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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Fractional Sobolev space on an interval

Consider a fractional Sobolev space $H_p^t(I)$ defined on an interval $I\subset \mathbb{R}$. When $I=\mathbb{R}$ the space can be defined via Fourier transform. Is it possible to do it when $I=(-1,1)...
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Sobolev spaces on non-compact manifolds — independence on charts

Are there some standard references where basic facts about fractional-order (or at least integer-order) Sobolev spaces on non-compact manifolds are treated? More precisely I would like to be able to ...
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28 views

Importance of Sobolev Spaces

Why Sobolev spaces are so important in study of partial differential equations? What could have light up the mind of researchers to use these spaces to analyze PDEs?
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1answer
34 views

An example of open set $\Omega$ in $\mathbb{R^n}$ for which $C^{\infty}_c(\Omega)$ is dense in $W^{l,p}(\Omega)$

We know that the statement $C^{\infty}_c(\mathbb{R^n})$ is dense in $W^{l,p}(\mathbb{R^n})$ is always true for any $l\in \mathbb{N}$ and $p\geq \infty$, $p\neq \infty$. My professor told me that it ...
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1answer
22 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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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
18 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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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 H^1(...
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22 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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1answer
168 views

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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16 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 $\...
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2answers
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$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
39 views

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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21 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), j=1,...
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1answer
49 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, 2\...
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23 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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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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26 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: $$\hat{u}(x)=\left\{\begin{...
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1answer
40 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), j=1,...
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16 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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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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23 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
17 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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1answer
40 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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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 \text{...
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1answer
32 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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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
30 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 $\{ u_{...
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25 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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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
42 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); v=\...
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1answer
29 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
48 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 C_s(\|u\|_0+\|Pu\|_{...
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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 v-...
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1answer
40 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), j=1,...
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1answer
93 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
40 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 - \...
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1answer
31 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 is,...
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1answer
25 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' \in ...
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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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36 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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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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11 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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24 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 ...
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1answer
34 views

Poincaré-inequality

I learned this week about sobolev-spaces and thereby my prof. claimed the truth of an inequality (without proof) that seemed to him very clear and easy to see but to me it was not clear at all. He ...
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1answer
102 views

Show that if $\,u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$, then $\,u \in W_{0}^{1,p}\left(I\right)$.

I want to show the following statement ($1 \leq p < \infty$), for an open interval $I$: If $u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$ then $u \in W_{0}^{1,p}\left(I\right) $. $W^...
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2answers
46 views

Construction of Sobolev space

I am reading about the construction of Sobolev spaces from $L^2$. In the book I read (Introduction to Partial Differential Equations, by Gerald B. Folland) that the operator used to construct those ...
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27 views

Help showing $u \in W_0^{1,p}(I)$ if and only if $u=0$ on $\partial I$

I am reading the proof the following statement provided in Functional Analysis, Sobolev Spaces and Partial Differential Equations, by haim Brezis: If $u \in W_0^{1,p}(I)$, then $u=0$ on $\partial ...
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0answers
15 views

Sobolev space membership of logarithmic function

Determine the largest $s\in(0,1)$ for which the following integral converges $$\int_0^1\int_0^1\frac{\Big|\log|x-\frac{1}{2}|-\log|y-\frac{1}{2}|\Big|}{|x-y|^{1+2s}}^{2}dxdy$$