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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Concerning the proof of regularity of the weak solution for the laplacian problem given in Brezis

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim brezis, and I am a bit confused about the proof. The theorem is stated as: Let $\Omega \subset ...
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6 views

Weak formulation 1-D PDE with non-homogenous robin boundary condition

Question: Question statement Worked Solution: Worked solution I have a few questions I hope you can help me with. Firstly, is my weak formulation correct? Why can't I use a test function in $H^1_0$ ...
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2answers
42 views

$u$ continuous and the weak derivative $Du$ continuous $\Rightarrow$ $u \in C^1$?

Supose we have $u \in W^{1,p}$ (i.e $u$ has weak partial derivatives, which we denote by $Du$), and that both $u$ and $Du$ are continuous (More precisely, there is a continuous representative in the ...
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1answer
34 views

Weak solution in $\mathbb{R}^{N}$

I'm bit confusing about definition of weak solution. If I have the following problem: $\begin{cases} \tag{P} -\Delta u = f \textrm{ in } \Omega, \\ u = 0 \textrm{ in } \partial\Omega, \end{cases}$ ...
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16 views

Does Green's (first) identity hold for Weak Derivatives?

Recall Green's First Identity: $$\int_{\Omega}v \Delta u \ (d\Omega) =\int_{\partial \Omega}v (\nabla u )\vec{n} \ d (\partial \Omega) - \int_{\Omega} \nabla u \nabla v \ (d \Omega)$$ Which ...
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1answer
42 views

Confused about the notation $||\nabla u||_{L^p(\Omega)}$

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis, and also Evans' PDE. I am confused about what $||\nabla u||_{W^{1,p}(\Omega)}$ precisely means. In ...
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1answer
19 views

Showing regularity $(u \in C^2(\overline{\Omega}))$ for the Laplacian Problem.

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim brezis, and I am a bit confused about the proof. The theorem is stated as: Let $\Omega \subset ...
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0answers
36 views

Weak solutions for elliptic equations

i have two questions about the study of weak solutions for elliptic equations? Why study these equations in divergent form? Why the minus sign in the principal part? Thanks very much
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25 views

Derivative of kinky function

I have the following result: $$-\frac{1}{2}y^2(\frac{\partial^2f}{\partial x^2}-\frac{\partial f}{\partial x}) = -\frac{1}{2}y^2K\delta_{x=\log(K)}$$ where $$f(x) = (K-e^x) \mathbb{1}_{\{x \leq ...
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1answer
44 views

Passing from Classical Formulation to Weak Forulation on PDEs. Integration by parts in $n$ dimensions?

I am reading about the Variational Method for solving PDEs and ODEs. The process to pass from a classical solution to a weak solution is pretty clear when we are dealing with ODEs, by using ...
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2answers
32 views

Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$

Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$. Further Show that $v_0$ is the solution of some differential equation with ...
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0answers
24 views

Question about the proof of $W^{1,p}_0(\Omega) \Rightarrow u=0 $ on $\partial \Omega$

I am reading the proof of Theorem 9.17 in the book Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. The theorem says: Suppose that $\Omega$ is of class ...
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1answer
82 views

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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22 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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13 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
878 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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38 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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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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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
34 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
48 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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25 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
217 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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50 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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29 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
50 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
30 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
212 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
28 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
23 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
61 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
33 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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19 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
78 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
45 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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1answer
99 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
80 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
45 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
35 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
37 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
37 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
65 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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3answers
448 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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1answer
29 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 ...
0
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1answer
29 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 $\mathbb R^n$, ...