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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37 views

Convergence in $L^2$ of difference quotients to derivative of function in $H^1$

Is it true that if $u\in H^1({\mathbb R})$, then $(u(x+h)-u(x))/h$ converges to $u'(x)$ in $L^2({\mathbb R})$, as $h\to 0$? It's hard for me to get a handle on this, since $u'$ doesn't have to be ...
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1answer
71 views

Imbedding theorem for Sobolev space $D^{k,p}$

I can't find any reference on the imbedding theorem for the Sobolev space $D^{k,p}$, which is defined by $$D^{k,p}(\Omega)=\{u\in L_{loc}^{1}(\Omega)|D^{k} u\in L^{p}(\Omega)\}.$$ Is it the same as ...
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0answers
8 views

How to check whether f belongs to H^(\beta+t)

In the Lemma 3 of the paper"Wendland H. Multi scale analysis in Sobolev spaces on bounded domains" How to check whether f belongs to H^(\beta+t)? Thank you!
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1answer
157 views

proof on Poincare's inequality.

This might be a silly question. So basically, I have proved the Poincare's inequality for $p=1$ case. That is, for $u\in W^{1,1}(\Omega)$, I have $||u-\bar{u}||_{L^1}\leq C||\nabla u||_{L^1}$. Here ...
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2answers
117 views

dual of $H^1_0$: $H^{-1}$ or $H_0^1$?

I have a problem related to dual of Sobolev space $H^1_0$. By definition, the dual of $H^1_0$ is $H^{-1}$, which contains $L^2$ as a subspace. However, from Riesz representation theorem, dual of a ...
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0answers
62 views

convergence sequence in L^1

Let an sequence $u_n$ such as $u_n$ converge to $u$ in $H^1_0(\Omega)$ weak, and $u_n$ converge to $u$ in $L^2(\Omega)$ strong and a.e $x \in \Omega$. Let $g_n(x,u_n)$ an Caratheodory function such ...
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1answer
28 views

Extension of a function from the edge.

How can I extend the function $f\in W^{1-\frac{1}{p},p}(\mathbb R\times\lbrace0\rbrace)$ up to $g\in W^{1,p}(\mathbb R\times(0,\infty))$?
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0answers
57 views

Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
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1answer
68 views

convergence in L^{1} strong

I search an proof of this lemma: First,we have this definition: we tell that an sequence $f^{\epsilon}$ is equi integrable if $$\forall \eta, \exists \delta > 0, |E|\leq \delta \implies ...
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0answers
16 views

Does the inequality $\int_{\Omega}(-\Delta)^{\frac 12}G(w(x))(u(x)-C)^+ \geq 0$ hold? If not, can we bound it from above in a particular way?

Let $G$ be a locally Lipschitz function such that $G(0)=0=G'(0)$ and $G$ is also increasing. I want to know if $$\int_{\Omega}(-\Delta)^{\frac 12}(G(w(x))(u(x)-C)^+ \geq 0$$ where $C$ is a constant. ...
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0answers
42 views

Variable density in the equation of motion

At a fixed point in time, consider the equation of motion $$ \nabla \cdot \boldsymbol \sigma(u) + \boldsymbol f = \rho \ddot{\boldsymbol u} \quad \text{in $\Omega \subset \mathbb R^d$} $$ for a ...
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1answer
34 views

Sobolev inequality in negative index

For $s>n/2$, is it true that $$ \int |fg| dx\leq ||f||_{H^s}||g||_{H^{-s}}?$$ This inequality is used on pg 398 of the Majda Bertozzi book on Vorticity and Incompressible flow but I can't make ...
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1answer
81 views

Why is $-\Delta+1$ an isomorphism between Sobolev space $W^{2,p}$ and $L^p\,$?

Consider the linear elliptic operator $L: W^{2,p}(\mathbb{R}^N)\to L^p(\mathbb{R}^N)$ defined by $Lu=-\Delta u + u$. Can we prove that $L$ is an isomorphism for all $p\geq 1$? It can be proved that it ...
2
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1answer
41 views

Product of Hölder and Sobolev functions

Here $C^{\kappa , \lambda} ( \overline{\Omega} ) = \left\{ h|_{\overline{\Omega}} :h \in C^{\kappa , \lambda} ( \mathbb{R}^{n} ) \text{ and } h \text{ has compact support} \right\}$ denotes $\kappa$ ...
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1answer
39 views

Somewhat L2 against H1 estimate; an inequality in H1

somehow I'm a little slow on this one: Let $\Omega = [0,1]^2 \subseteq \mathbb{R}^2$ and $\emptyset \neq D \subsetneq \Omega$. Do constants $c_1,c_2,c_2\in\mathbb{R}_{\geq 0}$ exist such that $$ c_1 ...
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0answers
33 views

Trouble understanding proof of the Poincare Inequality

In the proof of the Poincare Inequality in this article on page 4 the domain of integration is divided into two parts, $v< 0$ and $v>0$. I think this has to do with the absolute norm appearing ...
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0answers
35 views

Boundary value problem for two functions

The question is: Let $\mathcal{H}=H_0^1(\Omega)\times H^1(\Omega)$ and consider the solution $(u,v)\in\mathcal{H}$ to the differential problem \begin{equation} -\Delta u=f+a(v-u)\quad\text{in }\Omega ...
2
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2answers
51 views

Product of Sobolev functions

Suppose that $\Omega$ is 2-dimensional bounded open set with smooth boundary and $f\in W^{2,2}( \Omega) $ and $ g,h\in W^{1,2} ( \Omega) $. What can we say about the regularity of the product of ...
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1answer
29 views

Limiting argument when proving inequality in Sobolev space

I found this limiting argument very common in proving inequalities in Sobolev spaces. Basically, what people do is to observe that test functions (smooth functions with compact support) are dense in ...
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1answer
57 views

Notion for weak derivatives of $L^p(0,T,X)$-functions

A definition in Evan's PDE-book from chapter 5.9.2 says (let $X$ be a Banach space): Let $u\in L^1(0,T,X)$. We say $v\in L^1(0,T,X)$ is the weak derivative of $u$ provided $$\int_0^T \phi'(t)u(t)dt = ...
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0answers
53 views

Sobolev Spaces separable

How do I demonstrate that the Sobolev spaces $W^{1,\infty}$ is not separable? PS: I know that space $L^{1,\infty}$ is not separable but was unable to use this information.
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0answers
55 views

estimations of solutions

Let $\Omega=\mathbb{R}^2_+=\{(x,y) \in \mathbb{R}^2; y > 0\}$, $f \in L^2(\Omega)$, $\lambda \in \mathbb{R}^*_+$, $A=(a_{ij})_{1\leq i,j \leq 2}$, $a_{ij} \in \mathbb{R}$ and there exist $\alpha ...
2
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2answers
59 views

Density in $H^1_0$

Let $\Omega = \mathbb{R}^2_+=\{(x,y)\in \mathbb{R}^2; y>0\}$ and let $v \in H^1_0(\Omega)$. For $h \neq 0$ we define $D_h v = \dfrac{v(x+h,y)-v(x,y)}{h}$ such as $\forall \varphi \in ...
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1answer
80 views

Sobolev space-exercice [closed]

Let $\Omega = \mathbb{R}^2_+$. My question is: how we prove that if $v \in H^2(\Omega)$ such as $v(x,0)=0$, then $\dfrac{\partial v}{\partial x} \in H^1_0(\Omega)$ ?
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1answer
31 views

Composition with exponent Sobolev Space

I have a problem with following statement: We have $f \in W^{1,1}(B) (B-ball\ in\ \mathbb{R^n}), \ \nabla f \ e^{f} - $ integrable $\Rightarrow \ e^f \in W^{1,1}$ I've started with a sequence of ...
5
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1answer
178 views

Subdifferential boundary conditions: Testing with $L^2$ or $H^{1/2}$ functions

My question was essentially this: Does it make a difference if I test subdifferential boundary conditions with functions from $L^2(\Gamma)$ or $H^{1/2}(\Gamma)$? In the following, I will phrase the ...
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1answer
39 views

Some properties of capacity

Let $\Omega\subset\mathbb{R}^N$. For $K\subset \Omega$ we can define the $p$-capacity, $p\in (1,\infty)$ as the number $$\operatorname{cap}_p(K)=\inf \int_\Omega |\nabla u|^p$$ where the infimum is ...
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1answer
77 views

Definition of fractional Laplacian on a compact manifold?

How does one define the fractional Laplacian operator $(-\Delta)^s$ on a compact Riemannian manifold? In $\mathbb{R}^n$, it is defined $$ (-\Delta)^s f(x) = c_{n,s} \int_{\mathbb{R}^n} \frac{f(x) - ...
2
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1answer
65 views

Poincaré inequality for a subspace of $H^2(\Omega)$

Suppose that $\Omega\subset\mathbb{R}^d$ is a smooth, bounded, and connected domain. Let \begin{equation} H=\{u\in H^2(\Omega):\int_\Omega u(x) dx=0 ~\text{and}~ \nabla u\cdot v=0~ ...
2
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1answer
26 views

how to prove that this weak solution is subharmonic?

My question is about this article http://hal.inria.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf. My question is : Consider a smooth, bounded and convex domain $K$ in $R^n$ such that $K\subset \{ x_1 = ...
3
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1answer
43 views

Why is this estimate using a compact embedding in a sobolev space true?

Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz-domain. We then have, for $s\in[1,6)$ the compact embedding $H^1(\Omega)\stackrel{c}{\hookrightarrow}L^s(\Omega)$ ensuring the existence of a ...
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0answers
26 views

Why do the following regularisations of a function in Sobolev space exist?

Suppose $v_1, v_2$ satisfy $\mu \leq v_1(x,t), v_2(x,t) \leq M$ a.e. in $Q:=\Omega\times(0,T)$ and $$(v_1, \eta_1) \quad\text{and} \quad (v_2, \eta_2) \in L^2(0,T;H^1(\Omega)) \cap L^2(Q).$$ Define ...
2
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1answer
26 views

Functions of Sobolev space with asymptotic decay

Define a subspace of the Sobolev space $H^1(\mathbb R^d)$ to be $$ X=\{u\in H^1(\mathbb R^d), |u(x)|=O(|x|^{1-d}), \text{ as } |x|\rightarrow +\infty\} $$ Is there a norm $\|\cdot\|_X$ such that $(X, ...
3
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1answer
97 views

Having trouble understanding the finite element method

I am trying to read some sources on how to implement the finite element method and I am having difficulty putting all the concepts together. I can read and understand the Galerkin approach just fine. ...
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1answer
35 views

Isomorphism between Sobolev space and L^p

Let $L_1$ be an elliptic PDE operator $L_1:W^{2,p}\rightarrow L^p$ and $L_2=e^fL_1$ where f is a bounded function. I proved $L_1$ is an isomorphism, can I conclude $L_2$ is an isomorphism?
3
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1answer
35 views

What does it mean for a distribution to be in $L_2$?

I am new to Sobolev space and distribution theory. So here is what I know. Distributions are linear functionals on $C_0^\infty$. Let's look at the simplest Sobolev space. $H^1(\Omega)$ is equal to the ...
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1answer
43 views

minimizes the functional to solve a pde

I am trying to do this exercise: Let $\Omega$ a open bounded domain in $R^n$. Consider the Dirichlet problem $$ \left\{ \begin{array}{ccccccc} -\Delta u = \lambda \sin (u) + f , \ \text{in ...
3
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1answer
38 views

Identification of $L^2$ limits with distributional convergence

I just read the thread on "too much effort" and I would like to be more specific. Is the following reasoning correct: Let $g,g_\delta\in H^1(D)$, $D$ some domain in $\mathbb{R}^n$ with the following ...
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0answers
49 views

Extending by zero a Sobolev function

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set and $u\in W_0^{1,2}(\Omega)$. Define $B_R=B(x_0,R)$ for $x_0\in\partial\Omega$ and consider $\tilde{u}=u\chi_{\Omega\cap B_{2R}}$. Do we need some ...
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0answers
37 views

$(g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s}$ right or wrong?

I am proving something, and I may need the following inequality: $$ (g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s} $$ where $(g,f)_{L^2} $ is inner product and $f$ and $g$ have higher enough regularity ...
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1answer
34 views

Show that this functional is coercive - variational methods

For $u \in H^{1}_0({\Omega})$ ($\Omega$ is a domain open and bounded in $R^n$). Let $0 < \lambda < \lambda_1$ ($\lambda_1$ is the first eigen value of the laplacean) and a fixed $f \in ...
2
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1answer
70 views

Weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$

What does weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$ means? $\Omega$ is open, bounded, has boundary smooth and etc...
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0answers
28 views

Dual space of Sobolev functions with homogenous neumann BC

Let $\Omega$ be a bounded Lipschitz-domain with outward normal vector $\nu$ and let us take a look at the Sobolev-spaces $H^1:=H^1(\Omega)$, $H^1_0:=H^1_0(\Omega)$ and ...
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2answers
40 views

Show that a map is a continuous bilinear form on $H^1(0,1)$ space

Let $u,v \in H^1(0,1) = \{f : (0,1) \longrightarrow \mathbb{R}, f,f' \in L^2(0,1) \}$, show that $$a(u,v) = \int_0^1 (u'v' + uv)\; dx$$ is a continuous bilinear form.
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1answer
48 views

If $f \in H^1$ and $f=g$ a.e. or in $L^2$, is $g \in H^1$?

If $f \in H^1(\Omega)$ and either $$f=g \quad\text{a.e.}$$ or $$f=g \quad\text{in $L^2(\Omega)$},$$ is $g \in H^1(\Omega)$? I think since we identify functions that are equal almost everywhere that ...
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1answer
35 views

Does $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ give something useful?

If $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ is there any way to extract a strongly convergent subsequence of $u_n$ in some useful space for PDEs? Or ...
2
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2answers
48 views

Boundedness of a sequence of functions

Let $\Omega$ be a bounded domain and $f_n\in L^2(\Omega)$ be a sequence such that $$\int_\Omega f_nq\operatorname{dx}\leq C<\infty\qquad \text{for all}\quad q\in H^1(\Omega),\ ...
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0answers
20 views

Showing $\langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle$

Let $u \in L^2(0,T;H^1)$ have weak derivative $u' \in L^2(0,T;H^{-1}).$ Let $c$ be a constant. I want to show that $$\langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle$$ where $(f)^+ = ...
1
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1answer
36 views

Does $C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})=C^1([0,T];H^{s+1})$?

I have asked this question with other problems, but about this part nobody answers. So I want to ask again, and put some details in it. My question is whether the following equality is correct. If it ...
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1answer
24 views

Bounding an integral (Sobolev spaces and duality pairing), explanation needed

I don't understand how the inequality is derived here: I understand the equality but not how he gets the last line. I get a triple integral and I don't know to deal with that.