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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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 ...
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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 $C^{\infty}(\overline{U})$ ...
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1answer
38 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 ...
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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 ...
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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) = ...
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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 $f(0) = 0$ ??
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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 ...
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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 ...
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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 ...
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29 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 ...
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1answer
49 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 ...
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39 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 $f \in L^\infty$ $$u_t - \Delta u = f$$ $$u(0) = u_0$$ ...
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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 ...
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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: ...
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64 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 ...
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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 ...
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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 ...
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1answer
31 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 ...
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1answer
31 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 ...
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1answer
34 views

Sobolev's inequality implies isoperimetric inequality.

How to show that Sobolev's inequality implies isoperimetric inequality in $\Bbb R^d$? I tried to search it on the internet, but most of the results I got are only about the converse direction. So, ...
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2answers
54 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 ...
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26 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 ...
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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 ...
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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, ...
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2answers
47 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 ...
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1answer
64 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 ...
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1answer
20 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 ...
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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 ...
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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 ...
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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 $W^{1,p}(\mathbb{R}^n)$ ...
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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 ...
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34 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\\ ...
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1answer
32 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: ...
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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 $p>n$ ...
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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 $H_0^1(\Omega)$ ...
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2answers
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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 ...
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0answers
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Estimate for gradient

Notation: $B_{1}$ is the unit closed ball in $\mathbb{R}^{n}$ $<.>$ is the canonical inner product of $\mathbb{R}^{n}$ Let $u \in H^{1}(B_{1})$, $\xi \in C^{1}_{0}(B_{1})$. Set $v=(u-k)^{+}$ ...
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1answer
38 views

Nonlinear elliptic PDE - passing to the limit

In the notes I am trying to follow one can find the following argument (part of a longer proof on existence of a weak solution to a certain type of nonlinear elliptic pde): Let $V = H^1_0(\Omega)$ ...
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1answer
20 views

Poincaré's inequality for functions with prescribed boundary

Let $I=(0,1)$ and $u\in W^{1,2}(I)$. It is not difficult to see that there is a constant $C>0$ such that $$\|u\|_{2}\le C(|u(0)|+\|u'\|_2).\tag{1}$$ If we restrict the inequality $(1)$ to the set ...
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What is the difference between $H^1_{loc}$ and $H^1$?

I have started studying Sobolev spaces and I came across a space referred to as $H^1_{loc}$. I am not sure what the $loc$ subscript infers? What is it that makes this space different from $H^1$? Why ...
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1answer
96 views

The existence of minimizer in Sobolev space

Let $B\subset \mathbb R^2$ be a unit ball. let $v\in W^{1,2}(B)$ be given. We know that $0\leq v\leq 1$ and it is possible that $v=0$ on some positive $\mathcal L^2$ measurable set in $B$. Let $w\in ...
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0answers
35 views

$L^{2}$ convergence of sequence $|u_{j}|^{p}\nabla u_{j}$

Suppose a sequence $\{u_{j}\}$ is bounded in the Sobolev space $H^{2+\epsilon}(\Omega)$, for $\epsilon>0$, where $\Omega$ is say a bounded, $C^{\infty}$ domain. Here, the fractional space ...
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1answer
22 views

Product of $W^{1,p}_0$ functions

Let $p>n$, and let $f,g\in W^{1,p}_0(\mathbb{R}^n)$ be two sobolev functions. Prove that $fg\in W^{1,p}_0(\mathbb{R}^n)$. I was able to prove the Leibniz formula for weak derivativatives, but ...
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27 views

weak derivative of sign function

How the weak derivative of the sign function $$\begin{equation*} sign(x)=\left\{ \begin{array}{rl}1 & \text{if } x> 0,\\ 0 & \text{if } x=0, \\ -1 & \text{if } x<0 \\ ...
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2answers
58 views

Sobolev space definitions

By definition of the Sobolev space $W^{m,p}$ we have : $$W^{m,p}(\Omega)=\{u\in L^p(\Omega)\ |\ \forall \alpha \text{ such that } |\alpha|\le m, D^{\alpha}u\in L^p(\Omega)\}$$ Can someone give me a ...
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1answer
31 views

Sobolev inequality with a constant independent of the support of the function

I am revising our PDE module and I came across the following version of the Sobolev inequality: $$ \exists C>0 \forall u\in W^{1,p}_0(\mathbb{R}^n):\, \|u\|_{L^{p*}}\le \|Du\|_{L^p}\, , $$ where ...
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1answer
20 views

Lax Milgram, prove continuity for $a : {W^{2,2}(I)} \times {W^{2,2}(I)} \rightarrow\mathbb{R}$

Let $I = (0,1)$ and $b>0$. Let $f\in L^2(I)$. I need to show that there exists a unique $u \in W^{2,2} (I)$ such that $$a(u,\phi) = \int^1_0 (u''\phi'' + bu'\phi' + u\phi) dx = \int^1_0 f\phi \ ...
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1answer
21 views

Inner Product on Sobolev Space with p=2

Wikipedia defines the Sobolev Space: $H^{s,p}(\mathbb{R}^n)= \left\{f \in L^p(\mathbb{R}^n): \mathcal{F}^{-1}[(1+|k|^2)^{\frac{s}{2}} \mathcal{F}f] \in L^p(\mathbb{R}^n) \right\}$ Where $s \in ...
2
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0answers
19 views

Lebesgue Space/Bochner Space interpolation Theorem

I need the embedding, for $I\subset\mathbb{R}$ is a bounded intervall and $\Omega\subset\mathbb{R}^n$ is a bounded domain, $$L^{q_1}(I;L^{p_1}(\Omega))\cap L^{p_2}(I;L^{p_2}(\Omega))\hookrightarrow ...
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1answer
43 views

Linear functional

For $a<b \in \mathbb{R}$, let $G=(a,b)$ be a bounded interval. For every $x \in G$, let the generalized function $\delta_x$ be defined by $$ \int_G \delta_x \phi(x)dx = \phi(x) ~~ \text{for every} ...