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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1answer
25 views

Proof $\mathcal{C}^1(-1,1)$ is not a closed subspace of Sobolev space $H_0^1 \left[-1,1\right]$

Give a sequence of functions $\varphi_n\in \mathcal{C}^\infty(-1,1)$, Cauchy with respect to the Sobolev space $H^1_0$ norm $$|| \varphi||_1=\sqrt{\int_{-1}^1 (\varphi')^2+\int_{-1}^1 \varphi^2}$$ ...
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0answers
24 views

The eigenfunction of $-\Delta$.

Let $u_k$ be eigenfunctions for operator $-\Delta$ over domain $\Omega\subset \mathbb R^2$, open bounded, smooth boundary. Then, we know that $u_k$ forms a basis for $L^2$. Let $u\in H_0^1(\Omega)$ be ...
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0answers
17 views

Approximation of Sobolev function with convolution

As a homework exercise in Sobolev spaces course we have the following: Suppose $u \in W^{1,p}(\mathbb{R}^n_+)$ and $u_\varepsilon(x)=u(x+2\varepsilon e_n)$, where $e_n=(0,\dots, 0,1)$ and ...
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0answers
18 views

Inequality in the proof of Weak Harnack Inequality

Let $\Omega \subset \mathbb{R}^{n}$ a bounded domain s.t $B_{1} \subset \Omega$ , $u \in H^{1}(\Omega)$ a nonnegative supersolution in the weak sense of the equation $Lu=-D_{i}(a_{ij}(x)D_{j}u)$ ...
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1answer
19 views

about lower semicontinuity of a functional - PDE

Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^n$. Fix $u_0 \in H^1(\Omega) - \{ 0\}$. Define $$ J(u):= \int_{\Omega} \langle A(x) \nabla u(x), \nabla u(x)\rangle dx,$$ $$u \in ...
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0answers
17 views

Calculus of variations - existence of minimum of a functional-elliptic problem

Let $\Omega$ be a bounded domain with smooth boundary in $R^n$. Fix $u_0 \in H^1(\Omega) - \{ 0\}$. Define $$ J(u):= \int_{\Omega} \langle A(x) \nabla u(x), \nabla u(x)\rangle dx,$$ $$u \in K:=\{ v ...
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0answers
17 views

Weak derivative of a piecewise defined function

I am currently looking at these online notes on PDEs, page 59. How does it follow that if $f^R = \phi(x/R) f(x)$ $ \phi(x) = \left\{\def\arraystretch{1.2}% \begin{array}{@{}c@{\quad}l@{}} 1 ...
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0answers
61 views

About the gradient of a function in $H^{1}(\Omega)$

let $\Omega \in \mathbb{R}^{n}$ a bounded domain and $u \in H^{1}(\Omega)$ a real function. In the Leoni's Book - A First Course in Sobolev Spaces, the author define $\nabla u = (D_{1} u,\dots, ...
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0answers
40 views

If $u \in W^{1,p}(I) \bigcap C_c(I)$ then $u \in W_{0}^{1,p}(I) $ where $ W^{1,p}(I)$ is the Sobolev Space

I want to show the following statement: If $u \in W^{1,p}(I) \cap C_c(I)$ then $u \in W_{0}^{1,p}(I) $. $W^{1,p}(I) $ is the Sobolev Space, i.e. the space consisting of the functions that are ...
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0answers
23 views

Proving that $W^{1,2,0}(U)$ is dense in $H^1(U)$ for an open set $U\in\mathbb{R}$

I am trying to show that $W^{1,2,0}(U)$ is dense in $H^1(U)$ for an open, bounded set $U\subset\mathbb{R}.$ Where $W^{1,2,0}(U)$ is defined to be the space of smooth functions $f:U\to\mathbb{C}$ such ...
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1answer
31 views

Sobolev space interpolation

We are attempting to show that, for $t_0,t_1\in\mathbb{R}$, and $t_\sigma=(1-\sigma)t_0+\sigma t_1$ for any $\sigma\in[0,1]$, then the following inequality holds for any $u\in H^{t_1}(\mathbb{R}^d)$: ...
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0answers
33 views

Help proving a map between Sobolev spaces is continuous [duplicate]

We are trying to show that the map $f\mapsto|f|$ is a continuous (nonlinear) map from $W^{1,p}(\Omega)\to W^{1,p}(\Omega)$ for any bounded/open region $\Omega$ and for $p\in[1,\infty)$. We have tried ...
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0answers
53 views
+100

Proving that if $f\in\mathcal{F}C^{1}_{b}(X)$ then $f\in W^{1,p}(X,\gamma$) for $p>1$

Let $X$ be a separable Banach space endowed with a centered nondegenerate Gaussian measure $\gamma$ and $H$ the Cameron-Martin space. Then consider $f\in\mathcal{F}C^{1}_{b}(X)$. I want to prove that ...
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1answer
31 views

Showing that $(G' \circ u)u' \in L^p(I)$ where $G \in C^1(\mathbb{R})$ and $u \in W^{1,p}(I)$

I am trying to prove the rule of differentiation of a composition for weak derivatives in Sobolev spaces following the proof given in Corolary 8.11 in Functional Analysis, Sobolev Spaces and Partial ...
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1answer
31 views

Distributional derivative of $L^2$ function

If $f\in L^2(\Omega)$, where $\Omega$ is a domain in $\mathbb{R}^n$, why is it that the distributional of $f$, say with respect to $x_1$, is in $H^{-1}(\Omega)$, the dual space of $H^1_{0}(\Omega)$? ...
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0answers
30 views

When is it true that the Sobolev trace of a positive a.e. function is positive a.e?

Let $u \in H^1(\Omega)$ on a bounded smooth domain $\Omega$. Is it true that if $u \geq 0$ a.e., then $Tu \geq 0$ a.e. on $\partial\Omega$ where $T$ is the trace? I don't think it is, since $u$ can ...
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1answer
40 views

functional analysis [closed]

Please I have two questions: we take $ V = \{v \in H ^ 1 (0,1) ; v (0) = 0\}$ \ Q1/ the space $V \cap H ^ 2 (0.1 )$ is dense in $ V $ ? Q2/ the space $V \cap H ^ 2 (0.1)$ is dense in $H ^ 1 ...
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0answers
24 views

Fundamental solution of the Poisson equation with variable exponent

Let the variable exponent $p(x)$, where $p(x) \in C(\overline{\Omega})$, I want to know the fundamental solution of $$-(\Delta u)^{p(x)}=\delta_0.$$
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0answers
19 views

Convergence of the series $\sum_{k=1}^{\infty} \frac{1}{2^k} |x-y_k|^{-\alpha}$

Suppose $S=\{y_k\}_{k=1}^{\infty}$ is a countable dense subset of the open unit ball $B(0,1)$ in $\mathbb{R}^n$. We write \begin{equation} u(x) = \sum_{k=1}^{\infty} \frac{1}{2^k} |x-y_k|^{-\alpha} ...
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0answers
10 views

Homogenous Sobolev Space definition

Consider the expressions $$ \|u\|_{A}^2:=\sum_{|\alpha|=2}\|\partial^\alpha u\|_{L^2(\mathbb{R}^d)}^2 $$ and $$ \|u\|_{B}^2:=\|\Delta u\|^2_{L^2(\mathbb{R}^d)}. $$ I think I have seen both used to ...
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0answers
34 views

The inverse of Laplacian for different orders.

This question is related to my previous question here Let $u,v\in C_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. Let $\Delta$ denote the Laplacian operator, ...
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0answers
22 views

What is the “minimal” space one can embed $W^{1,p}$ into?

I am studying Sobolev embeddings at the moment, and I have run into a passage of my notes where, introducing the Sobolev-Gagliardo-Niremberg inequality, my teacher says: Ecco, come vi dicevo, ...
3
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2answers
61 views

Estimate of a Fourier Multiplier Operator

Let $m_t (\xi) = \cos (2\pi |\xi| t).$ Define the operators, for $t>0,$ $$ T_t f = ( m_t \widehat{f} )^{\vee}.$$ It is asked to prove that, whenever $f$ is sufficiently regular, $$ \| T_t ...
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1answer
47 views

Computing the inverse of Laplacian operator.

I am considering the following equation: $$ f(t):=\int_\Omega[(I-t\Delta)^{-1}\Delta(I-t\Delta)^{-1}u]\cdot u\,dx $$ where $u\in C_c^\infty(\Omega)$ and $t\geq 0$ a real number. $I$ is the identity ...
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1answer
14 views

Is L^2-norm of Laplace operator equivalent to 2-seminorm?

Let's say $\Omega$ is a bounded domain in $\mathbb{R}^2$. Is it true that $C_1\|\Delta u \|_{0,\Omega} \leq |u|_{2,\Omega} \leq C_2\|\Delta u \|_{0,\Omega}$? The left inclusion is obvious by ...
0
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1answer
31 views

Does the weak divergence exist for each $\mathcal L^2(\Omega;\mathbb R^d)$-function?

Let $\Omega\subseteq\mathbb R^d$ be open. $v:\Omega\to\mathbb R$ is called weak divergence of $u:\Omega\to\mathbb R^d$ $:\Leftrightarrow$ $$\int_\Omega v\varphi\;{\rm d}\lambda=-\int_\Omega\langle ...
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0answers
24 views

Existence of $u\in C^1[0,1]$ such that $u\notin H^1(0,1)$

Find a $u\in C^1[0,1]$ such that $u\notin H^1(0,1)$, where $H^1(0,1)=W^{1,2}(0,1)$, the Sobolev space. So finding an example for the case $C^0$ is pretty easy,$u(x)=x^\alpha, ...
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1answer
31 views

Convergence rate of functions [closed]

This problem needs more details but I can't write everything here. Please close this problem! Let $(v_\epsilon)\subset W^{1,2}(I)$ where $\epsilon>0$ and $v_\epsilon\to 1$ in $L^1$ and $0\leq ...
0
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1answer
34 views

Minimizing point for $L^2$ distance.

I am trying to study the following equation: $$ F(t):=\|u_t-u^*\|_{L^2(\Omega)}^2 $$ where $u^*\in H^1(\Omega)\cap L^\infty(\Omega)$ is fixed and $\Omega\subset \mathbb R^2$ is open bounded with ...
0
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1answer
25 views

Intuition for Sobolev spaces $H^{\frac{1}{2}}$ and $H^{-\frac{1}{2}}$

I'm looking for some intuition for Sobolev spaces $H^{\frac{1}{2}}$ and $H^{-\frac{1}{2}}$. Any explanation I've seen is very technical, I'm looking for the most simple explanation possible that gives ...
0
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1answer
30 views

What is the divergence $\operatorname{div}u$ of a $L^2(\Omega)^d$ function $u$?

Let $\Omega\subseteq\mathbb R^d$ be open. In the book Navier-Stokes Equations - Theory and Numerical Analysis by Roger Temam the author is using the divergence $\operatorname{div}u$ of a ...
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2answers
47 views

cutoff function vs mollifiers

Q1. What are cutoff function? What are mollifiers? I cannot distinguish the two. Could anyone give some concrete/simple examples of cutoff functions? And how they differ from mollifiers I did check ...
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1answer
15 views

What is the purpose of the requirement on mollification radius?

On page 66 of "Sobolev Spaces (Adams ed2)" in the proof of Lemma 3.16 (Mollification in $W^{m,p}(\Omega)$), it is mentioned that $\varepsilon < {\rm dist}(\Omega', \partial\Omega)$. However, I ...
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1answer
16 views

Relation between seminorm of Sobolev space and $L^2$ norm

Let we have the seminorm of second derivative of $u$ in $H^2(\Omega)$ i.e. $|u|_{H^2(\Omega)}=\int_{\Omega} \sum_{|\alpha|=2} D^{\alpha}u $. Can we derive that $|u|_{H^2(\Omega)}\leq C||\Delta ...
2
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1answer
40 views

Using Nash inequality to derive an inequality (from proof in paper)

We work on a domain $\Omega \subseteq \mathbb{R}^N$ with the Dirichlet Laplacian. Let $\lVert \cdot \rVert_p$ denote the $L^p$ norm. I am trying to understand why the following inequality is true: ...
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1answer
31 views

Second order elliptic equation with nonlinearity depending on the gradient

Let us consider the problem $$-\Delta u =f(x,u,\nabla u)\text{ in }\Omega$$ $$u=0 \text{ on }\partial\Omega,$$ where $\Omega\subset \mathbb{R}^n$ is a smooth and bounded domain. I have seen at many ...
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1answer
20 views

Sobolev Inequalities

I have problems with the following exercise using Sobolev Embedding Theorems. Let $B_1(0) \in \mathbb{R}^3$ the unit ball and consider the functional $$ F(u) = \int_{\omega} | \nabla u(x)|^2 + ...
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1answer
21 views

A Poincare inequality on fractional Sobolev space

Let $\Omega$ be a bounded smooth domain. Does the following inequality hold for all $u \in H^s_0(\Omega)$: $$\lVert u \rVert_{L^2(\Omega)} \leq C|u|_{H^s_0(\Omega)}$$ where the right hand side is the ...
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1answer
43 views

Where is that half coming from?

A few lessons ago, my professor proved Poincaré inequality in the following form: Let $\Omega$ be a domain contained in $\mathbb{R}^{N-1}\times(0,a)$ for some $N\in\mathbb{N},a>0$. Then for all ...
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1answer
19 views

Why $H^1(\Omega)\cap H^2(\Omega_-)\cap H^2(\Omega_+) \subset W^1_p(\Omega),\forall p>2$?

I'm reading a book and it says that: Let $\Omega=\Omega_-\cup\Omega_+$. Define $X(\Omega)=H^1(\Omega)\cap H^2(\Omega_-)\cap H^2(\Omega_+)$. Then by the Sobolev embedding theorem, $X(\Omega)\subset ...
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1answer
45 views

motivation for definition of the weak solution to elliptic equations

The following is an excerpt from the Partial Differential Equations by Evans (2nd edition chapter 6 p.314 ): In (7), we have the assumption that $$ a^{ij}, b^i,c\in L^\infty(U),\quad f\in L^2(U). ...
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1answer
22 views

is $u\in L_{loc}^1(\mathbb{R}^2)$ and 2. $u\in W_{loc}^{1,1}(\mathbb{R}^2)$?

Let $U\in C^1((0,\infty))$, $u:\mathbb{R}^2\to \mathbb{R}$ defined by $$u(x,y)=U(\sqrt{x^2+y^2}).$$ Under which conditions on $U$ is 1.$u\in L_{loc}^1(\mathbb{R}^2)$ and 2. $u\in ...
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2answers
35 views

Can $f\in W^{k,p}(U)$ be extended to a function in $W^{k,p}(\mathbb{R}^n)$ in general?

Let $U$ be a nonempty open subset of $\mathbb{R}^n$. Suppose $f\in L^p(U)$ for some $p$ with $1\leq p<\infty$. By extending $f$ to be identically zero outside $U$, one has $f\in L^p(\mathbb{R}^n)$. ...
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1answer
14 views

Why V is dense in the dual of $W_0^{m,p}$

On page 65 of "Sobolev Spaces (Adams ed2)", it is said that to prove $V=\{L_v: v \in L^{p'}(\Omega)\}$ is dense in $(W_0^{m,p}(\Omega))'$, it is sufficient to prove that if $F \in ...
1
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1answer
32 views

Trace operator and $W^{1,p}_0$

Let $W^{1,p}$ be the Sobolev space of $L^p$ functions with $L^p$ first derivatives. Let $W^{1,p}_0$ be the closure of the test functions in $W^{1,p}$. I am not explicitly writing the domain of the ...
1
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1answer
44 views

A convergence in sobolev spaces involving time

Suppose $\Omega $ is bounded and $$‎‎u^m‎‎\xrightarrow[]{w^*}‎ u ‎\quad \text{in} \,\, ‎L^{\infty} ‎(‎\mathbb{R}^+,H^2(‎\Omega‎)‎\cap ‎H_0^1(‎\Omega‎))$$ and ‎$$‎‎u^m_t\xrightarrow[]{w^*}‎ u_t ...
0
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0answers
11 views

Trace theorem in $H^{1/2,1}(\mathbb{R}^2)$

Let $E=\{u\in H^{1/2}(\mathbb{R}^2)|\partial_y u\in L^2(\mathbb{R}^2)\}$ with inner product $(u,v)_E=(u,v)_{H^{1/2}}+(\partial_y u,\partial_y v)_{L^2}$. Show that $\forall u\in E,u(x,0)\in ...
0
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0answers
66 views

extract a subsequence from $\limsup$ convergence

Let $I_\delta:=(-\delta,\delta)$ where $\delta>0$. Let $\{v_\epsilon(x)\}_{\epsilon>0} \subset W^{1,2}(-1,1)$ be a sequence such that $0\leq v_\epsilon\leq 1$ and $v_\epsilon(x)\to1$ a.e. (you ...
0
votes
1answer
41 views

Inequality Sobolev space

I am stuck with following exercise: Show that for $u \in H^1{(0,1)} = W^{1,p}(0,1)$ denoting the Sobolev space with $p = 2$ and with $u(0) = 0$: $$ \| u\|_{L^\infty{(0,1)}} \leq \| u' ...
1
vote
1answer
114 views

I want to prove $f\notin W^{1,1}(\mathbb{R},\gamma_{1})$

Let $\gamma_{1}=\mathscr{N}(0,I_{1})$ in $\mathbb{R}$ be the standard Gaussian measure. Consider the sequence $(f_{n})_{n\in\mathbb{N}}\in C_{b}^{1}(\mathbb{R})$ defined by $$f_{n}(x)=\begin{cases} ...