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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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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$\Delta u = f , \operatorname{div} f=0 \Rightarrow \operatorname{div}u=0$ 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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17 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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35 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
60 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
22 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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48 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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54 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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20 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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52 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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19 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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28 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)^{'}= ...
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2answers
47 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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20 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
62 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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152 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
69 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
37 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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30 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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21 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
16 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
45 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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38 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
327 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
33 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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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 ...
3
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1answer
71 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
80 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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24 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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27 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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41 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
76 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, ...
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1answer
59 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
26 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 ...
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1answer
73 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 ...
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1answer
83 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 ...
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1answer
86 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=(\...
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33 views

Compactness of Sobolev Space

Let $\Omega \subset \mathbb{R}$ be a bounded domain and $2<p<\infty$. Assume $u \in C^{2,1}(\Omega \times (0,\infty))\cap C^{1}((0,\infty);L^{2}(\Omega))\cap C([0,\infty);H_{0}^{1}(\Omega))$ ...
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1answer
47 views

Extension of weak derivatives in Bochner spaces

I am struggling to understand estimate $(15)$ from the following proof from the PDE book by Evans: He argues that estimate $(15)$ follows from difference quotients, but I can't understand this. In ...
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30 views

Limit cases in sobolev embedding

I am trying to find coumterexamples for the critic cases in sobolev embedding theorem for all $\mathbb{R}^N$. For instance, a function $u\in W^{1,p}(\mathbb{R}^N)$ s.t. $u\notin L^q(\mathbb{R}^N)$ for ...
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1answer
18 views

Existence of convergent sequence implies convergence of function as $t\to\infty$

Let $\Omega \subset \mathbb{R}$ be an unbounded domain and $u \in C([0,\infty);H_{0}^{1}(\Omega))$. Assume there exists a sequence $\{t_{n}\}_{n\in\mathbb{N}}$ such that $t_{n}\to\infty$ and $||u(t_{n}...
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1answer
30 views

Isometric embedding of $L^2$ onto $H^{-1}$

Let $X$ be a Banach space. Many sources in the literature identify $L^2(X)$ with $H^{-1}(X)$ through the identification $$ \varphi: L^2(X) \to H^{-1}(X); \quad \quad \varphi(u)(v) := (u,v)_{L^2}, \...
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50 views

Intuition Sobolev spaces and smoothing splines

With inputs $X_1, \dots, X_n$ in a closed interval $[a,b]$ and $a<b$ the smoothing spline estimate $\hat{f}$ of a given odd order $k$ is given by minimizing the following penalized residual sum of ...
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1answer
97 views

Confusions in Evans book regarding weak derivatives in Banach spaces

I am studying PDE using Evans' book and I have two main confusions (probably stupid questions to experts) regarding weak derivatives in Banach spaces. First confusion: $\def\u{\mathbf u}$ $\def\v{\...
3
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1answer
99 views

$ W_{0}^{2}(\Omega)=\{ f\in W_{0}^{1}(\Omega):\Delta f\in L^{2}(\Omega)\}? $

Let $\Omega\subset\mathbb{R}^{n}$ be an open bounded domain. Let $W^{2}\left(\Omega\right)$ be the usual Sobolev space $$ W^{2}\left(\Omega\right)=\left\{ f\in L^{2}\left(\Omega\right):f,\partial_{i}...
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28 views

Existance of $\phi \in L^2$ such as $L(\vec v)=\int_{\Omega}\phi \;\text{div} \vec v \;dx$

[I was working on Stokes pde, and I'm stuck in proving this, couldn't find it anywhere] Let $\Omega \in \mathbb{R}^N$ an open bounded connected set such as $\partial \Omega$ is $\mathcal{C}^1$. And ...
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1answer
37 views

what is $W^{-1,2}$?

I have been reading about the weak Poincare lemma in the book "Linear and Nonlinear Functional Analysis with Applications" by P.G. Ciarlet. Let $\Omega$ be a simply connected domain in $\mathbb R^n$...