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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46 views

How can I prove $\mathcal S$ is dense in $W^{s,2}$?

Let $\mathcal S (\Bbb R^n)$ be the Schwartz class and $W^{s,2}(\Bbb R^n)$ be the Sobolev space($s=0,1,\cdots$). In fact I know that $C_c^\infty(\Bbb R^n)$ is dense in $W^{s,2} (\Bbb R^n)$ and ...
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46 views

How to show that $u(t,x)\in C([0,T],H^{1}(\mathbb{R}^{n}))$?

If $u(t,x)\in L^{2}([0,T],H^{2}(\mathbb{R}^{n}))$, $\partial_{t}u \in L^{2}([0,T],L^{2}(\mathbb{R}^{n}))$, prove that $$ u(t,x)\in C([0,T],H^{1}(\mathbb{R}^{n})) $$
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34 views

Density of finite element functions in $W^{1,p}(\Omega)$

I would like to know if the following statement is true: For each $u \in W^{1,p}(\Omega)$ and $\varepsilon > 0$ there exists a piecewise affine function $u_{\varepsilon}$ and a triangulation of ...
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35 views

Minimum is attained in a subset of a Sobolev space

Let $\Omega \subset \mathbb R^n$. I have a functional of the form, $$\int_{\Omega}f(x,u,\nabla u)dx$$ where $u \in W^{1,p}(\Omega, M)$, $M \subset \mathbb R^d$ is a compact, smooth Riemannian ...
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223 views

A continuous embedding.

If $2 \le q \le \infty$ and $2s >n$, then is there a continuous embedding $ H^s (\Bbb R^n ) \hookrightarrow L^q(\Bbb R^n) $ ? Here $H^s$ means a general Sobolev space for $s = 0,1,\cdots$.
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46 views

One of conjectures of De Giorgi

conjecture: If $\exp(tw), \exp(tw^{-1}) \in L^1_{\operatorname{loc}}$ for each $t > 0$ then $w$ is regular weight. $w$ is regular if weighted Sobolev space $W^l_p(\Omega,w)$ is equal to the ...
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152 views

A weak convergence in Sobolev space.

Let $u \in C^0([0,T], H^{s-1}(\Bbb R^n)). $ Let $\{t_n \} \subset [0,T]$ such that $\lim_{n \to \infty} t_n = t_0$. Let $ u(t_n ) \to u(t_0) $ in the Sobolev space $H^{s-1} ( \Bbb R^n )$ for $s = ...
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67 views

Proving $ f \in C_b^1 ( [0,\infty) \times \Bbb R^n) $ by using the Sobolev inequality.

Let $s > 1 + n/2$ for $n \in \Bbb N$, and $s$ be an integer. If $f \in C^0 ( [0,\infty), W^{s,2} (\Bbb R^n )) \cap C^1 ( [0, \infty) , W^{s-1,2}(\Bbb R^n)) $ then how can I show that $$ f \in C_b^1 ...
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68 views

details in a proof

I saw this statements in a proof, but I would like to see the details. Let $U \subset \mathbb{R}^{n}$ and $u$ sperharmonic function in the sense that \begin{equation} \int_{D} \langle \nabla u(x) ...
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146 views

Is $C_0^\infty $ dense in $W^{m,p}$?

Is $C_0^\infty $ dense in $W^{m,p}$? Here $C_0^\infty$ = $C_c ^ \infty$ : $C^\infty$ with compact supports, and $W^{m,p}$ : Sobolev spaces.
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169 views

A question about weak lower semicontinuity

Let $\Omega \subset \mathbb{R^{n}}$ be a bounded domain and $u , u_j \in H^{1}(\Omega)$ such that $u_j \rightharpoonup u$ in $H^{1}(\Omega)$ $$ F_{1}(u) = \int_{\{ u > 0\}} \dfrac{1}{2}\langle A_1 ...
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209 views

Application of method of continuity in partial differential equations

Consider a differential operator $$L_t:= (1-t)(\Delta-\lambda) + t L,\qquad t\in[0,1].$$ For any $u\in C^2_0(\mathbb{R}^2)$, we have $$\lambda^2 \|u\|_2^2 + 2\lambda\sum_{i}\|u_i\|_2^2 + ...
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99 views

Eigenfunction associated to the smallest eigenvalue of an elliptic operator

Let $(T_n)$ be a sequence of elliptic operators defined in $H^2(\Omega)\cap H_0^1(\Omega)$ to $L^2(\Omega)$, with $\Omega$ being a bounded domain with smooth boundary. All of them have a smallest ...
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65 views

Show properties of elements of $\mathcal{H}^2$

I have problems with my homework. First of all: $\mathcal{H}^2$ means the Sobolev space, right? Because in the script we are using a $W$ for Sobolev spaces and the $\mathcal{H}^2$ can't be found ...
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32 views

$-\Delta u = f$ in $L^2(0,T;H^{-1}(\Omega))$ (as opposed to $H^{-1}(\Omega)$)

Why does nobody consider the equation $-\Delta u = f$ in the space $L^2(0,T;H^{-1}(\Omega))$? Eg. given $f \in L^2(0,T;L^2(\Omega))$ find a solution $u \in L^2(0,T;H^1_0(\Omega))$ such that ...
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54 views

Product rule in fractional Sobolev spaces

I have to prove the following inequality $$\Vert fg\Vert_{H^s(\mathbb{R}^3)}\leq C \Vert f\Vert_{H^\sigma(\mathbb{R}^3)}\Vert g\Vert_{H^{s+1}(\mathbb{R}^3)}$$ with $s\geq 0$, ...
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35 views

$\sup$ of a $C^s$ smooth function.

I want to prove that for a function $F\in C^k(\mathbb{R}^n)$ which vanishes at zero, and a function $u\in H^k(\mathbb{R}^n)$we get: $$\left\| \int_{r=0}^1 F'(ru)(\cdot)dr \right\|_{L^{\infty}} \leq ...
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25 views

This Sobolev function is continuous?

Let $\Omega \subset R^n $ $(n \geq 2) $ an open bounded domain with smooth boundary $u \in W^{ 1,p}(\Omega)\cap L^{\infty}(\Omega)$ ($p \geq 2$ fixed). Suppose that exist $M > 0$ such that $$ ...
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39 views

Proof of weak derivatives in Evans PDE?

In the textbook of Partial differential equation of Evans. Why from $\int_U(v-\overline v)\phi dx=0$ for all $\phi \in C_c^\infty (U)$, we can get $v-\overline v=0$ a.e.? How to prove it? ...
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30 views

Can we find $Y$ such that $Y^\star=W^{1,p}(\Omega)$?

Let $\Omega\subset \mathbb{R}^N$ be an open set and $p\in [1,\infty)$, $1/p+1/p'=1$. If we define $T:W^{1,p}(\Omega)\to L^p(\Omega)^{N+1}$ by $Tu=(u,\nabla u)$ then, $T$ can be viewed as an isometry. ...
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19 views

Local $H^1$ and continuity on borders implies global $H^1$

Let $\Omega_1,\Omega_2\subset\mathbb{R}^n$ open and disjoint sets with $F:=\overline\Omega_1\cap\overline\Omega_2\neq\emptyset$ (for example, $\overline\Omega_1$ and $\overline\Omega_2$ be two ...
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16 views

A function that is in $H^s$, what can be said on its boundedness?

assume we have a function $u\in H^s$, what can I say on its $\| u \|_{L^{\infty}}$? Where a function is in $H^s$ iff $ \|(1+|y|^s)\hat{u} \|_{L^2} < C < \infty$, where $\hat{u}$ is its Fourier ...
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27 views

chain rule for weak derivatives

First I apologise for not writing properly, but I'm using a cell phone. We know that having a Sobolev function $u$, if we have a good enough function $f$, then $f\circ u$ is Sobolev and the chain ...
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35 views

Basis of $H^1(\Omega)$ which is orthonormal wrt. $L^2(\partial\Omega)$ inner product?

Let $\Omega$ be a domain with $\partial\Omega$ bounded. Is it possible to find a smooth basis of $H^1(\Omega)$ and $L^2(\Omega)$ which is orthonormal wrt. the $L^2(\partial\Omega)$ inner product? ...
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24 views

Second order equation and regularity

Let $U$ be an bounded open set in $\mathbb R^3$ and ${V}\subset \subset{W} \subset \subset {U}$. Let $u \in {H^1}(U)$, and $f\in L^{2}(U)$ satisfies $$\int_{\mathbb R^3} \nabla u(x). \nabla \varphi ...
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36 views

Integration by parts in Sobolev space

I'm looking for a reference of the following fact (if it is true...): if $u\in W^{1,1}(\Omega)$ and $v \in W^{1,\infty}(\Omega)$ ($\Omega$ a open subset of $\mathbb{R}^n$ ($n \ge 1$) with a regular ...
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14 views

Verification and presentation of anisotropic sobolev space results

Hi I am interested anisotropic Sobolev spaces. Can someone with knowledge of this topic check if the following is correct in presentation. I am finding it hard to find a good book which deals with the ...
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20 views

Weak time derivative for functions $u \in L^2(0,T;L^2(\Omega))$

The weak time derivative of a function $u \in L^2(0,T;H^1)$ is defined to be $u' \in L^2(0,T;H^{-1})$ satisfying $$\int_0^T \int_\Omega u(t) \varphi'(t) = -\int_0^T \langle u'(t), \varphi(t) \rangle$$ ...
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25 views

Besov norm in $W^{1,2}(\mathbb{R}^n)$

A well known result on Besov spaces is that $\Lambda_1^{2,2}(\mathbb{R}^n)=W^{1,2}(\mathbb{R}^n)$. One way to define this Besov space (without Fourier transform) is to consider $$ ...
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23 views

Norm of linear functional in $W^{1,2}$

I want to find the norm of the following linear functional: $\phi:W^{1,2}_{[0,1]} \rightarrow\mathbb{R}$ $\phi(f)=\int^1_0t\cdot f(t)+ f^{'}(t) dt $ The norm is: ...
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14 views

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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40 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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18 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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16 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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26 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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12 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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18 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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11 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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22 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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35 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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30 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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10 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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32 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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20 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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40 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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22 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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34 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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16 views

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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22 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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0answers
23 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 ...