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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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
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2answers
463 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
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1answer
560 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 ...
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0answers
14 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 ...
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1answer
30 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 ...
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2answers
257 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
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2answers
30 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 ...
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45 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 ...
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1answer
39 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 ...
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1answer
48 views

Weak derivative question

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?
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2answers
48 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 ...
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2answers
75 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!
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1answer
34 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 ...
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1answer
54 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 ...
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0answers
37 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*} ...
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2answers
73 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?
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58 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 ...
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1answer
46 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 ...
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30 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 ...
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26 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 ...
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1answer
44 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 ...
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0answers
32 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 $$\int_0^T \int_\Omega u(t) \varphi'(t) = -\int_0^T \langle u'(t), \varphi(t) \rangle$$ ...
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1answer
45 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 $W^{1,2}(\Omega)$.?
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31 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 ...
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1answer
37 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 ...
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0answers
50 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 ...
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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 ...
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60 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
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1answer
123 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, ...
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0answers
64 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 ...
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0answers
30 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} ...
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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 ...
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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 ...
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2answers
75 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 ...
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1answer
70 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 ...
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1answer
32 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, ...
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1answer
124 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 ...
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2answers
179 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 $$\int_0^T u(t)\phi'(t) = -\int_0^T v(t)\phi(t)$$ ...
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0answers
23 views

A derivative identity, for any multi-index.

I'm trying to prove by induction on the multi-index $\alpha$, that, $$\sum\limits_{j=\frac{|\alpha|}{2}}^{|\alpha|}\sum\limits_{|\beta|=2j-|\alpha|}c_{\beta}x^{\alpha}[m_z(x)]^{j+1}=D^{\alpha}m_z(x)$$ ...
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1answer
61 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 ...
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1answer
74 views

Weak derivatives equals zero

Im just learning sobovel space, I was wondering if the weak derivate holds similar things of the original derivate. Let $U\subset \mathbb{R^n}$ is a open set, and $u\in W^{1,p}$ if $$Du=0 \ \ a.e$$ ...
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0answers
30 views

Weak stochastic integral

I recently encountered the following object, referred to as "weak stochastic integral" in the book of SPDE's by Prévôt/Röckner [PR07]: $$ \int_0^T \langle \Psi \,\mathrm dW(t), \Phi(t)\rangle $$ A ...
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1answer
214 views

Properties of weak derivatives in Sobolev spaces

In PDE Evans 2nd edition, pages 261-263, there is a theorem and its proof which concerns the four properties of weak derivatives. Unfortunately, I do not understand the fourth property, which I will ...
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1answer
24 views

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$?

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$, where $H^1(R;H^1(R^n))=\{f\colon R\to H^1(R^n)\colon \int \|f(t,\cdot)\|_{H^1}^2\,dt <\infty \quad \text{and} \int \|f'(t,\cdot)\|_{H^1}^2\,dt <\infty \}$ ...
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1answer
58 views

A weak derivative of a function of two variable which depends only on one variable

Assume that $f: \mathbb R^2 \rightarrow \mathbb R$ is locally integrable and has a locally integrable weak patrial derivative $\partial_1 f.$ Let moreover $f$ depends only on the first variable: ...
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1answer
43 views

Find a weak derivative of this function

Let $B(0,1)$ be the open unit ball in Euclidean space $\mathbb R^2$ and $(a_n)_{n=1}^\infty$ be a dense subset of $B(0,1)$. I wish to show for fixed $s\in (0,1)$ the function $$ ...
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1answer
282 views

Weak Derivative Heaviside function

I have to prove that the Heaviside function $$ H(x):=\begin{cases} 1 &\mbox{if } x \in [0,+\infty) \\ 0 &\mbox{otherwise}\end{cases} $$ doesn't admit weak derivative in ...
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1answer
41 views

If $u \in L^2(0,T;H^1)$ has a distributional derivative $u' \in L^2(0,T;H^{-1})$, does $(u^+)' \in L^2(0,T;H^{-1})$ make sense?

If $u \in L^2(0,T;H^1)$ has a distributional derivative $u' \in L^2(0,T;H^{-1})$, does $(u^+)' \in L^2(0,T;H^{-1})$ exist, i.e., is $u^+$ weakly differentiable in time? By $u^+$ I mean the positive ...
0
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1answer
86 views

negative Sobolev space contains $L^1$ for a compact domain

I'd like to use something like Aubin-Lions lemma for the following spaces: $$ C^{0, \alpha}(B) \subset L^1(B) \subset W^{-1, q}(B),$$ with $B \subset \mathbb{R}^n$ being a compact, say a closed ball ...
2
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1answer
54 views

Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$. Define the truncation function$$T_\epsilon(x) = \begin{cases} ...