For question about weak derivatives, a notion which extends the classical notion of derivative and allows us to consider derivatives of distributions rather than functions.

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Integral of weak derivative is finite

I want to show that the integral over the unit ball of the weak derivative of $u(x)=|x|^{-\gamma}, 0< \gamma< \frac{n-p}{p}$ is finite, in order to show that the function is in $W^{1,p}$. How ...
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2answers
21 views

Example of a function $u \in W^{1,p}(\Omega)$ whose extension $\hat{u}$ to be $0$ outside $\Omega$ $\hat{u} \notin W^{1,p}(\mathbb{R}^n)$

Let's consider a function $u \in W^{1,p}(\Omega)$, where $W^{1,p}(\Omega)$ is the Sobolev Space and $\Omega$ is an open set. When we extend $u$ to $\hat{u}$ like this: ...
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Given $u \in W^{1,p}(\Omega)$ then $\overline{\alpha u} \in W^{1,p}(\Omega)$, with $\alpha \in C^1 \cap L^{\infty},\nabla \alpha \in L^{\infty}$

Given $u \in W^{1,p}(\Omega)$ and $\alpha$ a function which verifies $\alpha \in C^1(\Omega)$, $\alpha \in L^{\infty}(\mathbb{R}^n)$, $\nabla \alpha \in L^{\infty}(\mathbb{R}^n)^n $ y $supp(\alpha) ...
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10 views

Existence of weak derivative of a translation of a function

Let $f$ a function in $L^1_{loc}(\mathbb{R^n})$ such that his weak derivative of order $\alpha$, $D^{\alpha}_wf$, exists. We consider a vector $h\in \mathbb{R^n}$ and we define $g(x)=f(x-h)$. I have ...
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1answer
864 views

function a.e. differentiable and it's weak derivative

Note - I am just starting to learn about theory of distributions, so this may be a trivial question, if so I'd be grateful for a reference, nevertheless the question is the following: suppose I have a ...
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30 views

Weak formulation of incompressible navier stokes and heat transfer differential equations for coupling in COMSOL

I am asked to simulate a 2-D coupled problem in COMSOL(Navier stokes with Heat transfer) of a simple room. I'm not sure if COMSOL already has preexisting physics for navier stokes and heat tranfer ...
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16 views

Prove, for an open $\Omega \subset \mathbb{R}^n$ with $x\in \Omega$, that $u\in W^{1,p}(\Omega-\{x\})\implies u\in W^{1,p}(\Omega).$

Let $n\geq 2$, $\Omega\subset \mathbb{R}^n$ open, $x\in \Omega$ and $p\geq1$ I want to prove the above implication. We just need to show that the weak derivative of $u$ on the punctured domain remains ...
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0answers
21 views

There is no Weak derivative of $\log|x|$ for $x \in (-1,1)$

I'm pretty new to this concept. I know that if it existed it had to be $1/x$ on $(-1,1) \backslash \{0\}$. However, I am not sure how to evaluate the integrals to get the non-existance
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1answer
33 views

Second distributional derivative of cosine

I need to compute second distributional derivative of the function $$ g(x) = cos|x-2|, $$ but I'm not sure about my solution. \begin{align} \left<g'', \varphi \right> = \left<g, ...
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1answer
44 views

Heat equation - Evans

I have the following question. In Evan's PDE book it is stated (p 345, section 6.61) that if we take the differnential operator: $$ Lu=-\Delta u +cu $$ then there exists a $\mu>0$ such that for all ...
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23 views

Dual-mixed formulation for Poisson eq. — how include Dirichlet BC, since solution only in $L^2$

When putting the Poisson Equation into weak form we usually get to solve this: Find $u\in H^1$ s.t. $$ \int_\Omega\textrm{grad}u \cdot \textrm{grad} v=\int_\Omega fv\quad \forall v \in H^1(\Omega) $$ ...
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2answers
204 views

How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$?

How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$? Since $|u|\in L_p$, I only need to show weak derivative of $|u|$ exists and $D|u| \in L_p$. Can anyone give me some hint? Thanks!
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43 views

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

I have a fairly simple question. Suppose that $p>n$ and $u,v\in W^{1,p}_0$, how can I prove that the product is also in $W^{1,p}_0$ ? Of course, we have to employ Morrey's inequality. My idea was: ...
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16 views

Can we show that each element of the Sobolev space $H^k(D)$, with $D\subseteq\mathbb R^d$ being a bounded domain, has a continuous representative?

Let $d\in\left\{2,3\right\}$ $D\subseteq\mathbb R^d$ be a bounded domain $\lambda$ be the Lebesgue measure on $D$ $H^k(D)$ be the Sobolev space Can we show that each element of $H^k(D)$ has a ...
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1answer
48 views

Distributional derivatives

I need to compute derivatives as distributions of following functions: $f(x) =$ $|x|$ $|x^2 - 1|$ $\mathrm{sgn}(x)$ $4$ Where $f : \mathbb{R} \to \mathbb{R}$. ad 1) $|x|$ is continuous, so it ...
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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 ...
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2answers
200 views

Exercise 4 chapter 7 Evans book

I want to solve the following problem: Suppose $H$ is a Hilbert space, if $u_{n}\rightharpoonup u$ in $L^{2}(0,T;H)$ and $u'_{n}\rightharpoonup v$ in $L^{2}(0,T;H^{'})$. I want to prove that ...
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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 ...
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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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2answers
56 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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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 ...
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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 ...
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0answers
77 views

About the gradient of a function in $H^{1}(\Omega)$

Let $\Omega \in \mathbb{R}^{n}$ a bounded domain and $u \in H^{1}(\Omega)$ a real function. In the Leoni's Book - A First Course in Sobolev Spaces, the author define $\nabla u = (D_{1} u,\dots, ...
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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 $H_0^1(\Omega)$ ...
3
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1answer
96 views

Solve the differential equation $x^2u'=0$ in the sense of distributions

Solve the differential equation in the sense of distribution: $$x^{2}\frac{du}{dx}=0$$ This is from "Principles of Applied Mathematics" by Keener, problem 4.1.5. The solution in the back of the ...
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1answer
79 views

Distributional solution of the heat equation

I want to show that $\frac{\partial{E}}{\partial{t}}-\Delta E=\delta(t,x) $. So it suffices to show that $\langle \frac{\partial{E}}{\partial{t}}-\Delta E, \phi \rangle=\phi(0,0) $. So far I have ...
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1answer
44 views

Identity having to do with weak derivative

For $a<b \in \mathbb{R}$, let $(a,b) = G \subset \mathbb{R}$ be a bounded interval in the real numbers. Show that there exists no $v \in L^2(G)$ and no $y \in G$ such that $$ \int_G v \varphi ...
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1answer
47 views

How can we apply the formula?

I want to calculate $\Delta \ln{||x||}$ for $x=(x_1, x_2)$. $$\langle \Delta \ln{||x||}, \phi \rangle= \langle \ln{||x||}, \Delta{\phi}\rangle=\int_{\mathbb{R}^2} \ln{||x||} \Delta{\phi(x)} dx= ...
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1answer
34 views

Derivative as distribution

We consider the Heaviside function $H(x)$. $H'(0)$ doesn't exist. The derivative exists if we define $H$ as a distribution $$H: \phi \to \int_{-\infty}^{+\infty} H(x) \phi(x) dx= \int_0^{+\infty} ...
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1answer
34 views

Integration of weak derivative of hat function for FEM.

Just curious, lets say I want to solve $u''=f$ $u(0)=0 \space\space u(1)=0$ In the weak form the equation looks like $<u',\phi ' > = <f,\phi>$ Now my question, if I test my test ...
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2answers
92 views

Can we just set it?

Suppose that we have $G(\overline{x},\overline{y})=-\frac{1}{4 \pi} \frac{1}{||\overline{x}-\overline{y}||}$ for $\overline{x}, \overline{y} \in \mathbb{R}^3$. I want to calculate ...
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1answer
41 views

Functions of the form $u(x,y) = f(x-y)$ are weak solutions of $u_x + u_y = 0$

This is a problem out of Logan's Applied Math book. Section 6.7, problem 2. Show that for any locally integrable function f on $\mathbb{R}$ the function $u(x,y) = f(x-y)$ is a weak solution to ...
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1answer
36 views

Showing that $(G' \circ u)u' \in L^p(I)$ where $G \in C^1(\mathbb{R})$ and $u \in W^{1,p}(I)$

I am trying to prove the rule of differentiation of a composition for weak derivatives in Sobolev spaces following the proof given in Corolary 8.11 in Functional Analysis, Sobolev Spaces and Partial ...
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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} ...
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1answer
54 views

Does the weak divergence exist for each $\mathcal L^2(\Omega;\mathbb R^d)$-function?

Let $\Omega\subseteq\mathbb R^d$ be open. $v:\Omega\to\mathbb R$ is called weak divergence of $u:\Omega\to\mathbb R^d$ $:\Leftrightarrow$ $$\int_\Omega v\varphi\;{\rm d}\lambda=-\int_\Omega\langle ...
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440 views

Weakly differentiable but classically nowhere differentiable

Is there any example of a function which is weakly differentiable but none of its versions are classically differentiable anywhere (or differentiable only on a set of measure 0) ? Thanks
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28 views

is $u\in L_{loc}^1(\mathbb{R}^2)$ and 2. $u\in W_{loc}^{1,1}(\mathbb{R}^2)$?

Let $U\in C^1((0,\infty))$, $u:\mathbb{R}^2\to \mathbb{R}$ defined by $$u(x,y)=U(\sqrt{x^2+y^2}).$$ Under which conditions on $U$ is 1.$u\in L_{loc}^1(\mathbb{R}^2)$ and 2. $u\in ...
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1answer
28 views

Doubts on distributional derivative: null derivative and non negative derivative

suppose we are in $\mathbb R^n$ and fix $E\subset \mathbb R^n$ and let $\chi_E$ be the indicator function of the set $E$. I'm reading an article that says that a vector field $X$ on $\mathbb R^n$, ...
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0answers
31 views

Distributional derivative $Xu=0$ then $u$ is constant over the flow lines of $X$

Suppose we have a Lie group on $\mathbb R^n$, something like $(\mathbb R^n, \cdot)$, where with $\cdot$ we denote the group law on $\mathbb R^n$. Call $\mathfrak g$ the Lie algebra of $(\mathbb R^n, ...
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47 views

an iff proof on the existence of weak derivative

I have trouble understanding the following proposition. Proposition $f,g\in L_{\text{loc}}^1(\Omega)$. Then $g=D^{\alpha}f$ iff. there exists $f_m\in C^{\infty}(\Omega)$ such that $f_m\to f$ in ...
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1answer
157 views

$C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$

As the title say, I want to prove that $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ i.e. $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n) \quad (\star)}$. In a book ...
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1answer
20 views

Are there $H^1_0(\Omega)$-functions in the plane that are discontinuous over curves?

Consider a bounded domain $\Omega \subset \mathbb{R}^2$ with Lipschitz-boundary and a curve $\gamma : [0,1] \rightarrow \Omega$. Is it possible to construct a function $f \in H^1_0(\Omega)$ which is ...
3
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1answer
28 views

$u\in W^{1,1}(U) \Rightarrow u^+\in W^{1,1}(U)$ , where $U\subset\mathbb{R}^n$ open

Im trying to show that for an open set $U\subset\mathbb{R}^n$ and a function $u\in W^{1,1}(U)$, also the positive part $u^+$ is in $W^{1,1}(U)$. My idea is the following: Let $E\subseteq U$ defined ...
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1answer
63 views

necessary and sufficient conditions for the existence of solution in the space $W^{k,p}$

I am learning about weak derivatives and sobolev space. In particular I need help to learn the proving strategy/technique. I have trouble proving on how to show a solution belongs to some sobolev ...
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0answers
35 views

$-u''+u=\delta_y$ where $u(0)=u(1)=0$

Let $0<y<1$ be arbitrary. What is the weak solution of the differential equation $-u''+u=\delta_y$ where $u(0)=u(1)=0$ then? The weak form of the equation above is given by ...
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1answer
22 views

Calculate the weak formulation of $\int_\Omega \left(\alpha v(x) + \beta\Delta\Delta v(x)\right)w(x)dx=\int_\Omega \gamma w(x)dx$

I would like to calculate the weak formulation of the following equation: $$ \alpha v(x)+\beta\Delta^2 v(x)=\gamma $$ Which brings me to this formula: $$ \int_\Omega \left(\alpha v(x) + \beta ...
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0answers
53 views

Is the image of gradient map from Sobolev space to Lebesgue space weakly closed?

Suppose f is a map defined between $W_0^{1,p}(\Omega)$ and $L^{p'}(\Omega)$ as follows - $u \mapsto |\nabla u|^{p-1}$. Is the range of this map weakly closed in $L^{p'}$?.
2
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1answer
202 views

Distributional derivative of absolute value function

I'm tying to understand distributional derivatives. That's why I'm trying to calculate the distributional derivative of $|x|$, but I got a little confused. I know that a weak derivative would be ...
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0answers
33 views

Weak chain rule the other way round.

If $f: \mathbb{R} \to \mathbb{R}$ is weakly differentiable and $u: \Omega \to \mathbb{R}$ is smooth, where $\Omega \subset \mathbb{R}^n$, can we show that for any test function $\varphi \in ...
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0answers
35 views

Why are these functions Locally Integrable?

Apparently the following are locally integrable: (a) $x^r\in L_1(0,\infty)$, $\forall r\in\mathbb{R}$. But this couldn't be right because $\frac{1}{x}\notin L_\text{loc}(\mathbb{R})$ (since ...