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Questions tagged [weak-derivatives]

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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What's the minimum regularity needed for Picard−Lindelöf?

Consider a system of first-oder ODEs of the form $$ \dot{x}(t) = \sum_{i = 1}^n f_i(t) V_i(x) $$ where $V_i$ are Lipschitz continuous functions $V_i: U \to \mathbb{R}^n$ on some open $U ⊆ \mathbb{R}^...
Carlos Esparza's user avatar
2 votes
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59 views

Bochner-Sobolev spaces with second time derivative and embeddings

In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these spaces as $$W^{1,p,q}(a,b,X,Y) = \left\{v, \ \...
Maths_GEES 's user avatar
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1 answer
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Doubt in proof of Second Existence Theorem for Weak Solutions

I have the same doubt as in here, but none of the answers there seem to make things clearer to me. When Evan proves the Second Existence Theorem for Weak Solutions, he asserts on Step 4 that: $v-K^*v=...
Kadmos's user avatar
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2 votes
1 answer
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Convergence in $L^1_\text{loc}$ of weak derivates

Let $I$ an open inteval in $\mathbb{R}$. I'm looking for an example of a sequence of functions $(u_n)$ and a function $u$ in $I$ such that $u_n$ for all $n$ and $u$ has weak derivates $u_n'$ and $u'$ ...
matdlara's user avatar
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1 answer
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Derivative of a sign-like function

Assume a function $f: \mathbb{R}^2 \to \mathbb{R} $ such that $$f(x_1,x_2)=\begin{cases} 1 & x_2\le h(x_1,t) \\ 0 & x_2 > h(x_1,t) \end{cases}$$. I am trying to find the derivative of f in ...
user1174736's user avatar
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Function that don't belong to any Sobolev space

Let $u:B(0,1)\subseteq \mathbb{R}^N$, with $N\geq 2$ such that $$ u(x) = \begin{cases} 1 & \text{if } x_N \geq 0,\\\\ 0 & \text{if } x_N <0. \end{cases} $$ I want to show that $u\not\in W^{...
matdlara's user avatar
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-1 votes
1 answer
52 views

Inverse of the Dirichlet Laplacian?

I have read somewhere that $(-\Delta)^{-1}u$ for $\Delta :H_0^1 \rightarrow H^{-1}$ is defined as the unique weak solution to $-\Delta v= u$ with $v=0$ on the boundary (assuming $U$ is some sufficient ...
Perelman's user avatar
  • 269
1 vote
0 answers
31 views

Chain rule with Sobolev functions

Let $f: \mathbb{R} \to \mathbb{R} $ be Lipschitz and let $u\in W^{1,p}(\Omega)$, with $1\le p <\infty$ and $\Omega\subset \mathbb{R}^N$ open. Assuming that $f(0)=0$ I have to show that $f\circ u\in ...
Shiva's user avatar
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When does the weak derivative of a function exist?

I've tried to find an answer but people use integrals or explain things with Sobolev spaces which I do not know how to use in this context and have not studied(not in my syllabus). I cannot understand ...
Diananas's user avatar
1 vote
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25 views

Weak derivative operator is not sequentially continuous in $L^1_{\text{loc}}(\Omega)$

Suppose we have a sequence $(u_n)_{n\in\mathbb{N}}$ in $L^1_{\text{loc}}(\Omega)$ and $u\in L^1_{\text{loc}}(\Omega)$ such that $u_n\to u$ in $L^1_{\text{loc}}(\Omega)$ and all the functions $u_n$ and ...
Nepal Draus's user avatar
1 vote
1 answer
30 views

Discretization of the heat equation: is the bilinear form $a(u,v) = (u,v)_{L^2} - \tau (u',v')_{L^2}$ coercive for every $\tau > 0$?

I'm working on the heat equation in 1D on the domain $\Omega = (0,1)$ $$ \partial_t u(t,x) - \partial_x^2 u(t,x) = 0 $$ with boundary conditions $u(t,0) = u(t,1) = 0$ and initial temperature ...
tim-kt's user avatar
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Prove that the derivative of the mollification approaches the strong $L^p$ derivative

Let $g \in C_c(\mathbb{R})$ be such that $\int_\mathbb{R} g dx = 1$. Suppose that $f \in L^p(\mathbb{R}$ has a strong $L^p$ derivative, i.e. there is some $h$ such that $\displaystyle\lim_{y \to 0} \...
Squirrel-Power's user avatar
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DiracDelta/x = -DiracDelta'? - Use and Correctness of Statement [duplicate]

One property of the Dirac Delta Distribution is $x \delta'(x) = -\delta(x)$ because of $\int x \delta'(x) f(x) dx = -\int \delta(x) (xf(x))' dx = -\int \delta(x) (f(x)+xf'(x)) dx = -\int \delta(x) f(x)...
theta_phi's user avatar
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1 vote
1 answer
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Weak convergence in $L^p$ spaces their dual

It is well known that the dual space of $L^p$ is $L^{p^\prime}$ for $p\neq \infty$ and that these spaces are reflexive. Now assume you have an operator $L^{p^\prime} \rightarrow (L^{p^\prime})^*, x\...
Perelman's user avatar
  • 269
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0 answers
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Does the Heaviside function belong to $W^{s,p}$ for some $s>0$?

Let us consider $n$-fold tensor product of Heaviside functions: \begin{equation} H(x_1, \cdots, x_n) := \prod_{i=1}^n \chi_{[0,\infty)}(x_i) \end{equation} Then, for any bounded open set $\Omega \...
Keith's user avatar
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1 vote
2 answers
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Trace theorem in $H^2(\Omega)$ and the affective restriction on the boundary

In Theorem 2.7.4 from Kesavan Topic in Functional Analysis 2003 says that there exists a map $\gamma = (\gamma_0, \gamma_1)$ from $H^2(\Omega)$ to $(L^2(\Omega))^2$ such that If $v \in C^\infty(\...
Lucas Linhares's user avatar
2 votes
0 answers
34 views

Understand the constant related to the continuity of the extension operator in Sobolev Spaces

We know that the extension theorem steams that for a given $\Omega$ bounded domain, $\Omega \subset \subset V$ and $u \in H^1(\Omega)$ there exists $E(u) \in H^1(\mathbb{R}^N)$ such that: $E(u) = u$ ...
Lucas Linhares's user avatar
1 vote
1 answer
37 views

Continuity almost everywhere of a function in $L^2(\mathbb{R}^N)$

Let $u \in L^2(\mathbb{R}^N)$ and take $\overline{u}$ a representant of $u$. Suppose $\overline{u}(x_0) < 0$ for some $x_0 \in \mathbb{R}^N$. Can we say that there exists $\delta > 0$ such that $...
Lucas Linhares's user avatar
1 vote
0 answers
29 views

Some additional conclusions on the Extension Theorem of Sobolev Spaces

Extension theorem says that for a given $\Omega$ bounded domain and $u \in H^1(\Omega)$ there exists $E(u) \in H^1(\mathbb{R}^N)$ such that: $E(u) = u$ almost everywhere in $\Omega$ $\text{supp}(E(u))...
Lucas Linhares's user avatar
1 vote
0 answers
34 views

The support of derivatives of the extension operator in Sovolev spaces

The Sobolev extension theorem states that for a given $\Omega$ bounded domain and $u \in H^1(\Omega)$ there exists $E(u) \in H^1(\mathbb{R}^N)$ such that: $E(u) = u$ almost everywhere in $\Omega$ $\...
Lucas Linhares's user avatar
2 votes
1 answer
46 views

A property of solution operator of a elliptic PDE involving positive part of a function

For a given $u \in L^2(\mathbb{R}^N)$, there is a unique $S(u) \in H^1(\mathbb{R}^N)$ such that $$ \int_{\mathbb{R}^N} \nabla S(u) \nabla \varphi + S(u) \varphi = \int_{\mathbb{R}^N} u \varphi, \quad \...
Lucas Linhares's user avatar
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1 answer
27 views

Weak derivatives and function composition

I know there are lots of questions about this on this website, but usually it is for the opposite order of composition as the one I have here. Suppose $u(t,x)$ is weakly differentiable (in both ...
qp212223's user avatar
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Positive part of the solution - relation between $B(u)$ and $B(u^+)$

Given $u \in L^2(\mathbb{R}^N)$, by Riesz Representation Theorem there exists a unique $B(u) \in H^1(\mathbb{R}^N)$ weak solution of $$ -\Delta v + v = u, \quad \mathbb{R}^N. $$ Given $u \in L^2(\...
Lucas Linhares's user avatar
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0 answers
36 views

Restriction of solution operator of a elliptical equation in $\mathbb{R}^N$

For a given $u \in L^2(\mathbb{R}^N)$, we know by Riez Representation Theorem that there exists a unique $B(u) \in H^1(\mathbb{R}^N)$ such that $$ \int_{\mathbb{R}^N} \nabla B(u) \nabla \varphi + B(u) ...
Lucas Linhares's user avatar
1 vote
0 answers
99 views

Why is $u\in H^1$ enough s.t. $\int \nabla u\cdot\nabla v$ is a well defined expression?

Definitions Consider a distribution $T\in \mathcal{D}'(\Omega)$, then we define the distributional derivative by "duality" (Adjoint operator etc.), meaning: $$T'(\phi):=-T(\phi')\ \ \forall\...
user1313292's user avatar
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1 answer
45 views

Showing $f \in W^{1,p}(\Bbb{R}^2)$ for $p \in [1,2)$ but $f$ is not continuous.

Fix $\psi \in C_c^\infty(\Bbb{R}^2)$ such that $\psi \vert_B \equiv 1$ where $B$ is the unit ball in $\Bbb{R}^2$. Define $f \in C_c^\infty(\Bbb{R}^2)$ by $$f(x_1,x_2)=\psi(x_1,x_2)\frac{x_1^2}{x_1^2+...
homosapien's user avatar
  • 4,225
1 vote
2 answers
126 views

How can one rigorously demonstrate that some function (not bounded) is $ H^1(\Omega)$

Let $\Omega = \{(x,y)\in\mathbb R^2: \sqrt{x^2+y^2}<\frac{1}{2}\}$ be a bounded domain and $v(x,y) = \ln\left|\ln\sqrt{x^2+y^2}\right|$. How to show that $v\in H^1(\Omega)$ I would like to know if ...
Noname's user avatar
  • 585
2 votes
0 answers
33 views

Characterising extrema with weak derivatives

Let $u\in H^1(U)\cap C(\overline U),$ where $U$ is a connected open set. Then, what can we say about $Du$ in a neighbourhood of a maximum $x_0$ of $u$ on $\overline U?$ I have never seen a discussion ...
Ma Joad's user avatar
  • 7,534
1 vote
1 answer
42 views

$\log \log (\frac{2}{||x||})$ in $W^{1,2} \setminus L^\infty$ of a ball in $\mathbb{R}^2$

Let $\Omega = B(0,1/2) \subset \mathbb{R}^2$, and let $$ u(x) = \log \log \left(\frac{2}{||x||}\right) \quad for \quad x \in \Omega. $$ I have to show that $u$ is in $W^{1,2}(\Omega) \setminus L^\...
Contrad's user avatar
  • 47
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1 answer
81 views

How to calculate the gradient of a radial function

Let $\Omega = B(0,1) \subset \mathbb{R}^N$. I have to determinate for which $\alpha > 0$ such that $$ u(x) = ||x||^{-\alpha} \quad \text{for} \quad x \neq 0 $$ is in $W^{1,p}(\Omega)$. I found a ...
Contrad's user avatar
  • 47
0 votes
0 answers
28 views

Modulus of a function in $H^2_0(\Omega)$ is also in $H^2_0(\Omega)$?

It is well known that for a bounded domain $\Omega \in \mathbb{R}^N$ and for a function $u \in W^{1,2}_0(\Omega)$ it is true that $|u| \in W^{1,2}_0(\Omega)$. I would like to know it the same happens ...
Lucas Linhares's user avatar
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0 answers
18 views

Minimal Requirements on the Boundary for the Existence of Extension Operators

If $\Omega$ is some open set in $\mathbb{R}^N$ with a bounded boundary, then if $\partial \Omega$ is sufficiently regular, there exists a continuous operator $P: W^{1,p}(\Omega) \rightarrow W^{1,p}(\...
Perelman's user avatar
  • 269
1 vote
0 answers
45 views

Is the Fourier series solution to the plucked string problem a weak solution to the wave equation?

I am looking at the Fourier series method applied to the plucked string problem: \begin{align} u_{tt} = u_{xx} & , \qquad 0 < x < \pi, t>0 ; \nonumber \\ u(0,t) = u(\pi,t) & , \qquad ...
ivan44's user avatar
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0 answers
25 views

$f_j \rightharpoonup f$ and $g_j \rightharpoonup g$ weakly in $H^1(\Omega)$ $\Rightarrow$ $\nabla(f_kg_k)\rightharpoonup \nabla(fg)$ in $L^p(\Omega)$

Suppose that $\Omega \subset \mathbb{R}^d$ is bounded with a Lipschitz boundary and $f_j \rightharpoonup f$ and $g_j \rightharpoonup g$ weakly in $H^1(\Omega)$. Show that, for a subsequence, $\nabla(...
Mr. Proof's user avatar
  • 1,575
1 vote
1 answer
72 views

Derivative of $H(c)=\mathbb E[(c-X)\chi_{X\leq c}]$

Let $c$ be a real consant, $X$ a real random variable and $H(c)=\mathbb E[(c-X)\chi_{X\leq c}]$. Show that $$H'(c)=P[X\leq c].$$ I already proved this if $X$ has a density function. But what if $X$ ...
Moritz's user avatar
  • 151
1 vote
1 answer
232 views

Derivative of expected value

Let $X\geq 0$ be a real random variable, $Y(t), t\geq 0$ a stochastic process and $A\geq 0$. I want to determine the derivative of $$F(z)=E[e^{X}(A-Y(X))\chi_{\{Y(X)\leq z\}}]$$ $X$ and $Y(X)$ have no ...
marc's user avatar
  • 242
0 votes
1 answer
63 views

Fundamental lemma of calculus of variations for higher order-derivatives

Consider an expression of the form $$0=\int_a^b f_0(t) \varphi(t) + f_1(t)\varphi'(t) + \ldots + f_n(t) \varphi^{(n)}(t) \, \mathrm{d}t, \qquad \forall \varphi \in C_c^\infty(a,b)$$ and $f_0, \ldots, ...
motionart's user avatar
  • 158
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0 answers
26 views

Properties of weak derivatives: Let $u\in W^{k,p}(U)$. Then, prove that $D^{\alpha}u \in W^{k-|\alpha|,p}.$

Properties of weak derivatives: Let $u\in W^{k,p}(U)$. Then, prove that $D^{\alpha}u \in W^{k-|\alpha|,p}.$ My attempt:- We know that $D^\alpha u$ exists in the weak sense and $D^\alpha u \in L^p(U), \...
Unknown x's user avatar
  • 849
0 votes
1 answer
71 views

$L^2$-convergence implies convergence in $H_0^k$

Suppose that $Mx_n$ converges to $M x$ in $L^2([0,1], \mathbb{R}^m)$ for $n \to \infty$ and $M x_n \in H_0^k([0,1], \mathbb{R}^m)$, for $M \in \mathbb{R}^{m\times m}$ and a $k \in \mathbb{N}$. The ...
motionart's user avatar
  • 158
0 votes
0 answers
21 views

Doubt about Pólya–Szegő inequality

Pólya–Szegő inequality states that if $u \in W^{1,p}_0(\Omega)$, then $$ (1) \,\,\,\,\,\,\,\,\,\ \int_{\Omega} |\nabla u|^p dx \geq \int_{\Omega} |\nabla u^\ast|^p dx, $$ where $u^\ast$ denotes the ...
Lucas Linhares's user avatar
0 votes
0 answers
11 views

Schwarz rearrangement for a function with one variable fixed

Decompose Euclidian space as $\mathbb{R}^N = \mathbb{R}^{n} \times \mathbb{R}^{m}$. If $u \in H^1(\mathbb{R}^N)$ I know it is well defined the Schwarz rearrangement of $u$ and the Pólya–Szegő ...
Lucas Linhares's user avatar
1 vote
1 answer
25 views

Fixing a variable of a $H^1_0(\Omega)$ function

Let $\Omega \subset \mathbb{R}^{N} = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$. Suppose that $\Omega \subset A_1 \times A_2$, where $A_i \subset \mathbb{R}^{N_i}$. If $u \in C^\infty_0(\Omega)$, I ...
Lucas Linhares's user avatar
2 votes
0 answers
60 views

Computing the distributional derivative of a particular function

Let us consider the function $$ f(x) = \begin{cases} 1 & \text{if } x \in \mathbb Q \\ -1 & \text{if } x \in \mathbb R \setminus \mathbb Q \end{cases} $$ I have a main question about the ...
zelda's user avatar
  • 21
2 votes
1 answer
127 views

Variational formulation for the heat equation

Let $J = (0,T)$, $T > 0$, $G = (a,b) \subset \mathbb{R}$ (finite interval), and $f \in C(J;L^2(G))$. I consider the heat equation with zero Dirichlet boundary conditions and with initial value $u_0 ...
julian2000P's user avatar
1 vote
0 answers
15 views

Reverse inequality for Riemann-Liouville fractional integral

Definition 2.3. The Riemann-Liouville fractional integral operator of order $\alpha \geq 0$, for a function $f \in C_\mu,(\mu \geq-1)$ is defined as $$ \begin{aligned} J^\alpha f(t) & =\frac{1}{\...
Grandes Jorasses's user avatar
0 votes
0 answers
45 views

Distributional and classical derivative

Let $\Omega$ be an open subset of $ \mathbb{R}^n$, and $f\in L^1_{loc}(\Omega)$. Given a multiindex $\alpha$, suppose that the $\alpha$-distributional derivative $D^\alpha T_f$ is a distribution ...
Shiva's user avatar
  • 133
0 votes
1 answer
152 views

Density of compactly supported functions in a Sobolev space

Denote by $W^{k,p}(\mathbb{R})$ the Sobolev space of functions in $L^p(\mathbb{R})$ whose all weak derivatives up to order $k$ belong to $L^p(\mathbb{R})$ with the norm $$ \| f \|_{W^{k,p}(\mathbb{R})}...
Tony419's user avatar
  • 797
0 votes
0 answers
30 views

A problem of continuity at an endpoint

Let $\alpha>0$ and $1\leq p \leq \infty$. We define a weighted Sobolev space $$X^{\alpha,p}(0,1):= \{u\in W_{loc}^{1,p}(0,1):u\in L^p(0,1), x^\alpha u'\in L^p(0,1)\},$$ where the notation $u\in W_{...
Sam Wong's user avatar
  • 2,307
0 votes
0 answers
89 views

Definition of weak directional derivative

I have recently come across the notion of weak directional derivatives in the context of Sobolev functions. Let $u\in W^{1,p}(\Omega)$ denote a Sobolev function for arbitrary exponent $p\geq 1$ and a ...
HelloEveryone's user avatar
3 votes
0 answers
85 views

Weak derivative different from the classical one.

If $f \in C^k(\Omega)$, with $\Omega\subset \mathbb{R}^n$ open and such that $\partial \Omega$ is regular, than by Gauss-Green formulas, we can show that the classical $\alpha$ derivative $D^\alpha f$ ...
Shiva's user avatar
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