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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2
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0answers
403 views

Showing that smoothing operators are compact

Suppose I have a bounded, linear map $T: H^1(X) \to H^1(X)$ such that $T(H^1(X)) \subset C^\infty(X)$. Is $T$ a compact operator? I'm guessing this depends on whether or not $X$ is (pre)compact, and ...
2
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523 views

Stampacchia Theorem: $\nabla G(u)=G'(u)\nabla u$?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $G:\mathbb{R}\to\mathbb{R}$ a Lipschitz function with $G(0)=0$. Stampacchia's Theorem states that if $u\in W_0^{1,p}(\Omega)$, then $G(u)\in ...
2
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105 views

$H^s(\mathbb{T})$ space is a Banach algebra with pointwise product

I have ran across the following theorem but the given proof does not convince me. Theorem Let $u, v \in H^s(\mathbb{T})$ with $s>1/2$. Then the pointwise product $uv$ is in $H^s(\mathbb{T})$ and ...
2
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43 views

“Compensated weak compactness” in $W^{1,1}$

In a problem I'm working on I want to show that a bounded sequence $\{u_n\} \subset W_0^{1,1}((0,1))$ converges weakly. Of course, since $W^{1,1}$ is not reflexive I don't get this for free. The ...
2
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99 views

A question on weakly convergence and norm convergence.

Let $2 \le p<\frac{2n}{n-2}$. Suppose that a sequence $\{u_k\}_k\subset H^1(\mathbb{R}^n)$ weakly converges to $u \in H^1(\mathbb{R}^n)$, and hence weakly converges to $u$ in $L^p(\mathbb{R}^n)$. ...
2
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0answers
125 views

A question on a bounded sequence in $H^1(\mathbb{R}^n)$.

Let $r>0$, and $2 \le q \le 2^*$. Suppose that $\{u_k\}_k$ is a bounded sequence in $H^1(\mathbb{R}^n)$ and $\lim_{k\to \infty} \sup_{y \in \mathbb{R}^n} \int_{B_r(y)}|u_k|^q dx \rightarrow 0.$ ...
2
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0answers
100 views

Sobolev spaces in $\mathbb R^n$ with functions having support on a closed set

I am interested in $H^s$ Sobolev spaces in $\mathbb R^n$ which have functions with support in a given closed set $K$ , denoted by $H^s_K$. Here $K$ is the complement of some bounded open set $\Omega$. ...
2
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0answers
105 views

Limit in norm of a Sobolev space

I consider, for $s>\frac{1}{2}$, the space $L^{2,-s}(\mathbb{R}^3)=\bigg\{f: \int_{\mathbb{R}^3}|f(x)|^2(1+|x|^2)^{-s}<\infty\bigg\}$ and I have to show that the function ...
2
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79 views

How to find a basis of $H^2(-N,N)$?

I need to find an orthonormal basis of $H^2(-N,N)$ where $N \in \mathbb{N}$ and $H^2$ denotes the Sobolev space $W^2_2$. I have no idea how to start. A hint to some literatur would be perfect.
2
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181 views

Why is this function space dense in this Bochner-type space? (Rogers and Renardy)

Let $\phi(t) = \sum_{i=1}^N\beta_i(t)\phi_i(x)$, where $\phi_i$ is basis for space $V$, and $\beta_i \in C_c^\infty(0,T).$ Renardy and Rogers says that 1) Functions of the above form are dense in ...
2
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71 views

Inequality involving Bessel potential.

I'm not able to prove the following inequality: Fix $s>0$ $$\|fg\|_{H^s}\lesssim \|fJ^sg\|_{L^2}+\|gJ^sf\|_{L^2},$$ where $\widehat{J^sf}(\xi)=(1+|\xi|^2)^{s/2}\hat{f}(\xi)$ (Bessel potential). ...
2
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0answers
78 views

Bound for a Sobolev function in an integral

For a compact bounded set $\Omega$, for the expression $$\int_\Omega \Delta u (\nabla u \cdot \nabla f)$$ where $u \in H^2$ and $f \in C^\infty$, is it possible to show that the expression is $\geq$ ...
2
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106 views

Convergence of Schwartz kernels implies convergence of operators

Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) ...
2
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0answers
49 views

Unusual Compact Embeddings

Can anybody give a reference to the following two facts? The embeddings $$H_0^{1,2}(\mathbb R^n)\to L^2(\partial B_1(0))$$ and $$H^{1,2}(\partial B_1(0))\to H^{1/2,2}(\partial B_1(0))$$ are compact? ...
2
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98 views

Weak derivative and homeomorphisms commute

Suppose $\phi:H^1(S) \to H^2(R)$ is a linear homeomorphism, where $S$ and $R$ are compact surfaces in $\mathbb{R}^n$. Let $\varphi \in D(0,T;H^1(S))$ be a $H^1(S)$-valued $C_c^\infty(0,T)$ function. ...
2
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0answers
157 views

Sobolev-type inequality.

Let $0<\alpha<n$, $1<p<q<\infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$\left\|\int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha}} \right\|_{L^q(\mathbb{R}^n)} \leq ...
2
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73 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 ...
2
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83 views

ODE with irregular coefficient

Here's a simple ODE \begin{align} &\frac{d}{dx}h(x)=a(h(x))\\ &h(0)=x_0 \end{align} I want the solution $h(x)$ to be (at least) continuous with its first and second order derivative exist only ...
2
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0answers
195 views

An exercise about Sobolev Spaces

Let $\Omega\subset\mathbb{R}^n$ be an limited open set of class $C^1$ and $1\leq p<\infty$. Show that $$\bigcap_{m=0}^{\infty}W^{m,p}(\Omega)=C^{\infty}(\overline{\Omega}).$$
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132 views

How can we glue Sobolev functions?

Let $u:A\cup B\to R$ be a function (where $A$ and $B$ are disjoint connected sets and $A\cup B$ is connected) such that $u$ restrict to $A$ and to $B$ are in $W^{1, p}$. Which result guarantees me ...
2
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356 views

Weak convergence and limit of this sequence

Let $f_n$ be bounded uniformly in the $H^1$ norm, so we have (weak convergence) $$f_n \rightharpoonup f \qquad \text{in} \qquad H^1(\Omega\times [0,T]).$$ Then by compact embedding, we have strong ...
2
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0answers
100 views

Compactness of an embedding between weighted spaces

I read somewhere that if: $N\geq 2$ is an integer, $p\in ]1,N[$, $r>N/p$, $m\in L^r(0,a)$ (with $a>0$) and $m>0$ a.e. in $(0,a)$, then the weighted Sobolev space $W^{1,p^\prime} ...
2
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0answers
89 views

Extension of Uncertainty Relations to a specific potential in Schrödinger Equation

Given some $\|\psi \|$ $\in$ $L^2 (\mathbb R^n) $ such that $\| \psi \|_2 =1$ and a function (potential) $V: \mathbb R^n \rightarrow \mathbb R$. The Schorödinger equation tells us that $-\triangle ...
2
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0answers
316 views

Adjoint of multiplication operator on Sobolev space

This is sort of an idle question, and I'll admit I didn't think very hard about it. Let $H^1 = H^1(\mathbb{R}^n)$ be the Sobolev space with norm $||f||_{H^1}^2 = ||f||_{L^2(\mathbb{R}^n)}^2 + ...
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0answers
12 views

Galerkin method in Sobolev space

I've got this problem to solve: Using Galerkin method, prove that there exists a weak solution of this differential equation: $$-\Delta u = a(x) \circ \triangledown u - u_t +f(x)$$ on $\Omega$ $$u ...
1
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0answers
31 views

When is it true that the Sobolev trace of a positive a.e. function is positive a.e?

Let $u \in H^1(\Omega)$ on a bounded smooth domain $\Omega$. Is it true that if $u \geq 0$ a.e., then $Tu \geq 0$ a.e. on $\partial\Omega$ where $T$ is the trace? I don't think it is, since $u$ can ...
1
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0answers
19 views

Convergence of the series $\sum_{k=1}^{\infty} \frac{1}{2^k} |x-y_k|^{-\alpha}$

Suppose $S=\{y_k\}_{k=1}^{\infty}$ is a countable dense subset of the open unit ball $B(0,1)$ in $\mathbb{R}^n$. We write \begin{equation} u(x) = \sum_{k=1}^{\infty} \frac{1}{2^k} |x-y_k|^{-\alpha} ...
1
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0answers
27 views

Chain rule in $\mathbb{R}^d$, with $d\ge 2$.

Given $\Omega\subset\mathbb{R}^d$ be an open bounded set with Lipschitz boundary, let $v\in (H^1(\Omega))^d$, $\psi\in H^1(\Omega)$ $T_K(x):=B_K x+b_K$, where $B_K$ is a non-singular invertible ...
1
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0answers
16 views

weak formulation of $u''=\psi'(u)+f$ with $ u\in W^{1,2}_0((a,b))$.

Let $u\in W^{1,2}_0((a,b))$, $(a,b)=I$ and $\psi'$ the derivative of a convex function $\Psi\in C^1(\mathbb{R})$. Consider $f\in L^2(I)$ and the differential equation $$u''=\psi'(u)+f.$$ I want to ...
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0answers
26 views

Equivalence of norms on $W^{1, p}(I)$

Let $I=(a, b)$, $a<y<b$ an arbitrary point and $J\subseteq I$ an open interval. Then the norms $\left\Vert \cdot \right\Vert_{\sharp}$, $\left\Vert \cdot \right\Vert_{\flat}$ on the Sobolev ...
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0answers
25 views

Confusion about the definition of Sobolev spaces on manifolds

Let $(M,g)$ be a manifold with metric $g$ parametrized by the mapping $S$ and parametric domain $\Omega$. The sobolev space of order one with respect to the $L_2(M)$-norm $H^1_2(M)$ is defined as ...
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0answers
41 views

positive and negative parts of a function?

I have this question that I found in a demonstration of a theorem: Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and $u^+,u^-$ the positive and negative parts of $u$ respectively. Why do ...
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0answers
30 views

Variational formulation curl-div equation

I want to prove that the following problem admits a unique solution $A_I\in H(curl;\Omega_I)\cap H(div;\Omega_I)$. $$ \begin{cases} curl(\varepsilon_I^{-1}curl A_I)=curl v_I\;\;\text{in}\,\,\Omega_I\\ ...
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0answers
32 views

Derivatives of mollified functions

I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following: Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
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0answers
23 views

Zero extension of a $W^{2,\infty}$ Sobolev function outside its domain

Let $O$ be a non empty open subset of a bounded open set $\Omega\subset \mathbb{R}^n$ and let $f\in L^2(\Omega)\cap W^{2,\infty}(O)$. Let $u: \Omega \to \mathbb{R}$ be a function such that the ...
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0answers
22 views

A limit about measure set

Assume $\Psi\in W^{1,\frac{3}{2}}_{loc}(\mathbb{R^2})$, satisfies: $$\lim\limits_{r\to 0}\frac{1}{r}\int_{B_r(x)}|\nabla\Psi(y)|^{3/2}dy=0$$ Then for each $(x_0,y_0)$, and for every $\epsilon>0$ ...
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0answers
17 views

Sum of normed function spaces equipped with two-parameter family of norms, Dual inequality

Let $(\cdot,\cdot)$ be the $L^2(\Omega)$ scalar product, and let $V=L^1(\Omega)$, $W=H^{-1}(\Omega)$ (the dual space of $H_0^1(\Omega)$). My question is if there exists a constant $C$, such that for ...
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0answers
28 views

Integrability, Sobolev space

Hello I have a question about Sobolev spaces. Let $n \in \mathbb{N}$, $\alpha >\frac{n}{2}$, $c :\mathbb{R}^{n} \to \mathbb{R}$ be a $\alpha$-integrable function. I want to show the following: ...
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0answers
21 views

Sobolev inequality on $\mathbb{R}^3$

I know that for $u \in H^q(\mathbb{R}^d)$ with $d>q$ we have for $p = \frac{qd}{d-q}$ that $\|u\|_p \le C \|u\|_{H^q}.$ Now, I have somewhere back in my mind that it is also in the unbounded case ...
1
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0answers
34 views

Weak differentiability and diffeomorphisms

Let $U,V\subset\mathbb{R}^n$ be open sets and assume the existence of a $\mathcal{C}^1$-diffeomorphism $\phi:U\rightarrow V$. Let $u\in W^{1,p}(U)$, $1\leq p\leq\infty$, and define ...
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0answers
37 views

Question about the interpret of Picone inequality for non-regular functions.

Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open. There is a well-known picone identity that says Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...
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0answers
30 views

Find the infimum value of a functional.

Consider the functional $$F(u)=\int_{0}^{1}x^{\alpha}|u'(x)|^pdx,\ \ \ u\in W^{1,p}(0,1)$$ where $\alpha\ge 0$ and $1<p<\infty$. Given $a<b$, find the value of $$\inf\{F(u): u\in ...
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0answers
22 views

Understanding the embedding of $W^{\infty, 2}$

I am trying to understand Sobolev's embedding theorem, more precisely to understand when a Sobolev generalized function of infinite order is smooth of some order. Consider the following statement ...
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0answers
23 views

Bessel potential action on a product of function

Let's consider the Bessel potential $$J^s:=(I-\Delta)^{\frac{s}{2}}$$ Does it exist some kind of Leibniz rule for $$J^s(fg)$$?
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0answers
25 views

Can one explicitely construct a sequence of functions of compact support approximating $u\in W_{0}^{1,p}(\Omega)$?

We define $W^{1,p}_{0}(\Omega)$ as the closure of $C_c^{\infty}(\Omega)$ in the $W^{1,p}$-norm (or equivalently as the closure of the $W^{1,p}$-functions with compact support). Given $u\in ...
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0answers
48 views

A question on vanishing viscosity in Evans PDE

In Evans book on PDE (2nd edition) he uses the vanishing viscosity method to prove existence of a weak solution to the following linear hyperbolic system ...
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0answers
40 views

Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in ...
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0answers
33 views

Under some regularity assumptions to the boundary $\partial\Omega$, the first weak eigenfunction of $-\Delta$ in $\Omega$ is also a strong one

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $H:=W_0^{1,2}(\Omega)$ be the Sobolev space, $C^{2,\alpha}(\Omega)$ be the Hölder space for some $\alpha\in (0,1]$ and ...
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0answers
23 views

The relation of the Homogeneous Sobolev norm and general Sobolev norm

I'm wondering if the inequality $$ \left\| F\right\|_{\dot H^k(\mathbb R^n)} \le C\left\| f\right\|_{L^\infty(\mathbb R^n)} \left\| f\right\|_{\dot H^k(\mathbb R^n)} $$ holds for $k\in[0,10]$ then $$ ...
1
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0answers
61 views

How to apply Sobolev inequalities?

I'm struggling with an application of Sobolev inequalities in Evans' book. He presents his argument like this: For $4<p<5$ we have $2(p-1)=2(p-4)+6=2(p-4)+ 2^*$ and therefore $$\left( ...