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Questions tagged [sobolev-spaces]

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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16 views

Derivation of weak formulation for surface Laplacian

For the surface Laplacian $\Delta_S= \nabla_S \cdot \nabla_s $ and a twice differentiable function $u$ on the surface $S$ we have $$\Delta_S u=\Delta u-\kappa\frac{\partial u}{\partial n}-n^TH(u)n,$$ ...
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9 views

Density of $C^\infty(\overline{\Omega})$ in $W^{k,2}(\Omega)$

First, $\Omega\subseteq \mathbb{R}^n$ and $$\Omega=\prod_{i=1}^n\left(a_i,b_i\right). $$ I know that the space of test functions $C^\infty_c(\Omega)$ is indeed dense in $W^{k,2}(\Omega)$. Is there ...
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34 views

Differentiability under the integral sign

Do we have some kind of Lebesgue theorem in the case of an integral defined over a boundary? $\frac{d}{dt}\int_{\partial\Omega}f(t,x)d\sigma$
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18 views

Differentiability of functional

Let $\Omega$ be a bounded domain with Lipschitz border. We define $G:W^{1,p} \rightarrow\mathbb{R}$ by $$G(u)=\int_{\partial\Omega}|u|^pd\sigma.$$ How can I show that $(G'(u),h)=\int_{\partial\Omega}|...
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11 views

Dual space of $W^{\frac{1}{p'},p}(\partial\Omega)$

Let $u\in W^{1,p}(\Omega)$ verifying \begin{eqnarray}\int_{\Omega}|D\widehat{u}|^{p-2}(D\widehat{u},Dh)_{R^N}dz+\widehat{\xi}\int_{\Omega}|\widehat{u}|^{p-2}\widehat{u}hdz+\int_{\partial\Omega}\beta(Z)...
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0answers
46 views

Inequality of a p-laplacian

Let $\Omega$ be a bounded domain with lipschitzian boundary.$2\leq p$ Why do we have $$\int_{\Omega}\langle|\nabla u|^{p-2}\nabla u- |\nabla v |^{p-2}\nabla v,\nabla v\rangle\leq \int_{\Omega}(|\nabla ...
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19 views

A boundary equality and dual of the trace's image [duplicate]

Let $\Omega$ be a bounded domain with lipschitzian boundary. How can I show that $\left\langle\frac{\partial u}{\partial n},h\right\rangle_{W^{\frac{1}{p' },p}}=0, \ \forall h\in W^{1,p}\implies \...
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25 views

Boundary equality [on hold]

Let $\Omega$ be a bounded domain with lipschitzian boundary. How can I show that $\left\langle\frac{\partial u}{\partial n},h\right\rangle_{W^{\frac{1}{p' },p}}=0, \ \forall h\in W^{1,p}\implies \...
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20 views

Maximal monotone functional

Given a bounded domain $\Omega$ $h:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ defined by $$ h(u)=\int_{\Omega}|Du|^{p-2}(Du,Dh)_{\mathbb{R}^N}dz$$ why is it continuous and maximal monotone ?
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1answer
45 views

Does derivative imply weak derivative? [on hold]

It is known that there are functions whose weak derivative exists but (classical) derivative does not exist. I want to confirm that "any differentiable function is weakly differentiable". Help me.
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1answer
31 views

Is $C^k$ function space is dense in Sobolev space?

It is known that $C^\infty$ function space dense in Sobolev space. But I want to know that is this statement true also for $C^k$ function space?
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1answer
27 views

Sequentially continuity

Given a map $E:L^\infty\rightarrow C^1(\overline{\Omega}), g_n\rightarrow g$ in $L^\infty(\Omega)$. $u_n=E(g_n), n\in \mathbb{N}$ and u=E(g) and show that every time we take a subsection we can find ...
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23 views

Inequality involving periodic functions and Sobolev space.

Set $n\in\mathbb{N}$. For $\psi=u+iv$ in $H^1_T(\mathbb{R}^n)=\lbrace \psi\in H^1_{loc}(\mathbb{R}^n,\mathbb{C}):\psi(x)=\psi(x_1+T,...,x_n+T)$, I wonder if $$ \int_Q |\nabla u|^2\geq C\int_Q (u-1)^3 ...
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35 views

Evans pde proof of global approximation theorem

I have a question regarding step 3 of the proof (shown in picture below). I am wondering if there is a need for us to consider the subset $V\Subset U$ instead of directly using $U$ when showing the ...
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1answer
29 views

Sequentially lower semicontinuity

Let $\Omega$ be a bounded open set with lipschitz boundary, How can we show that the functional defined by $f:W^{1,p}\rightarrow\mathbb{R}$ $f(u)=\int_{\Omega}|Du|_{\mathbb{R}^N}^p$ is sequentialy ...
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22 views

Reference request: Density of testfunctions in sobolev space $ W^{1}_{0}$

can someone help me out and name a good source where the following statement ist proven? $C^\infty_0(\Omega)$ is dense in $W^{1,p}_0(\Omega)$ with $\Omega \subset \mathbb{R}^n$ being an open, bounded ...
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9 views

well-posedness of elliptic pdes with gaussian weights

Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by $$ \gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}. $$ Let $d\gamma_n$ denote the following measure ...
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1answer
60 views

Convergence of sequence of function for a bounded sequence in the Sobolev space

Let $u_n$ be a bounded sequence in $W_{0}^{1,p}(\Omega)$. Then upto a subsequence one has $$ u_n\to u \mbox{ weakly in}\,W_{0}^{1,p}(\Omega). $$ How the following statement is true? $$ \int_{\Omega}|\...
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1answer
64 views

Sobolev Embedding into $L^{\infty}$

I tried to find the question here, but I couldn't. I'm a bit puzzled by Sobolev embeddings at the moment. In my lecture notes, I found the statement "If $\Omega \subseteq \mathbb{R}^d$ is ...
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1answer
19 views

Measurability of $\nabla \cdot (a \cdot \nabla u)$ implies measurability of $u$?

Suppose I know that $\nabla \cdot (a(t)\nabla u(t))$ is such that $\nabla \cdot (a\nabla u) \in L^2(0,T;L^2(\Omega))$ on some bounded domain $\Omega$. Here $a\colon [0,T] \to \Omega$ is such that $a \...
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1answer
56 views

Fractional Sobolev norm of a radial function

Let $f(x):\mathbb{R}\to \mathbb{R}$ be a smooth function, with $\operatorname{supp}(f) \subset [0,1]$. Consider the radial function $F:\mathbb{R}^d\to \mathbb{R}$, defined by $F(x) := f(|x|)$. A ...
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0answers
30 views

How to show this Sobolev inequality?

I am currently studying numerical analysis and stumbled upon the following task: Prove that for any $q \in [2, +\infty)$, there exists $C \gt 0$ such that $\Vert f\Vert_{L^{q}(\mathbb{R}^2)} \le C\...
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1answer
14 views

Weak Convergence of sequence in a Sobolov Space.

Consider the question asked in here. I understood most of the answer in the question but the part about the weak convergence I did not get. To show that $u'_{n_k}$ converges weakly to $u$ in $L^p$, ...
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0answers
20 views

Norm Equivalence in Sobolev Space

I am attempting to demonstrate that for all $u \in W^{2,p}(U) \cap W^{1,p}_0(U)$ \begin{align*} \int_U |u|^p \ dx \leq C\int_U |Du|^p \ dx \end{align*} where $U$ is some open subset in $\mathbb{R}^n$....
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59 views

How is “continuous dependence on initial conditions” defined? And how to prove it?

EDIT: I tried to prove the continuous dependence of the problem by somehow use a weak Maximum principle and posted a modified Version of this post on mathoverflow: https://mathoverflow.net/questions/...
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18 views

Equivalence of Gradient and Hessian Norms in Sobolev Space

I am attempting to demonstrate that for $u \in W^{2,p}(U) \cap W^{1,p}_0(U)$ \begin{align*} \int_U |Du|^p \ dx \leq C\int_U |D^2u|^p \ dx \end{align*} where $U$ is some open subset of $\mathbb{R}^n$. ...
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18 views

understanding Hilbert Sobolev spaces

We denote the space $\dot{H}^k$ for the homogeneous Sobolev space and $H^k$ for the inhomogeneous Sobolev space where $H^k=W^{k,2}$. It is true that de dual of $\dot{H}^k$ can be identified with $H^k$...
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1answer
26 views

Sobolev imbedding theorem $H^{1,p}(\mathbb{R}^n)$ contained in $L^{{np}/(n-p)}(\mathbb{R}^n)$ (Taylor Michael)

i. I can not make sense of the following: For (2.4) I imagine that it is the fundamental theorem of the calculation but I can not prove it formally. Neither will it be understood how to arrive at ...
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15 views

Using Hille-Yosida to show existence of solution to differential equation.

Suppose $I=(0,l)$ is an interval and $v\in C(\overline{I})$. Consider the differential equation $$\begin{cases}u_t=u_{xx}+v(x)u_x,\\ u(0,t)=u(l,t)=0,\;\;\\u(x,0)=u_0.\end{cases}$$ where $u=u(x,t):I\...
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Sobolev spaces on domains and manifolds, what is the difference?

What are the (technical) differences between Sobolev spaces on domains $\Omega \subset \mathbb R^n$ or (compact) manifolds such as two-dimensional spheres?
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1answer
25 views

Are 2 Hilbertspaces with different inner products identical if the associated norms are equivalent?

suppose we have the Sobolev space $H^1_0(\Omega)$ over a bounded domain $\Omega \subset \mathbb{R}^2$. With the standard inner product it sure is a Hilbert space. BUT: What if we equip $H^1_0$ with ...
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24 views

Schauder regularity

My question is: If $\Omega$ is bounded with smooth boundary... Let $f \in C^{0,\alpha}(\overline{\Omega})$, and let $u$ be a weak solution of the Poisson equation $-\Delta u = f$. Then $u \in C^{2,\...
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1answer
50 views

$f(x)=\sin(x)\in H_{0}^{1}(\Omega)$ with $\Omega=(0,\pi)$?

We know that $f\in C^{\infty}(0,\pi)$ and $f(0)=f(\pi)=0$. But $supp\ f=[0,\pi]$. From the definition, $H_{0}^{1}(0,\pi)$ is the clusear of $C_{0}^{\infty}(0,\pi)$ in $H^{1}(0,\pi)$. Since, $f$ is not ...
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1answer
65 views

We could identity $H^{-1}$ with $H_0^1$ but we don't. Why?

Today in our lecture on partial differential equations while discussing dual spaces of Sobolev spaces: We could identify $H^{-1}$ with $H_0^{-1}$ by the $H_0^1$ inner product (Riesz) But won't and ...
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32 views

How to understand the definition of $W_0^{1,2}(\Omega)$

We defined $W_0^{1,2}(\Omega)$ as the closure of $C_c^{\infty}(\Omega)$.If $u \in W_0^{1,2}(\Omega)$ then u is compactly supported , but is it also right for |u| ?
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49 views

Density of smooth functions in the weighted Sobolev space $W^{1,p}((0,1),w)$, where $w(x)=x^{\lambda}$, $\lambda>0$

Let $1<p<\infty$ and $w(x):=x^{\lambda}$, where $\lambda>0$. One way to define the weighted Sobolev space $W^{1,p}((0,1),w)$ is the following (see [1]): we say that $u\in W^{1,p}((0,1),w)$ if ...
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1answer
43 views

Why is $f_\epsilon(u) \in W_0^{1,2}(\Omega)$?

For $\epsilon>0$ let $f_\epsilon(u)=\sqrt{\epsilon^2+u^2}-\epsilon$ One calculates that $\nabla f_\epsilon(u)=\frac{u}{\sqrt{\epsilon^2+u^2}}\nabla u $ , for $\epsilon$ to 0 this term goes to $\...
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1answer
81 views

Do invariant functions form a Banach (sub)manifold in function spaces?

Let $G$ be a topological group, and $X$ some function space; preferably a Sobolev space $X=W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is some invariant subset ($g\Omega \subset \Omega$) or ...
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0answers
40 views

$u \in W_0^{1,2}(\Omega) \Rightarrow |u| \in W_0^{1,2}(\Omega)$

For $\epsilon>0 $ define $g_\epsilon(u)=\sqrt{\epsilon^2+u^2}-\epsilon$. One finds $ \nabla g_\epsilon(u)=\frac{u}{\sqrt{\epsilon^2+u^2}}\nabla u$ and $ g_\epsilon(u)\in W_0^{1,2}(\Omega)$ . Then ...
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1answer
38 views

Deriving Variational Formulation

Determine the variational formulation of \begin{cases} -\Delta u+u=xy \quad& \text{in } \Omega\\ \nabla u\cdot \vec{n}+2u=3 \quad& \text{in } \partial\Omega \end{cases} What I have tried: \...
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38 views

Is the embedding of $W^{2,p}$ onto $C^1(\overline{I})$ compact?

We know that when $I$ is a bounded interval and $1<p\leq \infty$ that the injection $W^{1,p}\subset C(\overline{I})$ is compact. The proof of this fact uses the Arzela-Ascoli theorem on the unit ...
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1answer
28 views

Is every linear finite element space over a bounded domain a subspace of the sobolev space H^1?

Since my knowledge of functional analysis, $L^p$-, Sobolev- and Hilbert spaces is not very good, I thought I could ask... Suppose we have a domain $\Omega \subset \mathbb{R}^2$ which is continuously ...
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28 views

$||.||_{\infty}$ in sobolev space

Let $u_n ∈ W^{1,1} (I), I=(0,1)$ defined by: $u'_n (x) = n$ if $x < 1/n$ $u'_n (x) = 0$ if $x > 1/n$ $u_n (0) = 0$. Find $||u_n − 1||_∞$. My attempt: We have $u_n(x)=u_n(0)+\int_0^{1/n}ndt=...
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0answers
36 views

Estimates for Poisson's equation

I am studying Calderón-Zygmund estimates ($L^p$ estimates) for Poisson equation. What I already know is: Let $\Omega$ be a bounded domain, $f \in L^p(\Omega)$, $1<p<\infty$, and let $w$ be the ...
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1answer
29 views

The dual space of the Sobolev space $W^{k,p}(\Omega)$.

Let $\Omega$ be a nice domain in $\Bbb R^n$. It is known that any element $T\in\left( W^{k,p}(\Omega)\right)^*$ admits a (possibly non-unique) representation of the form $$ Tu = \sum_{|a|\le k} \int_\...
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0answers
22 views

Orlicz space is complete under the Luxemburg norm

I want to prove that Orlicz space is complete. Here is my attempt Let $(X,\mu)$ be a measurable space and suppose $\phi(x)$ is a young function. Define $$L^{\phi}=\{f\hspace{0.2cm} \text{measurable:}\...
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0answers
31 views

Question on proper choice of test function

Let $\Omega$ be a $\mathcal{C}^2-$compact manifold in $\mathbb R^2$ and consider the non-homogeneous heat equation: $$ \partial_t v - \Delta v = f \quad \text{in } \ \Omega \times (0,T), $$ where $f \...
2
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0answers
32 views

function is convex in Sobolev space

Let $u ∈ W^{1,1} (]0, 1[)$ and $$J_\varepsilon(u) =\frac{1}{2}\int_0^1(\varepsilon+x^\alpha)u'^2dx+\frac{1}{4}\int_0^1u^4 dx-\int_0^¹uf dx.$$ Prove $J(u)$ is convex. My argument is sum of 3 ...
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2answers
37 views

Sobolev embedding when $p=n$: $W^{1,p}(\mathbb{R}^{n}) \hookrightarrow L^{q}(\mathbb{R}^{n})$ for $q: p \leqslant q < \infty$

In class, aside from the standard Gagliardo-Nirenberg-Sobolev and Morrey inequalities, my professor also covered the case when $p=n$. In particular, if $p=n$, then $W^{1,p}(\mathbb{R}^{n}) \...
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1answer
38 views

Compact embedding of fractional space

Is the space $H^\lambda((0,T); H^1(K))$ for $0 <\lambda <1$ where $K$ is compact subset of $\mathbb{R}^n$ compactly embedded in $L^2( (0,T) \times K)$? $H^\lambda((0,T); H^1(K))\hookrightarrow \...