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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3
votes
1answer
48 views

Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
5
votes
0answers
51 views
+50

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 } ...
2
votes
0answers
64 views
+50

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 $\mathbb{R}^{3}$ ...
0
votes
1answer
21 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 ...
1
vote
1answer
66 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 ...
0
votes
0answers
18 views

injection in sobolev space [on hold]

let $\Omega$ an open bounded on $\mathbb{R}^n$. There is an injection between $L^{\infty}(\Omega)$ and $H^1_0(\Omega)$? thank's for the help.
0
votes
0answers
25 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 ...
1
vote
1answer
116 views

How to prove $\int_{\Omega}\frac{1}{\kappa}uv\ +\ \int_{\Omega}\kappa\nabla u\cdot\nabla v$ is an inner product in $H^1$

I need help with this problem from my homework Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and let $\kappa : \Omega\rightarrow\mathbb{R}$ be a continuous function, such that there's ...
0
votes
1answer
24 views

Application of Sobolev inequality

I saw this inequality without knowing how to justify it. This is probably an application of Sobolev inequality (or not). Here it is: $u,v\in H^1(\mathbb{R}^d)$, then ...
0
votes
0answers
37 views
+100

Subdifferential boundary conditions

Let $\Omega \subset \mathbb R^d$ be a domain and $\Gamma$ be its boundary. Assume also that we have a convex (proper, lower semicontinuous) function $\phi \colon \mathbb R^d \to \mathbb R \cup ...
2
votes
1answer
63 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 ...
1
vote
0answers
45 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 ...
2
votes
0answers
42 views
+50

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) ...
0
votes
0answers
9 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}, ...
1
vote
2answers
99 views

How exactly does the constant $C$ in the Sobolev inequality depend on the domain?

The Sobolev inequality theorem -as stated here- says Let $U$ be a bounded open subset of $\mathbb{R}^N$, with a $C^1$ boundary. Assume $u \in W^{k,p}(U)$. If $k<n/p$ then $u \in L^q(U)$, where ...
2
votes
2answers
66 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) = ...
2
votes
1answer
42 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 ...
0
votes
0answers
22 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 ...
2
votes
0answers
47 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 ...
2
votes
0answers
21 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 $\Omega \subseteq \mathbb{R}^n$. ...
2
votes
1answer
19 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 ...
1
vote
1answer
32 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 $$f_x\in L^{3/2}(0,\infty).$$
4
votes
1answer
42 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 ...
1
vote
1answer
74 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 $\dfrac{\partial v}{\partial x} \in H^1_0(\Omega)$ ?
3
votes
0answers
34 views

How do I integrate $\langle\nabla u,\nabla v \rangle$ in arbitrary dimensions?

I am trying to show that if $u_n$ are eigenfunctions of the Laplacian operator that make up an orthonormal basis of $L^2$, then $u_n\sqrt{\lambda_n}^{-1}$ form an orthonormal basis of $H^1_0$. I ...
1
vote
1answer
26 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 ...
3
votes
1answer
40 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 ...
2
votes
1answer
31 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\; ...
0
votes
1answer
25 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 $$\nabla(uv)=u\nabla v+v\nabla u.$$ ...
1
vote
1answer
33 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 $p\geq 1$
1
vote
2answers
60 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) = ...
2
votes
1answer
33 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 ...
1
vote
1answer
64 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 ...
-1
votes
1answer
37 views

Show $W^{1,p}(a,b)$ is compactly embedded in $L^p(a,b)$ for any $1<p\leq \infty$

I always get these sorts of questions wrong. Any help would be appreciated. This is my answer: The problem assumes $(a,b)$ is any bounded interval in $\mathbb R$. Consider a bounded sequence ...
2
votes
0answers
61 views

Differential of a Sobolev map between manifolds

Let $\Sigma, M$ be compact Riemannian manifolds. By embedding $M$ isometrically into $\mathbb{R}^N$, one can define the Sobolev spaces $W^{k,p}(\Sigma, M)$ by $$W^{k,p}(\Sigma,M) = \{ u \in ...
1
vote
0answers
36 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 ...
0
votes
0answers
20 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 ...
2
votes
1answer
71 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 ...
0
votes
1answer
55 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 ...
0
votes
0answers
7 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!
2
votes
2answers
60 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 ...
0
votes
0answers
56 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 ...
1
vote
1answer
25 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 $g\in W^{1,p}(\mathbb R\times(0,\infty))$?
1
vote
0answers
46 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 ...
0
votes
0answers
14 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. ...
0
votes
0answers
16 views

A parabolic maximum principle (if initial value is bounded, so is solution)?

Let $u \in L^2(0,T;H^1)$ with $u_t \in L^2(0,T;L^2)$ solve the PDE $$u_t - \Delta u = f$$ $$u(0)= u_0$$ on $\Omega \times (0,T)$ where $\Omega$ is a bounded domain. We do NOT have the Poincare ...
3
votes
1answer
72 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. ...
1
vote
1answer
27 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 ...
2
votes
1answer
26 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 $\kappa$ ...
0
votes
1answer
22 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 ...