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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47 views

Functions of fractional-order Sobolev spaces

In fracture mechanics one might end up dealing with functions such as $$w(r,\theta) = \sqrt{r} \sin \frac \theta 2,$$ which is defined, for example, on a cracked unit circle, $\Omega = ...
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1answer
39 views

Proposed proof for Sobolev space result

I have the following result which seems that it must be true, but I would like to prove it: This is my proposed proof. If $U \subset \mathbb{R}^{n}$. Given $u \in W^{1,p}(U)$, where $u$ has compact ...
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1answer
24 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
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2answers
180 views

Extension and trace operators for Sobolev spaces

Given that $\Omega \subset \mathbb{R}^{n}$ is an open, convex, Lipschitz bounded set. Let $O \subset \Omega$ be open bounded set then consider. $$u_{m} \rightharpoonup^{*} u \text{ in } ...
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1answer
168 views

Stokes theorem and Sobolev spaces.

I am interested under which regularity condition is Stokes' theorem is still valid. For concreteness I am interested in the following problem Let's consider a domain $\Omega$ in $\mathbb{R}^{3}$ ...
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26 views

injection in H^{-1}

let $\Omega$ an open on $\mathbb{R}^n$. if $f \in H^{-1}(\Omega)$ and $g \in L^1(\Omega)$. Who is the Sobolev space $V$ who can contains $f-g$ such as $V$ is injected on $H^{-1}(\Omega)$? Thanks for ...
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13 views

what does well posdeness results tells us concerning non linear evolution equations?

Consider a nonlinear Shr\"odinger equation, $$iu_{t}+\bigtriangleup u + f(u)= 0, u(0)= u_{0}$$ where $u(t, x)$ is complex valued function of $(t,x) \in \mathbb R \times \mathbb R^{n}$, $i=\sqrt{-1}, ...
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1answer
67 views

Proving $u\mapsto |u|^2u$ is Lipschitz on bounded subsets of $H^2(\Omega)\cap H_0^1(\Omega).$

I'm reading a paper and am stumped verifying two details. Let $\Omega$ be a bounded region in $\mathbb{R}^2$ with smooth boundary. I'd like to show that the map $u\mapsto |u|^2u$ is a map from ...
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2answers
72 views

Variational methods : Why i can't apply this theorem?

Consider the following problem: Find a weak solution for $$u'' + u = \sin t ,\,\, 0 < t < \pi$$ $$u(0) = u(\pi)=0 $$ the corresponding functional for the problem is $\varphi(u) = ...
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0answers
28 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
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60 views

Extension of function from $W^{1,p}(\Omega - \{x\})$ to $W^{1,p}(\Omega)$

Suppose $\Omega\subset \mathbb{R}^n$ is open, $p\geq 1$, $n\geq 2$ and $u \in W^{1,p}(\Omega-\{x\})$. Show that $u$ extends to a function in $W^{1,p}(\Omega)$. So far I have; clearly $u$ and each ...
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0answers
49 views

Sobolev spaces necessary and sufficient condition

I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not. Problem Formulation: Let $A$ be a subset of ...
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0answers
27 views

Relationship between eigenvalues of differential operator and eigenvalues of its adjoint operator.

I am considering $L\phi = -\triangle \phi + u \cdot \nabla \phi$ and its "adjoint" operator $L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$ on a bounded domain $\Omega \subseteq \mathbb{R}^n$. ...
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0answers
105 views

How to prove comparison principle for parabolic PDE (nonlinear)

Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$. Consider the PDE $$u_t = \Delta F(u) ...
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1answer
23 views

What is the metric on ${(L^{p}(\Omega))}^N$? (prove that sobolev spaces uniformly convex)

Here what I've done to prove that Sobolev Spaces $W^{(m,p)}(\Omega)$ are uniformly convex for $1<p<\infty$ Given integers $n\geq 1,k\geq0$, we define $N(n,m)$ as the number of multi-indices ...
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1answer
33 views

Is 'f' belong sobolev?

I was trying to show that the function $$f(x) = \dfrac{x^{1/2}}{1+x^2} \in W^{1,3/2} (0,\infty)$$ that is, have to show that $$f\in L^{3/2}(0,\infty)$$ and $$f_x\in L^{3/2}(0,\infty).$$
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1answer
52 views

Chain rule for weak derivatives of $f(u)$ where $f'$ is not bounded but $u$ is?

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$. Suppose $u$ has a weak derivative $u_x$. I want the chain rule $$\partial_x (f(u)) = f'(u)u_x$$ to hold. We know this holds if $f'$ is bounded. But I don't ...
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1answer
76 views

First eigenvalue of Laplacian and Poincaré inequality

Any idea on how to solve: $\int_{\Omega} |\nabla u|^2 d^n x=\lambda_1\int_{\Omega}u^2 d^n x$, with $u\in H^1_0(\Omega)$ and $\Omega\subset\mathbb{R}^n$, and $\lambda_1$ the first eigenvalue of the ...
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1answer
43 views

bilinear continuous, coercive form

Let $k\in \mathbb{R}, k\neq 1$, consider the space $$ V = \{u\in H^1(0,1): u(0) = ku(1)\}$$ Let $$a(u,v) = \int_0^1 (u'v'+ uv)\; dx - \left(\int_0^1 u\; dx\right) \left(\int_0^1 v\; ...
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1answer
37 views

Is the set $\{ u \in H^1(\Omega) : 0 < a \leq u(x) \leq b\quad \text{a.e.}\}$ closed?

Is the set $\{ u \in H^1(\Omega) : 0 < a \leq u(x) \leq b\quad \text{a.e.}\}$ closed? $\Omega \subset \mathbb{R}^1$ is an interval. There is an embedding into $C^0(\Omega)$. But not sure if this ...
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1answer
54 views

Function in $H^1$, but not continuous

By the Sobolev embedding theorem, if $\Omega$ is bounded, $H^s(\Omega)\subset C(\Omega)$ for $s>1$, in $\mathbb R^2$. Where I can find a counterexample (if one exists) for the case $s=1$? I mean a ...
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1answer
34 views

Leibniz's rule in $W^{1,\,p}(\Omega)\cap L^\infty(\Omega)$

Is it true? If $\Omega\subset\mathbb{R}^n$ is bounded and $u,v\in W^{1,\,p}(\Omega)\cap L^\infty(\Omega)$, then $uv\in W^{1,\,p}(\Omega)\cap L^\infty(\Omega)$ and $$\nabla(uv)=u\nabla v+v\nabla u.$$ ...
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1answer
38 views

Show that this simple functional is not bounded below

Define $\varphi(u) = \displaystyle\int_{0}^{1} \displaystyle\frac{{|u'| }^2}{2} - \displaystyle\frac{{u }^2}{4} - hu \ dt$, $u \in H^{1}_{0}(0,1)$ where $h: [0,1] \rightarrow R$ is a continuous ...
3
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1answer
54 views

A literature reference for Sobolev mappings $W^{m,p}(M,N)$ for M, N smooth Riemannian manifolds

Anyone know a respectable reliable reference for the definition of Sobolev mappings $W^{m,p}(M,N)$ for M, N smooth compact Riemannian manifolds. It suffices for m natural and $p\geq 1$
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2answers
76 views

Discontinuous Sobolev Function

I'm trying to show that there's an $f \in H^1(\mathbb{R}^2)$ which is not ae equal to a continuous function. Per a couple of suggestions, I've decided to look at the function $f(x) = ...
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0answers
35 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
68 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
125 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
105 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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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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56 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
67 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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15 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
79 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 ...
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1answer
39 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
36 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
32 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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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 ...
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2answers
50 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
28 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
56 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
58 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 ...
1
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1answer
79 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 ...