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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15
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1answer
1k views

Are a weak derivatives and distributional derivatives are different?

For simplicity, given a real function $f\in L^1_{loc}(\Omega)$, we define both weak or distributional derivatives by $\int f'\phi = - \int f \phi'$ for all test functions $\phi$. Now, take $\Omega = ...
10
votes
1answer
741 views

Is a continuous function with continuous weak derivatives of class $C^1$?

Let $f$ be a real valued continuous function of many variables whose weak derivatives of first order are continuous. Is this function equals a.e. function of class $C^1$ ?
8
votes
2answers
1k views

Product rule of weak derivatives

I am working on proving the following proposition: If $u,v\in {W^1(\Omega)}$ and $uv,uDv+vDu\in L^1_{\operatorname{loc}}(\Omega)$, then we have the product formula $$D(uv)=uDv+vDu.$$ The definition I ...
7
votes
3answers
454 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
7
votes
1answer
263 views

Justification of formal derivative

there I'm having the following problem and so far I didn't find anything in the literature on this. $\Phi\in C^1(\overline{\Omega}\times[0,1])$ with $\Phi'(x,0)=\Phi'(x,1)=\Phi(x,0)=0$ for every $x\...
6
votes
1answer
266 views

Finite Element Method Weak Formulation

I have a question about the weak formulation of a PDE in finite element analysis. Suppose we have the following two-dimensional PDE: $$ \Delta \cdot u(x,y) = q(x,y) $$ where $q$ is given, $u$ is ...
6
votes
2answers
1k 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 ...
5
votes
2answers
180 views

Does convergence in H1 imply pointwise convergence?

I'm trying to figure out if convergence in $H^1(a,b)$ implies pointwise convergence (by the way: what is the usual name of this space?). It is defined to be Hilbert space of absolutely continuous ...
5
votes
2answers
247 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 $v=u'$....
5
votes
1answer
161 views

Sobolev spaces in one-dimensional vs multidimensional

Here in Wikipedia, it is said that in the one-dimensional case, it is enough to assume that the $(k-1)$-th derivative of the function $f$, is differentiable almost everywhere and is equal almost ...
5
votes
1answer
2k views

Finding Weak Solutions to ODEs

I'm wondering if anyone has a reference to a good set of notes on finding weak (distributional) solutions to ODEs, or has any tips or tricks. For example, $$ xy^\prime=0 $$ has a classical solution ...
5
votes
1answer
71 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 ...
5
votes
1answer
178 views

If $f_j \rightharpoonup f$ weakly in $W^{1,p}$ then $f_j \to f$ strongly in $L^p$?

Suppose $1<p<\infty$ and $\Omega$ is an open bounded set in $\mathbb R^n$ with nice boundary (say Lipschitz or even better). Let $(f_j)_j \subset W^{1,p}(\Omega)$ s.t. $f_j \rightharpoonup f$ ...
4
votes
2answers
309 views

Differences between $C_c^\infty[0,T]$ and $C_c^\infty(0,T)$

I believe it is true that: If $f \in C_c^\infty(0,T)$, then $f(T)=f(0)=0$. $C_c^\infty(0,T) \subset C_c^\infty[0,T]$ $C^\infty(0,T) \subset C_c^\infty[0,T]$ If $f \in C_c^\infty[0,T]$, it doesn't ...
4
votes
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^\...
4
votes
1answer
1k 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 $\...
4
votes
2answers
573 views

about weak derivative ( Sobolev Spaces )

the following afirmation is true ? Consider $\Omega $ a bounded and smooth domain . Let $u \in W^{1,p} ( \Omega)$ ( p>1). Supose that $u \geq 0$. let $\alpha >1$ . Then $\nabla u ^{\alpha} = \...
4
votes
1answer
273 views

Weak derivatives

How can I prove that $|\nabla u|=|\nabla|u||$ when $u$ is regular enough for example Lipschitz or $W^{1,1}_{loc}$. Other question is about the pointwise derivative when $f:[0,1]\to R$ is BV is that ...
4
votes
0answers
179 views

Differentiation of norm in Banach space (explanation of text needed)

Let $Y$ be uniformly smooth Banach space. Consider the convex $C^1$ functional $\Phi:Y \to \mathbb{R}$ defined $$\Phi(y) = \frac{1}{q}\Vert y \Vert^q_{Y}.$$ Its derivative $\varphi:Y \to Y'$ is a ...
3
votes
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 ...
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!
3
votes
1answer
490 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 ...
3
votes
1answer
338 views

Winding number for Lipschitz curves

For a closed curve $\Gamma: [0,1] \to \mathbb{C}$ (i.e. $\Gamma(0) = \Gamma(1)$) let $$ \iota_\Gamma(z) := \frac{1}{2\pi i} \int_\Gamma \frac{dw}{z-w} = \frac{1}{2\pi i} \int_0^1 \frac{\Gamma'(t)}{z-\...
3
votes
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 ...
3
votes
1answer
100 views

How does integration over $\delta^{(n)}(x)$ work?

For a math paper I need to be able to evaluate $\int_{-a}^{a}\delta^{(n)}(x)\ f(x)\ dx$ for differentiable $f$. I know that it is 'supposed' to equal $(-1)^nf^{(n)}(0)$: $$\int_{-a}^a\delta^{(n)}(x)\ ...
3
votes
2answers
107 views

Weakly convergence in $W^{1,q}, 1<q<\infty$

Let $(x_n)$ be weakly convergent against $x$ in the Sobolev space $W^{1,q},1<q<\infty$. Now I have to show, that $(\dot{x_n})$ converges weakly against $\dot{x}$ in $L^q$. (With the point I ...
3
votes
1answer
29 views

$u\in W^{1,1}(U) \Rightarrow u^+\in W^{1,1}(U)$ , where $U\subset\mathbb{R}^n$ open

Im trying to show that for an open set $U\subset\mathbb{R}^n$ and a function $u\in W^{1,1}(U)$, also the positive part $u^+$ is in $W^{1,1}(U)$. My idea is the following: Let $E\subseteq U$ defined ...
3
votes
2answers
222 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 ...
3
votes
1answer
426 views

Weak Derivatives and Lp spaces

I just want to know how to prove two properties of weak derivatives and $L^{p}$ spaces if they are true, the first one just involves weak derivatives: If we have a locally summable function $u: U \...
3
votes
2answers
410 views

Weak / Classical derivative

I know the definitions of both weak and classical derivative. But I am trying to see the classical derivative as a weak derivative. When we have $\int f' \varphi = -\int f\varphi'$ for all $\varphi\in ...
3
votes
1answer
820 views

Prove that $f$ does not have a weak derivative

Consider a function $f:\mathbb{R} \rightarrow [0,1 ]$ defined by: $\begin{equation*} f(x)=\left\{ \begin{array}{rl}0 & \text{if } x\leq 0,\\ 1 & \text{if } x\geq 1, \\ 1/2 & \text{if } x\...
3
votes
1answer
251 views

Consistency of derivative definitions in Sobolev spaces

Just for the sake of completeness, I begin defining the Sobolev space $H^m(\mathbb{R}^n), \; m \in \mathbb{N}$, as the following set: $H^m(\mathbb{R}^n) = \{u \in L^2 : P^{\alpha} F u \in L^2,\; \...
3
votes
0answers
44 views

How do I show that a distribution is a function?

While reading Grigor'yan's book on the heat kernel I have encountered the following definition of a Sobolev space on a Riemannian manifold $M$: $W^2 (M) = \{ u \in W^1 (M) : \Delta u \in L^2 (M) \}...
3
votes
2answers
73 views

Mistake in reasoning about Sobolev spaces

I am new to Sobolev spaces and, while trying to construct a proof, I make some subtle mistake that I cannot detect. The setting: let $C \subset \Bbb R^n$ be a closed, measure-$0$ set. Let $U = \Bbb R ...
3
votes
1answer
224 views

Weak derivative: Showing a function is equal to zero a.e.

I am very beginner in the theory of weak derivative. I am trying to fix the following problem: Suppose that $f\in{L_{loc}^{1}}$ and $\int_{a}^{b}f(x)\phi({x})dx=0$ for all $\phi\in{C_{0}^{\infty}}$ ...
3
votes
0answers
156 views

Generalized chainrule for Sobolev functions with a cut-off

let $\Omega\subset\mathbb{R}^n$ be a bounded domain and $f\in C^1(\bar\Omega \times (0,g(x)),[0,1])$ and $f(x,\cdot)$ increasing and $g(x)\in\mathbb{R}$ continuous (maybe better, Lipschitz?). I want ...
3
votes
0answers
298 views

When the weak derivative just is the strong (or classical) derivative?

When the weak derivative just is the strong (or classical) derivative? For instance, can we prove that weak derivate $Du\in C^\alpha$(or $C^0$) implies $u\in C^{1,\alpha}$(or $C^1$).
2
votes
2answers
2k views

Laplace transform of the derivative of the Dirac delta function

$$\int_{0}^{\infty} \delta'(t) e^{-st} \ dt = \delta(t)e^{-st} \Big|_{0}^{\infty} + s \int_{0}^{\infty} \delta(t) e^{-st} \ dt $$ $$= 0 - \lim_{t \to 0} \delta(t)e^{-st} + s e^{-st}\Big|_{t=0} = - \...
2
votes
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 $...
2
votes
1answer
620 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 $D(u\ast\rho_{\varepsilon})...
2
votes
2answers
911 views

Inverse Laplace operator $\Delta^{-1}$ and Sobolev spaces

I'm looking for some regularity results for the inverse Laplace operator. More precisely - we're set in $\mathbb{R}^3$ and we are looking at the operator $$ \Delta^{-1}f = \frac{x}{|x|^3} \ast f$$ I'd ...
2
votes
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 ...
2
votes
1answer
287 views

Distributional derivative of absolute value function

I'm tying to understand distributional derivatives. That's why I'm trying to calculate the distributional derivative of $|x|$, but I got a little confused. I know that a weak derivative would be $\...
2
votes
1answer
554 views

Chain rule for weak derivatives of $f(u)$ where $f'$ is not bounded but $u$ is?

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$. Suppose $u$ has a weak derivative $u_x$. I want the chain rule $$\partial_x (f(u)) = f'(u)u_x$$ to hold. We know this holds if $f'$ is bounded. But I don't ...
2
votes
1answer
74 views

Weak Laplacian of $\|x\|^\alpha$

Let $\alpha> 0$ and consider the function $\|\mathbf x\|^\alpha = (x^2 + y^2)^{\frac{\alpha}{2}}$ defined on $\mathbb R^2$. I want to compute the Laplacian $\Delta (\|\mathbf x\|^\alpha)$ in the ...
2
votes
1answer
900 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 ...
2
votes
1answer
102 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^...
2
votes
1answer
76 views

Is the regularity of $u$ necessary to deduce this result? (Evans PDE)

One of the exercises in Evans book on PDEs (at the end of chapter 7) is given as follows: Assume $$u_k\rightharpoonup u\quad\mbox{in}\quad L^2(0,T;H^1_0(U)),$$ $$u_k'\rightharpoonup v\quad\mbox{in}\...
2
votes
1answer
136 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 W^{1,...
2
votes
1answer
66 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} -\...