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.

learn more… | top users | synonyms

3
votes
0answers
117 views

Using a sequence of measures to create simple functions which approximate the Radon-Nikodym derivative of the limiting measure

I have a bunch of discrete probability measures with finite support: $\mu_1,\mu_2,\dots$, which strongly converge to an absolutely continuous probability measure $\mu$ in $\mathbf{R}^2$. That is, for ...
0
votes
2answers
67 views

Does the existence of weak derivatives require the lower order derivatives also to exist?

I just want to confirm that for weak derivatives you don't require the lower order derivatives to exist in order for the higher order derivatives to exist?
0
votes
0answers
14 views
1
vote
1answer
17 views

How to solve a simple differential equation in the way of weak solutions?

I want to prove that the solution equation $y'=y$ is $y=Ce^x$, where $C$ is a constant. Here $y$ belongs to the space of linear operators on $C_0^\infty(\mathbb{R})$, and $y'$ is its weak ...
0
votes
1answer
46 views

Interpolation between derivatives

I am trying to prove the following: Let $u \in W^{2,2}(\mathbb R)$. Then $\| u^\prime \|_{L^2}^2 \leq \| u \|_{L^2} \| u^{\prime \prime} \|_{L^2}$ holds (these are meant to be weak derivatives). ...
-2
votes
0answers
13 views

Subhessians for maximum eigenvalue of a matrix

I am trying to solve a non-linear, non-smooth convex optimization problem using a generic convex optimization solver. This solver requires and (sub)gradients of the objective and the constraints, as ...
0
votes
0answers
12 views

Definition of the Second Variational Derivative In terms of The first

I know that for functional $F$ the first variational derivative at $f$ with increment $h$ is defined as \begin{align*} \delta F[f,h]= \lim_{\alpha \to 0 } \frac{F[f+\alpha h]-F[f] ]}{\alpha }. ...
1
vote
1answer
15 views

Compact support of derivatives of $u$ in weak sense

I would like to know if let $u:\mathbb{R}^n\to \mathbb{R}$ be a function in $W^{k,p}(\mathbb{R}^n)$ such that $u$ has compact support in $\mathbb{R}^n$ then, for each $|\alpha|\le k,$ $D^\alpha u$ ...
0
votes
1answer
27 views

Weak derivative of $t^{\beta-1}$

Let $g(t)=t^{\beta-1}$, where $\beta\in (0,1)$. I am pondering, if there exists a weak derivative of $g$ in space $L^1(0,T)$, $T>0$. Firstly, we see that $g\in L^1(0,T)$. Now, we are looking for ...
1
vote
1answer
18 views

Dirac Delta function to solve PDE in the sense of distribution

For each Borel set $A$ and $x\in\mathbb{R}^n$, denote Dirac measure centered at $x$ as \begin{equation} \delta_x(A)=\begin{cases} 1 & \mbox{ if } x\in A \\ 0 & \mbox{ otherwise}. \end{cases} ...
2
votes
2answers
55 views

Continuous function without a weak derivative

Let $f:\Omega\to\mathbb{R}$ be a continuous function. Is it necessarily true that $f$ has a derivative in the weak sense? That is, is there some $v:\Omega\to\mathbb{R}$ such that for every test ...
0
votes
1answer
28 views

Show that this function is weakly differentiable

I need to show that the function \begin{equation} u(x,y) = 1-x_1^2 \quad x_1>0 \\ u(x,y)=1+x_1^2 \quad x_1 \leq 0 \end{equation} is weakly differentiable on the unit ball. It is clear what the ...
5
votes
2answers
534 views

Questions about weak derivatives

There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are ...
3
votes
1answer
631 views

Second derivative Dirac delta distribution times $(x-a)^2$, intepretation

I'm not sure if this calculation is correct and if I interpret it correctly (from old exam). Show that $ (x-a)^2 \delta ''_a = 2 \delta _a $. We have for distributions $f$ and test functions ...
0
votes
0answers
24 views

Energy functional for a differential equation

Is there a variational formulation for the following differential equation: $\frac{\partial}{\partial x}(D(u,x)\frac{\partial u}{\partial x})=0 $ $x$ varies over $[0,1]$, $D$ is bounded, is ...
1
vote
1answer
38 views

Weak derivative (Sobolev spaces)

I'm reading "Functional Analysis" - Michel Willem and I can't understand the definition of weak derivative from chapter 6, namely the definition of $$ \partial^\alpha f. $$ Can you give me a concrete ...
6
votes
2answers
288 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 nowhere differentiable (or differentiable only on a set of measure 0) ? Thanks
1
vote
2answers
33 views

A step of change of variables I don't understand when show derivative of a function in $L^n$?

How (16) can goes to (17) and how to show $loglog(1+\frac{1}{|x|})\in L^n$ for $n>1$? My attempt: Let $|x|=r$, then $(x^2)^{\frac{1}{2}}=r$. Differentiate on both sides, I get ...
0
votes
0answers
46 views

Proof-verification: $C^1$-functions are weak differentiable

I wanted to prove that the strong derivative coincides with the weak derivative if the function is regular enough, i.e.: Let $U \subset \mathbb{R}^n$ be an open subset and $f \in C^1(U)$, then for ...
1
vote
1answer
42 views

divergence form of the determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
0
votes
2answers
67 views

Question about this problem If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$

In this problem If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$ I don't understand this part: $$ \int_U \partial_i u^\epsilon\,v\,dx\to 0\tag w$$ for all $v\in L^2(U)$, but now we only ...
2
votes
2answers
84 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 $Du \in L_p$. Can anyone give me some hint? Thanks!
0
votes
1answer
42 views

How to prove the chain rule with respect to weak derivatives?

The following theorem is on the textbook "weak differentialble functions". I found it confusing from the absolutely continuous part. I am writing to ask is this the only way to prove it? Can anyone ...
1
vote
1answer
73 views

Difference quotient bounded in $L^1$ does not imply the weak partial derivative exists and is in $L^1$

Let $U\subset \mathbb{R}^{n}$ be open and $V \subset\subset U$ ($V$ is open and compactly contained in $U$). $C$ is a fixed constant. Given $u\in L^1(U)$. Show by example that if we have ...
1
vote
0answers
39 views

Proof of the Poincare inequality for $W_0^{1,2}((a,b))$.

I have a question about an exercise for which I already have the solution, which I do not unterstand completely. Let $a, b \in \mathbb R$ with $0 < a < b$. Then we have \begin{align*} ...
0
votes
2answers
94 views

first order weak derivative of function $ f(x) = |x| $

let $f(x)= |x| $ how can I calculate the first order weak derivative of this function in $x=0$? Does anyone have an idea on how to calculate this?
0
votes
0answers
101 views

chain rule for weak derivatives

First I apologise for not writing properly, but I'm using a cell phone. We know that having a Sobolev function $u$, if we have a good enough function $f$, then $f\circ u$ is Sobolev and the chain ...
5
votes
1answer
53 views

Geometric Interpretation of Weak Derivative

As we know, classic derivative $f'(x)$ of a function $f(x)$ can be interpreted as the rate of change of function $f$ in each point $x.$ How about weak derivative? Since it is defined through integral ...
1
vote
0answers
31 views

Prove that a function lies in $L^1$ and in $W^{(1,1)}$ for some parameter

I want to do the following tasks Let $G:=B_1(0)\subset \mathbb R^2$ be the open ball around $0$ with radius 2 in the norm $||\cdot||$ and $u_{\rho}(x)=||x||^{\rho}_2$, $x\in G$. Show the following ...
0
votes
0answers
27 views

Second order equation and regularity

Let $U$ be an bounded open set in $\mathbb R^3$ and ${V}\subset \subset{W} \subset \subset {U}$. Let $u \in {H^1}(U)$, and $f\in L^{2}(U)$ satisfies $$\int_{\mathbb R^3} \nabla u(x). \nabla \varphi ...
0
votes
1answer
49 views

calculate weak derivate of $|x-2|^2$

Let $u$ be a function with $u(x):=|x-2|^2$ on $I:=(-1,1)$. I want to test whether $u \in H^2(I) \backslash H^3(I)$. Let $\phi$ be in $C_0^\infty(I)$. Then: $T_u(\phi '') = \int_{-1}^1 |x-2|^2 ...
0
votes
0answers
48 views

Weak time derivative for functions $u \in L^2(0,T;L^2(\Omega))$

The weak time derivative of a function $u \in L^2(0,T;H^1)$ is defined to be $u' \in L^2(0,T;H^{-1})$ satisfying $$\int_0^T \int_\Omega u(t) \varphi'(t) = -\int_0^T \langle u'(t), \varphi(t) \rangle$$ ...
1
vote
1answer
54 views

Existence of weak derivative

Can a uniformly continuous function have a weak derivative?. In other words can $C_{unif.~cont.}$ be continuously be embedded in $W^{1,2}(\Omega)$.?
1
vote
2answers
45 views

Which Sobolev-Space to use to formulate weak biharmonic equation, $H^2_0$ or $H_0^1\cap H^2$?

For the weak formulation of the biharmonic equation on a smooth domain $\Omega$ $$ \Delta^2u=0\;\text{in}\;\Omega\\ u=0, \nabla u\cdot \nu=0\; \text{on}\; \partial\Omega $$ why does one take ...
0
votes
1answer
38 views

Choosing a clever “test function” in Sobolev spaces.

Given $\mathbf{f}$ with $f_1,...,f_N\in L^2(\Omega)$ $$\int_\Omega \mathbf{f} \cdot \nabla v = 0 \quad\forall v \in H_0^1(\Omega)$$ we have $\mathbf{f} = \mathbf{0}$ a.e. since $\mathbf{f} \in ...
0
votes
0answers
59 views

How can I prove that this function doesn't have a second weak derivative?

I'm trying to determine what weak derivatives the function $$ f(x)=\begin{cases} x&\mbox{if }0<x<1,\\ 1&\mbox{if }1\leq x<2, \end{cases} $$ has. I already managed to prove that it ...
1
vote
0answers
20 views

approximate a weak derivative

Let $\Omega$ be an open bounded set of $\mathbb{R}^N$ and $u\in H_0^1(\Omega)$. Suppose $|\nabla u|>1$ on a set of positive measure, then by inner regularity of Lebesgue measure, there exists a ...
2
votes
0answers
62 views

Weak convergence of determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
1
vote
1answer
138 views

Weak derivative as an $L^2$ limit of the difference quotient

Let $u \in H^1(\mathbb{R})$. Show that $$\left\| \frac{u(x+h)-u(x)}{h} - u' \right\|_2 \to 0\quad \text{ as } h \to 0, $$ where $u' \in L^2(\mathbb{R})$ is the weak derivative of $u$. In other words, ...
1
vote
0answers
82 views

Confused about weak derivatives in Evans

I'm a bit confused about how Evans refers to derivatives at some points and if he means weak derivative. In particular on page 301 he gives the definition that if $\textbf{u} \in L^1(0,T;X)$ and ...
1
vote
0answers
33 views

Weak differentiation and derivatives of test functions

I am currently working to see if $\frac{1}{x}$ is weakly differentiable on $(0,1)$. I have reached that conclusion via integration by parts that, if so, for all $ \phi\in C^{\infty}_c$: $\int_{0}^{1} ...
1
vote
0answers
24 views

Does existence and uniqueness of a classical solution impose uniqueness of weak solutions to a pde?

I wonder if one knows that there exists a unique classical solution of a pde (for instance: Fokker-Planck equation), is one able to conclude that there isn't any weak solution of the pde, which ...
0
votes
0answers
39 views

dense subspace of $C(0,T)$

I want to prove that the space H of functions which are continuous in [0,T] with weak derivative in $L^2[0,T]$ and their value in 0 is 0, is dense in the space of continuous functions in [0,T] with ...
0
votes
2answers
78 views

Sobolev space on union of two open sets

Let $\Omega_1,\Omega_2 \subset \mathbb{R}^n$ be open sets. Let $p \in [1,\infty]$. Let $u: \Omega_1 \cup \Omega_2 \to \mathbb{R}$ be a function such that $u|_{\Omega_1} \in W^{1,p}(\Omega_1)$ and ...
2
votes
1answer
79 views

Weak star convergence leads to weak convergence of derivative?

Let $u_\epsilon \in W^{1,p}(\Omega)$ be a sequence with $\Omega \subset \mathbb{R}^n$ bounded. Let $u_\epsilon \rightharpoonup^* u$ weakly* in $L^\infty (\Omega)$ (for a subsequence) with $u \in ...
1
vote
1answer
39 views

Is it a Sobolev function?

Let $U=(-1,1)\times(-1,1)$. Define $$ u(x)=\begin{cases}1-x_1 \;\;\text{if}\;\;x_1>0,|x_2|<x_1 \\1+x_1\;\;\text{if}\;\;x_1<0, ...
1
vote
1answer
171 views

Weak derivative zero implies constant function

Let $u\in W^{1,p}(U)$ such that $Du=0$ a.e. on $U$. I have to prove that $u$ is constant a.e. on $U$. Take $(\rho_{\varepsilon})_{\varepsilon>0}$ mollifiers. I know that ...
1
vote
2answers
201 views

Distributional/weak time derivative basic question

Suppose we have $u \in L^2(0,T;H^1(\Omega))$, and $v \in L^2(0,T;H^{-1}(\Omega))$ is the weak time derivative of $u$, so by definition it satisfies $$\int_0^T u(t)\phi'(t) = -\int_0^T v(t)\phi(t)$$ ...
1
vote
0answers
24 views

A derivative identity, for any multi-index.

I'm trying to prove by induction on the multi-index $\alpha$, that, $$\sum\limits_{j=\frac{|\alpha|}{2}}^{|\alpha|}\sum\limits_{|\beta|=2j-|\alpha|}c_{\beta}x^{\alpha}[m_z(x)]^{j+1}=D^{\alpha}m_z(x)$$ ...
1
vote
1answer
64 views

If $f$ is differentiable everywhere, is $f'$ the weak derivative?

Let $f \in C^0([0,T])$ be such that $f'$ exists in the classical sense everywhere, but $f'$ may not be continuous. Is it true that $f'$ is the weak derivative of $f$ too, if it exists? I know this is ...