For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

learn more… | top users | synonyms

1
vote
0answers
27 views

If $u \in W^{1,p}(I) \cap C_c(I)$ then $u \in W_{0}^{1,p}(I) $

I want to show the following statement ($1 \leq p < \infty$): If $u \in W^{1,p}(I) \cap C_c(I)$ then $u \in W_{0}^{1,p}(I) $. $W^{1,p}(I) $ is the Sobolev Space, i.e. the space consisting of ...
0
votes
2answers
41 views

Construction of Sobolev space

I am reading about the construction of Sobolev spaces from $L^2$. In the book I read (Introduction to Partial Differential Equations, by Gerald B. Folland) that the operator used to construct those ...
1
vote
0answers
25 views

Help showing $u \in W_0^{1,p}(I)$ if and only if $u=0$ on $\partial I$

I am reading the proof the following statement provided in Functional Analysis, Sobolev Spaces and Partial Differential Equations, by haim Brezis: If $u \in W_0^{1,p}(I)$, then $u=0$ on $\partial ...
1
vote
0answers
13 views

Sobolev space membership of logarithmic function

Determine the largest $s\in(0,1)$ for which the following integral converges $$\int_0^1\int_0^1\frac{\Big|\log|x-\frac{1}{2}|-\log|y-\frac{1}{2}|\Big|}{|x-y|^{1+2s}}^{2}dxdy$$
1
vote
2answers
39 views

Sobolev Space $W_0^{1,p}(I)$ and the boundary of $I$

Given $I \subset\mathbb{R}$ an open interval, the Sobolev Space $W_0^{1,p}(I)$ is defined as $W_0^{1,p}(I)=\overline{C^1_c(I)}^{W^{1,p}(I)}$ (The closure of $C^1_c(I)$ on the space $W^{1,p}(I)$) . ...
1
vote
2answers
29 views

Sobolev space and norms

Hello I'd like to know the solution to this question in my Numerics of PDE's class. Here goes: Given $V := \lbrace v \in H^1(0,1): v(0) = 0 \rbrace \subset H^1(0,1)$ Show that the $H^1$-seminorm ...
0
votes
0answers
24 views

The Sobolev Space $W^{m,p}$ is a closed subspace of $W^{1,p}$.

(This is a proof verification, I need to know if the procedure and steps are correct and if the proof is complete) I want to show that the Sobolev Space $W^{m,p}(I)$ is a closed subspace of ...
0
votes
0answers
25 views

Sobolev embedding fails for $p=n$

As everyone knows, the Sobolev embedding fails fails for $n\ge 2$ if we assume $p=n$. The standard example is the function $u(x)=\log \log \bigl(1+\tfrac{1}{x}\bigr)$. This function is obviously ...
2
votes
1answer
57 views

Find $y \in W_{2}^{1}[-1,1]$ s.t. $\forall x \in W_{2}^{1}[-1,1]$, $f(x)=\langle x, y \rangle$

Consider a Sobolev space $W_{2}^{1}[-1,1]$ with the following inner product: $\langle x, y \rangle = \int_{-1}^{1} [x(t)y(t)+x^{\prime}(t)y^{\prime}(t)]dt$. Let $f(x) = \int_{-1}^{1}e^{2t}x(t)dt$. ...
2
votes
2answers
38 views

Approximate Sobolev function by smooth function - error estimate?

I wondered if there is, in general, a way to estimate the error (in $L^2$) between a Sobolev function and its mollified version. Let's say $\Omega\subset\mathbb{R}^n$ is bounded with smooth boundary ...
1
vote
0answers
29 views

Are compactly supported functions in $W^{1,2}(\mathbb R^n)$ also in $W_0^{1,2}(\mathbb R^n)$? See for proof?

The question is stated clearly in the title. On the one hand, it seems obvious (and I give an argument below). On the other hand, after a quick search I haven't been able to find the statement ...
0
votes
0answers
37 views

vector-valued function space definition except for measure zero

I am wondering what's the correct way to mathematically describe the following problem. Say you have an object that can be defined as an open set $\Omega \in \mathbb{R}^d$, where the dimension $ ...
2
votes
0answers
24 views

Poincaré-like inequality

Let $\Omega\subset\mathbb{R}^3$ be an open bounded set. Let $\partial\Omega=\Gamma^1\cup\Gamma^2$, with $\Gamma^1\cap\Gamma^2=\emptyset$. We denote as $\Gamma^1_j$, $j=1,\dots,p_{\Gamma^1}+1$, the ...
1
vote
0answers
71 views

show that statements are equivalent (Sobolev Spaces)

Given that $u \in L^p(R), 1<p<\infty$ show that the following statements are equivalent: a) $u\in W^{1,p}(R)$ b) $\exists c>0 $ such that for $\forall h \in R$ the following inequality ...
1
vote
1answer
41 views

approximation in $W^{k,p}(\Omega)$ with $C^\infty_c(\Omega)$ functions

I read the following result in real analysis: Let $1 \leq p < \infty$ and $k \geq 0$. Then the space $C^\infty_c({\bf R}^d)$ of test functions is a dense subspace of $W^{k,p}({\bf R}^d).$ A ...
1
vote
0answers
28 views

Multi-index notation and differentation

For example, let $\Omega \subseteq \mathbb{R}^n$ open, and $C^\infty(\Omega):=\lbrace f: \Omega \longrightarrow \mathbb{C} : f$ $\mathrm{regular}\rbrace$. For $\alpha = (\alpha_1,...,\alpha_n) \in ...
0
votes
0answers
18 views

Question regarding Evan's proof of Global Approximation by $C^∞(\overline{U})$ functions

The page where the proof is is on Google Books. I reproduce the statement of the result: Suppose $U$ is bounded with $C^1$ boundary, and $u∈ W^{k,p}(U)$. Then there are $C^{\infty}(\overline{U})$ ...
2
votes
1answer
41 views

How is the $H^{1/2}$ norm of function defined on a subset of the boundary?

Let $\Omega\subset \Omega^d$, $d\in \{2,3\}$, be a bounded $d$-polyhedron with $n$ faces. Denote the faces of $\partial\Omega$ as $\{e_i\}_{i=1}^n$. Let $u\in H^{1/2}(\partial\Omega)$ Taking the ...
1
vote
1answer
27 views

Density of smooth positive functions

Let $\Omega$ be an open bounded set of $R^n$. For $f\in L^2(\Omega)$ such that $f>0$, a.e. in $\Omega, $ there is $(f_k)\subset W^{2,\infty}(\Omega)$ such that $f_k\to f$ in $L^2(\Omega)$. My ...
0
votes
1answer
17 views

Are piecewise constants in $H^{1/2}(\partial\Omega)$?

Let $\Omega\subset \mathbb{R}^d$ be a polyhedron. Denote the faces of $\partial \Omega$ as $\{e_i\}_{i=1}^n$ for some $n$. Define $u_i\colon \partial\Omega \to \mathbb{R}$ as $$u_i(x) = ...
0
votes
1answer
27 views

A fact about integration of $H^{1}(\mathbb R^{d})$ functions

For any $v \in H^{1}(\mathbb R^{d})$ how to show that $\int_{\mathbb R^{d}} f(v).\nabla v dx = 0$ ; where $f: \mathbb R \to \mathbb R^{d}$ is a Lipschitz continuous function such that $f(0) = 0$ ??
1
vote
0answers
27 views

Find a discontinuous function from $\mathbb R^n \to \mathbb R$ which is weakly differentiable. [duplicate]

For $k \in \mathbb N$, I use the notation $$H^k(\mathbb R^n) = \{ u \in L^2(\mathbb R^n) : D^\alpha u \in L^2(\mathbb R^n) \text{ for all multi-indices } \alpha \text{ with } \lvert \alpha \rvert \le ...
1
vote
1answer
19 views

Integral of a weak derivative

While reading chapter 6 of John Hunter's notes (https://www.math.ucdavis.edu/~hunter/pdes/pde_notes.pdf) I got stuck on some steps. I think they are all based on a similar idea as the following. Let ...
0
votes
1answer
18 views

Compactness Sobolev embedding for even functions on $\mathbb{R}$.

It is well-known from Lions's article,"Symétrie et compacité dans les espaces de Sobolev", that the subspace $H^s_r(\mathbb{R}^n)$ of the Sobolev space $H^s(\mathbb{R}^n)$ containing all radial ...
1
vote
0answers
31 views

Understanding multiindex notation and the Sobolev Space $W^{1,p}$.

The notation comes from Evans Partial Differential Equations. From Appendix A, we are given information about multiindex notation. Assume $ u : U \rightarrow R$, $ x \in U$. (a) A vector of the ...
3
votes
1answer
50 views

Find $v \in H^1(0,1)$ which satisfies the following

Determine the function $v \in H^1(0,1)$ which satisfies the equation $u(0)=\langle u,v \rangle_{H^1}$ for all $u\in H^1(0,1)$ . It is clear that on $H^1(0,1)$; $u(0)=\int_{0}^{1}(uv+u'v')$.What can ...
0
votes
0answers
40 views

Mistake in a PDE book regarding Lebesgue's differentiation theorem? To do with weak formulation

I'm reading "Elliptic and Parabolic Equations" by Wu, Yin and Wang. In Section 4.2, they consider the heat equation given $u_0 \in L^\infty$ and $f \in L^\infty$ $$u_t - \Delta u = f$$ $$u(0) = u_0$$ ...
0
votes
1answer
34 views

Why the need of Sobolev spaces in this proof of isoperimetric inequality?

I was reading the chapter about isoperimetric inequalities in DaCorogna's book "Introduction to The Calculus of Variations". The isoperimetric inequality is proved to be equivalent to Wirtinger ...
1
vote
1answer
47 views

If an $H^1$ function vanishes on a set of positive measure, its $L^2$ norm is controlled by the gradient

I am trying to solve question 15 from Evans' PDE book, chapter 5. You have a set of positive measure, subset of the unit ball $B$, such that $u$ is equal to zero on that set. Then, one can show that: ...
1
vote
0answers
65 views

Results about Hilbert-Sobolev space with homogeneous boundary condition.

I am currently reading works about numerical method for solving differential equations. The main setting of the work revolves on the space $H^m_0[0,1]$, which is defined by $$ H^m_0[0,1]:=\{f\in ...
2
votes
0answers
76 views

$\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$

Let $\Omega\subset \mathbb R^d$ ($d=2,3$) is a bounded Lipschitz domain. Question: Is it true that for each function $g(x)\in L^2(\Omega)$ one can find a sequence $\{g_n\}_1^\infty$ of ...
1
vote
0answers
41 views

Sobolev space: Prove a function is in $W^{1,\infty}$

I am reading the book: Fully nonlinear elliptic equations of Caffarelli and Cabre. In page 8 (Prop 1.2) they prove that if function $u$ in a convex domain locally has at least one paraboloid touching ...
0
votes
1answer
32 views

Constant weak differentaible functions?!

I have the following question. Suppose I have a function from $\mathbb{R}^2\to\mathbb{R}$ which only depends on the first coordinate. I know that the function viewed as a function from ...
0
votes
1answer
33 views

Entropy/Variance inequality

The following inequality is sometimes used as a building block to prove log Sobolev inequalities. Does anyone have a simple proof of it? $$ x\log x + y\log y - (x+y)\log \frac{x+y}{2}\leq (\sqrt ...
2
votes
2answers
57 views

Definition of weak time derivative

My quesion involves the weak time derivative. In the book: 'Partial Differential Equations' by Evans the time derivative $u'$ of a function $u: [0,T] \rightarrow H^1_0(U)$ is defined by an element ...
1
vote
0answers
27 views

The eigenvalue for mollified function

Let $u_k$ be eigenfunctions for operator $-\Delta$ over domain $\Omega\subset \mathbb R^2$, open bounded, smooth boundary. Then, we know that $E:=\{u_k\}_{k=1}^\infty$ forms a basis for $L^2$ and we ...
1
vote
0answers
29 views

Operators of order $k$ between Sobolev spaces

It may sound like tautology, but I have a problem in proving that differential operator of order $k$ is an operator of order $k$. What exactly I'm asking for? Let $M$ be a manifold and $E,F$ be two ...
0
votes
1answer
17 views

How to show that a function is in a Sobolev space

This question is about the solution of exercise 1.20 in Elman, Silvester, Wathen. Finite Elements and Fast Iterative Solvers. (The first Chapter of the book is open access and available, for example, ...
2
votes
2answers
48 views

What's the value of $\alpha$ satisfying $||f'||^2\ge \alpha||f||^2$? [duplicate]

I am reading a paper about numerical analysis of a certain method for solving operator equation. Let our Hilbert space be $L^2[0,1]$, we define the subspace $D\in L^2[0,1]$ by $$ D:=\{f\in ...
3
votes
1answer
65 views

Weak convergence in $W^{1,p}(\mathbb{R})$

Let $\varphi\in\mathcal{C}^\infty_0(\mathbb{R})$ a map with compact support. For all $n\in\mathbb{N}$, I define $u_n$ by $u_n(x)=\varphi (x+n)$. I would like to prove that $u_n$ converges weakly ...
1
vote
1answer
23 views

Classical solution satisfies weak formulation of Poisson equation

I have a domain $\Omega \subset \mathbb{R}^2$ and the Poisson equation with Dirichelet boundary condition: $$ \begin{cases} -\nabla^2 u &= f\qquad in\ \Omega \\ u &= 0\qquad on\ \partial\Omega ...
3
votes
1answer
39 views

What is the square root of the Laplace operator?

Let $\Delta$ be the Laplace operator $$ \Delta f = \sum_{i=1}^d \frac{\partial^2 f}{\partial x^2_i}$$ with $Dom(\Lambda) = H^1_0(\mathcal{O}) \cap H^2(\mathcal{O})$ where ...
0
votes
0answers
16 views

Test function for proof of moser harnack inequality

There exists a standard method to build test functions? I would like to construct a sequence of functions $\phi_{m} \in C^{\infty}_{c}(B_{1})$ such that $\phi_{m} \to \psi{u}^{-2}$, where $\psi \in ...
1
vote
1answer
29 views

Weak convergence preserver pointwise inequality

The proof of boundedness of Hardy-Littlewood maximal function in Sobolev spaces in Kinnunen's paper has the following argument: "... Hence $(v_k)$ is a bounded sequence in $W^{1,p}(\mathbb{R}^n)$ ...
0
votes
0answers
15 views

$H^1$-conforming finite elements

Let us consider $\Omega\subset\mathbb{R}^d$, $d=2,3$ polyhedral domain, i.e. $\bar{\Omega}$ is the union of a finite number of polyhedra. Let $\bar{\Omega}=\bigcup_{K\in\mathcal{T}_h}K$, where ...
0
votes
0answers
35 views

coercive bilinear form

Here I want to prove the uniqueness of the weak solution for the homogeneous Dirichlet boundary value problem: \begin{equation*} \left\{ \begin{array}{rl}3u′′ − 2u′ + 3u = f \\ u(0) = u(1) = 0\\ ...
0
votes
1answer
34 views

Sobolev space of a function

Let $f(x) = √x$ denote the square root function. For what m ∈ {1, 2, 3 . . . } and $p ∈ [1,∞)$ values is it true that $$f ∈ W^{m,p}(1,∞)?$$ I just tried like this by using integration by parts: ...
0
votes
1answer
23 views

Prove the product of two $W_0^{1,p}$ functions gives another $W_0^{1,p}$ function if $p>n$

Fix $\Omega \subset \mathbb{R}^n$ want to show if $p>n$ then $$u,v \in W_0^{1,p}(\Omega) \implies uv \in W_0^{1,p}(\Omega).$$ I think I have an answer but I'm not sure why the condition $p>n$ ...
1
vote
1answer
43 views

If $u_n \rightharpoonup u$ in $H_0^1(\Omega)$, what can we say about $\{\nabla u_n\}$?

I have a bounded sequence $\{u_n\}$ in $H_0^1(\Omega)$. Then, a general result about Hilbert spaces and weak convergence, implies that there exists a subsequence $\{u_{\sigma(n)}\}$ in $H_0^1(\Omega)$ ...
5
votes
2answers
78 views

Proving that the eigenfunctions of the Laplacian form a basis of $L^2(\Omega)$ (and of $H_0^1(\Omega)$)

I am studying the eigenfunctions and eigenvalues of the Laplacian on an open, bounded domain $\Omega \subset \mathbb{R}^n$ with homogeneous Dirichlet boundary conditions. I have read about the the ...