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Questions tagged [sobolev-spaces]

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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Am I allowed to integrate by parts?

I have the following situation: Consider a function $f\in H^1(\mathbb{R})$, and a uniformly bounded function $S\in\mathcal{C}^\infty(\mathbb{R})$ with well-defined limits at $\pm\infty$ and such that ...
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1answer
35 views

About an equality of fractional Laplacian on a bounded domain

Let $0<s<1$. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. We know that $$\|(-\Delta)^{s/2}u\|_{L^2(\mathbb{R}^n)}^2=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+...
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1answer
19 views

A convenient redefiniton of the Sobolev norm

I am dealing with the Sobolev space $W^{m,2}[0,1]$, i.e. the space of functions on $[0,1]$ with absolutely continuous $m-1$st derivative and square integrable $m$th derivative. I am using the norm $$|...
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1answer
11 views

Sobolev-Gagliardo-Nirenberg: Why is $|f|^q$ continously differentiable?

I wanna understand a proof of the Sobolev-Gagliardo-Nirenberg inequality. Therefore, I need to know why $|f|^q \in C_c^1(\mathbb{R}^n)$ for $f \in C_c^1(\mathbb{R}^n)$ and $q>1$. Can eventually ...
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24 views

Technique to prove existence?

I would like to prove the existence of a T-periodic function $v$ in $H^1(R^N)$ s.t. $v=|v|e^{i\theta}$ for some T-periodic $\theta \in H^1(R^N)$ and s.t. one certain functional $I_c$ turns negative at ...
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2answers
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If $v\in H^1(]0,1[)$, then $|v(\lambda)| \leq ||v||_{L^1(]0,1[)} + ||v'||_{L^2(]0,1[)}$

Let $v\in H^1(]0,1[)$. I want to prove for all $\lambda \in [0,1]$ that, $$|v(\lambda)| \leq ||v||_{L^1(]0,1[)} + ||v'||_{L^2(]0,1[)}$$ My idea : I defined $u(\lambda)= \int_0^{\lambda}v'(t)dt$ ...
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1answer
27 views

Is the projection onto the unit circle Sobolev?

Let $f(x,y)=\frac{x}{\sqrt{x^2+y^2}}$. Does $f \in W^{1,p}(B)$ for some $p \ge 1$, where $B$ is the open unit disk in $\mathbb{R}^2$? (I guess we can replace $B$ with a disk with arbitrarily small ...
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$\|f\|_{L^{\frac{10}{3}}(\mathbb T^3)} \leq C \|f\|_{H^1(\mathbb T^3)}^{3/5} \|f\|_{L^2(\mathbb T^3)}^{1-(3/5)}$? [on hold]

Let $\mathbb T^3$ be 3-dimensional torus. Can we expect that $\|f\|_{L^{\frac{10}{3}}(\mathbb T^3)} \leq C \|f\|_{H^1(\mathbb T^3)}^{3/5} \|f\|_{L^2(\mathbb T^3)}^{1-(3/5)}$?
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How is this operator well defined? $\frac{D}{(1+D^2)^{1/2} }$.

Let $D_+ = \partial_x +x, D_-=-\partial_x+x$. $$D= \begin{pmatrix} 0 & D_- \\ D_+ & 0 \end{pmatrix} $$ which acts on a dense subspace $C_c(\Bbb R) \oplus C_c(\Bbb R)$ of $L^2(\Bbb R) \...
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Discrete equivalent of Sobolev norms and numerical experiment

I am solving a boundary value problem (BVP) that involves a system of equations (similar to the Euler or Navier-Stokes equations) for which, at this moment, there exists no sufficient theory to define ...
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1answer
26 views

Convergence of Sequence of Solutions to Elliptic Equation

Consider the standard uniformly elliptic equation on a domain $\Omega \subset \mathbb{R}^d$: $$ \mathrm{div}(A(x)\nabla u) = f $$ for $u \in H^{1}(\Omega)$ , $f\in H^{1} (\Omega)$, $a_{ij}(x)$ ...
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compute $H^{3/2}(\partial\Omega)$-norm for smooth $u$ and $\Omega$

I am a little bit confused about different definitions of the trace space $H^{3/2}(\partial \Omega)$, and I hope I can find some simple examples on how to explicitly compute these norms for simple ...
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Heuristic on Sobolev and BV functions

Let $f: \Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a Sobolev or BV vector field. A heuristic that I've heard frequently is the following: $f$ is almost Lipschitz on a large "good" set but ...
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1answer
35 views

Inequality in Sobolev Space ($L^p$ norm)

I want to find a constant $C$ that depends on the parameters $a$ and $p$ that satisfies the inequality $$\|f\|_p \leq a\|f'\|_1+C\|f\|_1$$ for all $f \in W^{1,1}(0,1)$. This is for arbitrary $p \in [...
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$\langle curl \ u, curl \ v \rangle + \langle div \ u, div \ v\rangle=\langle f,v\rangle$ and $\operatorname{div} f=0 \Rightarrow div \ u=0$

We consider a solution $u \in H_0(curl)\cap H(div)$ of $\langle curl \ u, curl \ v \rangle + \langle div \ u, div \ v\rangle=\langle f,v\rangle$ (1). Here $f \in L^2(\Omega), div f=0, i.e. \langle ...
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1answer
24 views

Sobolev's inequality for higher derivatives

The following is from p.102 of Sobolev Spaces by R. Adams and J. Fournier. Here $\|\cdot\|_q$ is the $L^q$ norm, $C_0^\infty$ means compactly supported and $C^\infty$, and $$|\phi|_{m,p}:=\bigg(\sum_{...
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$\Delta u = f , \operatorname{div} f=0 \Rightarrow \operatorname{div}u=0$ on non convex domain.

I am specifically referring to this paper and why equation (7.7) is the weak formulation of (7.6). My question is why $ \operatorname{div} u=0$ is implied by formulation (7.7) on a general domain. If ...
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What is optimal constant in an inequality?

My questions are that What is 'optimal' constant in an inequality? and What is the difference between 'sharp' and 'optimal' constant? In a Theorem of a paper, it's given that Let $\...
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Eigenfunctions and Spectral Decomposition

(Theory 9.31, from Haim Brezis functional analysis Sobolev space and partial differential equations, P311, chapter 9) $\Omega$ is bounded open set. There exist a Hilbert basis $(en)_{n\geq 1} $ of $...
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1answer
31 views

Reference request: Laplace-Beltrami eigenfunction bases for Sobolev spaces

I'm working on a smooth $(d-1)$-dimensional surface $M\subset \mathbb{R}^d$. Let $(\phi_k)_{k\in\mathbb{N}}$ be an orthonormal basis of $L^2(M)$ consisting of the eigenfunctions of the Laplace-...
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1answer
20 views

Proof of a theorem on Sobolev multipliers on $(0,1)$ without extending to $\mathbb R$

So, I would like to prove "intrinsically" that if $g\in L^2(0,1)$ and $b > 0$, I can find $a\ge 0$ such that for all $f\in H^1(0,1) = W^{1,2}(0,1)$ $$ \|gf\|_2\,\le\,a\|f\|_2 + b\|f'\|_2. $$ I have ...
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42 views

Integrabilty of distibutions

Let $T : D(\Omega) \to \mathbb R$, be a generalised function. What can we say about its integrability? For example, does it belong to $L^p(\mathbb R)$? I am interested in the case of non-regular ...
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2answers
48 views

$L^2$ and Sobolev space

In Raymond's book on Pseudodifferential Operator page 18, he says , where $S'$ is the tempered distributions, we define sobolev space of exponent $s$ as $u \in S'$ with $\lambda^s \hat{u} \in L^2$. ...
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1answer
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$L^2$ and Schwartz Space

It is stated, Introduction to the theory of Pseudodiffernetial Operators, by Raymond, pg 9, Theorem 16, if $S$ is the Schwartz space If $\varphi \mapsto U(\varphi)$ is a semilinear form on $ S$ ...
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28 views

Relation between strong convergence in $L^{p}$ and weak convergence in $H_{0}^{1}(\Omega)$

Let $\{u_{n}\}_{n\in\mathbb{N}}$ be a bounded sequence in $H_{0}^{1}(\Omega)$ for a bounded interval $\Omega \subset \mathbb{R}$. By weak compactness of Hilbert Space, we can extract a subsequence of $...
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inequality in Sobolev spaces

Let $I\subset \Bbb R$ be a bounded interval and $u∈ W^{1,p} (I)$. Let $(u_n)_n\subset C^{\infty}(I)$ such that $u_n\to u\in W^{1,p} (I)$. I see in some proof that $||u_n'-u'||_{L^p} \le ||u_n-u||_{W^{...
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Meaning of Compactness

Let $\Omega \subset\mathbb{R}$ be a bounded domain (interval) and observe the following problem : \begin{align*} (P) \begin{cases} u_{t} = \Delta u + |u|^{p-1}u\, \quad x\in\Omega, t>0\\ u(0,x) = ...
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1answer
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computation of the norm

I try to understand the notions of weak derivative and Sobolev space I take this example: $f(x)= |x|\quad $ for $ \quad x\in [-1, 1]$ The derivative in the sense of distributions is $(T_f)^{'}= ...
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Fréchet derivative of the energy functional

Let $\Omega \subset\mathbb{R}^n$ be an open set and $$E(u)=\frac{1}{2}\int_{\Omega} | \nabla u|^2 \quad (u \in H_0^1 (\Omega)). $$ Then, what is the Fréchet derivative of the functional $E$? And why? ...
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Why Poincaré inequality works on $W^{1,p}_0(\Omega )$ and not on $W^{1,p}(\Omega )$

In wikipedia they say that : if $\Omega \subset \mathbb R^n$ is open and bounded, then there is $C>0$ s.t. for all $u\in W^{1,p}_0(\Omega )$, $$\|u\|_{L^p}\leq C\|\nabla u\|_{L^p}.$$ I agree that ...
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1answer
46 views

Distributional derivative of Weierstrass function

How can we compute the distributional derivative of the Weierstrass function $$W(x) =\sum_{k=1}^\infty \lambda^{(s-2)k}\sin(\lambda^k x)$$ where $s \in (0,2)$ and $\lambda$ are fixed parameters? We ...
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A proof for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$.

I believe the formula $$\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u $$ to be true for $u\in W^{1,p}(\Omega)$, where $\Omega$ is an open domain with nice boundary, but failed to find a good ...
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Question about local Sobolev spaces $H^s_{loc}(\mathbb{R}^n)$

We define the local Sobolev space $H^s_{loc}(\mathbb{R}^n)$ as $$ H^s_{loc}(\mathbb{R}^n)=\big\{f\in\mathcal{D}^{\prime}(\mathbb{R}^n): \forall \Omega\Subset\mathbb{R}^n \ \exists g_{\Omega}\in H^s(\...
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1answer
36 views

proof of sobolov inequality

could anyone help me to understand this: $$f(x) :=\left\{\begin{array}{ll}{u(x)-u_{x}(x),} & {x<\xi} \\ {u(x)+u_{x}(x),} & {x>\xi}\end{array}\right.$$ where ξ is determined by $$u(\xi)=\...
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1answer
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Coercivity - Weak Poisson's equation

Given the weak formulation of the Poisson equation, i.e. For given source function $f\in H^{-1}(\Omega)$ find $u \in H_0^1(\Omega)$ such that $$\int_{\Omega}\nabla u \cdot \nabla v \, dx= \int_{\Omega}...
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$L^\infty$ boundedness for solution of elliptic PDE with Neumann BC

On a bounded smooth domain in $\mathbb{R}^n$ consider the equation $$-\Delta u + ku = f$$ $$\partial_\nu u = 0$$where $k>0$ is a constant and $f \in L^2(\Omega)$. We know that $u \in H^2(\Omega)$. ...
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''Essentially discontinuous'' function on $G=(-1,1)^2$ in $W^{1,2}_0(G)$

In a paper by V Sverak (1988, Arch. Ration. Mech. Anal.; Example 1, p.119), the following function $u$ serves as the building block of an important counterexample: ''Consider a function $u \in H^{1}_{...
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1answer
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If $f\in C(U)$, then $f^{\epsilon }\to f$ uniformly on compact subsets of $U$

If $f\in C(U)$, then $f^{\epsilon }\to f$ uniformly on compact subsets of $U$ my Question how he says that $f$ is uniformly on $W$ i am so learner and the only hope i learn sobolev spaces is ...
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1answer
29 views

How to study SOBOLEV SPACES in Evans PDE [closed]

Im New here can you some one suggest me how to study sobolev spaces i am almost studied from one month im not getting any thing is any videos are there and is any one teach sobolev space (any tutor ...
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26 views

Understanding the Convolution and smoothing

here my question is what is mean by $f^{\epsilon}:=\eta_{\epsilon}*f$ in $U_{\epsilon}$ and how can we change form $U$ to $B(0,\epsilon)$ in the molification definition and what is use convolution ...
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282 views

Weak solutions to $\Delta u=f$ are in $W^{2,2}$

I believe the following statement is true. Let $\Omega$ be a smoothly, bounded domain in $\mathbb{R}^{n}$. The statement: Let $u\in H^{1}(\Omega)$ so that there exists $f\in L^{2}(\Omega) \;s....
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1answer
23 views

Direct sum of Sobolev spaces

We know that we may decompose $L^2(\mathbb{R})$ as $L^2(\mathbb{R^-})\oplus L^2(\mathbb{R^+})$. Can we write $L^2(\mathbb{R})=L^2(\mathbb{R_-^*})\oplus L^2(\mathbb{R_+^*})$? Now, let $H^1(\mathbb{R})$...
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1answer
25 views

Question on trace Sobolev's theorem for domain $\Omega \times (0,T)$

Let $\Omega \subset \mathbb R^3$ be an open,bounded subset with a $C^2-$boundary $\Gamma$. Fix $T>0$. Can we claim that $W^{1,2}(\Omega \times (0,T)) \hookrightarrow C([0,T];L^2(\Gamma))(*)$ ...
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1answer
36 views

Convergence of $f(u_n) \to f(u)$ when $u_n \to u$ in $L^p$/Sobolev spaces

I got a function $f\colon \mathbb{R} \to \mathbb{R}$ which is such that $0 \leq f \leq 1$ and it is a smooth function. Suppose $u_n \to u$ in $H^1(\Omega)$. $\Omega$ is a bounded smooth domain What ...
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1answer
68 views

Conditions on the domain for Sobolev embeddings

I am reading the proof of the Sobolev embedding theorem presented in the book Sobolev Spaces by Robert A. Adams and John J. F. Fournier. I could not understand the proof for part II of the theorem. ...
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1answer
67 views

How to prove this Holder- type inequality?

I don't understand why this inequality holds (i found it in some notes): Let $ v\in L^1 \cap C_c, $ then $$ \Vert v\Vert_{L^{q}} \leq \Vert v\Vert_{L^{1}}^{\frac{1}{q}} \cdot \Vert v\Vert_{L^{\infty}}...
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0answers
17 views

To show a function is in a Sobolev space, can we use weak spherical derivatives?

For example, say $\Omega=B(0,1)$ in $\mathbb{R^2}$, and I have a function represented by $$u(x)=\begin{cases}\ln\ln1/|x|& |x|\leq e^{-2}\\\ln(2) &e^{-2}<|x|\leq 1.\end{cases}$$ There is a ...
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0answers
21 views

References on equivalent characterization for Sobolev spaces of functions of one variable

I cited a result which characterizes Sobolev spaces of functions of one variable as $$H^p(a,b):=\{x\in C^{p-1}[a,b]:x^{(p-1)}(t)=α+\int^t_aΨ\,\mathrm ds,\ α\in\mathbb R,\ Ψ\in L^2\},$$where $p\in\...
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0answers
38 views

Can I pass to the limit?

Let be a sequence $u_n$ of $C_0^\infty (\mathbb{R}^N)$-functions converging to $u$ in $H^1(\mathbb{R}^N)$, which implies that $u_n\rightarrow u$ in $L^2(\mathbb{R}^N)$ and $\nabla u_n\rightarrow \...
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1answer
62 views

Integration by parts for $u \in H^{1}$ and $v\in H^{1}_{0}$

Let $\Omega$ be a smoothly, open bounded domain in $\mathbb{R}^{n}$. Assume that $u\in W^{1,2}\left(\Omega\right)$ and $v\in W^{1,2}_{0}\left(\Omega\right)$. Is the integration by parts always true, ...