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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44 views

What is the divergence of a distribution?

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ If $p\in \mathcal D'(\Omega)$, then $$\frac{\partial p}{\partial x_i}(\phi):=-p\left(\frac{\...
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1answer
46 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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2answers
50 views

If $p,q$ are distributions with $\partial_ip=\partial_iq$, then $p=q$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ $\mathcal D(\Omega):=C_c^\infty(\Omega)$ Let $$\frac{\partial p}{\partial x_i}(\phi):=-p\left(\frac{\partial \phi}{\partial x_i}\right)\;\;\;\...
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1answer
56 views

Has the distributional Laplacian $\Delta f:C_c^\infty(\Omega)'\to C_c^\infty(\Omega)'$ a unique extension in $H_0^1(\Omega)'$?

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D:=C_c^\infty(\Omega)$ and $$H=\overline{\mathcal D}^{\langle\;\cdot\;,\;\cdot\;\rangle_H}\tag 1$$ with $$\langle\phi,\psi\rangle_H:...
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0answers
23 views

For $u_h(t) = \frac 1h \int_t^{t+h}u(s)\;\mathrm{d}s$, is $[\partial_t (u_h(t))]_{x_j} = \partial_t[(u_{x_j})_h(t)]?$

Let $u$ belong to $L^2(0,T;H^1(\Omega))$ and $u_t \in L^2(0,T;(H^1(\Omega))^*)$ and define the function $$u_h(t) = \frac 1h \int_t^{t+h}u(s)\;\mathrm{d}s$$ Is it true that $$[\partial_t (u_h(t))]_{...
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0answers
36 views

Relationship between the distributional Laplacian and the weak Laplacian

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the $L^2(\Omega)$- or $L^2(\Omega,\mathbb R^d)$-inner product (depending on the context) $\mathcal ...
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1answer
31 views

Showing that $|u|$ is absolutely continuous for $u \in C^1(\overline{\Omega})$

Suppose that we have $u \in C^1(\Omega)$, for an open set $\Omega \subset \mathbb{R}^n$. I want to show that $|u|$ is differentiable almost everywhere. I intuitively understand that this is true, but ...
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0answers
23 views

Distribution Convergence in PDE theory

I'm trying to follow an Example (5.1) from a PDE book (Vasy). I was having trouble following the proof of the following question (modified): Define a bump function: ${\delta_j}^{-n}\phi(\frac{x}{\...
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6answers
1k views

Why is it useful to show the existence and uniqueness of solution for a PDE?

Don't get me wrong, I understand that it is important in mathematics to qualitatively study the problems given. But I would like to know to what extent this helps for example, to actually solve the ...
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1answer
72 views

Is a weakly differentiable function differentiable almost everywhere?

I am working with Sobolev spaces. Let's suppose $\Omega \subset \mathbb{R}^n$ is an open set. A function $u: \mathbb{R}^n \to \mathbb{R}$ in $L^1(\Omega)$ is said to be weakly differentiable if there ...
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1answer
35 views

Is Laplacian of product of functions (one is smooth) square integrable?

Let $\Omega\subset \mathbb{R}^d$: open, connected. Suppose $u\in L^2(\Omega)$ and $\Delta^n u\in L^2(\Omega)$ for $n=1,\dotsc,N$, where $\Delta$ is in the weak sense. Let $\zeta\in C_c^{\infty}(\...
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0answers
101 views

If $F∈H^{-1}(Ω,ℝ^d)$ and $∃p∈\mathcal D'(Ω):\left.F\right|_{\mathcal D(Ω,ℝ^d)}=∇p$, then $∃\overline p∈H^{-1}(Ω):F=∇\overline p$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ and $\mathcal D(\Omega,\mathbb R^d):=C_c^\infty(\Omega,\mathbb R^d)$ $H^{-1}(\Omega):=H_0^1(\Omega)'$...
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0answers
37 views

Why the use of a certain weak formulation?

For an open set $\Omega \subset\mathbb{R}^n$, consider the following problem: $$\left\{\begin{matrix} - \Delta u + u=f & \mbox{ in }\ \Omega \quad \quad \quad \quad \quad \\ \ \ \ \quad \...
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0answers
27 views

If $p$ is a distribution such that $\nabla p$ is regular, then $p$ must be regular too

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $q\ge 1$ Each $f\in L^1_{\text{loc}}(\Omega)$ can be identified with $\langle f\rangle\in\mathcal ...
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0answers
15 views

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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0answers
56 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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0answers
30 views

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
24 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
45 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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2answers
66 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
30 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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1answer
24 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 ...
1
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1answer
37 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
28 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 ...
3
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1answer
106 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^...
3
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1answer
42 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 ...
0
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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 $H^1_0$ ...
0
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1answer
44 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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2answers
47 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
49 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
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
48 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 ...
0
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1answer
22 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{...
0
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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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0answers
27 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 ...
1
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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 $C^1$...
0
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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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2answers
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
13 views

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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0answers
14 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 ...
2
votes
1answer
916 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 ...
0
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0answers
48 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 ...
0
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0answers
17 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 ...
0
votes
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
1
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1answer
36 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''\...
1
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1answer
59 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 ...
0
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0answers
36 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. $$ \int_\Omega\textrm{grad}u \cdot \textrm{grad} v=\int_\Omega fv\quad \forall v \in H^1(\Omega) $$ ...
3
votes
2answers
230 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!