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
63 views

weak solution for a simple boundary problem

Consider a smooth, bounded and convex domain $K$ in $R^n$ such that $K\subset \{ x_1 = 0 \}$ and $\Omega $ a bounded convex domain (not necessarily smooth) such that $\partial \Omega \supset K$. ...
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1answer
63 views

Using Cauchy Schwarz for functions in Sobolev Space

Hi I just want to confirm something simple and check that the following is allowed: The Cauchy-Schwarz inequality states if $A = ((a_{ij}))$ is a symmetric, non-negative $n \times n$ matrix then ...
5
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1answer
238 views

Folland PDE chapter 6.C problem 1

Problem 6.C.1: Suppose $0 \neq \phi \in C^\infty_c(\mathbb{R}^n)$ and $\{ a_j \}$ is a sequence in $\mathbb{R}^n$ with $|a_j| \to \infty$, and let $\phi_j(x) = \phi(x - a_j)$. Show that $\{\phi_j\}$ ...
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0answers
16 views

Bessel Potential spaces

Let $\Omega_1,\Omega_2 \subset \mathbb{R}$ be bounded. The mapping $F: \Omega_1 \rightarrow \Omega_2$ shall be bijective, continuously differentiable and such that $||DF(x)||$ and $||(DF(x))^{-1}||$ ...
4
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1answer
164 views

Weak solution $u(x,t)$ of heat equation converges as $t \in \infty$

Where can I find a proof that the weak solution $u \in L^2(0,T;H^1) \cap H^1(0,T;H^{-1})$ of the heat equation $$u_t -\Delta u = f$$ converges as $t \to \infty$ to the solution of the elliptic PDE ...
2
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1answer
266 views

The proof of Morrey's Inequality in Evans Book

The proof of Morrey's Inequality in page 266 of Evans's PDE book really puzzles me a lot. I cannot get the general idea of the proof. I know a simple proof just in the case of $n=1$: For any ...
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1answer
85 views

First order weak derivatives of $f(x)=|x|^r$

Let $f(x)=|x|^r$ for a given real number $r$. Show that $f$ has first order weak derivatives on the unit ball $B_1(0)\subset \mathbb{R}^n$ provided that $r > 1-n$. Does anyone have an idea on how ...
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0answers
79 views

Regularity theorem for Laplacian

Let $\Omega \subset \mathbb R^d$ be a bounded domain, $d>2$. Let $f \in C^\infty(\Omega)$. If $u \in L^2$ is a distributional solution of $\Delta u = f$ in $\Omega$ then $u \in C^\infty(\Omega)$ ...
5
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1answer
112 views

The second Friedrichs' inequalities?

In paper On the Validity of Friedrichs' Inequalities,$\Omega$ is a bounded convex domain of $\mathbb{R}^d$, $d=2,3$. Then $$ \tag{1}\qquad \|\mathbf{u}\|_{1,\Omega} \le ...
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1answer
51 views

Some chain rule in W1p

Question: I need to prove the following chain rule: Let $F:\mathbb{R}\rightarrow\mathbb{R}$, $F\in C^1$ with $F'$ bounded. Let $U$ bounded and $u\in W^{1,p}(U)$ with $1\leq p\leq\infty$. Show that ...
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1answer
150 views

Friedrichs's inequality?

Friedrichs's second inequality is stated as follows(see www.win.tue.nl/~drenth/Phd/friedrichs.ps): For all $\mathbf{u} \in H^1(\Omega)^2$ satisfying either $\mathbf{n}\cdot\mathbf{u} = 0$ or ...
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2answers
130 views

Inequality involving Lp space, Holder Space, Sobolev Space

Do we have the following inequality or some variation of this: $||u||_{L^{p}(U)} \leq ||u||_{C^{0,\gamma}(\bar{U})} \leq ||u||_{W^{1,p}(U)}$ for $n < p \leq \infty$ for $u \in W^{1,p}(U)$ where $U$ ...
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1answer
53 views

About convergence in Sobolev space

I have $u_{n}$ sequence of $H^{1}_{0}(\Omega)$ where $\Omega$ is open bounded and connected domain in $\mathbb{R}^{n}$ with $n>1$. $u_{n}\rightarrow u$ in $H^{1}_{0}(\Omega)$ norm. Let ...
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2answers
80 views

Using Results in Sobolev Spaces

I have two questions about using results in Sobolev Spaces and a last one on an inequality involving Holder spaces: 1.The Gagliardo-Nirenberg-Sobolev Inequality states the following: Assume $1 \leq p ...
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0answers
36 views

Estimative in Hs spaces

Consider $W(t)$ defined by $\widehat{W(t)} = (2 \pi)^{\frac{-n}{2}} \cos (t|\xi|)$. Consider $g \in H^{s-1} (R^{n-1})$. Show that: $$ \| W(t ) *g\|_{H^s} \leq c \sqrt2 (1 + |t|)\| g\|_{H^{s-1}}$$ ...
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1answer
47 views

Strong differentiability in Sobolev spaces

My question is: how can one prove that if $\phi\in H^{n}(a,b)$ for all positive integer $n$, then $\phi\in C^{\infty}(a,b)$? $H^{n}(a,b)$ denotes de Sobolev space of the real interval $(a,b)$. For my ...
2
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1answer
42 views

estimate on $| \nabla (u |u|^2) - \nabla(w|w|^2)|$ for $u,w \in H^1$

suppose $u, w \in H^1 (R^2)$. I'd like to know where does the following inequality come from (it appears in a proof I've been reading and I can't figure it out) $$ | \nabla (u |u|^2) - \nabla(w|w|^2)| ...
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2answers
81 views

What is the closure of $ C^\infty_c(\mathbb{R}^n\setminus\{0\})$ in Sobolev $ W^{1,p} $ norm?

For $1 \leq p < \infty, n\geq 1 $ my guess of the answer was $ W^{1,p}(\mathbb{R}^n)$ but I can't prove the inclusion $ \overline{C^\infty_c(\mathbb{R}^n\setminus\{0\})} \subseteq ...
2
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1answer
239 views

Question on Sobolev Space

In learning the Sobolev space, I have a question why the Sobolev space $W^{k,p}$ could be embedded in the Holder space $C^{k,\alpha}$. Can we find a function in Holder space but not in the Sobolev ...
2
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1answer
33 views

Quantifying Ill-posedness using Sobolev Space Estimates

I've been learning about ill-posed/inverse problems, and I'm having a hard time parsing/understanding the following, which seems crucial to the theory: Say we have an operator ...
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1answer
71 views

$C_c^{\infty}$ is dense in $W^{k,p}(\mathbb R^n)$

As the title say, I want to prove that $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ i.e. $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n) \quad (\star)}$. In a book ...
3
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1answer
84 views

Similar to Poincare inequality on Sobolev spaces

The following looks quite similar to Poincare's inequality: Let $\displaystyle{ 1 \leq p < \infty}$ and $\displaystyle{ U \subset \mathbb R^n}$ open and such that $\displaystyle{ U \subset ...
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0answers
32 views

$u \in H^n$ then also $|u|^2 \in H^n$ for complex valued function in Sobolev space

Suppose that a complex valued function $u$ is in the Sobolev space $H^n(R^n) = W^{n,2}$. Is it necessarily true that $|u|^2 \in H^n$ ? The result is true for real - valued functions since we know ...
2
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1answer
121 views

Show that $\lim_n \|\partial^s (f_n - g_n)\|_p = 0$ (no homework…)

the setting is as follows: Let $\Omega \subset \mathbb{R}^m$ be open and consider some $L^p(\Omega)$ which I will shortly write as just $L^p$ from now on. Furthermore let (for some $k \in ...
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1answer
53 views

Relation between Schwartz space and Sobolev space $H_{1}$

The Schwartz space, $S(\mathbb R): = \{f\in C^{\infty}(\mathbb R): \sup_{x\in \mathbb R} |x^{\alpha} D^{\beta}f(x)|< \infty , \forall \alpha, \beta \in \mathbb N \cup \{0\} \}$ and $S'(\mathbb R) ...
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0answers
23 views

estimate for a function in the H^s space

Consiser $f \in H^s(R^n)$ $(s> n/2)$. Show that : $$ |f(x) - f(y)| \leq \gamma_s(|x-y|) || f||_s , \ \ \forall x,y \in R^n .$$ Where $\gamma_s(|x-y|) = 2 (2 ...
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1answer
131 views

Function always continuous in a Sobolev Space?

Hy everybody got a quick question. I know that all function F in a Sobolev Space has a continuous representative called U such as U=F almost everywhere. Lets take for example: The Sobolev space on ...
4
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2answers
131 views

Using the Extension Operator Theorem for Sobolev Spaces

I want to know if certain conditions hold after applying the Sobolev Extension Theorem: Assume $U$ is a bounded open subset of $\mathbb{R}^{n}$ and $\partial U$ is $C^{1}$. Suppose $1 \leq p < n$. ...
2
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1answer
64 views

basic exercise distribution theory

Consider $f \in L^{2}(R^n)$ with $\Delta f \in L^{2}(R^n) $. Show that ${\partial}^{|\alpha| } , (|\alpha| \leq 2 )f \in L^{2}(R^n)$. (the derivatives is in the distribution sense). My book gives ...
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1answer
58 views

Behavior of $u\in W_0^{1,p}(\Omega)$ near the boundary.

Assume that $\Omega\subset\mathbb{R}^N$ is a bounded regular domain. Let $1<p<\infty$ and take $u\in W_0^{1,p}(\Omega)$. Is true that given $\epsilon>0$ there is a neighbourhood $V$ of ...
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0answers
39 views

An weak type sobolev multiplication theorem

Suppose that $f_i\rightharpoonup f$ in $W^{1,p}$ and $g_i\rightharpoonup g$ in $W^{1,q}$ with $q=\frac{np}{n-p}$, what the best $r$, for which we have $f_ig_i\rightharpoonup fg$ in $L^r$ (maybe ...
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0answers
24 views

norm of $x \in \mathbb R^d$ is in Sobolev space

For which values of $\alpha, k,p,d$ is $$ \|x\|^\alpha \in \textrm{W}^{k,p} (B(0,1)) \quad ? $$ where $\displaystyle{ \textrm{B}(0,1) = \{x \in \mathbb R^d : \|x\|<1\}}$ This is an ...
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3answers
538 views

Weak convergence and strong convergence

Suppose $f_i$ is uniformly bounded in $W^{1,p}$ for some $+\infty>p>1$, then by passing to a sub-sequence, we can suppose $f_i$ is weakly convergent to $f$ in $W^{1,p}$. Assume furthermore that ...
2
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1answer
55 views

Does $H_0^1(\Omega)$ embed into $H_0^1(R^d)$?

Given a domain $\Omega$ in $\mathbb{R}^d$ and a function $f\in H_0^1(\Omega)$, the closure of the test functions on $\Omega$, does the extension of f by 0 to all of $\mathbb{R}^d$ necessarily lie in ...
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0answers
79 views

In which space does my (weak) solution exist?

I am reading through some numerical analysis papers, and was wondering what characteristics of a problem define which function space the weak solution of a problem belongs to. For example, for the ...
3
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1answer
144 views

Gradient of weak solutions of elliptic PDE at interfaces

Let $\Omega$ be a $C^1$-domain partitioned into two open sets $\Omega_1$ and $\Omega_2$ with interface $I=\partial\Omega_1 \cap \partial\Omega_2$. Now, say $u \in H^1(\Omega)$ is a weak solution of a ...
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0answers
20 views

Weak convergence version of sobolev multiplication theorem

Suppose that $1/n-1/p+1/q=0$, and $f_i$ weakly convergent to $f$ in $W^{1,p}$, $g_i$ weakly convergent to $g$ in $W^{1,q}$, can we conclude that $f_i*g_i$ weakly convergent to $f*g$ in $W^{1,p}$?
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66 views

$W^2_p$ regularity of solutions of linear elasticity

I want to prove the following statement: Let $\Omega$ be bounded, polygonal, and convex, then for the solution of the linear elasticity (elliptic) equation with inhomogenous Dirichlet boundary ...
2
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1answer
190 views

Evans PDE p.308 Exercise 16 (2nd ed)

Here is the statement of the problem (Evans PDE 2nd Ed., p.308, exercise 16) Show that for $n \geq 3$ there exists a constant $C$ so that $$ \int_{\mathbb {R}^n} \frac{u^2}{\vert x ...
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0answers
48 views

A Sobolev norm inequality

Assume $\mu\in H^{1}$ is compactly supported in $\Omega$ and $\Delta_{g}\mu\in H^{m}$ for some $m\ge 0$. Then if $K\subset \Omega$, there is a constant $C=C(K,m)$ such that $$|\mu|_{H^{m+2}(K)}\le ...
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1answer
177 views

Definition of weak solution in $W^{1,2}(\Omega)$.

I have a problem: For $\Omega$ be a bounded domain in $\Bbb R^n$. We consider $$\left\{\begin{matrix} \Delta u-\lambda u =f \ \rm in \ \Omega & \\ u\mid_{\partial {\Omega}} =0 ...
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1answer
116 views

Using Sobolev-Nirenberg-Gagliardo

I am currently studying a proof of a General Sobolev Inequality. I have the following question: Consider the Sobolev Space $W^{k,p}(U)$. With the added assumption that $k > \frac{n}{p}$. Let $l = ...
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1answer
360 views

Gagliardo-Nirenberg interpolation inequalities on a bounded domain

In the original paper of Nirenberg, the following inequality is proven : if $\frac{1}{p} = \frac{j}{n} + \left( \frac{1}{r} - \frac{m}{n} \right) \alpha + \frac{1 - \alpha}{q}$ and $\frac{j}{m} \leq ...
4
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1answer
162 views

Is $L^2(\Omega)$ dense in $H^{-1}(\Omega)$?

Is it true that $L^2(\Omega)$, identified with its own dual, is dense in $H^{-1}(\Omega)$? $H^{-1}(\Omega)$ is the dual of $H^1_0(\Omega)$ and $H^1_0(\Omega)$ is the $H^1$-closure of smooth functions ...
2
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0answers
142 views

Using the Sobolev-Nirenberg-Gagliardo inequality in a proof

If $1 \leq p < n$. The Gagliardo-Nirenberg-Sobolev inequality states that there exists a constant C such that $||u||_{L^{p^{*}}(\mathbb{R}^{n})} \leq C||Du||_{L^{p}(\mathbb{R}^{n})}$ for all $u ...
2
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0answers
76 views

(Sobolev space) A star domain in $\Bbb R^n$

We know the theorem: If $\Omega$ is a star domain in $\Bbb R^n$, then $C^{\infty}(\overline{\Omega})$ is dense in $W^{k,p}(\Omega)$. It means that ...
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1answer
463 views

Prove Friedrichs' inequality

I'm trying to show that the theorem (Friedrichs' inequality) in my book: Assume that $\Omega$ be a bounded domain of Euclidean space $\Bbb R^n$. Suppose that $u: \Omega \to \Bbb R$ lies in the ...
0
votes
1answer
37 views

Showing that function are equal almost everywhere in Sobolev Spaces

Consider the Holder space $C^{0,1-\frac{n}{p}}(\mathbb{R}^{n})$ and the Sobolev Space $W^{1,p}(\mathbb{R}^{n})$. Take $u_{m} \in C_{c}^{\infty}(\mathbb{R}^{n})$ such that Morrey's Inequality we have ...
2
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0answers
25 views

Time continuity of a solution to the generalized porous medium equation

Let $\Phi(x,s)$ be a smooth, non-negative function with $\Phi(\cdot,0)=0$ and $\Phi$ increasing in $s$ and $\partial_s \Phi(x,0)=0$ and $\Omega$ be a bounded smooth domain. Now assume there is a weak ...
4
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1answer
126 views

find a weak solution in an intersection of Sobolev spaces

In using lax-milgram to find a weak solution in an intersection of sobolev spaces the weak solution for $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ was ...