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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15 views

Convergence of $\partial_{x_j} u(x,t)$ when $u$ converges in $L^2$ norm.

I hope you can help me with this question. We take $u(x,t)\in L^\infty_{loc}(\mathbb{R},H^1(M))\cap Lip_{loc}(\mathbb{R},L^2(M))$, the derivatives $\partial_{x_j} u $ exist and are continuous, i.e ...
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0answers
34 views

A question in PDE/Sobolev spaces [on hold]

I'll be grateful if you help me or give me hints to answer the following: Suppose $\Omega$ is a bounded domain in $\mathbb R^N$ with $C^1$ boundary $f\in L^2(\Omega)$. Prove that for every ...
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1answer
20 views
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1answer
48 views

Evans PDE: Chapter 5, Problem 9 - Clarification

I've been trying to work out the solution to Question 9 in Chapter 5 of Evans, and I'm having some difficulties. I've been looking at the solution posted here: question 9 - chap 5 evans PDE And I ...
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1answer
29 views

Sobolev Embedding and Uniform $C^1$ bound

I am currently reading a paper and am a little confused about the following, which for clarity, I distill into the following question: Suppose $\{w_i\} \subset C^2(\mathbb{R}^n)$ is a sequence of ...
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0answers
36 views
+50

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
2
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1answer
23 views

Functions belonging to $L^p(\mathbb{R}^n ; \mathbb{R}^m)$ if and only of their norm belongs to $L^p(\mathbb{R}^n)$

The definition I have of $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m)$ is that we require each component function to be in $L^p(\mathbb{R}^n)$. Is is true that $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m) ...
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1answer
21 views

Variant of Ladyzhenskaya’s inequality

I am trying to show that if $\Omega \subset\subset \mathbb{R}^2$ with $C^1$ boundary and $ u \in W^{1,2}(\Omega)$ then \begin{equation*} \int u^4 < C \left(\int u^2 \right)^2 + C \left(\int u^2 ...
0
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1answer
22 views

Show that this function is weakly differentiable

I need to show that the function \begin{equation} u(x,y) = 1-x_1^2 \quad x_1>0 \\ u(x,y)=1+x_1^2 \quad x_1 \leq 0 \end{equation} is weakly differentiable on the unit ball. It is clear what the ...
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0answers
69 views

Elliptic regularity in an unbounded domain

Let $\Omega \subset \mathbb{R}^N$ be an unbounded domain with nonempty boundary. Let $$\mathcal{D}^{1,2}(\mathbb{R}^N) = \{ u \in L^{2^*}(\mathbb{R}^N) : \nabla u \in L^2(\mathbb{R}^N ; \mathbb{R}^N) ...
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1answer
25 views

sobolev space question?

My attempt: By the Fourier inversion formula, $$u(x) = (2\pi)^{-n}\int_{\mathbb{R}^n} \hat{u}(\xi) e^{i x \cdot \xi} ~d\xi,$$ $$(2\pi)^{n}|u(x)| = |\int_{\mathbb{R}^n} \hat{u}(\xi) e^{i x \cdot \xi} ...
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1answer
20 views

Fractional Sobolev spaces on closed manifolds

Let $M$ be a closed manifold and $0<s<1$. How is the fractional Sobolev space , $H^s(M)$ defined? In particular if $M$ is a closed smooth simple curve in the place how is $H^{1/2}(M)$ ...
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1answer
41 views

When is the $L^{2}$ norm smaller than the $H^{-1}$ norm?

If $u\in L^{2}$ then we can define the functional: $$u(\phi)=\int \phi u $$ for all $\phi \in H^{1}_{o} $. which means that $u$ is a linear functional in $H^{-1}$. Now for any $f\in H^{-1}$ ...
2
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0answers
26 views

A linear functional of $H^k$

Define $H^k$ as usual: $$H^k=\left\{f; \int (1+|\xi|^2)^k|\widehat{f}(\xi)|^2d\xi<\infty\right\}$$ Define $l(u)=\int (1+|\xi|^2)^k\widehat{u}(\xi)d\xi$, how to show that $l\in (H^k)^*$. I tried ...
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1answer
27 views

For which $p$ the sequence $x^n$ converges in the Sobolev space $W^{1,p}(I)$?

I would like to know for which $p$ the sequence $u(n)=x^n$ converges in the Sobolev space $W^{1,p}(I)$. Is it true that converges only for $p=1$? I find out this looking for which $p$ the Sobolev ...
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0answers
27 views

Question about representation of the eigenvalues of second order elliptic operator

let $Lu:=-\text{div}(A\cdot\nabla u)$, where $A$ is symmetric. Eigenvalues of $L$ is $\lambda_1<\lambda_2<\cdots$. By definition. (If exists a nontrivial solution $w$ such that $Lw=\lambda w$, ...
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0answers
49 views

Direct method in the calculus of variations

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$. $$ \mathcal{F(u)} = \int_{\Omega}\frac{1}{2}|Du|^2dx $$ $$ u \in \mathcal{A}: = \{v \in W_{0}^{1,2}: \int_{\Omega}v^2dx = 1\} $$ Does this ...
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1answer
42 views

A trace inequality with epsilon in Sobolev spaces

We know the standard trace inequality: for a bounded domain with certain boundary regularity, there is a $C>0$ such that $$ \|Tu\|_{L^2(\partial \Omega)}\leq C\|u\|_{H^1(\Omega)}, \quad \quad u\in ...
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1answer
24 views

Approximating $u \in H^1$ s.t. $u(T)=0$ with $u_n \in H^1_0$ in the gradient norm?

Let $u \in H^1(0,T)$ with $u(T)=0$. Is it possible to find a sequence $u_n \in H^1_0(0,T)$ such that $\nabla u_n \to \nabla u$ in $L^2$? I only need the convergence in the gradient.. not the full ...
6
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1answer
159 views

An interesting semi-linear PDE problem

Assume $u\in H^1(\mathbb{R}^n)$ has compact support, and assume that it is a weak solution of the semi linear equation $$ -\Delta u+c(u)=f\;\;\text{in}\;\mathbb{R^n} $$ where $f\in ...
1
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1answer
54 views

If $f \in L^1(M)$, is it true that $f(x) < \infty$ for almost all $x$?

If $M$ is a measurable space (eg. $M$ is a Riemannian manifold which is compact) and if $f \in L^1(M)$, is it true that $|f(x)| < \infty$ for almost all $x$? I am trying to figgur out if $u \in ...
4
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0answers
32 views

Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$ for some $\overline{p}>p*'$

Let $\Omega \subset \mathbb{R}^{N}$ be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function such that $g(x,t)=0$ for $t\leq0$ . Suppose that ...
3
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1answer
23 views

A trouble with the existence of an $C_{0}^{\infty}$- function $v,v>0$ such that $\int_{\Omega}hv^{\alpha}>0$

Let $\Omega$ be a smooth bounded subset of $\mathbb{R}^{n}$ , an $L^{\sigma_{\alpha}}$ -function $h$ with $h^{+}\neq0$ , $\dfrac{1}{\sigma_{\alpha}}+\dfrac{\alpha}{p*}=1$ , does there exist ...
0
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0answers
39 views

Proof that $C^{\infty}_0$ is dense in $W^{1,p}(\mathbb{R}^n)$

I have taken $u \in W^{1,p}(\mathbb{R}^n)$ and a countable cover of $\mathbb{R}^n$ by open balls of increasing radius. I was hoping to mollify and use a partition of unity to deduce that ...
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3answers
39 views

$W^{1,\infty}$ function that cannot be approximated by Smooth Functions

I am trying to find a function $u \in W^{1,\infty}$ that cannot be approximated by smooth functions. It seems like it should be an easy construction but I am blanking. Thanks.
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0answers
14 views

RKHS vs RKHS from sobolev spaces.

Which is more desirable in terms of solving a differential equation? Constructing an RKHS from sobolev space (essential reproducing kernels are of infinite support). Or directly choosing ...
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0answers
37 views

Existence of weak solution to $-\Delta u =0$, $u|_{\Gamma_1} = g$ and $u_{\Gamma_2} = 0$?

Let $\Omega$ be a compact Riemannian manifold with $\partial\Omega = \Gamma_1 \cup \Gamma_2$ a union of disjoint sets. Let $g \in H^s(\Gamma_1)$ and consider $$-\Delta u = 0 \text{ on $\Omega$}$$ ...
0
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1answer
23 views

Does the trace operator commute with partial differentiation?

Let $u\in H^2(\mathbb R^n)$. According to the trace theorem, $u$ has a trace $Tu\in H^1(\mathbb R^{n-1}\times\{0\})$ (at least locally). Suppose you know more about that trace, for example that $Tu$ ...
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0answers
31 views

How to prove Poincare-like inequality? [duplicate]

Suppose $u\in W^{1,1}$ and $\partial u$ is $C^1$. I want to prove the following: $\int_{\partial\Omega}|u-\bar u|\leq A\int_{\Omega}|\nabla u|$, where $\bar u=\dfrac{1}{|\Omega|}\int_{\Omega}u$ and ...
4
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1answer
42 views

Bochner-Sobolev space definition

I take course in PDE and I'm a little bit puzzled with space $W^{1,p}(0,T,X)$. Evans defines in §5.9.2 $W^{1,p}(0,T,X)$ as follows $$ W^{1,p}(0,T,X) = \{ u \in L^p(0,T,X): u'\in L^p(0,T,X)\} $$ but ...
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1answer
37 views

Trace operator on $W^{1,\infty}$

My question is rather short and simple really: Is the trace operator well defined on $W^{1,\infty}(\Omega)$ for some bounded Lipschitz domain $\Omega$? The reason I ask is because I have seen ...
0
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1answer
104 views

How to prove Poincaré-like inequality for the integral over the boundary?

Suppose $u\in W^{1,1}$ and $\partial u$ is $C^1$. I want to prove the following: $$\int_{\partial\Omega}|u-\bar u|\leq A\int_{\Omega}|\nabla u|$$ where $\bar u=\frac{1}{|\Omega|}\int_{\Omega}u$ and ...
3
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0answers
141 views

Complicated convergence of nonlinear term

Let $1<p<\infty$, $\Omega\subset\mathbb{R}^m$ be open, bounded with $\partial\Omega\in C^1$. Assume that $u_k\to u$ weakly in $W^{1,p}(\Omega;\mathbb{R}^n)$. We know that $u_k\to u$ strongly in ...
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2answers
48 views

Examples of Sobolev Spaces

I would like to apologize in advance for this trivial question! Does constant functions $u \equiv C$ and, in partucular, $u \equiv 0$ belong to $W_{0}^{1,2}(\Omega)$? Update 2: $\Omega \subset ...
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0answers
30 views

$L^{\infty}$ estimate on the boundary by means of Sobolev norms in the interior

I am studying this paper http://143.248.27.21/mathnet/paper_file/washington/gunther/cpde.pdf and I have a doubt I have not been able to solve since yesterday. In page 14, just before the last bunch of ...
2
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1answer
30 views

On the weak closedness of a closed ball with fixed $L^2$-norm in a periodic Sobolev space

Preliminaries: Let $\mathrm{L}_P^2$ denote the Hilbert space of $P$-periodic, locally square-integrable functions $f\colon \mathbb{R} \to \mathbb{C}$ with Fourier series representation $$f(x) \sim ...
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1answer
26 views

What kind of functions lie in $L^2(\Omega )-H^{1/2}(\Omega )$?

It's well known that a discontinuous function does not lie in $H^{1/2}(\Omega )$ if $\Omega\subset\mathbb{R}$ is one dimension. My question is, For $\Omega\subset \mathbb{R}^3$, are there any simple ...
2
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0answers
123 views

Prove difference quotient converges to weak derivative in $L^p$

I am trying to solve the following exercise: Let $U$ be an open set in $\mathbb{R}^n$ and let $V$ be a compact subset with Lipschitz boundary. Assume that $f$ is in $L^p(U)$ with $1<p<\infty$ ...
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0answers
42 views

Negative index Sobolev spaces, properties

I was hoping to find some references for the following facts: Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^n$. (i) If $\nabla p$ is a distribution in $H^{-1}(\Omega)$ then ...
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1answer
24 views

Bounded operator from $H^1$ to $L^2$

I am working through the nice Finite Element notes by Douglas N. Arnold at the moment. They can be found here. A teeny tiny detail in the proof of Lemma 7.4 gives me a headache and I am looking for ...
4
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1answer
52 views

Poincare-like inequality

I would like to prove that there exists $C>0$ such that $$\| u \|^2_{L^2(B(0,1))} \leq C \left ( \| \nabla u \|^2_{L^2(B(0,1))} + \| u \|^2_{L^2(\partial B(0,1))} \right )$$ for every $u \in ...
4
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1answer
38 views

$\||f|^{2}f\|_{H^{s}}\leq C \|f\|^{2}_{L^{\infty}} \|f\|_{H^{s}}$ for $s>0, f\in H^{s}(\mathbb R)$?

We let $H^{s}(\mathbb R^{n}), (s\in \mathbb R)$ the Sobolev spaces. It is well known that: the space $H^{s}(\mathbb R^{n})$ is an algebra with respect to pointwise multiplications, for $s>n/2.$ ...
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0answers
24 views

Prove weak derivative commutes with difference quotient

Let $U$ be an open set in $\mathbb{R}^n$,$f:U\to \mathbb{R},f\in W^{1,p}(U)$. Let $\tau_{h,i}f(x)=\frac{f(x+he_i)-f(x)}{h},h>0$ Given any compact $V\subset U$, show there exists $h_0>0$ such ...
2
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1answer
36 views

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; ...
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1answer
22 views

An application of integration by parts in Sobolev Spaces

Let $u \in H^2([0, 1])$ with $u(0) = u'(1) = 0$. Is the following calculus correct? $$\underset{0}{\overset{1}{\int}} u''(x) u(x) dx = u'(x) u(x) \underset{0}{\overset{1}{|}} - ...
1
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1answer
31 views

Is the set of vanishing $k$ derivatives of smooth functions in a null set dense in $W^{1,p}$?

Let $\Omega$ be an open set with compact closure denoted by $\overline{\Omega}$ and a null set $N\subset\Omega$ with respect the Lebesgue measure. Then consider the two sets ...
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0answers
23 views

Does Gagliardo-Nirenberg inequality in unbounded domain still hold?

Often we have the following Gagliardo-Nirenberg inequality: Let $1\leq p_1, p_2\leq \infty$, $0\leq r<l (r, l\in Z_+)$. Suppose that the number $$ \theta=\frac{n/p-n/p_1-r}{n/p_2-n/p_1-l} $$ ...
1
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1answer
36 views

Euler functional associated to a $p-$laplacian bvp

I have this BVP: \begin{cases} -\Delta_{p}(u)=\lambda_1 |x|^{\theta}|u|^{q-2} u+f(x,u)-h ~~\text{in}~\Omega,\\ u\in W^{1,p}_0(\Omega). \end{cases} where $\Delta_p$ is denoting the $p-$Laplacian ...
1
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0answers
18 views

Compute the value of the Sobolev norm in $H^{-1/2}$

I am working with finite elements using domain decomposition in 2D and one of the solutions I need to obtain is the co-normal derivative of the solution along a segment that is the intersection of two ...
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0answers
40 views

When is the convolution of a product the product of convolutions?

Although the convolution of the product is not the product of the convolution, i.e. $$fg*h\neq (f*h)(g*h).$$ I am wondering if this true (for a suitable class of functions) in the limit when one ...