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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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Question on the dimension of the boundary of a bounded domain

Motivated by this post Dimension of boundary of a bounded domain; what to use for Sobolev inequalities and since the provided answer didn't help me that much, I thought I should just ask once more. ...
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0answers
21 views

Approximation of a $W^{1,2}(\Omega,\mathbb{S}^2)$ function

Suppose $\Omega \subset \mathbb{R}^3$ is bounded domain with smooth boundary and $u \in W^{1,2}(\Omega,\mathbb{S}^2)$ such that $|u|=1$ a.e. in $\Omega$. It is known that there is a sequence $u_n \in ...
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1answer
25 views

Is continuous increasing function in $H^1([0,1])$

Consider a function $f(x):[0,1]\rightarrow \mathbb{R}$. If $f(x)$ is continuous and increasing, is $f(x)$ in $H^1(\Omega)$, the Sobolev space with norm $\sqrt{\int_0^1 (|f(x)|^2 + |Df(x)|^2) dx}$?
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Understanding iterated covariant derivatives to define Sobolev spaces on manifolds

I'm having big troubles understanding the definition of Sobolev spaces on manifolds. Ok, so we have a Riemannian manifold $(M, g)$, and then we can define a natural riemannian measure (which I will ...
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2answers
41 views

Does strong convergence in $H$ (or $L^2$) imply convergence in $V$ (or $W_0^{1,2}$)?

Suppose: $\mathbf{V}$ and $\mathbf{H}$ are Hilbert spaces. $\mathbf{V} \hookrightarrow \mathbf{H}$ is compact embedding. $\mathbf{V}$ is dense in $\mathbf{H}$. For example $\mathbf{V} = W_0^{1,2}(\...
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0answers
22 views

Integrability of the Fourier transform in Sobolev space

I believe that the statement below is a standard fact but I haven't figured out yet: Suppose $f\in L^{1}(\mathbb{R}^{n})$ has integrable partial derivatives of order $n+1$ and $D^{\alpha}f\in L^{1}(...
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Trace of $H^1(\Omega)$ function onto regular hypersurface inside $\Omega$ (and not the boundary)?

Let $\Omega$ be a smooth domain in 2D and let $S$ be a closed smooth surve inside $\Omega$ Do functions in $H^1(\Omega)$ have a trace on $S$ and does it follow that $u \in H^1(\Omega) \implies u|_{S}...
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31 views

Can Analytic Functions on the Circle be Characterized by Sobolev Norms?

In Exercise I.4.4 of Katznelson's book An Introduction to Harmonic Analysis we learn that the Fourier series $$f(z) = \sum_{k=-\infty}^{+\infty} f_k z^k$$ defines an analytic function in the ...
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25 views

Weak convergence of weak derivative implies strong convergence

Note that here both $w_n$ and $\phi_n$ are vector functions. The following is given : $ w_n \rightarrow w$ weakly in $ L^q(B; {\mathbb{R}}^M)$ & $\{curl[w]\}_{n=1}^{\infty}$ is precompact in $ ...
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1answer
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Approximating Sobolev functions in $W^{1,p}(\mathbb{R}_+^n)$

Let $p \geq 1$. I know that there exists a continuous and linear extension operator $$ E : W^{1,p}(\mathbb{R}_+^n) \to W^{1,p}(\mathbb{R}^n) .$$ I read that from the existence of such an extension ...
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1answer
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Estimate of a weak solution in a nonhomogeneous equation

$\textbf{Problem}$ Let $\Omega \subset \mathbb{R}^n$ be open, bounded and connected with $\partial \Omega\in C^1$. For each $i,j=1,\cdots,n$, assume that $a_{ij},b_i,c \in L^{\infty}(\Omega)$ (real ...
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24 views

Existence of minimizers in Evans chapter 8

I am trying to understand the following point in Evans PDE book. In chapter 8, page 465, he proved the existence of minimizer to the following problem, let $U$ be open, bounded with connected, $C^1$ ...
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2answers
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Why is $L³$ continuously embedded into $H^{-1}$, the dual of $H¹_0$?

Why is $L³$ continuously embedded into $H^{-1}$, the dual of $H_0^1$? In this article https://drive.google.com/open?id=0B0MDtuRPAebZQ3NvRFVQekJ4VEpTMldiRWxmbU9GLVM4MDNF the authors claim, in the ...
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1answer
29 views

Closed subset of $W^{1,2}([a, b], \mathbb{R})$

In the context of weak solutions of boundary value problems, I want to show that the set $$\{u \in W^{1, 2}([a, b], \mathbb{R}) \; : \; u(a) = 0 = u(b) \}$$ is closed in $W^{1,2}([a, b], \mathbb{R})$. ...
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1answer
50 views

Bound gradient in $H^2_0(\Omega)$ by Laplacian

Let $\Omega\subseteq \mathbb{R}^n$ be an open set and show that $$ \lVert{\nabla u\rVert}_{L^2}^2 \leq \epsilon\lVert{\Delta u\rVert}_{L_2}^2 + \frac{1}{4\epsilon}\lVert{u\rVert}_{L^2}^2 $$ for any $\...
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22 views

Example 3 - Section 5.5.2 - Partial Differential Equations by Evans (2)

Could somebody help me: there is example from book:Example And I don't understand why: $ u_{x_i}=-\frac{\alpha x_i}{|x|^{\alpha+2}} $. I thought that $|x|=|x_1+x_2+...+x_n|$. And here is my ...
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1answer
25 views

prove $\rho\in K\to ||\sqrt{\rho}'||_{L^2}^2$ is convex, where $K$ is a convex cone

$$K = \{\rho\in H^1(0,1);\rho \ge 0,\sqrt{\rho}\in H^1(0,1)\}$$ I have proved that $K$ is a convex cone. Now I'm asked to prove that $\rho\in K\to ||\sqrt{\rho}'||_{L^2}^2$ is convex. I tried ...
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1answer
30 views

Continuous but not compact Sobolev embedding

Let $U \subset \mathbb{R}^n$ be an open ball and $p^* = \frac{pn}{n-p}$ be the Sobolev conjugate. Show that, for $p=2$, $W^{1,p}(U)$ cannot be compactly embedded in $L^{p^*}(U)$. Now, from the ...
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2answers
70 views

Why weak convergence and a.e. convergence imply the convergence of this integral?

In the proof of the Brezis-Nirenberg theorem on the book "Nonlinear Analysis and Semilinear Elliptic Problems" by Ambrosetti and Malchiodi, it is used lemma 11.11 at pag. 183, in whose proof we have a ...
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Distributional second-order derivatives of $\frac{e^{-|x|}}{4\pi |x|}$ to show the solution of $u -\Delta u=f$ is in $H^2$

In Brezis's book "Functional Anlaysis" it is proven that the solutions of the Helmotz equation $u - \Delta u=f$ where $f \in L^2 (\mathbb{\Omega})$ belong to $H^2 (\mathbb{\Omega}) \cap H^1 _0 (\Omega)...
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34 views

Compactly supported functions are dense in local Sobolev spaces

Let $\Omega\subset\mathbb{R}^n$ be an open set. I would like to know why $\mathcal{D}(\Omega,\mathbb{C}^l)=C^{\infty}_{\mathrm{c}}(\Omega,\mathbb{C}^l)$ is dense in $H^s_{\mathrm{loc}}(\Omega,\mathbb{...
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31 views

How to show that the embedding is compact.

If $U\subseteq \mathbb{R}^n$ is a "nice" bounded domain, then we know that the embedding $H_0^1(U) \hookrightarrow L^2(U)$ is compact. I want to show that the the embedding $H_0^1(U) \hookrightarrow ...
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0answers
25 views

Weak derivative of radial function

I want to prove that some radial function u is weakly differentiable in R^n, how do I translate that to a condition over the real function f(r) =u(r), for r>0?
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0answers
12 views

Composition with Lipschitz map is Lipschitz on Sobolev spaces

Suppose that $F: \mathbb{R}^d \rightarrow \mathbb{R}^d$ is Lipschitz with some constant $L$ and that $F(0)=0$. Then it is clear that $F$ defines a Lipschitz continuous map $L^2(\mathbb{R}^d) \...
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1answer
25 views

Theorem in Adams Sobolev spaces book requires $u\in L^p(\Omega) \cap L^r(\Omega)$ but we only have $u \in C^\infty$ so how can theorem be applied?

Let $\Omega$ be a (open) domain in $\mathbb{R}^n$. In Theorem 4.19 of Adams book on Sobolev spaces he makes use of Theorem 2.11 (An Interpolation Inequality). Theorem 2.11 requires that if we have $1\...
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1answer
38 views

completion of $C^\infty_0(D)$ w.r.t $\|\cdot\|_\nabla$

Let $D$ be an unbounded domain in $\mathbb{R}^n$. Consider the set $C^\infty_c(D)$ with two different norms: $\|\cdot\|_\nabla$ and $\|\cdot\|_\nabla + \|\cdot\|_{L^2}$. It is known that when $D$ is ...
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1answer
45 views

Is there a bounded domain on which Poincaré's inequality does not hold?

Suppose that $U$ is a bounded domain in $\mathbb{R}^n$. Poincaré's inequality states that (for $U$ sufficiently "nice") there exists a constant $C>0$ such that if $u\in H^1(U)$ satisfies $\int_U u =...
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1answer
19 views

an upper bound of a $d$ dimensional vector under a length constraint.

Let $k = (k_1, \cdots, k_d) \in \mathbb{N}^d$ and $s = (s_1, \cdots, s_d) \in \mathbb{R}_+^d$. define $$k^s = (k_1^{s_1},\cdots,k_d^{s_d})\in \mathbb{R}_+^d.$$ Let $|\cdot|_p$ denote the $p$-norm on ...
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1answer
34 views

Riesz representation and inverse operator.

In class, my professor went through the following construction: Let $\Omega$ be a bounded domain and define $X$ to be $H_0^1(\Omega)$ or $H^1(\Omega)$. We also define $A: X \to X^\prime$ (where $X^\...
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1answer
29 views

Sobolev spaces for vector-valued functions

Let $ \Omega \subseteq \mathbb{R}^3$ be open. How is defined the space $W^{1,p}(\Omega)$ for vector valued functions $f:\Omega \to \mathbb{R}^3 $? What is the norm in $W^{1,p}(\Omega)$?
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How to prove that a linear differential operator generates a semigroup?

I am working on a nonlinear PDE and I should now prove that the linear operators $L:=\partial_x^3+\partial_x^2$, $S(u):=u L$ and $T(u):=L+S(u)$ are generators of a $C_0$ semigroup such that $\|e^{-s A}...
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0answers
36 views

Clarification for a proof about Lipschitz approximation in $W^{1, p}(\mathbb{R}^n)$

I was reading a proof about the Lipschitz approximation of functions $u\in W^{1, p}(\mathbb{R}^n)$. There the author defines a set $$E_{\lambda}=\{x\in\mathbb{R}^n:M|\nabla u|(x)\leq \lambda\}, \quad ...
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0answers
61 views

Order of convergence for spline interpolation in a Sobolev norm

We are interested in a result stating the order of convergence of a spline interpolation for a given function in $W^{k,p}_{\text{loc}}(\mathbb{R})$. Let us be more precise: Let $h\in \mathbb{R}_{>...
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1answer
43 views

A definite integral inequality

Suppose $f(x)$ has continuous derivative on $[-\pi, \pi]$, $\,f(-\pi)=f(\pi)\,$ and $\,\int_{-\pi}^{\pi}\, f(x)\, dx=0$. Then prove that: $$ \int_{-\pi}^{\pi} [\,f'(x)]^2\, dx \ge \int_{-\pi}^{\pi} f^...
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0answers
18 views

Unable to determine Sobolev space

I have the following function: $T ( \textbf{r}) = \frac{q}{4 \pi k} \int_{s_i=0}^{L_i} \frac{1}{\left| \textbf{r}-\textbf{r}'\right|} ds_i = \frac{q}{4 \pi k} \log\left( \frac{\tan(\theta_2/2)}{\tan(\...
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25 views

$H^1(\Omega')$ is in genral not a subspace of $H^1 (\Omega)$ for bounded domains $\Omega' \subset \Omega$

Reading about Sobolev spaces I found the following statement: $H^1 ({\Omega}')$ is not a subspace of $H^1(\Omega)$ for $\Omega'\subset \Omega$. $\left(\text{However } \ H^1_0 (\Omega ')\subset H^1_0(...
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1answer
21 views

How to choose the Testfunction-Space?

Every time I want to transform a boundary-value-problem of the form $-u'' + ... = f$ to it's weak formulation, I have problems choosing the testfunction-space. I have the feeling that in scripts the ...
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1answer
30 views

How can a discontinuous function belong to $C_B^1(\Omega)$, the space of continuous functions $u$ with bounded derivatives?

Let $\Omega = \{(x,y) \in \mathbb{R}^2 \ : \ 0 < |x| < 1, \ 0 < y < 1\}$ and consider the function $u$ defined on $\Omega$ by (Sobolev Spaces by Adams, page 68, Example 3.10) $$ u(x,y) = \...
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0answers
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Equivalent definition for capacity in Sobolev spaces.

Define the capacity of set $E\subset\subset\Omega$, $\Omega$ bounded, as $$ \text{cap}_p(E, \Omega)=\inf_{u\in \mathcal{F}_p(E, \Omega)}\int_{\Omega}|\nabla u|^p, $$ where the family of functions is $...
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0answers
15 views

Continuous embedding of weighted Lebesgue space

Let $w$ belong to the class of Muckenhoupt weight $A_p$ for some $1<p<\infty$ and define the weighted Lebsgue space $L^p(\Omega,w):=\{u:\Omega\to\mathbb{R} \text{ measurable }: ||u||=(\int_{\...
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1answer
58 views

What is the orthogonal complement of $H^1_0$ in $H^1$?

Let $\Omega$ be a closed domain with smooth boundary in $\mathbb{R}^n$. Let $H^1_0(\Omega)$ be the closure of compactly supported smooth functions under the norm $\|u\|_1 = \int_\Omega u^2 + |\nabla u|...
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1answer
51 views

Variational problem of Laplace equation

We know that: $\Delta u=-f, \in\Omega$ $u=0$ ,on $\partial\Omega$ Is equivalent to this variational problem: find the minimum of $J(u)$ in $C^2(\Omega)\cap C^1(\bar\Omega)$, where $J(u)=\int_\Omega[...
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Writing explicitly the difference of two indicator-like functions (with “non-fixed domain”)

How can I rewrite explicitly the following difference of indicator functions? $$\int_{0}^A \mathbf{1}_{\{\int_0^x f(s) \ \mathrm{d}s + y + \epsilon h> g(x) \}}(x,y,z) - \mathbf{1}_{\{ \int_0^x ...
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1answer
28 views

a question about Differential quotient operator in sobolev spaces

Define $\nabla_h u=\frac{\tau_h u-u}{h}$, where $\tau_h u=u(x_1+h,x_2,...,x_n)$. We use $\|\cdot\|$ to denote the norm in $L^2(R^n)$. We have the following lemma: Lemma 1. If $u$ belongs to $H^1(...
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0answers
22 views

Weak convergence in $H^1(\mathbb R^3)$ implies convergence of integrals

Suppose that $f_n \rightharpoonup f$ in $H^1(\mathbb R^3)$ (weak convergence). Then $$\int_{\mathbb R^3} \frac{\lvert f_n(x) \rvert^2}{\lvert x \rvert} dx \stackrel{n\to \infty}{\longrightarrow} \int_{...
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0answers
31 views

prove that the space $C^{\infty}(R_{+}^n)\cap W^{1,2}(R_{+}^n)$ is dense in $W^{1,2}(R_{+}^n)$

I already know that as $C^{\infty}_c(R^n)$ is dense in $W^{1,2}(R^n)$, but I don't know why the space $C^{\infty}(R_{+}^n)\cap W^{1,2}(R_{+}^n)$ is dense in $W^{1,2}(R_{+}^n)$ Is the statement true ...
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0answers
29 views

The spaces $L^p$ with $0<p<1$

Assume that there exists a constant $ M> 0$ such that $/\hat{u}/ >M$ . Is the following inequality true? $\left \| \hat{u}\right \|_{L^{p}} $ $\leq $ $\left \| \hat{u}\right \|_{L^{2}} ...
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0answers
32 views

Why is the laplacian a closed operator in $W^{2,p}(\mathbb{R}^n)$?

I have read that the laplacian is a closed operator in $W^{2,p}(\Omega)$,(that is, $\Delta : W^{2,p} \to L^p$) where $\Omega$ satisfies some conditions (I need the case $\Omega = \mathbb{R}^n$ so ...
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1answer
18 views

Weak Formulation's Well-Posedness with Poincare-Friedrichs Inequality

I have arrived at the weak form for a PDE which has the following form. Let $V=\{ v\in H^1(0,1):v(1)=0 \}$, $F$ is a bounded linear functional on V. $$\left\{\begin{array}{ll} \...
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1answer
48 views

Prove that $W_0^{1,p}$ is a Banach space

$\textbf{Problem}$ Prove that $W_0^{1,p}(\Omega)$ is a Banach space where $\Omega$ be an open and bounded set in $\mathbb{R}^n$ $\textbf{Proof}$ $\quad $Let $\{u_n\}$ be the Cauchy Sequence in $W_0^{...