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
67 views

$H^1$ function with smallest seminorm

Let $\Omega\subset\mathbb R^d$ be a Lipschitz domain and $u\in H^1(\Omega)$. Find a $w\in H^1(\Omega)$ with the same boundary values but minimal seminorm on $H^1(\Omega)$. I've read that harmonic ...
2
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2answers
111 views

Improvement of $W^{1,p}$ regularity of a elliptic equation solution.

I'm looking for some reference for results like $$ \mbox{div}(A(x) \nabla u) = 0, \ \ u \in H^1=W^{1,2} \Rightarrow u \in W^{1,p}, p>2 $$ where $A(x)$ is elliptic, this is, $Id\lambda \le A(x) \le ...
1
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0answers
57 views

Sobolev spaces - about weak derivative [duplicate]

Let $U$ a bounded and open subset of $R^n$. Let $u \in H^{1}(U)$ a bounded function , $v \in H^{1}_{0}(U)$ a non negative function. Consider $\varphi : R \rightarrow R$ a convex and smooth ...
1
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1answer
67 views

Distributional derivative of a square integrable function and dual of Sobolev space

This is a simple question: Let $f\in L^2(\mathbb{R})$, $f'$ be its distributional derivative, then is $f'$an element of $H^{-1}(\mathbb{R})$, the dual of Sobolev space $H^1(\mathbb{R})$? Also, if I ...
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1answer
287 views

sobolev spaces - product of two functions

I am working in a exercise, to my solution works I need the following affirmation is true: Let $\varphi : R \rightarrow R$ a convex and smooth function. Let $u \in H^{1}(U)$ a bounded function and $v ...
2
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2answers
57 views

Is the following version of the fundamental lemma of the calculus of variations valid?

Let $U$ be an open subset of $\mathbb{R}^n$ with smooth boundary $\partial U$. Consider a function $f$ in $L^2(u)$. Suppose that for every $h$ in the Sobolev space$ H^2_0(U)$ it holds that $$\int_U f ...
3
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2answers
108 views

Finding a strong enough solution to a specific PDE problem.

Let $U\subset \mathbb{R}^n$ with smooth boundary $\partial U$. And consider the expression $$\Delta u = f.$$ $$\text{+"convenient boundary conditions"}$$ In my specific case $f\in H^2_0$. Under ...
3
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1answer
114 views

Gauss–Ostrogradsky formula for Distributions

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $u\in W^{1,p}(\Omega)$, $v\in L^{p'}(\Omega)^N$ with $p\in (1,\infty)$. Let $\operatorname{div}(v)$ be the divergence of $v$ in the sense of ...
2
votes
2answers
63 views

Definition of Sobolev Space

I have a definition that says that the space of functions that satisfy$$\|u\|_{H^m}^2=\sum_{k\in\mathbb{Z}}(1+|k|^2)^m|\hat{u}_k|^2<\infty$$is called Sobolev Space and when $m=1$, this is ...
3
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0answers
192 views

Weak solution to PDE boundary value problems

Take $\lambda>0$ and let $\Delta$ be the Laplacian operator in dimension $n$ and let $\Delta^2:=\Delta\circ\Delta$. Consider the boundary value problem $\Delta^2 u+\lambda u=f$ on some open subset ...
5
votes
1answer
72 views

Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary?

Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary? What smoothness is required of the boundary? I would be grateful for some references to this.
1
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1answer
223 views

What is the Sobolev Lemma?

In the paper I am reading the authors state that $|\nabla u|_\infty$ can be replaced by $|u|_3$ using the Sobolev Lemma. I am trying to find this lemma but its turned out to be very difficult. The ...
0
votes
2answers
75 views

Relationship between sobolev spaces

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. I would like to understand something about the relationships between these two spaces: $$H^1_0(\Omega) \cap H^2(\Omega)\quad \text{and}\quad ...
11
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1answer
488 views

Hille Yosida theorem application

Disclaimer: pretty long and specific (contraction semi groups involved). I have fourth order parabolic equation $$ u_t + \Delta^2 u = 0 $$ on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...
2
votes
1answer
80 views

problem from PDEs, H. Brezis

today I read book Sobolev space, PDEs of H.Brezis, and when I read chapter 8, I don't know why following remark is easy: Remark 11 (page 214). Let $I$ be a bonded interval, let $1\leq p\leq \infty$, ...
1
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1answer
61 views

Easy question about $H_0^1$ space

I have some trouble with proper understanding of $H_0^1(0,1)$ space. Consider the following space $$H_D = \{u\in H^1(0,1): u(0) = u(1) = 0\}.$$ What can we say about the connection between $H_D$ and ...
2
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0answers
232 views

Showing that smoothing operators are compact

Suppose I have a bounded, linear map $T: H^1(X) \to H^1(X)$ such that $T(H^1(X)) \subset C^\infty(X)$. Is $T$ a compact operator? I'm guessing this depends on whether or not $X$ is (pre)compact, and ...
2
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1answer
166 views

dominated convergence theorem

I am studying the proof of a theorem and in a part of the proof I have the following situation: Let $u : \Omega \rightarrow R$ a nonnegative measurable function, with $\Omega$ open and bounded. ...
-1
votes
2answers
117 views

finite elements-exercice

We consider in $\mathbb{R}^2$ the set of points $$\{M_1(-1,1),M_2(0,1), M_3(2,1),M_4(-1,0),M_5(1,0),M_6(2,0)\}$$ Let $\Omega$ a rectangular structure consisting of the heads $\{M_4(-1,0),M_6(2,0), ...
0
votes
1answer
30 views

$g=(g_1,…,g_N)$ $Q$ periodic implies $\int_Q \operatorname{div} g=0$?

Let $g_i:\mathbb{R}^N\to\mathbb{R}$ ($i=1,...N$) with $g_i\in W^{1,\infty}(\mathbb{R}^N)$ and define $g=(g_1,...,g_N)$. Let $G=\operatorname{div}g$, where $\operatorname{div}g=\frac{\partial ...
0
votes
1answer
305 views

norm of sobolev space $H^{1/2}$

Let $\Omega\subset\mathbb R^d$ a Lipschitz domain and $\Gamma:=\partial\Omega$. For $u\in C^{\infty}(\Gamma)$ we define $$||u||_{H^{1/2}(\Gamma)} = \inf_{\substack{v\in H^1(\Omega) \\ v|_\Gamma =u}} ...
2
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1answer
55 views

Proving that a certain function is in $W^{1,n}(B(0,1))$

Fix $0<\alpha<1-\frac{1}{n}$ and let $f\colon\mathbb{R}^n \rightarrow \mathbb{R}$ be the function $f(x)=(\log(\frac{1}{|x|}))^{\alpha}$. How can I prove that $f\in W^{1,n}(B(0,1))$?
2
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1answer
94 views

An example of self-adjoint and positive operator

Let $\Omega\subset R^2 $ and let $H=H_0^1(\Omega)\cap H^2(\Omega)$ with inner product : $\langle u,v\rangle_H =\langle u,v\rangle_{L^2(\Omega)} + \langle\Delta u,\Delta v\rangle_{L^2(\Omega)}$. I am ...
6
votes
2answers
70 views

Decay of $H^1(\mathbb{R}^n)$ functions

Is it true (is there a commonly known theorem) that says: $f \in H^1(\mathbb R ^n)$ $\Rightarrow$ $\displaystyle \lim_{|x| \to \infty} f(x) = 0$ pointwise (where $H^1$ denotes the Sobolev space ...
2
votes
0answers
189 views

Stampacchia Theorem: $\nabla G(u)=G'(u)\nabla u$?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $G:\mathbb{R}\to\mathbb{R}$ a Lipschitz function with $G(0)=0$. Stampacchia's Theorem states that if $u\in W_0^{1,p}(\Omega)$, then $G(u)\in ...
2
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1answer
66 views

Difference Quotient Proof

Theorem: Let $u \in W^{1,p}(U)$ and let $V \subset \subset U$ (I.e. there is a compact set containing $V$ that is in $U$). Then for $1\le p <\infty$ there exists a constant $C$ such that for $0 ...
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1answer
38 views

An inequality : $ \| u^3 \|_{H^3} \leq C ( \| u \|_{L^\infty}^2 + \| \partial_x u \|_{L^\infty}^2 ) \| u \|_{H^3}$

Let $u=u(x)$ be a real-valued function defined on $\mathbb R$. How does this inequality hold? $$ \| u^3 \|_{H^3} \leq C ( \| u \|_{L^\infty}^2 + \| \partial_x u \|_{L^\infty}^2 ) \| u \|_{H^3}$$ ...
1
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0answers
79 views

adjoint operator in Sobolev space

Let $g\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega),$ and let us define the operator $B : y \to g y$ from $H:=H_0^1(\Omega)\cap H^2(\Omega)$ to $H$, which we endowed with inner product : $<u,v>_H ...
1
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1answer
34 views

Extention of functions in Sobolev spaces

Let $\omega$ a subset of a domain $\Omega\in R^n,$ and let $f\in H^2(\omega)\cap H_0^1(\omega)$. It is known that a function $u\in H_0^1(\omega)$ admits an axtention $U\in H_0^1(\Omega)$. Does ...
1
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1answer
50 views

for what values $\alpha\in \Bbb{R}$ we have $v_{\alpha}\in H^{1}(\Omega)$?

If $\Omega$ circle with radius $\frac{1}{2}$ centred at the origin, for what values $\alpha\in \Bbb{R}$ for function $$v_{\alpha}(x,y)=|log(x^2+y^2)|^{\alpha}$$ we have $v_{\alpha}\in H^{1}(\Omega)$?
4
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1answer
73 views

Distributional traces

Let $\Omega \subset \mathbb R^n$ be a domain with a smooth boundary $\partial \Omega$. I know that for $s > 1/2$, one can define a (zeroth order) trace operator $$ \gamma \colon H^s(\Omega) \to ...
0
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1answer
169 views

Simple heat equation, solution regularity

I have a small problem with a regularity result for a simple parabolic heat equation: Given a $C^2$ open subset of $\mathbb R^n$ called $\Omega$, and a time $T > 0$, i have the following heat ...
5
votes
1answer
200 views

Why is $H^1 \neq H_0^1$?

I've been doing some reading on Sobolev-spaces and one remark said that $H_0^1$, i.e. the space of $H^1$-functions with zero-boundary values, is not the same as $H^1$. This seems clear to me, but when ...
2
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2answers
86 views

What happens when you change space of test functions associated with weak derivatives?

Recall that $u \in L^2(0,T;H^1)$ has weak derivative $u' \in L^2(0,T;H^{-1})$ iff $$\int_0^T uv' = -\int_0^T u'v$$ holds for all $v \in C_0^\infty(0,T).$ What happens if we only require that this ...
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1answer
186 views

Proof that function is in $H^1$

Let $\Omega = \{(x,y)\in\mathbb R^2: \sqrt{x^2+y^2}<\frac{1}{2}\}$ be a bounded domain and $v(x,y) = \ln\left|\ln\sqrt{x^2+y^2}\right|$. I'm trying to show that $v\in H^1(\Omega)$ where ...
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1answer
45 views

if the region $\Omega'\subset\Omega$ and $f\in H^k (\Omega)$ then$ f\in H^k (\Omega')$ [closed]

prove that if the region $\Omega'\subset\Omega$ and $f\in H^k (\Omega)$ then $f\in H^k (\Omega')$
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1answer
78 views

There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$?

There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$? Justify your answer! Thanks.
0
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1answer
62 views

Mean value theorem in sobolov space under integral

Sorry this seems like a basic question, but I'm having trouble figuring out the answer. Let g(x) be the step function over [-1,1] and f(x) a function with $f\in H^1[-1,1]$, that is it has a square ...
2
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0answers
63 views

$H^s(\mathbb{T})$ space is a Banach algebra with pointwise product

I have ran across the following theorem but the given proof does not convince me. Theorem Let $u, v \in H^s(\mathbb{T})$ with $s>1/2$. Then the pointwise product $uv$ is in $H^s(\mathbb{T})$ and ...
1
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1answer
87 views

Sobolev spaces - embedding - exercise

I have to show that $H^{2}(\Omega) \subset \subset H^1(\Omega)$. I think the Arzela - Ascoli theorem can help.. .I dont know how to start this exercise . I am beginner in Sobolev spaces.. someone can ...
5
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1answer
216 views

Proving an alternative norm on Sobolev space is equivalent to usual norm

I have this exercice and my problel is only in item 4, and i will desespere. Let $f \in L^2(\mathbb{R}^n).$ 1- Why the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admits a unique ...
5
votes
2answers
147 views

Is $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$ if $g$ is a Sobolev function on the real line?

If we are given a function $g\in W_2^k(\mathbb{R})$ (even consider $k=1$ for simplicity), then is it true or not that $\{g(n)\}_{n\in\mathbb{Z}}\in\ell_2$? That is, do we have ...
6
votes
1answer
207 views

Why are weak solutions to PDEs good enough?

Obviously solutions in $C^k$ are nicer but it seems people are happy with obtaining weak solutions in some Sobolev space that only satisfy the weak formulation. Why should this, in the real world, be ...
1
vote
1answer
107 views

Use difference quotient with not uniform bound to appoximate weak derivative

Suppose U is an open set,not necessarily bounded or has Lipschitz boundary, $f\in L^p(U)$ ,define the difference as usual: $$D^h_i f=\frac{f(x+he_i)-f(x)}{h},\ \ \forall x\in U'\subset\subset U$$ ...
1
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0answers
28 views

“Compensated weak compactness” in $W^{1,1}$

In a problem I'm working on I want to show that a bounded sequence $\{u_n\} \subset W_0^{1,1}((0,1))$ converges weakly. Of course, since $W^{1,1}$ is not reflexive I don't get this for free. The ...
1
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1answer
122 views

How to prove $\int_{\Omega}\frac{1}{\kappa}uv\ +\ \int_{\Omega}\kappa\nabla u\cdot\nabla v$ is an inner product in $H^1$

I need help with this problem from my homework Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and let $\kappa : \Omega\rightarrow\mathbb{R}$ be a continuous function, such that there's ...
2
votes
2answers
96 views

Sobolev space boundary value in PDE

I often read this: Let $\Omega$ be a open bounded set. There is a unique $u \in H^1(\Omega)$ such that $$-\Delta u = f \text{ on $\Omega$}$$ $$u|_{\partial \Omega} = g$$ But how can we write ...
3
votes
3answers
250 views

How to prove that $(u-v)^+\in W_0^{1,2}(\Omega)$, if $u\in W_0^{1,2}(\Omega)$, $v\geq 0$.

Let $\Omega$ denote a open subset of $\mathbb{R}^n$, and $W^{1,p}(\Omega)$ the Sobolev space of weakly differentiable functions $u\in L^p(\Omega)$ (that is, for which $D_iu$ exists and belongs to ...
1
vote
2answers
87 views

When does $v \in H_0^1$ and $|v|_{L^2}=0$ imply that $\|v\|_{H_0^1}=0$?

Let us assume that the boundary of the domain in the definition of the Sobolev spaces $L^2$ and $H_0^1$ is sufficiently smooth. Let $|\cdot |$ denote the norm in $L^2$. Then for a function $v$ in ...
1
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1answer
149 views

sobolev space-equivalence of scalar product

Let $f \in L^2(\mathbb{R}^n).$ Why the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admits a unique solution $u \in H^1(\mathbb{R}^n)$? 2- Prove that there exist a constant $C \geq 0$ ...