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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1answer
25 views

How do I prove the solution operator for my elliptic PDE is bounded?

For all $f\in L^2(0,1)$, I know tat there exists a unique $T(f) \in H^1(0,1)$, such that $$-T(f)''(x)+xT(f)'(x)+T(f)(x)=f(x) \quad \forall x\in(0,1).$$ I am wondering how to prove that $T(f)$ is ...
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19 views

Prove u has a version u^*∈C^(0,γ) (U┴-) , for γ=1-n/p, with the estimate ‖u^* ‖_(C^(0,γ) (¯U))≤C‖u‖_(W^(1,p) (U)) [closed]

Let U be a bounded , open subset of R^n, and suppose ∂U is C^1. Assume n<p≤∞,and u∈W^(1,p) (U). Prove u has a version u^*∈C^(0,γ) (U┴-) , for γ=1-n/p, with the estimate ‖u^* ‖(C^(0,γ) (¯U))≤C‖u‖(...
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1answer
45 views

PDE Sobolev Space, prove the equation holds weakly

Prove for all $f\in L^2(0,1)$, there exists a unique $T(f) \in H^1(0,1)$, such that $$-T(f)''(x)+xT(f)'(x)+T(f)(x)=f(x) \quad \forall x\in(0,1).$$ I know we need to use Lax-Milgram theorem and find a ...
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1answer
44 views

Basic questions about the sobolev space $H^\infty(\mathbb{R})$

Let's consider $H^\infty(\mathbb{R})$ to be the intersection of all Sobolev spaces $H^s$ for $s\geq0$, that is, $$ H^\infty(\mathbb{R}):=\bigcap_{s\geq 0}H^s(\mathbb{R}). $$ I am wondering some ...
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1answer
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+50

A Sobolev space on a perforated domain with extension by a solid vector field

Presentation : Let $\Omega \subset \mathbb{R}^3$ an open bounded regular set and $B=B(a,r)$ a ball such as $\bar{B} \subset \Omega$. I'm studying the following space : $$V=\{v|_{\Omega \setminus B} \...
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32 views

If $u \in H^1(\Omega)$ with $\Delta u \in L^2(\Omega)^*$, is $u \in H^2(\Omega)$?

Suppose $\Omega$ is a bounded smooth domain. If $u \in H^1(\Omega)$ is such that $\Delta u \in L^2(\Omega)^* \subset H^1(\Omega)^*$, does it follow that $u \in H^2(\Omega)$? By $-\Delta u$ I mean the ...
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23 views

Integration by parts for mixed partial derivatives.

I'm trying to find the weak form of a PDE, and I've gotten stuck. My problem is: Let $\Omega \subseteq \mathbb R^2$ be smooth and bounded, and $n = (n_1,n_2)$ be the unit normal pointing outward. Is ...
2
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1answer
51 views

If $u\in H_0^1(\Lambda,\mathbb R^2)\cap H^2(\Lambda,\mathbb R^2)$, is $\nabla^\perp\cdot\Delta u\in L^2(\Lambda)$?

Let $\Lambda\subseteq\mathbb R^2$ be open (and sufficiently regular for the subsequent consideration), $u\in H_0^1(\Lambda,\mathbb R^2)\cap H^2(\Lambda,\mathbb R^2)$ and $$w:=\nabla^\perp\cdot\Delta u,...
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1answer
48 views

If $\nabla\cdot u=0$ and $w=\operatorname{curl}u$, then $\int w=0$

Let $\Lambda\subseteq\mathbb R^2$ be open, $u\in C^1(\Lambda,\mathbb R^2)$ with $\nabla\cdot u=0$ and $$w:=\frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2}.$$ How can we show that $...
2
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0answers
30 views

Convergence of difference quotient in $L^{p}(\mathbb{R}^{n})$

Let $f \in W^{1,p}(\mathbb{R}^{n})$, where $p \in (1,\infty)$. Let us define $f^{i}_{h}$ as $$ f^{i}_{h}(x) := \frac{f(x+he_{i}) - f(x)}{|h|}. $$ Prove that $$ ||f^{i}_{h} - D_{i}f||_{L^{p}(\mathbb{R}^...
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0answers
40 views

Show that for all $w$, there is a unique $v$ with $\nabla\cdot v=0$ and $w=\nabla\wedge v$

Let $\Omega\subseteq\mathbb R^2$ be open and $w:\Omega\to\mathbb R$. I've read that there is a unique $v:\Omega\to\mathbb R^2$ with \begin{align}\nabla\cdot v&=0,\tag1\\\int v(x)\:{\rm d}x&=0,\...
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35 views

Terms in the sum

Given a multi index $\alpha = (\alpha_1,\alpha_2,...,\alpha_d)$, the order $|\alpha|$ of $\alpha$ is defined by $$|\alpha|=\sum_{i=1}^d\alpha_i$$ Let $D^{\alpha}u = \prod_{i=1}^{d}(\frac{\delta}{\...
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38 views

Two problems about Sobolev Space

Problem $1 .$ Show that the function $u: B(0,1) \rightarrow \mathbb{R},$ defined by $$u(x)=u\left(x_{1}, \ldots, x_{N}\right):=\left\{\begin{array}{ll} 1 & \text { if } x_{N}>0 \\ 0 & \text ...
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26 views

test spaces and variational formulation in finite elements for elliptic problems

I am an engineer trying to understand the basics behind the FEM for Helmholtz problems in acoustics. I'm studying a bit of functional analysis from the book of Hackbusch on elliptic differential ...
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1answer
48 views

weak solution / variational calculus

We have a one dimensional boundary value problem $$ - (\omega u_x)_x = f \, \text{for} \, -1<x<1 \\ u(-1)=u(1) = 0 $$ with $$ \omega(x): = \sqrt{(1-x^2)} \, \text{and} \, f(x) := x \, \text{for} ...
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2answers
134 views

Must continuous $H^1(\mathbb{R}^2)$ function tend to zero at infinity?

Here, $H^1(\mathbb{R}^2)$ is the standard Sobolev spaces for $L^2(\mathbb{R}^2)$ functions whose weak derivative belongs to $L^2(\mathbb{R}^2).$ My question in the title comes from calculus of ...
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1answer
41 views

weak solution/variational method and the Lax Milgram Theorem

I am trying to understand the variational method and the connection to the Lax-Milgram-Theorm. I don't know how to use the theory to solve this exercise. Let $\epsilon > 0$ and we have the ...
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2answers
59 views

Weak derivative in $L^1$

Consider $f:(0,1)\mapsto L^1(0,1)$ defined as $f(t)(x)=t\chi_{[0,t]}(x)$. I want to show that $\frac{f(t+h)-f(t)}{h}$ does not converge in $L^1(0,1)$. I have $\frac{f(t+h)-f(t)}{h}=\frac{(t+h)\chi_{[0,...
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32 views

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

Let us consider $f\in {L^2(\mathbb{R}^N)}$ such that there exists the weak derivative $\partial_{x_1}(|f|^2)\in {L^2(\mathbb{R}^N)}$, can we ensure that $\displaystyle \int_{\mathbb{R}^N}\partial_{x_1}...
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0answers
43 views

Show that a function $u$ has weak derivatives which satisfy the transport equation. [duplicate]

Let $v \in L^1_\text{loc}(\mathbb{R})$ weakly differentiable, $b\in \mathbb{R} \setminus \{0\} $ constant and $u:\mathbb{R}^2 \to \mathbb{R}$ defined by $u(x,t):= v(x-bt)$. Show that $u$ has weak ...
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0answers
37 views

Show that a weakly differentiable function $u$ on $(0,1)$ has a representative in the set of continuous functions

Let $u \in H^1((0,1))$ ($H^1(\Omega)$ being the set of weakly differentiable functions over $\Omega \subset \mathbb{R}^d$, also known as Sobolev-space) which satisfies the following inequality: $$\...
2
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0answers
21 views

Regularity of the one dimensional Poisson equation

Let $-\infty < a < b < \infty$ and set $U = (a,b)$. A weak solution of the Poisson's equation $\Delta u = f$ subject to $u = 0$ on the boundary with $f \in L^2(U)$, is a function $u \in H_0^1(...
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0answers
37 views

Not compact embeddings of Sobolev spaces in $L^p$

We know that if $1\leq p < \infty$, the following embbedings of Sobolev spaces in $\mathbb{R}^k$ are true: $\text { if } 1 \leq p<k, W^{1, p}\left(\mathbb{R}^{k}\right) \subset L^{q}\left(\...
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1answer
71 views

Weak derivative of $\log\left(\log\left(1+\frac{1}{|x|}\right) \right)$

I want to prove that if $n\geq 2$, the function of $u(x)=\log\left(\log\left(1+\frac{1}{|x|}\right) \right),\;|x|<\frac{1}{e}$ admits weak derivative respect of each component. I expose my ...
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1answer
52 views

A question in the proof that Lipchitz continuous functions implies $W^{1,\infty}$. [duplicate]

In $\S 5.8.2$ of Evan's PDE book, there is a theorem about characterization of $W^{1,\infty}$. Here it says On the other hand assume now $u$ is Lipschitz continuous; we must prove that $u$ has ...
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0answers
67 views

Show that the rarefaction solution is a weak solution.

Suppose $F: \mathbb{R} \rightarrow \mathbb{R}$, strictly convex function and consider the Riemann problem $u_{t}+(F(u))_{x}=0$, and $u_{L}, u_{R}$ are constants and satisfying $u_{L}< u_{R}$. Find ...
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12 views

Is the bound of different quotients the same as that of weak derivatives?

In $\S 5.8.2$ of Evan's PDE, there is a theorem relating to different quotients and weak derivatives. Theorem 3 (ii) Assume $1 < p < \infty$, $u \in L^p(V)$, and there exists a constant $C$ ...
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0answers
31 views

positive part of Banach space-valued Sobolev map

I refer to: W. Arendt, M. Kreuter - Mapping theorems for Sobolev spaces of vector-valued functions (https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/kreuter/adoi-sm8757-4-2017.pdf) ...
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19 views

The difference between Gateaux differentiable and Hadamard differentiable in the contest of linear functional

I have function in Rn plane in direction $βx−F$ , where $βx$ is the point probability with cumulative distribution function of $B(m)=1\{m>x\}$. Now, I can’t understand the difference between ...
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1answer
35 views

Integration by Parts for Weak Derivative?

We have integration by parts formula: $$\int_U u_{x_i}v\ dx=-\int_{U}u v_{x_i}\ dx$$ for any $u,v\in C^1_c(U)$. Now I try to generalize it to weak case, that is, the formula holds for $u,v\in H^1_0(U)$...
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1answer
35 views

Weak Derivative of a Function

I am working on fundamental solutions and I need to solve the distributional/weak derivative of the following which I have written in the formula for weak derivative, $\int_{-\infty}^{\infty}\frac{1}{...
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0answers
23 views

Proving that the dirac delta function is equivalent to the derivative of Heaviside function $H(x)$ using integration and measure

The dirac delta function is $ {F_\epsilon} x := \begin{cases} 0 & : x < 0 \\ \dfrac 1 \epsilon & : 0 \le x \le \epsilon \\ 0 & : x > \epsilon \end{cases}$ Integrating this function ...
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0answers
58 views

Do we need to complex-conjugate in the definition of a weak derivative?

Let $U \subset \mathbb{R}^n$ be an open set. If $p, q$ are conjugate exponents, then the duality pairing $$ \langle -,- \rangle_{\mathsf{L}^q,\mathsf{L}^p}: \mathsf{L}^q(U;\mathbb{C}) \times \mathsf{L}...
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1answer
38 views

Heaviside Function Derivative

I came across this link that proves how the derivative of the heaviside function is the delta function, but I would like to ask whether -H'(-x) = $\delta$(x) in a distributional sense of course. Also, ...
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1answer
21 views

Explicit weak solution of a first-order evolution equation with periodic condition

I am reading the topic on weak solutions of first-order ODEs, and I am wondering whether there is a closed-form formulas for the solution in some concrete scenarios. For example, given $f\in L^2(\...
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0answers
16 views

Fundamental theorem of calculus for multidimensional Lipschitz function

Let $f:\mathbb{R}^n\to \mathbb{R}$ be a Lipschitz function, and $u,v\in \mathbb{R}^n$. Does the following formula hold: $$ f(u)-f(v)=\int_0^1 D f(tu+(1-t)v)^T (u-v)\,dt, $$ where $Df$ denotes the ...
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0answers
16 views

Weak variational formulation with non-homogeneous bondary condtitions

I am trying to find the weak variational form of the PDE \begin{align*} - \Delta u + u &= f(x,y), \hspace{.5cm} (x,y) \in T \\ u &= g_1(x), \hspace{.5cm} (x,y) \in T_1 \\ u &= g_2(y), \...
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1answer
32 views

Why is $\int_\Omega(\Delta u)\cdot u^*\;dx=-2\int_\Omega \text{tr}\;Du^*Du\;dx$?

I'm trying to read an "An Introduction to Stochastic PDEs" (see http://hairer.org/notes/SPDEs.pdf). In the second chapter "Some Motivating Examples", they write down the Navier-Stokes equation for ...
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1answer
22 views

Moser Trudinger Inequality

Sobolev Embedding says that $W^{1,p}_0\rightarrow L^q$ continuously for $1<q<p^{*}$ and $1<p<N$. For the case $p=N$, $W^{1,N}_0\rightarrow L^{\infty}$ is not true. My confusion is the ...
2
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0answers
84 views

Weak formulation and application of Lax-Milgram

I am working with the following PDE: \begin{align*} -\nabla \cdot (a(x) \nabla u) + b(x) u = f(x) &, \hspace{.5cm} x = (x_1,x_2) \in \Omega, \\ u = 0&, \hspace{.5cm} x \in \partial \Omega_1 ...
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0answers
44 views

Confusion about absolutely continuous function

I want to prove that every absolutely continuous function on $[a,b]$ admits weak derivative, and it coincides with its derivative a.e. . So I pick $f\in AC([a,b])$, and $\phi\in \mathcal{C}^{\infty}...
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1answer
38 views

Example that Sobolev space is the completion of $C^k$

I know that the completion of $C^k$ under the sobolev norm is the sobolev space $W^{k,p}$, and I was trying to find an example to show that $C^k$ is not complete with respect to the sobolev norm ...
2
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0answers
36 views

Uniqueness of entropy solution

I am starting to study hyperbolic equations of conservation laws. I have read that there exists a unique entropy solution of the problem $$ \left\{\begin{array}{l} u_t+f(u)_x=0~,~~(x,t)\in\mathbb{R}\...
2
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1answer
27 views

Weak $W^{k,p}$-convergence implies weak $W^{m,p}$ convergence of the $\nabla^{k-m}$-sequence

I have the following question: given $m$, $p$, and $\Omega \subset \mathbb R^n$ bounded or unbounded, if the sequence $$u_i \rightharpoonup u$$ weakly in $W^{k,p}(\Omega)$, then is it true that $$\...
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0answers
36 views

Existence of higher weak derivative(s) implies lower

Let $\Omega\subset\mathbb R^2$ be an open set, and let $f\in L^1_{loc}(\Omega)$ have a weak derivative $f_{xx}\in L^1_{loc}(\Omega)$. Does this imply the existence of $f_x\in L^1_{loc}(\Omega)$? If ...
2
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1answer
63 views

The recovery of $W^{1,p}(\Omega)$ to $C^1(\Omega)$?

In Chapter 9 of Brezis' Functional Analysis, there is a Remark without proof. ($\Omega \subset \mathbb{R}^N$) "... Conversely, one can show that if $u \in W^{1,p}(\Omega)$ for some $1 \le p \le \...
3
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1answer
94 views

Solutions of p-Laplace equation

I found that for the following problem \begin{cases} -\Delta_p u = 1,&x\in B_1(0)\\ u = 0,\quad &x\in\partial B_1(0) \end{cases} where $B_1(0)$ is the unitary ball of $\mathbb{R}^N$ and $\...
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0answers
29 views

Prove $\nabla u=Du$ if $u\in\mathcal{C}^1(\Omega)$

I'm studying an introduction to Sobolev spaces and I found this remark which I can't prove: If $u\in\mathcal{C}^1(\Omega)$, then $u$ admits weak gradient $Du$ and $Du=\nabla u$ a.e. in $\Omega$. So ...
1
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1answer
20 views

Weaker ACL property if only one weak derivative, with respect to one variable, exists?

Consider $f:R^n\to R$, integrable and weakly differentiable with respect to the $n$-th variable only. We do not know whether weak partial derivatives exist as functions with respect to $x_1,...,x_{n-1}...
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1answer
41 views

Why distributional derivative of $\frac{\sin nx}{n}\to 0$

I come across following example But I done following calculation $<DI_{f_j},\phi>=-<I_{f_j},D\phi>=-\frac{1}{j}\int_{\mathbb R}\sin(jx)\phi'(x)dx=-\frac{1}{j}[\sin(jx)\phi(x)]_{-\infty}^{...

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