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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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
21 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?
2
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1answer
44 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 ...
1
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0answers
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 ...
2
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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
56 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
87 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 \| ...
2
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0answers
34 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
28 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
25 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 ...
0
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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 ...
1
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1answer
24 views

Lower semicontinuity of ${\dot{H}}^1$ norm

I have a in $H^1(\mathbb{R^N})$ uniformly bounded sequence $u_n \in H^1$. I also know $u_n\to u$ in $L^p$ for every $2\leq p < 2^\ast$, where $\ast$ means the Sobolev exponent. Can I conclude that ...
1
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1answer
55 views

Fundamental theorem of calculus in Sobolev Space $H^1$

I would like to know whether the the Fundamental theorem of calculus (Part II) can be applied in the following setting. Let $(a,b)$ be an open interval in $R^1$. Let $u \in H^1((a,b))$ with $u(a)=0$ ...
3
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1answer
28 views

The range of Sobolev spaces to which the function $r^\beta\sin\beta\theta$ belongs

I am learning about Sobolev space, and I am working on the following problem from "The mathematical theory of finite element methods" by Brenner. To make the problem a little bit easier for me, I ...
3
votes
1answer
76 views

Is $\|\nabla u\|_2$ a norm on $W^{1,2}(\mathbb{R}^N)$

Is $\|\nabla u\|_2$ a norm on $W^{1,2}(\mathbb{R}^N)$? I know when $\Omega$ is bounded, $\|\nabla u \|_2$ defines a equivalent norm on $W_0^{1,2}(\Omega)$ with $\|u\|_{1,2}$. And to me it seems ...
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0answers
22 views

Is $W(0,T;H^1, L^2) \cap L^\infty(0,T;L^\infty(M))$ dense in $W(0,T;H^1, H^{-1})$?

Let $M$ be a compact Riemannian manifold that is closed. Define $$W(0,T, H^1, L^2) = \{ u \in L^2(0,T;H^1(M)) \mid u_t \in L^2(0,T;L^2(M))\}$$ $$W(0,T, H^1, H^{-1}) = \{ u \in L^2(0,T;H^1(M)) \mid u_t ...
0
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1answer
51 views

Weak derivative question

I just want to confirm that for weak derivatives you don't require the lower order derivatives to exist in order for the higher order derivatives to exist?
3
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1answer
63 views

What does $|Du| \leq 1 $ a.e. mean for $u\in L^2(\mathbb{R}^n)$.

What does $|Du| \leq 1 $ a.e. mean for $u\in L^2(\mathbb{R}^n)$. In the paper that I am reading, the authors used the notation $|Du| \leq 1 $ a.e. I know when $u\in W^{1,p}(\mathbb{R}^n)$ then ...
1
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1answer
51 views

Is the set of smooth functions with zero mean dense in the space of Sobolev functions with zero mean?

Given a Lipschitz domain $\Omega$ in $\mathbb{R}^n$, I know that the space of infinitely differentiable functions with compact support $C^\infty_0 (\Omega) $ is dense in the Sobolev space ...
0
votes
2answers
48 views

Question about this problem If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$

In this problem If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$ I don't understand this part: $$ \int_U \partial_i u^\epsilon\,v\,dx\to 0\tag w$$ for all $v\in L^2(U)$, but now we only ...
2
votes
2answers
32 views

$ \int_{\mathbb R^d}\int_{\mathbb R^d}b(x,y)\,f(x)\,f(y)dx\,dy \leq ||b_+||_{L^2(\mathbb R^d\times \mathbb R^d)}||f||_{L^2(\mathbb R^d)}^2 $

We are given the following, $$ b:\mathbb R^d \times \mathbb R^d \rightarrow \mathbb R,\;\; f:\mathbb R^d\rightarrow \mathbb R $$ and $$ f\in L^2(\mathbb R^d)\; ,\;b\in L^2(\mathbb R^d\times \mathbb ...
1
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1answer
46 views

weak* convergence in Sobolev space

Consider $\Omega \subset \mathbb{R}^{N}$, open and bounded. If $u_{n} \rightharpoonup^{*} u$ in $W^{1,\infty}_{0}(\Omega)$, then does it follow that $u_{n} \rightharpoonup^{*} u$ in ...
2
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0answers
30 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
1answer
22 views

Characterization of Sobolev space $H^m(\mathbb R^n)$ with $m\in\mathbb N_0$?

I want to show that if $m\in\mathbb N_0:=\mathbb N\cup\{0\}$ then $$H^m(\mathbb R^n):=\{u\in\mathscr{S}^\prime(\mathbb R^n): \exists f\in L^2(\mathbb R^n); \partial^\alpha f\in L^2(\mathbb R^n)\ ...
0
votes
1answer
98 views

Compact embedding of anisotropic Sobolev space

I am interested to know if the following result for anisotrpic Sobolev spaces is correctly presented. Also, does anyone have a good reference for this? Or can maybe confirm this result in its current ...
0
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1answer
44 views

The minimizing problem over a sequence of shrinking balls

Let $B(0,r)\subset \mathbb R^3$ be a ball centered at $0$ with radius $r$. Define $$ \mathcal A_r:=\{u\in H_0^1(B(0,r)),\,\,\|u\|_{L^{q+1}}=1\}$$ where $1<q<5$. Hence we know that each ...
1
vote
1answer
38 views

Question of $u\in L^p(U)$ does not have a trace on $\partial U$. [duplicate]

Let $U$ be bounded, with $C^1$ boundary. Show that a "typical" function $u\in L^p(U)$ does not have a trace on $\partial U$. More precisely, prove that does not exist a bounded linear operator ...
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0answers
42 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 ...
0
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1answer
20 views

If given $F(u_k)\to F(u)$ and $F'(u_k)\nabla u_k \to F'(u)\nabla u$ in $L^p$, why we have $F(u) \in W^{1,p}$ and $\nabla F(u)=F'(u)\nabla u?$

If $F$ continuous, $u_k \in C^{\infty}(\overline U)$, $u \in W^{1,p} (U)$. If given $F(u_k)\to F(u)$ and $F'(u_k)\nabla u_k \to F'(u)\nabla u$ in $L^p$, why we have $$F(u) \in W^{1,p}$$ and $$\nabla ...
2
votes
1answer
31 views

Approximation of Sobolev space functions by smooth functions

I am learning about the Sobolev spaces from the book "Theoretical numerical analysis : a functional analysis framework" by Atkinson and Ham. Here is the excerpt from the book that I do not ...
0
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1answer
37 views

How to prove the chain rule with respect to weak derivatives?

The following theorem is on the textbook "weak differentialble functions". I found it confusing from the absolutely continuous part. I am writing to ask is this the only way to prove it? Can anyone ...
0
votes
1answer
21 views

Is $H^1(0,\infty) \subset C^0([0,\infty))$?

Is it true that $H^1(0,\infty) \subset C^0([0,\infty))$ is a continuous embedding? How would I prove it? I do know this holds for bounded domains in one dimension but here we have the half line. ...
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0answers
31 views

Why is this integral involving the mean value function zero?

Let $u$ and $v$ belong to $H^1(\Omega \times (0,\infty))$ on a bounded domain $\Omega$. Define $$(Au)(y) := \frac{1}{|\Omega|}\int_\Omega u(x,y)\;\mathrm{d}x.$$ We have that $Au \in H^1(0,\infty)$. ...
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1answer
24 views

How to compute the $H^{-s}(\Omega)$-norm of a function?

Suppose to have a sufficiently regular domain $\Omega\subseteq\mathbb{R}^d$. I know that, for $s\in\mathbb{R}_+$, the space $H^{-s}(\Omega)$ is defined as the dual of $H^s_0(\Omega)$, endowed with the ...
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0answers
43 views

$-\Delta u = f$ in $L^2(0,T;H^{-1}(\Omega))$ (as opposed to $H^{-1}(\Omega)$)

Why does nobody consider the equation $-\Delta u = f$ in the space $L^2(0,T;H^{-1}(\Omega))$? Eg. given $f \in L^2(0,T;L^2(\Omega))$ find a solution $u \in L^2(0,T;H^1_0(\Omega))$ such that ...
1
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0answers
30 views

Show that elements $u \in W^{1,\infty}(U)$ have continuous representatives

Let $U$ be a bounded, open subset of $\mathbb{R}^n$, and suppose that the boundary $\partial U$ is of class $C^1$. Suppose that $u \in W^{1, \infty}(U)$. I wish to prove that there exists a ...
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0answers
38 views

Inclusion of Sobolev spaces with fractional order

Let $W^{k,p}\mathbb(R^n)$ the usual Sobolev space. We know that if $k>l$ and $1\leq p<q<\infty$, $(k-l)p<n$ and $$\frac{1}{q}=\frac{1}{p}-\frac{k-l}{n}$$ then ...
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0answers
13 views

what is the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE?

I want to know the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE when right hand side of the elliptic operator is $L^1.$ Also how does right hand side with ...
0
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0answers
64 views

Product rule in fractional Sobolev spaces

I have to prove the following inequality $$\Vert fg\Vert_{H^s(\mathbb{R}^3)}\leq C \Vert f\Vert_{H^\sigma(\mathbb{R}^3)}\Vert g\Vert_{H^{s+1}(\mathbb{R}^3)}$$ with $s\geq 0$, ...
1
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1answer
15 views

Equivalent Solovec norms (atypical)

I have $s\in\mathbb{R}$ and I'm looking at the space $H^s$ of functions $f$, such that $$\lVert f\rVert_{H^s}^2:=\int_\mathbb{R} (1+\lvert x\rvert^2)^s\lvert f(x)\rvert^2\,dx<\infty.$$ I want to ...
1
vote
2answers
47 views

Infimum of $\frac{||u'||^p_{L^p}}{||u||^p_{L^p}}$ for $u \in W^{1,p}_0((0,1))$

Good afternoon everyone! It is very easy to show that the infimum mentioned in the title is strictly positive, but it seems much more difficult to show that it is attained within the Sobolev space of ...
2
votes
1answer
32 views

Are trace function embedded in $L^\infty$?

Consider a bounded domain $\Omega \subset \mathbb R^d$ with a Lipschitz boundary (could also be a smooth boundary). Is the trace space $H^{1/2}(\partial\Omega)$ embedded in $L^\infty(\partial\Omega)$? ...
1
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0answers
38 views

Boundedness of Helmholtz projection

Let's cosider the Helmoltz projection $P$ into solenoidal subspace defined as $$P=I-\nabla div\Delta^{-1}$$ What can I say about its boundedness as an operator in $H^s(\mathbb{R^3})$?
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0answers
61 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
28 views

log-sobolev inequalities for infinite measures.

I was wondering if we could have log-sobolev inequalities for infinite measures, most notably Lebesgue measure. I presume this is false, but I haven't been able to construct one. I tried playing ...
1
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0answers
14 views

Relationships among different definitions of Sobolev spaces

In Tsybakov's book(Page 51), Sobolev space (or Ellipsoid) for positive smoothness parameter $s$ is defined with sequential model, i.e. the series of the Fourier coefficients is finite. On the other ...
2
votes
1answer
75 views

Laplacian and Hodge Laplacian

I am new to the theory of differential forms, but there is one thing that I don't get at all. Imagine that you are on the sphere $\mathbb{S}^2$, then the Laplacian $- \Delta$ is known to be a ...