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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8 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 ...
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1answer
15 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
30 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
18 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
36 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 ...
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0answers
24 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
34 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
9 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 ...
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0answers
57 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 $s\geq 0$, ...
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1answer
14 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 ...
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2answers
33 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 ...
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1answer
22 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 $L^\infty(\partial\Omega)$? ...
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0answers
34 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
47 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
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0answers
25 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 ...
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0answers
7 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
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1answer
45 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 ...
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1answer
37 views

weak convergence of $L^2$ implies weak convergence of $W_0^{1,2}$ (up to a subsequence)?

In the paper that I am reading, it says that if $\{u_n\}$ are bounded in $W_0^{1,2} (\Omega)$ (bounded $\Omega\subset \mathbb{R}^N$) and $u_n \rightharpoonup u$ weakly in $L^2 (\Omega)$, then there ...
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1answer
34 views

Differential operator a bounded operator or not?

Is the operator $T$ a bounded operator mapping $T: H^n([0,\pi]) \rightarrow H^{n-1}([0,\pi])$ ($H^n$ is the n-th Sobolev space with respect to $L^2$) or not? The operator itself is given by ...
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1answer
43 views

Stuck with this function

im trying to find the values of $\alpha \in \mathbb{R}$ for which the function $f: B_2(0,\frac{1}{2}) \rightarrow \mathbb{R} $, $ x \mapsto |\log(\|x\|_2)|^\alpha$ is in $L^2(B_2(0,\frac{1}{2}))$ and ...
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0answers
36 views

$\sup$ of a $C^s$ smooth function.

I want to prove that for a function $F\in C^k(\mathbb{R}^n)$ which vanishes at zero, and a function $u\in H^k(\mathbb{R}^n)$we get: $$\left\| \int_{r=0}^1 F'(ru)(\cdot)dr \right\|_{L^{\infty}} \leq ...
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1answer
59 views

Chain rule in a Sobolev space

(Chain rule) Assume $F : \mathbb{R} \to \mathbb{R}$ is $C^1$, with $F'$ bounded. Suppose $U$ is bounded and $u \in W^{1,p}(U)$ for some $1 \le p \le \infty$. Show $$v :=F(u) \in W^{1,p}(U) \quad ...
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2answers
48 views

first order weak derivative of function $ f(x) = |x| $

let $f(x)= |x| $ how can I calculate the first order weak derivative of this function in $x=0$? Does anyone have an idea on how to calculate this?
4
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0answers
58 views

why $ \nabla v_n \to \nabla v \ \ (a.e.)$ and $ v_n \to v $

Can someone see the 10th line of page 9 in this article and give a hint that why $$ \nabla v_n \to \nabla v \ \ (a.e.)$$ and $$ v_n \to v $$ and how with theorem 2.1 we could conclude there exists $ ...
2
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0answers
35 views

if $\Delta u = |\nabla u|^{3/2}$ then $u \in C^{\infty}$

Suppose $u \in H^1_{\text{loc}}(\mathbb{R^2})$. I want to show that if $\Delta u = |\nabla u|^{3/2}$ in the distributional sense then $u \in C^{\infty}$. I know that since $\nabla u \in L^2$ (I'll ...
0
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1answer
58 views

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

Provide details for the following alternative proof that if $u \in H^1(U)$, then $$Du = 0 \text{ a.e. } \, \text{ on the set } \{u=0\}. $$ Let $\phi$ be a smooth, bounded and nondecreasing ...
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1answer
34 views

If $u \in W^{1,p}(U)$, prove that $Du=0$ a.e. on the set $\{u=0\}$.

Assume $1 \le p \le \infty$ and $U$ is bounded. (a) Prove that if $u \in W^{1,p}(U)$, then $|u| \in W^{1,p}(U)$. (b) Prove $u \in W^{1,p}(U)$ implies $u^+,u^- \in W^{1,p}(U)$, and ...
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2answers
50 views

About an uncommon theorem of Morrey

I am searching for a reference with the following result: Let $\Omega \subset R^n$ an open bounded domain with smooth boundary . Let $2\leq p < n$ and $u \in W^{1,p}(\Omega)\cap ...
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1answer
39 views

help to prove $||u||_{W^{2,2}(\Omega) }\le C ||\Delta u ||_{L^2(\Omega )} $

Can some one give a reference or hint for proving $$||u||_{W^{2,2}(\Omega)} \le C ||\Delta u ||_{L^2(\Omega )} $$
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0answers
26 views

This Sobolev function is continuous?

Let $\Omega \subset R^n $ $(n \geq 2) $ an open bounded domain with smooth boundary $u \in W^{ 1,p}(\Omega)\cap L^{\infty}(\Omega)$ ($p \geq 2$ fixed). Suppose that exist $M > 0$ such that $$ ...
3
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1answer
81 views

$-\Delta u - \alpha u^{1/3} = 0$ implies $u \equiv 0$ if $\alpha$ is small

Let $\Omega$ be a domain in $\mathbb{R}^{d}$ with smooth boundary. Let $u(x)$ be a $H^{1}(\Omega)$ solution of the equation $-\Delta u - \alpha u^{1/3} = 0$, $u|_{\partial \Omega} = 0$. The problem I ...
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0answers
20 views

Compactness of Pseudo-differential Operators on $H^s(\mathbb R^n)$?

The Sobolev space of order $s\in\mathbb R$ in $\mathbb R^n$, denoted by $H^s(\mathbb R^n)$, is defined as follows: $$H^{s}(\mathbb R^n):=\{u\in\mathscr{S}^{'}(\mathbb R^n): \exists f\in ...
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1answer
36 views

weak* convergence definition in Sobolev space

I have a question which might quite trivial but I would appreciate any assistance. Why does it follow that for Sobolev spaces, say $W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$, it follows ...
2
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0answers
34 views

Verify that the unbounded function belongs to $W^{1,n}$ [duplicate]

Verify that if $n > 1$, the unbounded function $u = \log \log \left(1+\frac 1{|x|}\right)$ belongs to $W^{1,n}(U)$, for $U=B^0(0,1)$. This is PDE Evans, 2nd edition: Chapter 5, Exercise 14. ...
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1answer
20 views

Bounded Right Inverse

If a linear operator between two Banach spaces is surjective and bounded, can we get any information about a right inverse? For example, is it bounded? Thanks, trying to understand trace operator ...
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1answer
48 views

Delta distribution as a bounded linear functional

Let $\Omega \subset \mathbb{R}^n$ be a domain with $0 \in \Omega$. For which $n \in \mathbb{N}$ is the Dirac-delta distribution a bounded linear functional on $H_{0}^{1}(\Omega)$, that is to say, an ...
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0answers
41 views

Proof of weak derivatives in Evans PDE?

In the textbook of Partial differential equation of Evans. Why from $\int_U(v-\overline v)\phi dx=0$ for all $\phi \in C_c^\infty (U)$, we can get $v-\overline v=0$ a.e.? How to prove it? ...
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1answer
36 views

Why does this completion of a Sobolev space contain constant functions? Please explain text.

Below, $\mathcal{C} = \Omega \times (0,\infty)$, $x$ refers to the variable in $\Omega$ and $y$ to the variable in $(0,\infty)$, and $\Omega$ is a bounded smooth domain. $tr_\Omega:H^1(C) \to ...
3
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1answer
34 views

Using Mac Shane's Lemma

Let $I \subset \mathbb{R}^{N}$ be a convex, bounded open set with Lipschitz boundary $\partial I$. Let $\lbrace u_{n} \rbrace_{n}$ and $u$ be such that $$ u_{n} \rightharpoonup^{*} u~~ \text{ in }~ ...
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1answer
74 views

Integrate by parts to prove this inequality

Prove $$\|Du\|_{L^{2p}(U)} \le C\|u\|_{{L^\infty}(U)}^{1/2} \|D^2 u\|_{L^p(U)}^{1/2}$$ for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$. This is PDE Evans, 2nd edition: Chapter 5, Exercise ...
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1answer
42 views

$\int_\Omega |\nabla u^+|^2 \, dx$ is not differentiable with respect to $u$ in $W_0^{1,2}(\Omega)$

Let $u \in W_0^{1,2}(\Omega)$, where $\Omega$ is some domain in $\mathbb{R}^N$, $N \geq 1$. Denote $u^+ := \max\{u, 0\}$. (It is know that $u^+$ also belongs to $W_0^{1,2}(\Omega)$ (see, e.g., ...
2
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0answers
32 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 ...
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1answer
36 views

Deriving $|u(x)-u(y)|\le|x-y|^{1-\frac 1p}\left(\int_0^1 |u'|^p \, dt \right)^{1/p}$

Assume $n=1$ and $u \in W^{1,p}(0,1)$ for some $1 \le p < \infty$. (a) Show that $u$ is equal a.e. to an absolutely continuous function and $u'$ (which exists a.e.) belongs to $L^p(0,1)$. ...
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1answer
34 views

The Trace Theorem for $W^{1,p}$ functions

I'm trying to understand the proof of the trace theorem. This is from a course I am taking, so I will write out what we have done explicitly. $\textbf{Trace Theorem}$ Suppose $\Omega ...
3
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1answer
49 views

Convergence in dual of Sobolev space

Hi please view the following question: Consider Sobolev space $W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^{n}$ is bounded. We also have a mapping $a: \Omega \times \mathbb{R} \times ...
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1answer
83 views

The averages, near the boundary, of a function $u\in W_0^{1,p}(\Omega)$, converge to zero.

Let $E_\delta =[0,1]^{N-1}\times [0,\delta]$, $p\in [1,\infty)$ and $1/p+1/p'=1$. Let $\varphi\in C^1(E_\delta)$ such that $$\varphi(x)=0,\ \forall \ x\in [0,1]^{N-1}\times \{0\}.$$ By the ...
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0answers
37 views

Space $W_0^{1,p}$ is weakly closed, and some related problems.

Take $1\leq p<\infty$ and $\Omega\subset\mathbb R^N$ is open, not necessary bounded or have smooth boundary. Define the admissible set $$ \mathcal A:=\{u\in W_0^{1,p}(\Omega),\, \lvert \nabla ...
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0answers
25 views

A generalization of the problem: "$\|\Delta u\|_{L^2}$ is an equivalent norm for standard $H^2$ norm in space $H_0^1\cap H^2$

We know the norm $\|\Delta u\|_{L^2(\Omega)}$ is an equivalent norm of $H^2$ norm in space $H_0^1(\Omega)\cap H^2(\Omega)$ where $\Omega$ open bounded with smooth boundary. Now let's generalize this ...
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0answers
18 views

Proof of existence of trace map $T:H^1(\mathbb{R}^n_+) \to H^{\frac 12}(\mathbb{R}^{n-1})$ not using the Fourier transform

I'm looking for a proof of the existence of the trace map $T:H^1(\mathbb{R}^n_+) \to H^{\frac 12}(\mathbb{R}^{n-1})$ which does not use the Fourier transform. In particular, I want to prove the ...
0
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1answer
47 views

The Sobolev Space $H^{1/2}$

this is a very stupid question. In my course of linear PDEs, the professor used $H^{1/2}$ without defining, and I have looking on google to find a definition, but the only related thing I found was ...