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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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
29 views

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
52 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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1answer
42 views

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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0answers
10 views

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
32 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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0answers
40 views

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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61 views

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
51 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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44 views

$\langle curl \ u, curl \ v \rangle + \langle div \ u, div \ v\rangle=\langle f,v\rangle$ 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
27 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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0answers
36 views

$\Delta u = f , \operatorname{div} f=0 \Rightarrow \operatorname{div}u=0$ 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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0answers
12 views

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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28 views

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
49 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
21 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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0answers
45 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
51 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
19 views

$L^2$ 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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1answer
38 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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0answers
18 views

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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0answers
27 views

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
46 views

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)^{'}= ...
2
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2answers
42 views

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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16 views

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
54 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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2answers
149 views

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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1answer
51 views

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
37 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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2answers
29 views

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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29 views

$L^\infty$ 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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0answers
18 views

''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
15 views

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
38 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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1answer
34 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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3answers
314 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
26 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
31 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
38 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
70 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
23 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
25 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
40 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
72 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, ...
2
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1answer
56 views

$ f\in C_{0}^{\infty} \Rightarrow f\in L^m $?

Let $ C_{0}^{\infty} $ be the subspace of $ C^{\infty} $ functions with compact support in $ R^n $. It's true that if $ f \in C_{0}^{\infty}, $ then $ f \in L^{m} \cap H^{s} $ where $ 1\leq m \leq 2$ ...
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1answer
25 views

Norm in Sobolev Spaces.

In an exercise, i would like to use the next afirmation. Si $L=\delta_0$ y $v\in H^1_0$ entonces $|<L,v>| =|v(0)| \le ||v||_\infty \le K_1 ||v||_{H^1} \le K_2 ||v||_{H^1_0}.$ But, i do not ...
6
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1answer
55 views

Is this Poincaré-type inequality valid?

It is proved that the Poincaré inequality is still true for functions with zero mean boundary traces. Motivated by this, I have the following question: Let $\Omega$ be an open,bounded and connected ...
3
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1answer
78 views

What Soboblev embedding i could use in this case?

I'm trying to understand why this inequality is true $$ \Vert f\Vert_{L^2}^{p}\cdot h\left(\Vert f\Vert_{\infty}\right) \leq c \Vert f\Vert_{k+1}^{p} \cdot h\left(\Vert f\Vert_{k+1}\right), $$ where ...
1
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1answer
62 views

Gauss - Green theorem for Sobolev $H^1$ space

I know the Gauss-Green theorem: Let $U \subset \mathbb{R}^n$ be an open, bounded set with $∂U$ being $C^1$. Suppose $u ∈ C^1(\bar U)$, then $$∫_U u_{x_i} dx = \int_{∂U} u \nu^i dS,$$ where $\nu=(\...