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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1answer
47 views

Can we conclude that $v_{n}\rightarrow v$ in $L^{\infty}\left(\Omega\right)$ if $p>N$

Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain, $v_{n}\rightharpoonup v$ in $W_{0}^{1,p}\left(\Omega\right)$ , $\left\Vert v_{n}\right\Vert _{W_{0}^{1,p}}=1$ $\forall n$ . So we ...
0
votes
2answers
65 views

is a convex continuous function absolutely continuous

Does a continuous convex function $\mathbb{R} \to \mathbb{R}$ belong to $W^{1,1}_{loc}$ ? thank you.
1
vote
1answer
210 views

Does weak convergence in $W^{1,p}$ imply strong convergence in $L^q$?

Does weak convergence in $W^{1,p}(\Omega)$ imply strong convergence in $L^q(\Omega)$ when $\Omega$ is bounded? If $f_j$ converges weakly to $f$ in $W^{1,p}$, what can we say about the $L^q$ ...
2
votes
1answer
148 views

Related question to : If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$

This problem was asked in here. I need to ask something. Provide details for the following alternative proof that if $u \in H^1(U)$, then $$Du = 0 \text{ a.e. } \, \text{ on the set } \{u=0\}. $$ ...
9
votes
0answers
216 views

prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
2
votes
1answer
80 views

Showing Sobolev space $W^{1,2}$ is a Hilbert space

I have the Sobolev space $W^{1,2}$ consisting of all continuous functions $f \in L^2(\mathbb{R})$ such that there exists an $f'$ with $f(b) - f(a) = \int_a ^b f'(t) dt$. $W^{1,2}$ has inner product ...
1
vote
1answer
112 views

Extension of bounded operators between norm spaces

Let $X$ and $Y$ be two Banach Spaces and $X_1$ be a subspace of $X$. If $T$ is a bounded linear operator from $X_1\to Y$, then, this is an extension of $T$ from $X\to Y$ such that $\|T\|_{X}\le ...
5
votes
1answer
121 views

Regularity theorem for PDE: $-\Delta u \in C(\overline{\Omega})$ implies $u\in C^1(\overline{\Omega})$?

I stumbled over this question in the context of PDE theory: Let $U$ be connected,open and bounded in $\mathbb{R}^n$ and $u \in C^0(\overline{U}) \cap C^2(U)$ and $\Delta u \in C^0(\overline{U})$ with ...
1
vote
2answers
110 views

Example 3, Sobolev space Evans

In the following example p.260 Evans. I think I understand everything except for one calculus fact in the second last equation: $$ \int_{\partial ...
0
votes
0answers
105 views

Elliptic W^{2,p}-estimates for a Neumann problem.

Consider the simplest elliptic-Neumann problem in $\Omega\subset \mathbb{R}^n$: $$ -\Delta u+u=f\quad \text{in } \Omega, \quad \frac{\partial u}{\partial \nu}=0\quad \text{on } \partial \Omega. ...
2
votes
2answers
166 views

Definition of Sobolev spaces: Fourier transform of tempered distribution

I consider in "McLean - Strongly Elliptic Systems and Boundary Integral Equations" the definition of the Sobolev space for $s \in \mathbb R$ $$ H^s(\mathbb R^n) := \{u \in \mathcal S^*(\mathbb R^n) ...
2
votes
1answer
58 views

Construction of a function $u$ such that $u \in W^{2,2}(\Omega) \cap W_0^{1,2}(\Omega)$ and $u \not\in W_0^{2,2}(\Omega)$

I'm wondering about an example of a function $u \in W^{2,2}(\Omega) \cap W_0^{1,2}(\Omega)$ such that $u \not\in W_0^{2,2}(\Omega)$. Clearly $W_0^{2,2}(\Omega) \subset W^{2,2}(\Omega) \cap ...
3
votes
1answer
26 views

If $u \in H^{\frac 12}(\Omega)$ and $c \in \mathbb{R}$ is $(u-c)^+ \in H^{\frac 12}(\Omega)$ too?

If $u \in H^{\frac 12}(\Omega)$ and $c$ is a constant, is the function $$(u-c)^+ \in H^{\frac 12}(\Omega)?$$ Here $(x)^+$ is $x$ when $x > 0$ and $0$ otherwise. If it were $H^1$ then it is a true ...
0
votes
1answer
47 views

Sobolev/Lebesgue norm estimates in $\mathbb{R}^3$

I'm currently working on a project in which I have to establish some estimates for some global Sobolev and Lebesgue norms. We know that if we have a bounded domain $\Omega$, then for any $q \leq p^*$ ...
4
votes
1answer
70 views

How would I show a functional is linear and bounded?

Using the following results, for any $f \in H^1(a,b)$, $f$ is continuous on $[a,b]$, and therefore, $$ \int_a^b f(x) dx = f(\zeta)$$ for some $\zeta \in (a,b)$. In addition $$f(c) = f(\zeta) + ...
0
votes
1answer
43 views

Lebesgue integration by parts in Sobolev space $W^{1,2}(\mathbb{R})$

Let $\phi, \psi \in W^{1,2}(\mathbb{R}) \subset L^2(\mathbb{R})$ and we want to integrate by parts the following piece: $$\int_{\mathbb{R}}\phi(x)\psi'(x)dx$$ Supposedly, it should look like this: ...
1
vote
1answer
45 views

logarithmic sobolev inequalities

I'm reading a paper about PDE in fluid dynamics, where it used something called logarithmic sobolev inequalities: $$||\nabla u||_{\infty}\leq C||\omega||_{\infty} (1+\ln ||u||_m)$$ Where ...
0
votes
1answer
45 views

How does $\inf_{c \in \mathbb{R}} \lVert u - c \rVert_{L^2} \le \lVert \nabla u \rVert_{L^2}$ imply this inequality?

Let $M$ be a compact Riemann manifold with boundary. I want to know, given the inequalities $$ \vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2(M)} + \lVert u ...
1
vote
1answer
47 views

Canonical projection of $W^{1,p}(\mathbb{R}^N)$ onto $W_0^{1,p}(\Omega)$

Suppose we have a bounded domain $\Omega \subset \mathbb{R}^N$ with sufficiently smooth boundary $\partial \Omega$. The Sobolev spaces $W^{1,p}(\mathbb{R}^N)$ and $W_0^{1,p}(\Omega)$ are defined as ...
2
votes
1answer
63 views

Confusion about inclusions of dual spaces

We have the triple inclusions $H^1_0(\Omega)\subset L^2(\Omega)\subset H^{-1}(\Omega)$, where the second inclusion is not literal but in the sense of distributions. Related to this answer, why is the ...
0
votes
1answer
38 views

What is the norm of $H^4(0, 1) \cap H_0^2(0, 1)$?

Let $I = (0, 1)$ and $H_0^2(I) = \{u \in L^2(I) : u', u'' \in L^2(I), u = u' = 0 \;\; \text{on} \;\; \partial I\}$. What is the norm of $$H^4(I) \cap H_0^2(I)?$$ $\|u\|_{H^4(I)} = \Big[ ...
1
vote
1answer
52 views

Introducing an operator by a bilinear form

Let $I = (0, 1)$ and $H_0^2 (I)$ the closure of $C_c^\infty (I)$ in $W^{2, 2} (I)$. Consider $$a : H_0^2 (I) \times H_0^2 (I) \to \mathbb{R}$$ defined by $$a(u, v) = \int\limits_I u''(x) v''(x) \, ...
0
votes
1answer
62 views

What is the dual of $H^{-1}(\Omega)$?

The dual of $H^1_0(\Omega)$ is defined to $H^{-1}(\Omega)$. But what is the dual of $H^{-1}(\Omega)$? Is it $H^1_{0}(\Omega)$? I am solving a problem which requires me to use the dual of ...
0
votes
1answer
134 views

Holder continuous functions embedded in Sobolev

For simplicity, I will consider $u\in W^{1,p}(\Omega)$, where $1<p<\infty$ and $\Omega\subset\mathbb{R}$ is open and bounded. I am able to show that \begin{equation} |u(x)-u(y)| \leq ...
3
votes
0answers
37 views

Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles.

Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...
1
vote
1answer
44 views

Fourier transform on fractional Sobolev spaces

We say that a tempered distribution $f$ satisfies $f \in H^s(\mathbb R)$ for some $s \in \mathbb R$ if $(1+|\xi|^2)^{s/2} \hat f \in L^2(\mathbb R)$. Here, $\hat f$ denotes the Fourier transform of ...
0
votes
1answer
40 views

Showing that a bilinear form is coercive

Let $I = (0, 1)$ and $H_0^2 (I)$ the closure of $C_c^\infty (I)$ in $W^{2, 2} (I)$. Consider $$a : H_0^2 (I) \times H_0^2 (I) \to \mathbb{R}$$ defined by $$a(u, v) = \underset{I}{\int} u''(x) v''(x) ...
2
votes
1answer
53 views

About fractional Sobolev space

I'm reading a paper used Sobolev space $H^s(\Omega)$ , I only know the definition of these space when $\Omega=R^n$ which used Fourier Transform, what if $\Omega$ is a bounded open set? And I also ...
2
votes
1answer
43 views

Continuity of functionals on Sobolev space

Let $U$ be a bounded set in $R^n$ and $W^{1,p}(U)$ denote a Sobolev space. Suppose $\{w_n\}\subset W^{1,p}(U)$ converges to $w \in W^{1,p}(U)$. Let $I[w]=\int_U F(Dw,w,x)dx$ for $w\in W^{1,p}(U)$, ...
1
vote
0answers
26 views

A calculation for an open ball in $\mathbb{R}^N$ and function space.

Let $B_r(x)$ denoting the ball of center $x$ and radius $r>0$. We denote by $\lambda_{1,\,B_\rho(y)}$ the first eigenvalue of $-\Delta$ in $W^{1,\,2}_0\left(B_\rho(y)\right)$ and by ...
0
votes
1answer
53 views

Convergence of $\partial_{x_j} u(x,t)$ when $u$ converges in $L^2$ norm.

I hope you can help me with this question. We take $u(x,t)\in L^\infty_{loc}(\mathbb{R},H^1(M))\cap Lip_{loc}(\mathbb{R},L^2(M))$, the derivatives $\partial_{x_j} u $ exist and are continuous, i.e ...
1
vote
1answer
31 views

Boundedness of a sequence in $L^\infty(I,H^1(M))\cap\mbox{Lip}(I,L^2(M))$ implies that its temporal derivative is bounded as well

I asked my question in mathoverflow, but it seems to be inappropriate there, so I try my luck here. ...
1
vote
1answer
631 views

Evans PDE: Chapter 5, Problem 9 - Clarification

I've been trying to work out the solution to Question 9 in Chapter 5 of Evans, and I'm having some difficulties. I've been looking at the solution posted here: question 9 - chap 5 evans PDE And I ...
1
vote
1answer
47 views

Sobolev Embedding and Uniform $C^1$ bound

I am currently reading a paper and am a little confused about the following, which for clarity, I distill into the following question: Suppose $\{w_i\} \subset C^2(\mathbb{R}^n)$ is a sequence of ...
9
votes
0answers
193 views

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
2
votes
1answer
27 views

Functions belonging to $L^p(\mathbb{R}^n ; \mathbb{R}^m)$ if and only of their norm belongs to $L^p(\mathbb{R}^n)$

The definition I have of $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m)$ is that we require each component function to be in $L^p(\mathbb{R}^n)$. Is is true that $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m) ...
0
votes
1answer
35 views

Variant of Ladyzhenskaya’s inequality

I am trying to show that if $\Omega \subset\subset \mathbb{R}^2$ with $C^1$ boundary and $ u \in W^{1,2}(\Omega)$ then \begin{equation*} \int u^4 < C \left(\int u^2 \right)^2 + C \left(\int u^2 ...
0
votes
1answer
44 views

Show that this function is weakly differentiable

I need to show that the function \begin{equation} u(x,y) = 1-x_1^2 \quad x_1>0 \\ u(x,y)=1+x_1^2 \quad x_1 \leq 0 \end{equation} is weakly differentiable on the unit ball. It is clear what the ...
0
votes
0answers
208 views

Elliptic regularity in an unbounded domain

Let $\Omega \subset \mathbb{R}^N$ be an unbounded domain with nonempty boundary. Let $$\mathcal{D}^{1,2}(\mathbb{R}^N) = \{ u \in L^{2^*}(\mathbb{R}^N) : \nabla u \in L^2(\mathbb{R}^N ; \mathbb{R}^N) ...
1
vote
1answer
35 views

sobolev space question?

My attempt: By the Fourier inversion formula, $$u(x) = (2\pi)^{-n}\int_{\mathbb{R}^n} \hat{u}(\xi) e^{i x \cdot \xi} ~d\xi,$$ $$(2\pi)^{n}|u(x)| = |\int_{\mathbb{R}^n} \hat{u}(\xi) e^{i x \cdot \xi} ...
1
vote
1answer
66 views

Fractional Sobolev spaces on closed manifolds

Let $M$ be a closed manifold and $0<s<1$. How is the fractional Sobolev space , $H^s(M)$ defined? In particular if $M$ is a closed smooth simple curve in the place how is $H^{1/2}(M)$ ...
-1
votes
1answer
66 views

When is the $L^{2}$ norm smaller than the $H^{-1}$ norm?

If $u\in L^{2}$ then we can define the functional: $$u(\phi)=\int \phi u $$ for all $\phi \in H^{1}_{o} $. which means that $u$ is a linear functional in $H^{-1}$. Now for any $f\in H^{-1}$ ...
3
votes
0answers
37 views

A linear functional of $H^k$

Define $H^k$ as usual: $$H^k=\left\{f; \int (1+|\xi|^2)^k|\widehat{f}(\xi)|^2d\xi<\infty\right\}$$ Define $l(u)=\int (1+|\xi|^2)^k\widehat{u}(\xi)d\xi$, how to show that $l\in (H^k)^*$. I tried ...
1
vote
1answer
38 views

For which $p$ the sequence $x^n$ converges in the Sobolev space $W^{1,p}(I)$?

I would like to know for which $p$ the sequence $u(n)=x^n$ converges in the Sobolev space $W^{1,p}(I)$. Is it true that converges only for $p=1$? I find out this looking for which $p$ the Sobolev ...
1
vote
1answer
126 views

A trace inequality with epsilon in Sobolev spaces

We know the standard trace inequality: for a bounded domain with certain boundary regularity, there is a $C>0$ such that $$ \|Tu\|_{L^2(\partial \Omega)}\leq C\|u\|_{H^1(\Omega)}, \quad \quad u\in ...
0
votes
1answer
31 views

Approximating $u \in H^1$ s.t. $u(T)=0$ with $u_n \in H^1_0$ in the gradient norm?

Let $u \in H^1(0,T)$ with $u(T)=0$. Is it possible to find a sequence $u_n \in H^1_0(0,T)$ such that $\nabla u_n \to \nabla u$ in $L^2$? I only need the convergence in the gradient.. not the full ...
6
votes
1answer
243 views

An interesting semi-linear PDE problem

Assume $u\in H^1(\mathbb{R}^n)$ has compact support, and assume that it is a weak solution of the semi linear equation $$ -\Delta u+c(u)=f\;\;\text{in}\;\mathbb{R^n} $$ where $f\in ...
2
votes
1answer
57 views

If $f \in L^1(M)$, is it true that $f(x) < \infty$ for almost all $x$?

If $M$ is a measurable space (eg. $M$ is a Riemannian manifold which is compact) and if $f \in L^1(M)$, is it true that $|f(x)| < \infty$ for almost all $x$? I am trying to figgur out if $u \in ...
4
votes
0answers
41 views

Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$ for some $\overline{p}>p*'$

Let $\Omega \subset \mathbb{R}^{N}$ be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function such that $g(x,t)=0$ for $t\leq0$ . Suppose that ...
3
votes
1answer
26 views

A trouble with the existence of an $C_{0}^{\infty}$- function $v,v>0$ such that $\int_{\Omega}hv^{\alpha}>0$

Let $\Omega$ be a smooth bounded subset of $\mathbb{R}^{n}$ , an $L^{\sigma_{\alpha}}$ -function $h$ with $h^{+}\neq0$ , $\dfrac{1}{\sigma_{\alpha}}+\dfrac{\alpha}{p*}=1$ , does there exist ...