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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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Unboundedness of the Sobolev norm of a sequence of functions.

Let $\{f_n\}$ be a sequence of functions that are continuous and lying in $H^k(\mathbb{R}^m)$. Assuming $k>\frac{m}{2}$, and if $f_n \to f$ pointwise, where $f\in H^k(\mathbb{R}^m)$, such that $f$ ...
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Solution of nonlinear heat equation decreases in time?

Let $f$ be a smooth decreasing function which is non-negative and bounded above, and consider the heat equation on a smooth bounded domain $$u_t - \Delta u = f(u)$$ with Dirichlet BCs and some initial ...
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A statement may use regularization theorem (sobolev space)

Here is the statement: suppose $u\in W^{m,p}(U)$(sobolev space), $supp{u}\subset U$, then $u_epsilon(x)=\frac{1}{n}\int{\alpha(\frac{x_1-x_1'}{\epsilon})……}\alpha(\frac{x_n-x_n'}{\epsilon})u(x')dx'$ ...
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1answer
21 views

The definition of generalized function (pde)

The definition of generalized function in my books is as follows: generalized function is linear continuous function on fundamental spaces, such as $\mathbb{D}(R^n),\varphi(R^n)$. But in ADAMS, the ...
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23 views

Does divergence in $H^1_0$ implies uniform divergence in finite dimensional subspaces of $H^1_0$ whose elements are continuous?

If $\Omega$ is a bounded open subset of $\mathbb{R}^n$, $\mathcal{B}_\Omega$ is the set of Borel subsets of $\Omega$, $\mu$ is the Lebesgue measure of $\mathbb{R}^n$, $H^1_0(\Omega)$ is the closure of ...
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1answer
25 views

Trace in $L^2$ space

If the sequence $\{u_n\}\in C^\infty(\bar{\Omega})$ and $\|u_n\|_{L^2(\Omega)}\leq \frac{1}{n}$, what could I say about $\|u_n\|_{L^2(\partial\Omega)}$? $\partial \Omega$ is the boundary of $\Omega$. ...
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1answer
26 views

a theorem about regularization of function

Theorem: Let $u(x)$ is locally integrable function on $R^n$, $\phi= e^{\frac{1}{|x|^2-1}}, when |x|<1;\phi= 0, when |x|\geq 1 $ $\alpha(x)=\frac{1}{C}\phi$ where $C=\int_{R^n}\phi(x)dx$ and $\...
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0answers
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Characterization of radial cone to set of non-negative $H^1(\Omega)$ functions

Let $\Omega$ be a bounded smooth domain and consider $$K=\{ v \in H^1(\Omega) : v \geq 0 \text{ a.e.}\}.$$ The radial cone at a point $v \in K$ is defined as the set $$R(v) := \{ w \in H^1(\Omega) : ...
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Truncation function

Define the truncation function $ T_k(s)=s$, if $|s|\leq k$ and $T_k(s)=k\frac{s}{|s|}$, if $|s|\geq k$. Let $w\in A_p$ (the class of Muckenhoupt weight, https://en.wikipedia.org/wiki/...
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$\underset{v \in S_h}{\Vert u - v_h \Vert_{1,\Omega}} \leq ch\Vert u \Vert_{2,\Omega} $ implies $H^2(\Omega) \hookrightarrow H^1(\Omega)$

We have the following that is derived from the Bramble-Hilbert Lemma $\underset{v \in S_h}{\Vert u - v_h \Vert_{1,\Omega}} \leq ch\Vert u \Vert_{2,\Omega} $ implies $H^2(\Omega) \hookrightarrow H^1(\...
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Prove a Gagliardo-Nirenberg inequality in $W^{2,p}$

$\textbf{Problem}$ Integrate by parts to prove: \begin{align*} \int_{\Omega} \vert Du \vert^p dx \leq C\Vert u \Vert_{L^p(\Omega)}^{1/2}\Vert D^2u\Vert_{L^p(\Omega)}^{1/2} \end{align*} for $2\leq ...
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$u,v \in W^{1,2}(\Bbb{R})$, then $\int_\Bbb{R}{uv'}dx=-\int_\Bbb{R}{u'v}dx$

So I need to show that, given $u,v \in W^{1,2}(\Bbb{R})$, then \begin{equation} \int_\Bbb{R}{uv'}dx=-\int_\Bbb{R}{u'v}dx \end{equation} My attempt has been this: If $u,v \in W^{1,2}(\Bbb{R}) \...
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1answer
21 views

A specific problem on : Does bounding of the Sobolev norm can cause bounding of a higher derivative?

Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|_{...
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1answer
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Sobolev inequality cannot hold for all compactly supported smooth functions

I am on a course on Sobolev Spaces and we had this as an exercise: Let $1\leq p<n$ and $q\leq p^*$, where $p^*=(pn)/(n-p)$. Show that $||u||_{L^q(\mathbb{R}^n)}\leq C(q,p,n)||\nabla u||_{L^p(\...
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1answer
29 views

A inequality between $||u||_{p}$ and $||\gamma (u)||_{p, \partial \Omega}$, where $\gamma$ is the Trace Operator?

Does someone know any inequality between $||u||_{p}$ and $||\gamma (u)||_{p, \partial \Omega}$, where $\gamma$ is the Trace Operator? I need to find something like $||u||_{p}\leq C||\gamma (u)||_{p, \...
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1answer
32 views

Is it true that if $f\in W^{1,p}(\mathbb{T})$ then the difference quotient of $f$ converges in $L^p(\mathbb{T})$ to the weak derivative of $f$?

Let $\mathbb{T}$ be the 1-torus and if $1\le p \le+\infty$ define the Sobolev space $$W^{1,p}(\mathbb{T}):=\left\{f\in L^p(\mathbb{T})\ |\ \exists g\in L^p(\mathbb{T}), \forall\varphi\in C^\infty(\...
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1answer
28 views

Sobolev Inequality on Interval

Consider the open interval $I = (0,1)$ in $\mathbb{R}$, and fix $p \in[1,\infty)$. For a function $u \in C^1(\bar{I})$ we have the standard $W^{1,p}$-norm: $$ \left\Vert u \right\Vert_{W^{1,p}} := \...
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Weak derivative and Locally summable functions

I have three question regarding the appearance of the space of locally summable functions in the definition of weak derivatives and sobolev spaces. The deifinition of weak derivatives from Evans: ...
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Density of a subset of Sobolev Space H1

I have a question about density. It's probably trivial but I am just learning functional analysis so nothing is trivial to me. Here is my question. Let $$ \mathcal{X}\colon=\mathcal{H}^1(0,1;\mathbb{...
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1answer
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Why is $\mathcal K=\{ v \in V: v \le \psi \}$ closed?

I ’m studying about obstacle problems at the moment and I came across with this statement: Let $V$ be a closed linear subset of $W^{1,2}(\Omega)$ where $\Omega$ is a bounded open connected subset ...
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How to prove the relationship between krasnoselskii's genus and the dimenson of a vector space?

I have to work with this definition for Genus: Let us denote by $U$ the class of all closed subsets $A ⊂ X- \{0\}$ that are symmetric with respect to the origin, that is, $u ∈ A$ implies $−u ∈ A$. ...
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$\frac{\partial f}{\partial x_{i}}$ is bounded variation implies $ f\in L^{\frac{n}{n-1}}\left(\Omega\right). $

Suppose $f(x_{1},x_{2},\cdots,x_{n})$ satisfies that $\frac{\partial f}{\partial x_{i}}$ is bounded variation in $\Omega$ for $1\leq i\leq n$. Then $$ f\in L^{\frac{n}{n-1}}\left(\Omega\right). $$ ...
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14 views

Solution and Test space Euler bernoulli Beam

Let $\Omega := (0,1)$ be the domain, with boundary $\partial \Omega = \lbrace 0, 1 \rbrace$, such that $\bar{\Omega} = \Omega \cup \partial \Omega$. Let $\alpha \in \mathbb{R}$ be a constant, we ...
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33 views

Show minus Laplacian operator is densely defined on $L^2$

I am trying to solve the following problem: Let $U \subset \mathbb R^n$ be a bounded and smooth domain, and $L$ be the minus Lapacian operator with zero boundary condition. Prove that $$L : L^...
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Interpolation inequality on $L^p$ space

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. If I already know the interpolation inequality that said $$ \|u\|_{L^q(\Omega)} \leq \|u\|^{\lambda}_{L^p(\Omega)} \|u\|^{1-\lambda}_{L^r(\Omega)} $$...
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1answer
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Orthonormal basis for Sobolev Space $H^1[0,1]$ in one dimension

Let $H$ be the space of all absolutely continuous functions $f:[0,1]\to\mathbb R$ with $f'\in L_2[0,1]$, equipped with the norm \begin{align*} ||f||_H=\sqrt{\int_0^1f(x)^2dx}+\sqrt{\int_0^1f'(x)^2dx}. ...
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1answer
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A density result in $W^{1,p} (\mathbb{R}^n) \cap C^{1}(\mathbb{R}^n)$

is the following result valid?: If $ u \in W^{1,p} (\mathbb{R}^n) \cap C^{1}(\mathbb{R}^n)$, then $\forall \epsilon > 0 ~ \exists f \in C_{c}^{\infty}$ s.t. $\|u-f\|_{W^{1,p}(\mathbb{R}^n)} < ...
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1answer
40 views

An inequality for Sobolev functions

Let $\Omega$ be a smooth bounded domain and $f\in L^p(\Omega)$, $u\in W^{1,1}(\Omega)$ such that $p\geq 1$. Then $$ \int_{\Omega}|fu|\,dx\leq C\,\int_{\omega}\{|f(x)|dx(\int_{\Omega}\frac{|\nabla u(\...
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$W_0^{m,2}$ estimate for weak solution of Laplacian

If $U$ is an open region of $\mathbb{R}^n $ with smooth boundary, $u\in W_0^{1,2}(U)$ satisfies $$\int_U \nabla u \nabla v dx =\lambda \int_U uvdx $$ where the test function $v\in W_0^{1,2} (U)$ (...
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2answers
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Sobolev-function $u\in W^{1, p}(\mathbb{R}^n)$ which is unbounded on every open set $U\subset\mathbb{R}^n$

I know that there is a function $u\in B^n(0,1)$ s.t. $u$ is unbounded on every open set of $B^n(0, 1)$. The usual approach is to pick a countable dense set $\{q_n\}\subset B^n(0, 1)$ and set $$u(x)=\...
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Fractional embedding inequality with $L^{\infty}$ norm

Here we consider the fractional Sobolev spaces and suppose $u$ is a vector function in $\mathbb R^2$. Is the following always true? $$\Vert Du \Vert_{L^{\infty}(\mathbb R^2)} \leq C\Vert Du \Vert^{1-\...
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Cauchy sequence but not essentially bounded

I have trouble understanding how can a sequence $\{u_k\} \in C^1{(\mathbb{U})}$, with $\mathbb{U} \subset \mathbb{R}^2$ bounded and $\|u_k\|_{H_1(\mathbb{U})} < \infty$, can be Cauchy with respect ...
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Fractional Hardy inequality

From classic literature, I know the following result. Let $\Omega\subset\mathbb{R}^d$ be a bounded open set of class $C^1$. Then there exists $C>0$ such that \begin{equation}\label{1} \|\frac{...
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Question on deducing the limit of a sequence in Sobolev space

Let $\Omega \subset \mathbb R^n$ be bounded and connected and for a given compact subset $E \subset \Omega$ define the set $ \mathcal K := \{v \in H^1_0 (\Omega) : v \ge 1 \;\;on\;E\;in\;H^1 (\...
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1answer
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Exact value of constant C in Sobolev seminorm

Let $T=[0,h]$ for some $h>0$. Show that, given $p\in[1,\infty]$ and $0\le m \le r$, there exists a constant $C$ such that for any $u\in C^\infty(T)$, there exists a polynomial $v\in P_r(T)$ ...
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Definition of homogeneous Sobolev spaces $\dot W^{-s,q}$

For the homogeneous Sobolev spaces $\dot H^{-s}$, we can define its norm like this: $$\Vert f \Vert_{\dot H^{-s}}=\Vert \Lambda^{-s} f\Vert_{L^2}$$, i.e., we use fractional derivatives to define it. ...
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28 views

Specific reference request for some important Functional Analytic results

Does there exist a consolidated reference which discusses most of the important inclusion/embedding/density results of infinite dimensional spaces. For example: $\mathcal{D}(\Omega)$ is dense in $W^{1,...
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Use ACL-characterization to show that $\nabla u=0$ implies $u=$constant a.e. in $\Omega$ when $u\in W^{1,1}(\Omega)$ and $\Omega$ is a domain.

I would like to use ACL-characterization of Sobolev functions to show that if $u\in W^{1,1}(\Omega)$ and $\nabla u=0$ then $u=$constant a.e. when $\Omega$ is a domain. This is usually proved using ...
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1answer
49 views

Limit of a sequence of functions in the Sobolev space

Let $\mathbb{D} \subset \mathbb{R}^2$ be the unit disk and consider the Dirichlet problem of the Laplace equation $\Delta u = 0$ in $\mathbb{D}$ with $u = 0$ on $\partial \mathbb{D}$. Then we know ...
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1answer
29 views

Approximation in a Sobolev Space

Consider the open unit disk $\mathbb{D}$ in $\mathbb{R}^2$. In my analysis course, we defined the Sobolev space $H^1(\mathbb{D})$ in a somewhat unusual way. More precisely, $H^1(\mathbb{D})$ was ...
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Determining domain of differential operators in a concrete semigroup problem

Let s>3/2, $H^s=H^s(\mathbb{R})$ be the Sobolev space of order $s$, $B$ be the set of bounded operators from $H^{1/2}$ to itself, $u\in H^s$ and $A(u):=u\partial_x$ an operator. How can I determine ...
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1answer
20 views

Multi-index derivatives in Sobolev spaces

I am studying inequalities in Sobolev spaces. In particular, this one: For all $m\in\mathbb{Z}^+\cup{0}$, there exists $c>0$ such that, for all $u,v\in L^{\infty}\left(\mathbb{R}^n\right)\cap H^m\...
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1answer
34 views

Weak solution of heat equation decreases in time

Let $u_t - \Delta u = f$ hold with some appropriate initial condition and Dirichlet boundary condition on some smooth bounded domain. We take $f$ to be negative. Does anyone know a citation that ...
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1answer
19 views

Does solution of heat equation increase in time if the source term is positive?

Let $u_t - \Delta u = f$ hold with $u(0) = u_0 \geq 0$ on a bounded domain where $u_0$ is in $H^1$. We take Dirichlet boundary conditions. If $f \geq0 $, is it true that $u$ is increasing in time?
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Continuity of the map associated with Neumann to Dirichlet map.

Consider the wave equation $$u_{tt}-\Delta u+qu = 0 \qquad (x, t) \in \Omega \times (0, T)$$ $$u=u_t=0 \qquad x \in \Omega, \ t=0$$ $$\frac{\partial u}{\partial \nu} = f, \qquad (x, t) \in \partial \...
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Reference Sobolev and distribution

I've studied distribution in dimension 1 seriously (maybe 20 hours). And recently we've learnt Sobolev space in dimension 1 in the context of distribution. I would like to learn more about it. I ...
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21 views

Sobolev space dense in L2 and L2 dense in dualspace

In Bresiz "Functional analysis" it is stated that if $I$ is a bounded interval in $\mathbb{R}$ then: $W_0^{1,p}(I) \subset L^2(I) \subset W^{-1,p}(I)$. (where $W_0^{1,p}$ is the closure of $C_0^{\...
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1answer
24 views

Let $u(x_1, x_2)=\min\{1, |x_2|/|x_1|\}$. For which $p$ it holds that $u\in W^{1, p}_{\text{loc}}(\mathbb{R}^2)$?

Here $W^{1, p}_{\text{loc}}(\mathbb{R}^2)$ is the usual locally integrable Sobolev space. I got as a result that $p>2$ and I would like to know if it is correct. Clearly $u$ is itself locally ...
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1answer
31 views

Equality of Sobolev spaces

It is well known that if $\Omega\subset \mathbb{R}^{N}$ open, $F\subset\Omega$ closed, such that $\mathcal{H}^{N-1}(F)=0$,where $\mathcal{H}^{N-1}$ denotes (N-1) dimensional Hausdorff measure, then $W^...
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0answers
28 views

Example of a function in $W^{1,2}(\Bbb R^2) = H^1(\Bbb R^2)$ that is not $L^\infty(\Bbb R^2)$ [duplicate]

Can anyone come up with such an example? I know that $W^{1,2}(\Bbb R)$ is continuously embedded in $L^\infty(\Bbb R)$, but I am not very familiar with Sobolev functions in higher dimensions. I cannot ...