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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If $\nabla \cdot (|\nabla u|^{p-2}\nabla u) \in L^2$ what space is $u$ in?

Define $\Delta_p u = \nabla \cdot (|\nabla u|^{p-2}\nabla u)$. I want to know, if $\Delta_p u \in L^2(\Omega)$, then what space is $u$ in? I am having trouble figuring it out. Take $p=2$. Then ...
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37 views

Neumann eigenvalue problem for the Laplacian

Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$. Consider the Neumann eigenvalue problem $$ \Delta u =\lambda u \, \text{at} \, \Omega \\ \frac{\partial u}{\partial \vec{\nu}}=0 ...
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16 views

Standard examples of operator

In a text I am reading it says that we can consider an operator $A: X \rightarrow X^{*}$ (where $X := W^{1,p}(\Omega)$) which is defined as $$Au = -\text{div}(a(x,u,\nabla u))$$ where $a: \Omega ...
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12 views

anisotropic vector valued function definition

Does anyone have insight into the significance of the following statement which defines an anistropic function $$a:\Omega \times \mathbb{R} \times \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$$ The ...
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24 views

Existence of strong solutions to parabolic p-Laplace equation

Can I find a reference to where the existence of strong solution $u \in L^2(0,T;W^{1,p})$ with $u_t \in L^2(0,T;L^2)$ is proved to the equation $$u_t - \Delta_p u = f$$ $$u(0) = u_0$$ ...
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10 views

How do we define fractional Sobolev spaces on manifolds?

Fix $s > 1/2.$ The trace operator is surjective from $H^{s}(\mathbb{R}^{n+1})$ to $H^{s-\frac{1}{2}}(\mathbb{R}^{n}).$ If $\Omega$ is a bounded open set of $\mathbb{R}^{n}$ with smooth boundary, ...
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15 views

Something like a trace inequality of $H^1(\Omega)$

I have the following question: Let $\Sigma$ an a surface inside of an open domain $\Omega\subseteq\mathbb{R}^3$, where $\Sigma$ divides $\Omega$ in 2 open domains (for example: $\Sigma$ could be a ...
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8 views

div(curl(v))=0 on $D(\Omega)'$

it is easy to prove that for all $v\in D(\Omega)$ (smooth functions with compact support in $\Omega$: $C^\infty_0(\Omega)$): $div(curl(v))=0$ but, how can I prove this result for all $v\in ...
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20 views

Equivalence discrete H^2 Sobolev norms

My aim is showing the equivalence of two discrete Sobolev norms. On $\mathbb{Z}^d$, $d\ge 2$, one defines the discrete derivative in the direction of the coordinate vector $\vec e_j$ as $$ ...
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41 views

Sobolev spaces - a few questions about them.

I just read that somebody used the sobolev space $W^{1}_2(\partial B^n)$ which confused me cause I always thought that the domain of a sobolev space has to be open? Then the norm was written down as ...
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34 views

Gelfand triples for Product Spaces

For $V = H^1(\Omega)$ and $H=L^2(\Omega)$. If we identify H with it's dual space $H^*$, then we have the following relation: \begin{equation} V \subset H \subset V^* \end{equation} Does this also ...
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29 views

Composition of a compact support function with a increasing one

Let $u\in H^1(\mathbb{R}^n)$ have compact support and $c:\mathbb{R}\to\mathbb{R}$ is smooth, with $c(0)=0$ and $c'\geqslant 0.$ I am trying to prove $c(u(x))\in L^2(\mathbb{R}^n)$ or $c'(u(x))\in ...
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8 views

orthonormal basis or Parseval frame for Sobolev spaces

Consider the uni-variate Sobolev space of order $m$: $$ \mathcal{W}_2^m=\{ f:[0,1] \rightarrow \mathbb{R}|f,f^{(1)},\dots ,f^{(m-1)}\, \text{are absolutely continuous and}\, f^{(m)}\in L^2\}.$$ It is ...
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28 views

Proving a Sobolev-type ineqauality

Given $I=(0,1)$ and $u\in W^{2,p}(0,1)$ for $p>1$. I am trying to prove that for any $\epsilon>0$, the following hold: $$ \|u\|_{L^\infty(I)}+\|u'\|_{L^\infty(I)}\leq ...
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14 views

Sobolev spaces of maps between manifolds and the Palais-Smale Condition

I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts: Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean ...
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32 views

Another way to show $\nabla u =0$ a.e. on $u=0$ for $u\in H^1(\Omega)$

This is an exercise on Evans PDE book, Ch5. It provides another way to prove $\nabla u =0$ a.e. on $u=0$ for $u\in H^1(\Omega)$ then the one in Evans & Griapy's book. The statement is as ...
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20 views

Embedding of fractional Sobolev spaces

I have a question regarding fractional Sobolev spaces. Given an open bounded set $\Omega\subset \mathbb{R}^{N}$ (Lipschitz, for instance), $s\in (0,1)$ and $1\leq p<q<\infty$, does the following ...
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30 views

dense subspace of $C(0,T)$

I want to prove that the space H of functions which are continuous in [0,T] with weak derivative in $L^2[0,T]$ and their value in 0 is 0, is dense in the space of continuous functions in [0,T] with ...
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reference request about sobolev space and BV space

I am studying Sobolev Space and BV space by using Leoni's and Evans & Gariepy's book. I was wondering that where can I find some explicit example and some computational question of those space. A ...
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21 views

Question about Sobolev spaces. Controlling Divergence.

I have a question about Sobolev spaces, I think I just need a reference. For $\Omega$ an open and bounded subset of $\mathbb{R}^d$, and $\vec{\Phi}\colon \Omega \to \mathbb{R}^d$ a vector valued ...
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22 views

Find an operator $Z$ in $H^1(0, \infty)$ with $\langle u,Zv\rangle = \int \bar{u}v dx$

I'm working with operators associated to bilinear forms. What I need to find is a continous, linear operator $T$ defined on $H^1((0, \infty))$ [note that $H^1 = W^{1,2}$ is the Sobolev space] such ...
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20 views

A function in $W^{1,p}(U)$, $U=B^0(0,1)$

Let $\{r_k\}_{k=1}^{\infty}$ be a countable, dense subset of $U=B^0(0,1)$ and a given function by $$u(x) = \sum_{k=1}^{\infty}\frac{1}{2^k}|x - r_k|^{-\alpha},\,\, x \in U$$ For which values of ...
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14 views

finding the solution space of helmholtz equation with mixed boundary?

Let $\Omega\subset \mathbb{R}^2$ be a bounded set and $\Omega' \subset \Omega$ such that $\partial\Omega \cap \partial\Omega' = \emptyset$ $$ \left\{ \begin{align} k^2 u + \Delta u &= 0 \quad ...
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29 views

bounding supremum with Sobolev embedding theorem

I consider $d \geq 2$ and the domain $ D = [0,1]^d$. I consider a function $f : D \to \mathbb{R}$ that is $k$ times continuously differentiable. [$k$ can be specified as large as possible.] I am ...
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36 views

Traces of Sobolev functions in an unbounded domain

I have a doubt concerning the trace of Sobolev functions. Let $C=\Omega\times(0,\infty)$ an infinite cylinder of basis a smooth domain $\Omega$ of $\mathbb{R}^{N}$ and consider the classical Sobolev ...
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64 views

Characterization of $H^{-1}(U)$

I am trying to understand the proof of characterization of $H^{-1}(U)$ as in Evan's Partial Differential equations. My questions is different to that of this link. In the theorem regarding the same, ...
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25 views

Find $u:[0,T]\to H^2$ such that $u(0)=u_0\in H^2$ and $u_t(0)=u_1\in H^1$.

Let $u_0\in H^2$ and $u_1\in H^1$. If we define $$ \begin{align*}u:[0,T]&\longrightarrow L^2\\ t&\longmapsto u_0+\int_0^tu_1\;ds \end{align*}$$ then $u(0)=u_0$. Furthermore, the weak ...
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104 views

About the image of the trace operator for Sobolev spaces.

Let $\Omega\subset\mathbb{R}^N$ be a bounded convex domain. Once every convex function is locally Lipschitz, we have that $\partial\Omega$ is Lipschitz, therefore, the trace operator $T: ...
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36 views

Fractional order Sobolev space over manifold

I stuck with a problem: If $\Omega\subset\mathbb{R}^{n}$ is open and bounded with a sufficiently smooth boundary $\Gamma$. Then how to prove dual space of $H^{1/2}(\Gamma)$ is $H^{-1/2}(\Gamma)$ ? ...
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49 views

When can we interchange summation with $L^2$ inner product?

(This question concerns a step in the solution given to Eignvalues of Laplacian operator and Sobolev spaces.) Why can we interchange the sum and the $L^2$ inner product in the following? $$(\sum_n ...
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48 views

Question with tried Eigenvalues of Laplacian operator and Sobolev spaces III.

Let $\Omega$ be a bounded subset of $\mathbb R^d$ and let $g\in L^2(\Omega )$. Let $(\lambda_n)_{n\in \mathbb{N}}$ be the eigenvalues of the Laplacian operator $\Delta$, and $(e_n)_n$ the ...
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23 views

eignvalues of laplacian operator and Sobolev spaces -II

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, Let $F=(F_t) \in C^0(I,L^2(\Omega))$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od the ...
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42 views

Question about functions in Sobolev space.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. If I consider a function $g:\mathbb{R}\rightarrow\mathbb{R}$ which has the following properties: $$ |g(x)|\leq M \qquad |g(x)-g(y)|\leq ...
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28 views

Dual of a Sobolev space (ex)

Consider $f(x_1,x_2)=\chi_{B_1(0,0)}(x_1,x_2)$. 1) Is $\nabla f\in (H^1_0(\mathbb{R}^2;\mathbb{R}^2))^*$? 2) $\langle \nabla f , u \rangle= ?$ with $u\in H_0^1(\mathbb{R^2})$? For the first ...
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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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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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30 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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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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63 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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19 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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38 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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40 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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27 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 ...
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114 views

A different weak formulation for parabolic PDE problem (test function space $L^2(0,T;H^2(\Omega))$).

Consider the PDE $$u_t - \Delta u = f$$ $$u(0) = u_0$$. Instead of the usual weak form, let me take this one: for every $\varphi \in L^2(0,T;H^2)$, $$\int_0^T \langle u_t, \varphi \rangle - \int_0^T ...
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28 views

Minimizing the homogenuos Sobolev norm for a given trace

Suppose that $\Omega$ is a bounded domain with regular boundary (think $C^1$). We have a function $f_b:\partial\Omega\to\Bbb R$ and we can expand it to the whole $\Omega$ in the sense of ...
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33 views

inequality for linear functions in sobolev space

Ist the following Statement true for $f$ and $g$ linear? $\vert fg \vert_{H^2} \leq C \Vert f \Vert_{H^1} \Vert g \Vert_{H^1}$, where $\vert \cdot \vert_{H^2}$ denotes the seminorm. My Idea: It is ...
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64 views

Sobolev spaces in polar coordinates

I need some properties about Sobolev spaces in polar coordinates. To be precise, let $U = \{(x,y)\in\mathbb R^2 : x^2 + y^2 < R\}$ be an open disc and let $H_0^1(U)$ be the usual Sobolev space ...
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25 views

Regularising a function that is constant on an interval (related to Heaviside)

Define the function $f:\mathbb R \to \mathbb R$ by $$f(x) = \begin{cases} x &\text{for $x < 0$}\\ 0 &\text{for $x \in [0,1]$}\\ x-1 &\text{for $x > 1$}& \end{cases} $$ Note that ...
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25 views

Hydrogenhamiltonian self-adjoint in one or two dimensions

let $d\in\{1,2\}$. I'd like to know if the operator $H=-\Delta - \frac{1}{|x|}$ is self-adjoint as an operator acting on a dense subset of $L^2(\mathbb R^d)$. In particular I'd like to know how its ...
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36 views

Simple question about the Gagliardo Niremberg interpolation inequality

Consider the Gagliardo Niremberg interpolation inequality : (Gagliardo Niremberg interpolation inequality)Let $q,r$ be any numbers satisfying $1 \leq q, r \leq \infty $ and let $j,m$ be any ...