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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Characterization of a set occurring in the Helmholtz-Hodge decomposition

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $q\ge 2$ Each $f\in L^1_{\text{loc}}(\Omega)$ can be identified with $\langle f\rangle\in\mathcal ...
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35 views

Proof of the Helmholtz-Hodge decomposition

Let $\Omega\subseteq\mathbb R^3$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ Let $$G^2(\Omega):=\left\{\nabla p:p\in L^2_{\text{loc}}(\Omega)\text{ with }\nabla p\in L^2(\Omega)^3\right\}$$...
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36 views

If $p$ is a distribution, what is the meaning of the claim $\nabla p\in L^p(\Omega)^d$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $q\ge 1$ I've seen the following Lemma (without a proof) in a paper and don't understand how I ...
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How can we extend $\operatorname{div}:C_c^\infty(\Omega)^d\to L^p(\Omega)$ to $W_0^{1,\:p}(\Omega)^d$?

Let $d\in\mathbb N$ and $p\ge 1$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be open and $$W_0^{1,\:p}(\Omega):=\overline{C_c^\infty(\Omega)}^{\left\|\;\cdot\;\...
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1answer
21 views

Questions about the proof: Continuous function with weak derivatives $\Rightarrow$ $C^1$

For an open set $\Omega$ of class $C^1$, suppose we have $u \in W^{1,p}(\Omega)$ and that $u$ is continuous and all the partial derivatives of $u$ are continuous. I want to show that $u$ is $C^1(\...
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1answer
43 views

Different approaches to differentiability in $L^2$

We can use different approaches to differentiability of $L^2(\mathbb{R})$ functions, e.g. we can say that $f\in L^2(\mathbb{R})$ is differentiable iff $f$ has a differentiable version (representative)....
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1answer
28 views

Passing from classical formulation to weak formulation for a general PDE

I am reading a paper dealing with a general elliptic PDE that I need to transform from classical formulation to weak formulation: $$\left\{\begin{matrix} - \sum_{i=1}^n \sum_{j=1}^n (a_{ij} u_{x_i})_{...
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2answers
61 views

Does existence of the second weak derivative of $f\in L^2$ imply existence of the first?

Let's consider a function $f\in L^2(\mathbb{R})$ for which the second weak derivative exists and lie in $L^2(\mathbb{R})$, i.e. there exists $f''\in L^2(\mathbb{R})$ such that for all $\varphi\in C_0^\...
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1answer
21 views

Distributional derivatives for functions that is continuous but nowhere differentiable

It is well known that the Brownian motion is an example of functions that is continuous but nowhere differentiable. In addition, its distributional derivative can be interpreted in the way mentioned ...
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1answer
35 views

Are weak (Sobolev) solutions to a linear ODE a classical ones?

Let $\Omega$ be an open subset of $\mathbb{R}$ and let $L$ be the differential operator $$ Lf = \sum_{k=0}^{n-1} a_k f^{(k)} + f^{(n)}, $$ where $a_k$ are reals. I would like to show that every ...
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1answer
27 views

Question concerning the proof of regularity for the Laplacian $f \in H^m(\Omega) \Rightarrow u \in H^{m+2}(\Omega) $

I am stuck at the proof of Theorem 9.25 in Haim Brezis' Sobolev Spaces, Functional Analysis and Partial Differential Equations. This theorem deals with the regularity for the Dirichlet Problem for ...
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1answer
40 views

If $\{\nabla u_j\}$ is Cauchy in $L^p(\mathbb{R}^n)$ and $\int_{B(0,1)} u_j dx = 0$, does $\{u_j\}$ converge in $L^p_{\text{loc}}(\mathbb{R}^n)$?

Let $1 < p < \infty$. Let $\{u_j\}_{j=1}^\infty$ be a sequence of functions in $W^{1,p}_{\text{loc}}(\mathbb{R}^n)$ such that $\nabla u_j \in L^p(\mathbb{R}^n)$ for all $j$, $\int_{B(0,1)} u_j ...
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1answer
35 views

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 \mathbb{...
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1answer
15 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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$u$ 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
42 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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1answer
46 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
21 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
21 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 \mathbb{...
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0answers
40 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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26 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 \log(K)...
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1answer
47 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
39 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
28 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 $C^1$...
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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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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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26 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: $$\hat{u}(x)=\left\{\begin{...
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0answers
42 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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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
35 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, \varphi''\...
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1answer
52 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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34 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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1answer
87 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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53 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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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
55 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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1answer
104 views

Show that if $\,u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$, then $\,u \in W_{0}^{1,p}\left(I\right)$.

I want to show the following statement ($1 \leq p < \infty$), for an open interval $I$: If $u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$ then $u \in W_{0}^{1,p}\left(I\right) $. $W^...
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2answers
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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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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
31 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
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 $\mathbb{R}\to\...
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2answers
65 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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0answers
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 H^...
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1answer
51 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
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 ...
3
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1answer
105 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
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 \text{...
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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= \lim_{...
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1answer
36 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} \...