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
20 views

Descomposition on temporal sobolev space

Let $\Omega$ an open subset of $\mathbb{R}^2$ with Lipschitz boundary. Can I descompose in a unique way any $u\in L^2(0,T;L^2(\Omega))$ such that for all $t\in [0,T]$, $u(t)=u_1(t)+u_2(t)$ with ...
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0answers
45 views

Estimate difference of solutions of an equation with a bilinear symmetric continuous form

My question refers to differential equations in Sobolev spaces. It is as follows: Let $\Omega \subset \mathbb{R}^n$ be a bounded open set. Let $a: H_0^1(\Omega) \times H_0^1(\Omega) \rightarrow ...
3
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1answer
117 views

Using Nemytskii Theorem for Sobolev Spaces

The Nemytskii mappings in Lebesgue spaces theorem is as follows: If $a: \Omega \times \mathbb{R}^{m_{1}} \times \cdots \times\mathbb{R}^{m_{j}} \rightarrow \mathbb{R}^{m_{0}}$ is a Caratheodory ...
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1answer
51 views

limit of function in Sobolev space

Let function $f(x)$ and $f(x)g(x)$ belong to $\mathcal{W}^{s+1}$ with $s\ge 1$, where $\mathcal{W}^{s+1}$ be the Sobolev space of regularity $s+1$ in $\mathbb{L}^2$-norm. We also have $g\in ...
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2answers
258 views

The normal derivative in PDE problems (how is the weak form defined?)

For smooth functions, we know that $$-\int_{\Omega}\Delta v w = \int_\Omega \nabla v \nabla w - \int_{\partial \Omega}\nabla v \cdot \nu w.$$ Thus we can define a generalised normal derivative ...
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2answers
103 views

The Poisson problem $-\Delta u =0$ with $u=g$ on the boundary where $g \in H^{\frac{1}{2}}$

Consider $$ \begin{align} -\Delta u =0 & \text{on $\Omega$} \\ u = g & \text{on $\partial\Omega$} \end{align} $$ where $g \in H^{\frac{1}{2}}(\partial\Omega)$. It seems there exists a ...
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2answers
61 views

Some doubts on the Trace Theorem

The Trace Theorem in Evan's Book (1st edition) says that, Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exsits a bounded linear operator $T$, $$T:W^{1,p}(U)\rightarrow L^p(U)$$ such ...
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1answer
62 views

A problem about Sobolev Spaces

I meet a problem in homework: Assume $\Omega\subset\mathbb{R}^n$ is bounded and $\partial\Omega$ is $C^1$. Does $H_0^2(\Omega)$ equal to $H_0^1(\Omega)\cap H^2(\Omega)$? Obviously, ...
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0answers
76 views

Product rule for Banach space-valued differentiable functions?

Let $\Omega \subset \mathbb{R}^n$ be a bounded open set and let $f(\cdot,\cdot)$ and $g(\cdot,\cdot)$ be functions from $[0,T]\times \Omega$ into $\mathbb{R}$. Suppose that $f \in ...
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0answers
55 views

The space $C^1([0,T]\times \Omega)$ for $\Omega$ open and bounded

Let $\Omega$ be open and bounded. Is there anything nice I can say about the space $C^1([0,T]\times \Omega)$ and its inclusion in some Bochner like spaces? If $f \in C^1([0,T]\times \Omega)$ then ...
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1answer
129 views

Integration by parts with few regularity

I'm having problems proving an integration by parts formula presented in the work of Alt and DiBenedetto on porous media flow (Remark 3.4.2). Essentially, the problem is the following. Let $s\in ...
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0answers
112 views

If $u$ has a weak derivative and $f$ is $C^1$ does $fu$ have a weak derivative (fractional Sobolev space and weak time derivatives)

Let $\Omega$ be an open bounded set. Let $s \in (0,1)$ and $H^s(\Omega) := W^{s,2}(\Omega).$ Let $f \in C^1([0,T]\times \Omega)$ and $u \in L^2(0,T;H^s(\Omega))$ with weak derivative $u' \in ...
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1answer
45 views

Showing $u \in H^s$ and $\varphi \in C^1$ implies $u\varphi \in H^s$ (product rule)

Let $\Omega$ be bounded and open set in $\mathbb{R}^n$. As a start, I pose this question: For $u \in H^s(\Omega)=W^{s,2}(\Omega)$, define the Holder seminorm type quantity $$F(u) = ...
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1answer
43 views

Please explain this notation of mapping into a set and product space (related to Sobolev spaces)

So does this mean that I can say that, for example, $\gamma \frac{\partial u}{\partial \nu}$ has a unique continuous extension as an operator from $W^s_p(\Omega)$ onto $W^{s-1-{\frac 1 ...
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1answer
128 views

A question about a Sobolev space trace inequality (don't understand why it is true)

Let $\Omega$ be an open set with boundary $\partial\Omega$. Let $u \in H^1(\Omega)$. There exists a $\lambda \in \mathbb{R}$ such that $$\int_\Omega |\nabla u |^2 + ...
2
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2answers
331 views

Is fractional Sobolev space $H^s$ Hilbert?

For $s \in (0,\infty)$ a fractional number, define $H^s(\Omega) = W^{s,2}(\Omega)$ on good domain $\Omega$. Every textbook doesn't say that $H^s$ is Hilbert. Is it? I have only seen this fact when ...
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1answer
46 views

Show that $L$ it is a continuous operator.

Let $L(.)$ a linear operator in $W$ so that $$L(\phi)=\int_0^T<f,\phi>_{H^{-1}, \ H_0^1}dt+(u_0,\phi(0))_{L^2}.$$ Then $L(.)$ it is continuous in $W$. Where $f \in L^2(0,T; H^{-1}(\Omega))$ ...
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2answers
74 views

Show that the function $u(x)=1 \in W^{m,\ 2}(\Omega)$, but not in $W_0^{m,\ 2}(\Omega)$

I have a problem: For $\Omega$ be a domain in $\Bbb R^n$. Show that the function $u(x)=1 \in W^{m,\ 2}(\Omega)$, but not in $W_0^{m,\ 2}(\Omega)$, for all $m \ge 1$. ...
2
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1answer
85 views

Establishing a relationship between weak solution in $L_2(\Omega)$ and weak solution in $W^{1,\ 2}(\Omega)$ with classical solution

I have a problem: For $\Omega$ be a bounded domain in $\Bbb R^n$. Denote $L_{2,\ 0}(\Omega)=\overline{C_{0}^{\infty}(\Omega)}$, with norm in $L_2(\Omega)$. We consider: ...
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1answer
176 views

Show that a Bilinear form is Coercive

I'm reading through Brezis' book on functional analysis, Sobolev spaces and PDE, and I'm having trouble showing that the Bilinear form: $a(u,v) = \int_{0}^{2}u'v'dx+\left(\int_{0}^{1}u dx\right) ...
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1answer
62 views

Show that $b(\ ., \ .)$ it is not coercive.

Let $b(\ .,\ .)$ a bilinear operator so that $$b(u, \phi)= \int_0^T((u(t), \ \phi(t)))dt- \int_0^T(u(t), \ \phi'(t))dt,$$ where $((\ ., \ .))$ it is the inner product of $H_0^1(\Omega)$ (Sobolev ...
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1answer
130 views

Gagliardo Nirenberg Sobolev inequality for n >= 2

I have a quick question regarding the Gagliardo-Nirenberg-Sobolev inequality. It states that: Assume $1 \leq p < n$. There exists a constant $C$, depending only on $p$ and $n$, such that ...
0
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1answer
68 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
67 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
263 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
19 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
175 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
388 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
128 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 ...
4
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0answers
89 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
125 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
67 views

Chain rule in $W^{1,p}$

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 ...
2
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1answer
170 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
194 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
56 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
95 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
38 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
49 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
44 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
84 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
321 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 ...
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0answers
16 views

Can $u\in W^{1,2}_0$, $\Delta u\in L^2$, $u\geq 0$ be approximated by a sequence smooth function $u_k\geq 0$

Assume that $\Omega\in R^3$ is a bounded Lipschitz domain. $u\in W^{1,2}_{0}$, $\Delta u\in L^2$, $u\geq 0$. Is it possible to approximate u by a sequence of nonnegative smooth functions $u_k$, ...
2
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1answer
34 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
77 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
108 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 ...
2
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1answer
124 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 ...
1
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1answer
61 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) ...
0
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0answers
27 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 ...
1
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1answer
164 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
votes
2answers
164 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 $1 \leq p < n$. ...