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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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Properties of Functions in $W^{1,p}(\Omega)$ and Their Weak Derivatives

Let $\Omega \subset \mathbb{R}^{N}$ and $W^{1,p}(\Omega)$ be Sobolev Space. Then we let $u\in W^{1,p}(\Omega)$ and define (i) $u^{+} := \max\{0,u\}$ (ii) $u^{-} := \max\{0,-u\}$ Then, we claim that $u^...
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Distributional derivative and non-distributional derivative

In which points is the following function differentiable (in non-distributional sense)? $$ f(t)= t \theta(t) + (1/2t^2-t+1/2)\theta(t-1)+(t-2)\theta(t-2)$$ My solution: \begin{align*} f'(t)&= \...
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Convergence of Mollifier in $W_0^{l,p}(\Omega)$

I want to prove that for $u \in W_0^{l,p}(\Omega)$ we have convergence of mollifiers $u_\rho \rightarrow u$. I appreciate that $u_\rho \rightarrow u$ in $L^p(\Omega)$ so just need to show convergence ...
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Partial integration on hypersurfaces - Why is there no “boundary integral”

In https://homepages.warwick.ac.uk/staff/C.M.Elliott/DziEll13a.pdf on page $299$ the notion of weak derivatives on hypersurfaces is introduced via For a $2$-dimensional hypersurface $\Sigma \subset \...
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25 views

Weak derivatives and self-adjoint operators

I have a question concerning an operator based on weak derivatives: Let $\Omega \subset \mathbb R ^3 \times \mathbb R ^3$ be an open and bounded set with smooth boundary and $0 \in \Omega$. Then, the ...
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Definition of weak derivative.

For $u \in C^1(\Omega)$ and $\phi \in C_0^{\infty}(\Omega)$ one has $$ \begin{split} \int_{\Omega}\nabla|u| \phi dx &=-\int_{\Omega}|u|\nabla \phi dx\\ &=-\int_{[u>0]}u\nabla \phi dx +\...
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If $f\in C^1$, how can we determine the weak derivative of $x\mapsto\min(1,f(x))$?

Let $f\in C^1(\mathbb R)$ and $$g(x):=\min(1,f(x))\;\;\;\text{for }x\in\mathbb R.$$ How can we determine the weak derivative of $g$? Let $\varphi\in C_c^\infty(\mathbb R)$ and $r\ge0$ with $$\...
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Is there a version of Taylor's theorem with integral remainder which is valid if the function is only differentiable in a weak sense?

Short question: Is there a version of Taylor's theorem with integral remainder which is valid if the function ($\mathbb R\to\mathbb R$) is only differentiable in a weak sense? Or at least if the ...
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Integrabilty of distibutions

Let $T : D(\Omega) \to \mathbb R$, be a generalised function. What can we say about its integrability? For example, does it belong to $L^p(\mathbb R)$? I am interested in the case of non-regular ...
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Why are test functions in the definition of weak derivatives required to be $\mathcal{C}^{\infty}$?

Recall that a function $u \in L^1_{\text{loc}}(a,b)$ is said to be weakly differentiable with weak derivative $\nu$ if the equation \begin{align} \int_{a}^{b} u(x) \phi'(x) dx = - \int_{a}^{b} ...
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To show a function is in a Sobolev space, can we use weak spherical derivatives?

For example, say $\Omega=B(0,1)$ in $\mathbb{R^2}$, and I have a function represented by $$u(x)=\begin{cases}\ln\ln1/|x|& |x|\leq e^{-2}\\\ln(2) &e^{-2}<|x|\leq 1.\end{cases}$$ There is a ...
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Does the weak derivative satisfy the product rule?

For a locally integrable function $f$, a weak derivative $f'$ satisfies the following relation: $\int{f\phi^{\alpha}}=(-1)^{\alpha}\int{f'\phi}$. Does a weak derivative also satisfy the product rule? ...
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strong convergence in $W^{1,p}$ if $L^p$ convergence of function and all derivatives

Let $\{u_n\}_{n \in \mathbb{N}} \subset W^{1,p}(\Omega)$ for $p \in (1, \infty)$ and $\Omega \subset \mathbb{R^n}$ open, bounded. If $u_n \to u^* \in L^p(\Omega)$ and $\nabla u_n \to 0 \in L^p(\...
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Is a weak derivate of $f$ always a classical derivative of some $g$?

Let $\Omega \subseteq \mathbb{R}^n$ be open, $p \in [1,\infty]$, $\alpha$ be a multi-index with $n$ entries. If $v,w\in L^p(\Omega)$ we call $w$ the weak-$\alpha$-derivative of $v$ if $$ \forall \...
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Shallow water equation entropy concept

I am a beginner in shallow water equation. I am interested in the equation $$h_t+(hu)_x=0$$ $$(hu)_t+(hu^2+\frac{1}{2}gh^2)_x=0$$ I have the following doubts 1)Weak solutions are not unique in ...
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Uniqueness of weak derivative

In this result, I understand almost everything but I don't understand why we using $\Omega' \Subset \Omega$ what is the major role of this why not we directly use $\Omega$. thank you
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Find the weak derivative

I want to find the weak derivative of $f(x)=1$ for $x\in(0,1)$ and $f(x)=0$ for $x\in(1,2)$. So basically it is constant ae. I was expecting the weak derivative to be $0$. However, when calculating, ...
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Distributional derivative of absolutely continuous function

In $\textit{Rudin, Functional Analysis, p. 148} $, the example 6.14 says that if $\Omega \subset \mathbb{R}$ is an interval and $f$ is a function of bounded variation which is left continuous at every ...
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Weak derivative of a Sobolev function in unbounded domains

I have following setup: $f\in W^{1,1}(\mathbb{R}^3)$ with $f\geq 0$ and $\int_{\mathbb{R}^3}f=1$, how can I show that the weak derivative of $\tilde{f}(x):=\int_{\mathbb{R}^2}f(x,y,z)\,\text{d}y\text{...
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Solving a distribution differential equation

The exercise is to solve $$x u'(x) = \delta(x).$$ By using the definitions $$\begin{cases}(u'|\varphi) = (-u|\varphi') \\ (fu|\varphi) = (u|f\varphi) \end{cases}$$ we get to solve $$(-u|\varphi + ...
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Show that the variational formulation has at most one solution

We have the problem: $$ -u''(x) + u(x) = f(x) ,\quad \quad x \in [0,L] $$ $$u(0) = 0 $$ $$u'(L) + u(L) = 4 $$ I then put it into variational form (hopefully correctly done) with introduction of $v \...
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Question about to Weak derivative of $|x|$

As I know that the function $f(x)=|x|$ is not differentiable.but in the weak sense it has weak derivative my question is it again weak derivative exists for this function I.e., suppose $f_1$ is ...
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What is the motivation of defining weak derivative as it is?

I've been reading lately about reproducing kernel Hilbert spaces (RKHS) and Gaussian processes (GP) and during my studies I came across with the concept of weak derivative and Sobolev spaces. I have ...
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Question on defining $L^p$ weak solution

Let us consider quasilinear second order PDEs. The following can have a weak formulation which depends only on $u$, not $u',u''$, so we can consider $L^p$ weak solution of this problem. $$u_t-u_{xx}=...
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Weak derivative of $u(x)=|x|$ not belong in $W^{1,p}(-1,1)$

Consider the functión $u \in W^{1,p}(-1,1) $, defined by $u(x)=|x|$, we know its weak derivative is $$g(x)=\left \{ \begin{matrix} 1 & \text{if }x\in(0,1) \\ -1 & \text{if } x \in (-1,0) \...
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Weak differentiability vs differentiability almost everywhere

Is there any relation between weak differentiability and differentiability almost everywhere? Does one imply the other? Does Lipschitz continuity imply both of them? Thank you.
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Truncation of a Sobolev function is still Sobolev

Let $\Omega \subset \mathbb{R}^d$ a regular domain (compact boundary $C^1$) e let $u \in H^1(\Omega)$. Let $tr(u) \equiv \alpha$. I would like to prove that the truncated function $$\bar{u}=u \chi_{\{...
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Definition of the weak derivative involving the mean curvature

Source: https://homepages.warwick.ac.uk/staff/C.M.Elliott/DziEll13a.pdf Definition 2.11 of the weak derivative: A function $f \in L^1(\Gamma)$ has the weak derivative $v_i=D_if \in L^1(\Gamma)$, $i \...
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Weak derivative question

I am confused of this example in Evans. Why is $|Du|$ calculated in this example?
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Weak Jacobian of Proximal Operator

Given a convex function $g(x):\mathbb{R}^n\rightarrow \mathbb{R}$, the proximal operator of $g$ is defined as $P_g(x)=\underset{u}{\arg\min}\quad \frac{1}{2}||x-u||_2^2+g(u)$. Since $g(x)$ is ...
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Properties of the solution of conservation laws

I learnt that if $u\in L^{\infty}$ is a weak solution of the IVP $u_t +f(u)_x=0$ with $u(x,0)=g(x)$ $\forall x \in \mathbb{R}$ Then $\int_{\mathbb{R}}u(x,t)dx=\int_{\mathbb{R}}g(x)dx$ $\forall t>...
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What are $\delta$-shock solutions?

I studied that a measure valued solution of a conservation law $u_t+f(u)_x=0$ is a measurable map $\eta : y \rightarrow \eta_y \in Prob(\mathbb{R^n})$ which satisfies $\partial_t(\eta_y, \lambda) +\...
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Differentiation in $L^p$ and Weak Differentiation.

In Weiss and Stein's book on Fourier analysis in Euclidean spaces, they define the notion of a 'derivative in $L^p$'. To be precise, they define the difference operators $$ (\Delta_h f)(x) = \frac{f(...
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Weak derivative

I really need help with this exercise: Let $f \in L_2 (\mathbb{R})$. Show the equivalence of the following statements: (a) $f \in H_1 (\mathbb{R})$. (b) The function $\xi \mapsto \xi \hat{f}(\xi) \...
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Weak derivative of radial function

I want to prove that some radial function u is weakly differentiable in R^n, how do I translate that to a condition over the real function f(r) =u(r), for r>0?
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Sobolev spaces for vector-valued functions

Let $ \Omega \subseteq \mathbb{R}^3$ be open. How is defined the space $W^{1,p}(\Omega)$ for vector valued functions $f:\Omega \to \mathbb{R}^3 $? What is the norm in $W^{1,p}(\Omega)$?
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weak solution of PDE and apply Lax-Milgram

Can someone help me for this problem? Write the weak formulation of: $$\left\{\begin{align} -\frac{\partial^2u}{\partial x^2}-5\frac{\partial^2u}{\partial y^2}=f\quad&\text{in}\quad\Omega\...
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Locally closed in the sense of distributions implies closed?

Let $F \in L^{p}(\mathbb{(-1,1)^{n}}; \Lambda^{2}\mathbb{R}^{n})$ be an $L^{p}$ $2$-form on the open cube $(-1,1)^{n}.$ Suppose we know that for every $x \in (-1,1)^{n},$ that there exists $0 < r = ...
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How is the differential of a Sobolev function on a manifold regarded as an a.e. defined section of $T^*M$?

Let $(M,g)$ be a smooth compact Riemannian manifold, and let $f \in W^{1,p}(M)$ for $p\ge 1$. (I don't assume $p>\dim M$). I have seen in various sources that people refer to the weak derivative ...
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Weak derivative of absolute value of function

Let $\Omega \subseteq \mathbb{R}^n$ be a domain. Suppose $u$ is locally integrable (i.e. $u\in L_{loc}^1(\Omega)$) and has a locally integrable weak derivative $\partial_i u$. Is there a way to find ...
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Solve $xu = 0$ in the sense of distribution

I got stuck at showing $\delta_0$ solves $xu=0$ in the sense of distribution (up to some constant). The hint states to decompose $\phi = \phi(0)g(x)+x\varphi(x)$ for some function $g(x),\varphi(x) \...
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1answer
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The restriction of a variable in a Sobolev function is a.e. Sobolev

Let $I,J$ be bounded open intervals in $\mathbb{R}$. How to show that $W^{1,1}(I\times J) $ is embedded in the Bochner space $L^1(I, W^{1,1}(J))$? Is there a standard reference where this is ...
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1answer
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Is weak derivative a bounded operator from $H^k(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$?

I'm struggling to become comfortable with the concept of the weak derivative and Sobolev space. In my textbook, it is proved that the Sobolev space $H^k(\mathbb{R}^n)$ is a Hilbert space, and it is ...
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1answer
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Is the function $f(x,y)=y \cdot \text{sign}(x)$ Sobolev?

Is the function $f(x,y)=y \cdot \text{sign}(x) \in W^{1,p}\big((-1,1) \times (-1,1)\big)$ for some $p \ge 1$? I think not but I am not sure if my reasoning below is correct. I would like to get some ...
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1answer
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Is there a Sobolev map $\mathbb{R}^4 \to \mathbb{R}^4$ whose differential zig-zags between a value and its negative?

Let $\Omega \subseteq \mathbb{R}^4$ be open subset. Does there exist a non-smooth Sobolev map $f \in W^{1,p}(\Omega,\mathbb{R}^4)$ such that $df \in \{A,-A\}$ almost everywhere, where $A \in \text{GL}(...
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Elliptic PDE- solution in weak sense

I looking for some literature, where I can find strategy how to solve elliptic PDE in weak sense - define notion of weak solution. Can anyone recommend me something ? Or some links to solved examples. ...
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$u,v \in W^{1,2}(\Bbb{R})$, then $\int_\Bbb{R}{uv'}dx=-\int_\Bbb{R}{u'v}dx$

So I need to show that, given $u,v \in W^{1,2}(\Bbb{R})$, then \begin{equation} \int_\Bbb{R}{uv'}dx=-\int_\Bbb{R}{u'v}dx \end{equation} My attempt has been this: If $u,v \in W^{1,2}(\Bbb{R}) \...
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2answers
55 views

Weak derivative and Locally summable functions

I have three question regarding the appearance of the space of locally summable functions in the definition of weak derivatives and sobolev spaces. The deifinition of weak derivatives from Evans: ...
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30 views

Convergence of energy estimates of weak solutions.

I have a question that is somewhat related to PDEs. Take a sequence of functions $\{t\mapsto R_k(t,\cdot)\}_k$ where for each $k$, $R_k(t,x)\in C^{1}(\mathbb{R}_{\geq 0}\times (0,1))$. Let $V$ be a ...
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Weak derivative Gronwall’s inequality

I was wondering whether anyone knows about the existence of a Gronwall’s inequality for the weak derivative. Thanks.