L^2(R^n)$? Furthermore, if we can have $\nabla u_n$ weak ...
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1answer
40 views

a priori estimates involving Sobolev norms

Let $\sigma,$ $f$ and $g$ be $C^{2}(\overline{\Omega})$ functions, with $0<\frac{1}{M} < \sigma < M.$ We have the Dirichlet problem: $\text{div}\sigma \nabla u=f, \hspace{3mm} \text{in} ...
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1answer
14 views

Inequation in Sobolev

Let $u\in W^{1,2}(\Omega)$. I need to prove that: $\|(\nabla u)u\|_{W^{0,2}(\Omega)}\leq \|\nabla u\|_{W^{0,4}(\Omega)}\;\|u\|_{W^{0,4}(\Omega)}$ I'm using the usual Sobolev notation (see for ...
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1answer
30 views

Weak derivative (Sobolev spaces)

I'm reading "Functional Analysis" - Michel Willem and I can't understand the definition of weak derivative from chapter 6, namely the definition of $$ \partial^\alpha f. $$ Can you give me a concrete ...
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25 views

Is the embedding $H^1(\Omega)^d\to H(div,\Omega)$ compact?

Let $\Omega\subseteq\mathbb{R}^d$, $d\in\mathbb{N}$, be a Lipschitz domain. I wonder what one can say about the embedding $H^1(\Omega)^d:=\{u=(u_i)_{i=1}^d \in L^2(\Omega)^d: \nabla u_i\in ...
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1answer
45 views

Sobolev spaces and Fourier transform

Does the equation $$\partial^{\alpha} (Ff)(y) = (-i)^{|\alpha|} F(x^{\alpha} f)$$ still hold for $|\alpha| \le m$ and $f \in H^m(\mathbb{R}^n) = W^{m,2}(\mathbb{R}^n)$$? $F$ denotes the Fourier ...
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0answers
21 views

Dual of $L^\infty(I,H^1(M))$

What is the dual of $L^\infty(I,H^1(M))$? Any references? Where $H^1(M)$ is Sobolev space, and $I$ is some interval in $\mathbb{R}$, and $M$ is a compact manifold, like the $n$-dimensional torus.
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0answers
23 views

Do $H^1(\mathbb R^d)$ functions vanish at $\infty$? [duplicate]

Do $H^1(\mathbb R^d)$ functions vanish at $\infty$? I have a feeling it must be true, but I am not able to prove it rigorously. I am worried especially about $d=1$
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1answer
23 views

If $(u(x)v(x)), (u'v') \in L^1$ why are $u, v \in H^1$?

Given functions $u(x), v(x)$ and given that $\int uv\:\mathrm{d}x < \infty$ and $\int u'v'\:\mathrm{d}x < \infty$ (that is, their product and the product of their derivatives are in $L^1$) why ...
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1answer
18 views

does exists a Liouville theorem in this case?

Let $u \in H^1 (\Omega)$ ($\Omega$ a smooth bounded domain in $R^n$) with $u $ bounded and $\Delta u = 0$ in $\Omega $ in the weak sense. Then u is constant? I am trying to find in the internet some ...
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1answer
45 views

Show that bilinear form is $H^1(0,l)$-elliptic/coercive

Let $$a(u,v) := \int_0^l \partial u(x) \partial v(x) + cu(x)v(x)\, \mathrm{d}x + \alpha u(l)v(l)$$ Show that $a(\cdot,\cdot)$ is $H^1(0,l)$-elliptic if either $c > 0$ or $\alpha > 0$. My ...
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0answers
29 views

Regularity of weak solution for $u_t - \Delta u = f$ with $u(0) = u_0 \in L^1(\Omega)$

Let $\Omega$ be a bounded domain, and consider the equation $$u_t - \Delta u = f$$ $$u(0) = u_0 \in L^1(\Omega)$$ with Neumann BCs (or Dirichlet if convenient) where $f$ is smooth. Using ...
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1answer
23 views

Gradient is zero on the preimage of a negligible set

Let $f\in W^{1,1}(\Omega)$, where $\Omega\subseteq\mathbb{R^n}$ is open, and $E\subset\mathbb{R}$ a negligible subset, i.e. $\mathcal{L}^1(E)=0$. Is it always the case that $Df\equiv 0$ a.e. on ...
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0answers
41 views

sobolev spaces integral estimation

if I have a sequence $f_k\in W_{1,p}(\Omega)$ which converge weakly to some function $ f $ and I know that $\nabla f_k-\nabla f\to 0$ in $L_{p}^{loc}(\Omega)$ I try to estimate the integral ...
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1answer
20 views

sobolev space- notation

if I have a sequence $u_k\in W_{1,p}$ what does it mean that : " every subsequence of $\nabla u_k$ which converge in the sense of measure...." what is a convergence is the sense of measure? thank ...
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1answer
30 views

The integration of radially symmetric function

Suppose $u\in C^\infty(\overline{B(0,1)})$ is radially symmetric. i.e., there exists a function $v$: $\mathbb R^+\to\mathbb R$ such that $u(x)=v(|x|)$. Here we take $B(0,1)\subset \mathbb R^2$. ...
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1answer
24 views

About a “gluing lemma” for Sobolev functions and a relation with elliptic equations

Let $\Omega$ and open bounded set in $R^n$ with smooth boundary. Suppose that $\overline{B(0,1)} \subset \Omega$. Let $h_1 \in H^{1}(B(0,1) - B(0,1/2))$ such that $\Delta h_1 = 0 $ in $B(0,1) - ...
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1answer
28 views

$\|u\|_{L^{3}(\mathbb R)} \leq C \|Du\|_{L^{2}(\mathbb R)}^{\alpha} \|u\|_{L^{2}(\mathbb R)}^{1-\alpha}$?

Let $u\in H^{1}(\mathbb R).$ Is Gagliardo–Nirenberg interpolation inequality valid for the $p=3, q=r=2, m=1, 0< \alpha < 1$ ; and $j=0$ ? That is, is it true that,$\|u\|_{L^{3}(\mathbb R)} ...
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2answers
65 views

$\|f\|_{L^{3}(\mathbb R)}^{4} \leq C \|f\|_{L^{2}(\mathbb R)}^{3} \|\nabla f\|_{L^{2}(\mathbb R)}$ ; for some constant $C$?

Let $f\in H^{1}(\mathbb R).$ (Sobolev space) My Question: Is it true that: $\|f\|_{L^{3}(\mathbb R)}^{4} \leq C \|f\|_{L^{2}(\mathbb R)}^{3} \|\nabla f\|_{L^{2}(\mathbb R)}$ ; for some constant ...
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1answer
56 views

$\|f\|_{L^{3}(\mathbb R)}^{3} \leq C \|f\|_{L^{2}(\mathbb R)} \|\nabla f\|_{L^{2}(\mathbb R)}^{3}$ ; for some constant $C$?

Let $f\in H^{1}(\mathbb R)$ (Sobolev space). My Question: Is it true that: $\|f\|_{L^{3}(\mathbb R)}^{3} \leq C \|f\|_{L^{2}(\mathbb R)} \|\nabla f\|_{L^{2}(\mathbb R)}^{3}$ ; for some constant ...
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0answers
28 views

Hardy inequality punctured space

given the minimization problem: $inf \ \frac{\int_{\Omega} |\nabla u|^p }{ \int_{\Omega} \frac{|u|^p}{|x|^p} } ,\ \ p>1$ infimum taken on all smooth functions with compact support in the ...
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0answers
107 views

Question 13 in Taylor's PDE vol III section 16.1.

my question comes from Taylor's PDE textbook, volume III. Consider a semilinear hyperbolic system, $u_t=Lu+g(u)$, $u(0)=f$, where $Lu=\sum_j A_j \partial_{x_j}u$, $g(0)=0, \ |g'(u)| \le C$, take ...
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1answer
18 views

weak convergence and local integrabillity

let $\Omega$ be a domain in $R^n$ and let $\delta(x)=dist(x,\partial \Omega)$ and assume that $u_k \to 0$ weakly in$ W_{1,p}^\circ$ (sobolev space with trace =zero). I can conclude that $u_k\to0$ in ...
2
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2answers
60 views

Product of functions in $H^1(B)$ where $B \subset \mathbb{R}^2$

I'm rather new to Sobolev spaces and finding myself rather deficient of intuition. So when given a problem like the below where I need to "prove or disprove", I'm finding myself stuck. Suppose $B$ is ...
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1answer
32 views

A special approximation of BV functions

Suppose $\Omega:=B(0,1)\subset \mathbb R^2$. Let $u\in BV(\Omega)$ be a radially symmetric, i.e., $u(x)=u(Rx)$ for all $R\in SO(2)$. In addition, suppose $w$ to be an affine function,. i.e., ...
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0answers
55 views

Sobolev embedding theorem in the homogeneous case

We know that if $s>\frac{n}{2}$ the following inclusion holds $$H^s(\mathbb{R}^n)\subset L^\infty(\mathbb{R}^n)$$ Is it also true in the case we deal with the homogeneous space ...
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0answers
25 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 ...
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1answer
22 views

Can we have $\|\nabla u \|_{L^1(B(0,1))}\leq \|\nabla u-\tilde{c} \|_{L^1(B(0,1))}$ for radially symmetric function?

Suppose $B(0,1)$ is the unit ball in $\mathbb R^2$ and $u\in C^\infty(\overline{B(0,1)})$. Suppose $u$ is radially symmetric, i.e. $u(x)=u(Rx)$ for any $R\in SO(2)$. My question is, do we have ...
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0answers
15 views

Interchanging limits in $\lim_{n \to \infty}\lim_{j \to \infty}\int_0^T \langle u_n', w_j \rangle $ (weak time derivative)

Let $V$ be a Hilbert space which is separable. Let $u_n \in L^2(0,T;V)$ with $u_n(t,x) = \sum_{i=1}^n u_{in}(t)w_i(x)$ where $u_{in}$ are absolutely continuous on $(0,T)$ and $w_i$ are a smooth basis ...
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1answer
31 views

Show that $2\nabla \sqrt f\,+\,x \sqrt f=0$ (a.e.). $\implies$ $\sqrt f\in \mathcal C_0$. (Derivatives are in weak sense)

Show that $2\nabla \sqrt f\,+\,x \sqrt f=0$ (a.e.). $\implies$ $\sqrt f\in \mathcal C_0$. (Derivatives are in weak sense) Given that $f\in L^1(\mathbb R^d),f\geq 0,\int_{\mathbb R^d}f=1, ...
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1answer
57 views

Riesz representation theorem in Banach spaces

My question is about functionals on $W_{1,p}(\Omega)$ spaces, $\Omega$ is contained in $\mathbb R^n$ I am trying to figure out if there is a way to characterize all linear functionals on the above ...
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0answers
70 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 ...
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1answer
37 views

Continuity of a nonlinear operator on fractional-order Sobolev spaces

Let $N\colon \mathrm{H}^s(\mathbb{R}) \to (\mathrm{H}^s(\mathbb{R}))^*$, where $s > \frac{1}{2}$, be an operator given by $N(u) = \langle u^p, \cdot \rangle_{\mathrm{L}^2(\mathbb{R})}$ for a fixed ...
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0answers
26 views

Existence of solutions to this ODE arising from Faedo-Galerkin method?

Let $\{w_j\}$ be a basis of $H^1_0(\Omega)$ and let $\phi(x) = \frac{x}{|x|^{1-{\frac 1p}}}$ (for $2 < p < 3$). Define $$v_m(t) = \sum_{i=1}^m \zeta_i(t)w_i$$ where the coefficients ...
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80 views

Sobolev norm of a convolution

Let $\eta$ be a rapidly decaying function such that it is radial and $(\mathscr{F}\eta)(\xi)=1$ for $\vert\xi\vert\leq 1$. (Here $\mathscr{F}$ is the Fourier transform). Let's put ...
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2answers
50 views

Properties of Sobolev spaces $W^{k,\infty}(\Omega)$

I'm looking for different properties of spaces $W^{k,\infty}(\Omega)$ for bounded domain $\Omega \subset \mathbb R^n$ and $k \geq 1$ that I couldn't find in literature. References are wery welcome. ...
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1answer
40 views

Boundedness of solutions for the Laplacian

A solution to the equation $-\Delta u+u=f$ for $f\in L^2(\mathbb R^n)$ belongs in $H^2(\mathbb R^n)$. Is it possible to obtain a solution in $H^2\cap L^\infty(\mathbb R^n)$ if $f\in L^2\cap ...
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0answers
20 views

Can $u\in W^{1,\infty}\cap H^1(\mathbb R^n)$ be approximated by a sequence $u_k\in C_0^{\infty}(\mathbb R^n)$ with $\|u_k\|_{1,\infty}$ bounded?

This problem is relevant to this but I am not really able to prove it or find a counterexample. Could anyone give a hint?
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1answer
43 views

Regularity properties of radially symmetric functions in Sobolev spaces.

Let $u\in W_0^{1,1}(B)$, where $B=\{x\in \mathbb{R}^N:\ |x|<1\}$. Assume that $u$ is radially symmetric, that is, $u(x)=u(y)$ if $|x|=|y|$. Define $f:[0,1]\to \mathbb{R}$ by $f(r)=u(x)$ where ...
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29 views

C^1 is not dense in Holder space

Let $\overline{\Omega}$ be a bounded, closed and convex set of $\mathbb{R}^n$. Prove that $C^1(\overline{\Omega})$, the space of continuously differentiable functions on $\overline{\Omega}$, is not ...
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0answers
22 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} ...
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1answer
52 views

Sobolev embedding $W^{1,2}(\Omega)\subset L^p(\Omega)$ where $\Omega$ is a halfplane

I would like to ask when the following Sobolev embedding holds true $$W^{1,2}(\Omega)\subset L^p(\Omega)$$ where $\Omega\subset \mathbb{R}^2$ is any open set and $1 < p < \infty$. All book ...
2
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1answer
83 views

Compactness Sobolev embedding for radial functions on $\mathbb{R}^N$

I want to show: Let $N\geq 2$ and $2< q <2^\ast$. Then the embedding \begin{align} H^1_{\text{rad}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N) \end{align} is compact. I was able to show ...
2
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1answer
33 views

Nash inequality : does $f\in L^1$ and $\nabla f \in L^2$ implies $f\in L^2$?

Let $f$ be any function that belongs to $L^1(\textbf{R}^d)\cap H^1(\textbf{R}^d)$ ($d$ a positive integer). Nash inequality applies in this case and gives us $$\| f\|_{L^2}\leq C \| f\|_{L^1}^r \| ...
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0answers
32 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}\}, $ ...
2
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0answers
27 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 ...
2
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1answer
67 views

how to use an embedding theorem to show existence of a solution

Can someone look at the 14th line of page 9 in this article and give a hint that how with the embedding theorem we could find $u\in W^{2,q}(\Omega)$ and how $u_n$ coverge strongly to $u$ in ...
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0answers
22 views

How to integrate functions in n dimensional space?

I have a question about an example of functions in Sobolev space. But I think you can give a hint without knowing the Sobolev space because I just want to know how to integrate a function with ...
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0answers
41 views

Doubt about convergence of a sequence in $H^1(\mathbb{R}^3)$

Let's consider a sequence $\{f_n\}_n$ of $C^\infty_0(\mathbb{R}^3)$ complex-valued functions and suppose thet $f_n\to f$ strongly in $H^1(\mathbb{R}^3)$. What can I say about the convergence of the ...
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1answer
15 views

sequence in $H_0^1(0,1)\setminus D(0,1)$

Consider $D(0,1)=\{f\in C^{\infty}(0,1); supp(f)\subset (0,1)\;\text{compakt}\}$ and $H_0^1(0,1)=\overline{D(0,1)}^{\|\cdot\|_{1,2}}$, with the norm ...