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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Weak formulation of a nonlinear problem with test functions in a dense subspace of $H_0^1$

I am reposting a question from Math Overflow, because it seems it gets no attention. Let $\Omega\subset \mathbb R^{d=3}$ is a bounded and Lipschitz domain. Let $u\in H_g^1(\Omega)$ satisfy the weak ...
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12 views

Estimation of gradients in Poisson's equation

I am trying to show the following result. Let $D\subset\Bbb R^3$ be a bounded open set with smooth boundary. For any $f\in H^{-1}(D)$, let $\phi$ be the unique weak solution to the following ...
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11 views

Counterexample for the density of smooth functions in Sobolev spaces on a manifold

I'm in desperate need for help understanding a counterexample for the assertion that for (appropriate) manifolds $M,N$ the space $C^\infty (M,N)$ is dense in $L^p (M,N)$ if $\dim(M) > p$. (The ...
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1answer
15 views

Does uniformly boundness in $W^{1,1}$ implies strong convergence in $L^{1}$?

Suppose $f_i$ is uniformly bounded in $W^{1,1}$. My question is, can we conclude that there exists a sub-sequence of $f_i$ convergent strongly to some $f$ in $L^{1}$? I am just reading a paper, this ...
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11 views

Heat Equation stationary convergence

Consider the heat equation: $$u_t -\Delta u=0 \quad \text{in} \quad Q_T=\Omega \times(0,T) $$ $$u(\sigma,t)=0 $$ $$u(x,0)=g(x) \quad \text{in} \quad \Omega $$ a weak formulation is: find $u \in ...
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21 views

How do we define $H^{-1}$? [duplicate]

In class we defined $H^{-n}$ on $\mathbb{R}^n$ via the Fourier transform of tempered distributions. But unfortuntely, on subset $\Omega \subset \mathbb{R}^n$ there are Schwartz functions. So let ...
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15 views

Bochner integrability and analytic semigroup

For a general strongly elliptic second order operator of the divergence form $$A=\partial_j\big(a^{ij}\partial_{i}\big)+b^i\partial_i,$$ with smooth enough coefficients on a smooth bounded domain ...
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9 views

Question on weighted Sobolev spaces

Let us define a weighted Sobolev space $W^{k,p}_\delta(\Omega)$ as \begin{equation} W^{k,p}_\delta(\Omega) = \left \{ u \in L^p(\Omega): (1+r^2)^{\frac{1}{2}(-\delta-\frac{3}{p}+|\beta|)}D^{\beta}u ...
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2answers
11 views

$u_n \rightarrow u$ in $W^{1,2}$ implies $u_n \rightarrow u$ and $u'_n \rightarrow u$ in $L^2$

I report the following excerpt from a textbook: "By the usual density argument we can find for every $u \in X = \left\{ u \in W^{1,2}:u(-1)=u(1) \text{ and } \int_{-1}^1 u = 0 \right\}$ a sequence ...
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1answer
16 views

Hyperbolic energy estimate in Evans PDE book

Before Theorem 6 in Chapter 7.4 in Evans' PDE book Evans claims that there exists $\beta > 0$ such that $$ \beta\|u\|_{H^1(\Omega)}^2 \leq B[u,u]\,, \quad \forall u \in H_0^1(\Omega)\,. $$ From how ...
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18 views

Approximation, Truncation argument, Sobolev space

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), ...
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1answer
38 views

Rellich's theorem fails at the end point.

If $\{f_k\}\subseteq H^s(\Bbb{R}^d)$ is a bounded sequence and support of each $f_k$ is contained in a common compact set, then there exists a subsequence that converges in $L^q$ for $\forall$ $q, ...
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15 views

Bounded sequence in Sobolev space has a convergent subsequence

Suppose $\{\phi_k\}$ are a bounded sequence of distributions on in $H^s(\Bbb{R^d})$, where $0<s<\frac{d}{2}$ and $H^s$ is the Sobolev space. Then the sequence has a subsequence that converges in ...
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1answer
36 views

About two subspaces of (1,2)-Sobolev space

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), ...
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0answers
10 views

Given $u \in W^{1,p}(\Omega)$ then $\overline{\alpha u} \in W^{1,p}(\Omega)$, with $\alpha \in C^1 \cap L^{\infty},\nabla \alpha \in L^{\infty}$

Given $u \in W^{1,p}(\Omega)$ and $\alpha$ a function which verifies $\alpha \in C^1(\Omega)$, $\alpha \in L^{\infty}(\mathbb{R}^n)$, $\nabla \alpha \in L^{\infty}(\mathbb{R}^n)^n $ y $supp(\alpha) ...
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2answers
20 views

Example of a function $u \in W^{1,p}(\Omega)$ whose extension $\hat{u}$ to be $0$ outside $\Omega$ $\hat{u} \notin W^{1,p}(\mathbb{R}^n)$

Let's consider a function $u \in W^{1,p}(\Omega)$, where $W^{1,p}(\Omega)$ is the Sobolev Space and $\Omega$ is an open set. When we extend $u$ to $\hat{u}$ like this: ...
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0answers
12 views

A approximation problem of bounded Sobolev function

Given a function $u\in L^\infty(\Omega)\cap H_0^1(\Omega)$, where $\Omega$ is a bounded domain in $R^n$, could we select a sequence $\{u_k\}_k\subset C_c^\infty(\Omega)$ such that $u_k\rightarrow u$ ...
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1answer
861 views

function a.e. differentiable and it's weak derivative

Note - I am just starting to learn about theory of distributions, so this may be a trivial question, if so I'd be grateful for a reference, nevertheless the question is the following: suppose I have a ...
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0answers
19 views

Function Singularity in a Sobolev Space

For which $t$ and $p$ does function of the form $$ f(x) = 1/|x|^{\alpha} ,\quad 0<\alpha<1$$ belong to the fractional Sobolev space $H_p^t(\mathbb{R})$, $p>1,t\geq 0$? UPD: actually I am ...
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1answer
45 views

What kind of functions lie in $L^2(\Omega )-H^{1/2}(\Omega )$?

It's well known that a discontinuous function does not lie in $H^{1/2}(\Omega )$ if $\Omega\subset\mathbb{R}$ is one dimension. My question is, For $\Omega\subset \mathbb{R}^3$, are there any simple ...
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0answers
15 views

Regularity of $u$ and $v$ satisfying $u_t - \Delta u = \Delta v - v_t$

I have that there are two functions $u$ and $v$ in $H^1(0,T;L^2)\cap L^2(0,T;H^2)$ satisfying weakly the equation on a bounded domain $\Omega$ $$u_t - \Delta u = \Delta v - v_t$$$ given some ...
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1answer
16 views

Examples in $W^{1,p}(U)\setminus C(\overline{U})$ and $C(\overline{U})\setminus W^{1,p}(U)$

The following is the trace theorem in Partial Differential Equations by Evans: Let $U$ be a domain (open connected subset) of $\mathbb{R}^n$. Suppose $U$ is bounded and $\partial U$ is $C^1$. Then ...
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0answers
80 views

Difference between $C^\infty (U)$ and $C^\infty (\overline U)$

I am learning Sobolev spaces. There seem to be a difference while approximating a function in $W^{k,P}(\Omega)$ by smooth function $C^\infty (\Omega)$ and $C^\infty ( \overline \Omega)$, where we use ...
2
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1answer
29 views

Poincare's inequality with difference quotient

For the classical Poincare's inequality, if $u \in H^1_0(\Omega)$, then $$\int_\Omega u^2 \,dx \le C \int_\Omega |\nabla u|^2 \,dx.$$ Do we have something similar with the difference quotient? That ...
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1answer
29 views

Intuition of weak solutions of elliptic equations in divergence form

Let $\Omega \subset \mathbb{R}^{n}$ a domain, and consider the following equation (1) $-D_{j}(a_{ij}D_{i}u) = 0$ (Einstein notation) The function $u \in H^{1}(\Omega)$ is a weak solution of (1) if ...
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0answers
13 views

Weak formulation of non-local Neumann problem

Consider the following probleblem: $$ -\Delta u +a(x)\int_{\Omega}b(z)u(z)dz = f \qquad \text{in $\Omega$} $$ $$ \partial_{\nu}u=0 \qquad \text{in $\partial\Omega $} $$ where $$\Omega\quad ...
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1answer
20 views

Spectrum of the Laplace operator with Neumann condition on intervals

Let $-\Delta f=-f'' $ be the Laplace operator on $[0,l]$ with domain consisting of functions on $[0,l]$ which have are in $H^2([0,l])$ and satisfy $f'(0)=f'(l)=0$. Then $-\Delta$ is self-adjoint and ...
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1answer
257 views

If $u \in W^{1,p}(U)$, prove that $Du=0$ a.e. on the set $\{u=0\}$.

Assume $1 \le p \le \infty$ and $U$ is bounded. (a) Prove that if $u \in W^{1,p}(U)$, then $|u| \in W^{1,p}(U)$. (b) Prove $u \in W^{1,p}(U)$ implies $u^+,u^- \in W^{1,p}(U)$, and ...
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14 views

Prove, for an open $\Omega \subset \mathbb{R}^n$ with $x\in \Omega$, that $u\in W^{1,p}(\Omega-\{x\})\implies u\in W^{1,p}(\Omega).$

Let $n\geq 2$, $\Omega\subset \mathbb{R}^n$ open, $x\in \Omega$ and $p\geq1$ I want to prove the above implication. We just need to show that the weak derivative of $u$ on the punctured domain remains ...
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1answer
24 views

Bounded sequence in $W^{1,p}$ converging to a non-differentiable function in $L^p$

Let $U = B(0,1)$ be the unit ball in $\mathbb R^n$, $p>1$ and $\{u_k \}$ a bounded sequence in $W^{1.p}(U)$. The Rellich-Kondrachov compactness theorem tells us that there is a subsequence $\{ ...
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21 views

$B=B(0,1)\subset\mathbb{R}^N$ and $\Omega=B\setminus\{0\}$. Does $H^1_0(\Omega)=H^1_0(B)$?

$B=B(0,1)\subset\mathbb{R}^N$ and $\Omega=B\setminus\{0\}$. (i) Assume $N=1$ and prove $H^1_0(\Omega)\neq H^1_0(B)$. (ii) Take $N\ge 2$. Does $H^1_0(\Omega)=H^1_0(B)$? I don't even know where to ...
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0answers
13 views

Prove that the following function is in $H^1(\Omega)$

Let $\Omega$ be such that $$ \overline{\bigcup_{k=1}^\infty \{b_k\}}^{|\cdot|} =\Omega:=\left\{(x_1,x_2)\in\mathbb R^2; \sqrt{x_1^2+x_2^2}<1/2\right\}, $$ where $|\cdot|$ denotes the Euclidean norm ...
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1answer
40 views

$|v|_{2,\Omega}=0$ implies $v=0$

I am stuck on this computation: let $\Omega$ be a domain in $\mathbb R^2$ and let $\Gamma_0$ be a relatively open proper subset of $\Gamma:=\partial\Omega$. Define $$ V=\{v \in H^2(\Omega); ...
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1answer
72 views

How to prove that $\mathcal{D}(\Omega)$ is dense in $\mathcal{D}'(\Omega)$?

How to prove that $\mathcal{D}(\Omega)$ is dense in $\mathcal{D}'(\Omega)$? Where $\mathcal{D}(\Omega)$ is the space of test functions with support compact and $\mathcal{D}'(\Omega)$ is the ...
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1answer
67 views

How to proof $C_0^\infty(\mathbb{R}^n)$ is dense in $H^s(\mathbb{R}^n)$ by using mollifier

Since the definition of $u\in H^s(\mathbb{R}^n)$ is $\left(1+|\lambda|^2\right)^{s/2}\hat{u}(\lambda)\in L^2(\mathbb{R}^n)$ I find it difficult to give an constructive prove that use mollifier. let ...
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1answer
252 views

Equivalents norms in Sobolev Spaces

I know that this is classical but I have never do the calculations to show that the norms in the sobolev space $W^{k,p}(\Omega)$ \begin{equation} \|u\|_{k,p,\Omega}= \Bigl(\int_{\Omega} ...
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1answer
23 views

Neumann Laplacian heat kernel or semigroup representation

I have the equation $$u_t - \Delta u = f\text{ on $\Omega$}$$ $$\partial_\nu u = g\text{ on $\partial\Omega$}$$ $$u(0) = u_0$$ for $f \in L^2(0,T;H^1)$, $g \in L^2(0,T;H^1(\partial\Omega))$ and $u_0 ...
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1answer
29 views

Coercitivity of an elliptic operator with constant coefficients

We are given an elliptic operator $P=\sum_{|\alpha|\leq m}a_\alpha\partial^\alpha$ that is elliptic in $\Omega$. $a_\alpha$ are constants. I am supposed to show that $$\|u\|_s\leq ...
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1answer
31 views

Show that there is an operator on $H^{2}$ and it's compact.

Let $H^{2}=W_{0}^{2,2}(\Omega)$. Define $(u,v)=\int_{\Omega} (\triangle u\triangle v+2v\triangle u)\mathrm{d}S$ as an inner product on $H^{2}$. Define $a(u;v)=\int_{\Omega} (\nabla u\cdot \nabla ...
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1answer
30 views

Sobolev space, continuous function

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), ...
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2answers
58 views

Sobolev space definitions

By definition of the Sobolev space $W^{m,p}$ we have : $$W^{m,p}(\Omega)=\{u\in L^p(\Omega)\ |\ \forall \alpha \text{ such that } |\alpha|\le m, D^{\alpha}u\in L^p(\Omega)\}$$ Can someone give me a ...
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1answer
25 views

Sufficient conditions that guarantee the integrate by part formula?

Suppose, for simplicity, that $f,g$ are functions defined on $[0,1]$. Under suitable hypotheses on $f$ and $g$, the integrate by parts formula yields $$ \int_0^1 f'(t)g(t)dt = f(t)g(t)|_{t=0}^1 - ...
2
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1answer
56 views

Find $y \in W_{2}^{1}[-1,1]$ s.t. $\forall x \in W_{2}^{1}[-1,1]$, $f(x)=\langle x, y \rangle$

Consider a Sobolev space $W_{2}^{1}[-1,1]$ with the following inner product: $\langle x, y \rangle = \int_{-1}^{1} [x(t)y(t)+x^{\prime}(t)y^{\prime}(t)]dt$. Let $f(x) = \int_{-1}^{1}e^{2t}x(t)dt$. ...
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0answers
12 views

Is this an elements of the Sobolev-Space $W^{1,2}\left(0,T,H^1(\Omega),H^1(\Omega)'\right)$?

Definition Let $p\in \mathbb{Z}$, $a<b \in \mathbb{R}$, and $X$, $Y$ be real Hilbert spaces. We define $$ W^{1,p}(a,b,X,Y) := \left\{ \varphi \, \Big| \, \varphi \in L^p(a,b,X),\, \varphi' ...
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0answers
40 views

Product of two $W^{1,p}_0$ functions

I have a fairly simple question. Suppose that $p>n$ and $u,v\in W^{1,p}_0$, how can I prove that the product is also in $W^{1,p}_0$ ? Of course, we have to employ Morrey's inequality. My idea was: ...
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0answers
16 views

Can we show that each element of the Sobolev space $H^k(D)$, with $D\subseteq\mathbb R^d$ being a bounded domain, has a continuous representative?

Let $d\in\left\{2,3\right\}$ $D\subseteq\mathbb R^d$ be a bounded domain $\lambda$ be the Lebesgue measure on $D$ $H^k(D)$ be the Sobolev space Can we show that each element of $H^k(D)$ has a ...
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1answer
48 views

positive and negative parts of a function?

I have this question that I found in a demonstration of a theorem: Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and $u^+,u^-$ the positive and negative parts of $u$ respectively. Why do ...
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0answers
10 views

Interpolationspace for H^{-1}\cap H^{1}

For which $\theta\in[1,\infty)$ does hold $(H^{-1}(\Omega),H^1(\Omega))_{1-\frac{1}{\theta},\theta}=L^2(\Omega)$ if $\Omega$ is a bounded domain with smooth boundary and is three dimensional. I don't ...
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0answers
8 views

Norms of Slobodekii and classical Sobolev space

For r to be a non-negative integer, we have $$\|u\|_{W^{r,p}(\Omega)} = \left( \sum_{|\alpha|\le r} \int_\Omega |\partial^\alpha u(x)|^p dx \right)^{1/p}.$$ For $0 < \mu < 1$, let $s = r + \mu$. ...
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0answers
21 views

Help showing compactness of the support of a function in the Sobolev Space $W^{1,p}$

In Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis, in the proof of Theorem 8.12, it is needed to show that the support of a function is compact. The function ...