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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2
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0answers
29 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
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0answers
11 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 ...
0
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0answers
11 views

Weak stochastic integral

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

Show $W^{1,p}(a,b)$ is compactly embedded in $L^p(a,b)$ for any $1<p\leq \infty$

I always get these sorts of questions wrong. Any help would be appreciated. This is my answer: The problem assumes $(a,b)$ is any bounded interval in $\mathbb R$. Consider a bounded sequence ...
1
vote
1answer
30 views

Notion for weak derivatives of $L^p(0,T,X)$-functions

A definition in Evan's PDE-book from chapter 5.9.2 says (let $X$ be a Banach space): Let $u\in L^1(0,T,X)$. We say $v\in L^1(0,T,X)$ is the weak derivative of $u$ provided $$\int_0^T \phi'(t)u(t)dt = ...
2
votes
2answers
56 views

Density in $H^1_0$

Let $\Omega = \mathbb{R}^2_+=\{(x,y)\in \mathbb{R}^2; y>0\}$ and let $v \in H^1_0(\Omega)$. For $h \neq 0$ we define $D_h v = \dfrac{v(x+h,y)-v(x,y)}{h}$ such as $\forall \varphi \in ...
1
vote
1answer
70 views

Sobolev space-exercice [closed]

Let $\Omega = \mathbb{R}^2_+$. My question is: how we prove that if $v \in H^2(\Omega)$ such as $v(x,0)=0$, then $\dfrac{\partial v}{\partial x} \in H^1_0(\Omega)$ ?
3
votes
1answer
57 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)\ ...
2
votes
0answers
22 views

How to interpret $\langle (f(u))', v \rangle = \langle f'(u)u', v \rangle$?

We know that if $u$ has a weak derivative $u'$, and if $f \in C^1$ with $f'$ bounded then $(f(u))'= f'(u)u'$. But how interpret the duality pairing $$\langle (f(u))', v \rangle = \langle f'(u)u', v ...
1
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0answers
14 views

Showing $\langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle$

Let $u \in L^2(0,T;H^1)$ have weak derivative $u' \in L^2(0,T;H^{-1}).$ Let $c$ be a constant. I want to show that $$\langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle$$ where $(f)^+ = ...
0
votes
0answers
15 views

PDE weak solution , how could i prove it?

let be $ n \ge 3 $ and U a bounded set including the point $0 $ then i must show that a) the vector $ u= \frac{x}{|x|} $ belongs to $ H^{1}(U, R^{n})$ and it is an harmonic map into the sphere $ ...
0
votes
1answer
90 views

Establishing the n-th order weak derivative.

Consider $$AC^{n}:=\{f\in{AC}:f^{(k)}\in{AC}, 1\leq k \leq n-1\}$$ where $AC$ stands for the space of absolutely continuous functions. Now, let $f,g\in{L_{loc}^{1}(a,b)}$ and ...
2
votes
1answer
74 views

Approximation of a function with a polynomial of degree n.

Let $f\in{L_{loc}^{1}}$ such that $\int_{a}^{b}{f(x)}{\phi}^{(n)}(x)dx=0$ for all $\phi\in{C_{0}^{\infty}}(a,b)$. Then how do we show there exists a polynomial $P(x)$ of degree less or equal to $n-1$ ...
2
votes
1answer
54 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 ...
3
votes
1answer
76 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 $\phi\in{C_{0}^{\infty}}$ ...
1
vote
0answers
51 views

some sobolev norm estimation

I would like to show this inequality. I need help to show this inequality Let $F(\Phi)=\left|\Phi\right|^{\alpha}\Phi$ with even integer $\alpha>0$. Let $k$ be a positive integer satisfying ...
0
votes
1answer
33 views

Does this function has weak derivatives?

Let $\Omega\subset\mathbb{R}^n$ be a bounded "smooth" domain and $\Omega_\delta=\{x\in \Omega:\ d(x,\partial\Omega)<\delta\}$. Let $U_\delta$ denote the part of the boundary of $\Omega_\delta$ ...
0
votes
1answer
41 views

Basic Query On Sobolev Space

Okay, I am very new to the premise of Sobolev Spaces and there is one exercise here that's really grinding my gears. The premise is this- to what Sobolev spaces for each real number $\alpha$ does ...
7
votes
1answer
237 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 ...
1
vote
2answers
77 views

Distributional derivatives on hypersurface?

In a paper I was reading, the define a set $Q=(0,T)\times \Omega$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain, and then they write $$\langle \frac{d}{dt}u - \Delta u, \varphi ...
3
votes
1answer
132 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. $f_j \rightharpoonup f$ ...
1
vote
1answer
58 views

weak differentiability of log log function

I want to understand why the following function has a weak derivative in two or three dimensions: $w(x) = \ln |\ln|x|| , x \in B_{1/2}(0)$. Can I say that if I have a strong derivative (except for ...
2
votes
1answer
68 views

For which real values of $\alpha$ PDE $\Delta u(x,y)+2u(x,y)=x-\alpha$ has at least one weak solution?

Problem. Consider boundary value problem: \begin{cases} \Delta u(x,y)+2u(x,y)=x-\alpha, & \text{in $\Omega$,} \\ u(x,y)=0, & \text{on $\partial\Omega$,} \\ \end{cases} where $\alpha$ is ...
0
votes
1answer
36 views

Uniqueness of the solution of the continuity equation with discontinuous vector field

I'm trying to prove the uniqueness of the following continuity equation with discontinuous vector field: $ \begin{cases} m_t(t,x) + [(-\alpha x + \sigma u^{*}(x) + c) \ m(t,x)]_x = 0 \qquad (t,x) ...
1
vote
0answers
37 views

Estimate difference of solutions of an equation with a bilinear symmetric continuous form

My question refers to differential equations in Sobolev spaces. It is as follows: Let $\Omega \subset \mathbb{R}^n$ be a bounded open set. Let $a: H_0^1(\Omega) \times H_0^1(\Omega) \rightarrow ...
1
vote
1answer
67 views

First order weak derivatives of $f(x)=|x|^r$

Let $f(x)=|x|^r$ for a given real number $r$. Show that $f$ has first order weak derivatives on the unit ball $B_1(0)\subset \mathbb{R}^n$ provided that $r > 1-n$. Does anyone have an idea on how ...
1
vote
1answer
41 views

Strong differentiability in Sobolev spaces

My question is: how can one prove that if $\phi\in H^{n}(a,b)$ for all positive integer $n$, then $\phi\in C^{\infty}(a,b)$? $H^{n}(a,b)$ denotes de Sobolev space of the real interval $(a,b)$. For my ...
1
vote
1answer
57 views

$C_c^{\infty}$ is dense in $W^{k,p}(\mathbb R^n)$

As the title say, I want to prove that $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ i.e. $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n) \quad (\star)}$. In a book ...
1
vote
0answers
21 views

norm of $x \in \mathbb R^d$ is in Sobolev space

For which values of $\alpha, k,p,d$ is $$ \|x\|^\alpha \in \textrm{W}^{k,p} (B(0,1)) \quad ? $$ where $\displaystyle{ \textrm{B}(0,1) = \{x \in \mathbb R^d : \|x\|<1\}}$ This is an ...
3
votes
1answer
182 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 ...
1
vote
1answer
87 views

Finding the weak derivatives

Finding the weak derivatives: a/ $f(x)=\left|x_1 \right|$, for all $x=(x_1,\ldots, x_n)$; b/ $f(x)=\operatorname{sign} (x_1)$, where $\Omega=\{\left|x \right|<1\}$. ...
1
vote
0answers
48 views

Self-adjointness of differentiation operator

Let's say $\mathcal{H}=L^2([0,1])$ and $p$ is the operator $-i\frac{d}{dx}$ defined on $\mathcal{D}(p)=\{f\in L^2([0,1])\ |\ f'\in L^2, f(1)=e^{i\theta}f(0)\}$. I have to prove that $p$ is ...
2
votes
1answer
98 views

Has a probability density function a weak derivative

Assume I have a probability density function $\rho$ on $R$. (e.g $\rho \geq 0$ $\int \rho dx =1 $ $\rho \in L^1(R)$ ...). So $\rho$ is the density wrt the lebesgue measure. Now I try to understand if ...
0
votes
0answers
39 views

Chainrule: “piecewise smooth” and Sobolev functions

since we had a lengthy discussion on my last question (Generalized chainrule for Sobolev functions with a cut-off I didn't find an answer to it yet, I'm going to post a related more specific question ...
2
votes
0answers
94 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 ...
1
vote
1answer
192 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 $C^1$ ?
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0answers
51 views

Does existence of weak spatial derivative imply existence of classical time derivative in this situation?

Let $f(x_1,...,x_n,t)$ be a function, where $(x_1,...,x_n) \in \mathbb{R}^n$ and $t \in [0,T].$ Denote by $f_{x_i}$ the weak (partial) derivative of $f$ wrt. $x_i.$ Is it possible for ...
1
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0answers
98 views

If a sequence of smooth functions converge in the Sobolev norm, what can one say about the limiting function?

Let $W^{k,p}(\mathbb{R}^n)$ be the Sobolev space of $k$ times weakly differentiable functions on $\mathbb{R}^n$, having finite Sobolev $p$ norm. Suppose $\{f_i\}$ is a Cauchy sequence in ...
-3
votes
1answer
88 views

$H^2$ and $C^1$ - Sobolev space

Let $\Omega$ be a domain of $\mathbb{R}^d$, $u\in H^2(\Omega)$ and let $k$ be a constant. Can we tell that $k \nabla u \in C^1$n with $u$ solution of the problem $$-\mathrm{div}(k\nabla u)=f \quad ...
1
vote
1answer
43 views

Weak derivative of extended function

Let a locally integrable function $u: D \rightarrow \mathbb R$ has weak derivative $Du$ and let $K:=\operatorname{supp} u \subset D$. Let's define $v(x)=u(x)$ on $D$ and $v(x)=0$ on $\mathbb R^n ...
1
vote
1answer
101 views

Steklov Average in time and spatial gradient can be interchanged?

I'm trying to understand a proof on Steklov average and weak derivatives. Let $\Omega$ be a bounded domain of $\mathbb{R}^n$, $T>0$, $h\neq 0$ and $u\in L^2(0,T;H^1(\Omega))$ and extend $u$ by ...
4
votes
1answer
545 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 ...
7
votes
1answer
242 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 = ...
3
votes
2answers
163 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 ...
5
votes
2answers
391 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 ...
2
votes
1answer
123 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 ...
0
votes
1answer
106 views

Prove Corollary of the Fundamental lemma of calculus of variations

From the Fundamental lemma of calculus of variations we know the following: Let $u \in L_{loc}(a,b)$ and for every $\phi \in C_0^\infty$, $$\int_a^b u(x) \phi(x) dx = 0$$ holds, then $u(x) = 0$ ...
1
vote
1answer
146 views

Find a weak derivative

Find a weak derivative of the function $u:(-1;2)\rightarrow \mathbb{R}$ defined as follows: $$u(x)=\begin{cases}|x|, & \text{if}\,\,\, x<1 \\ 1-x^2, & \text{if}\,\,\, x\ge1 \end{cases}$$ I ...
2
votes
1answer
272 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
2answers
173 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 ...