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.

Filter by
Sorted by
Tagged with
2
votes
0answers
26 views

How to prove that extensions of Sobolev functions if Sobolev?

Problem: Let us introduce this definition: Definition: given an open set $\Omega \subset \mathbb{R}^d$ and $u \in L_{loc}^1(\Omega)$ we say $v \in L_{loc}^1(\Omega)$ is $\frac{\partial}{\partial x_i}u$...
1
vote
0answers
26 views

Do the sum of some function belong to specific spaces

Consider the following function $f(x)$ on $[a,c]$ for $a,c \in \mathbb{R}$ \begin{cases} h (x) &a< x<b \\ g(x) & b\leq x<c \end{cases} Assume h and g are $C^1$ on ...
0
votes
0answers
36 views

Weak derivative of positive part of Sobolev function

I've been looking through the notes on Sobolev Spaces here: https://www.math.ucdavis.edu/~hunter/pdes/ch3.pdf Proposition 3.22 states that if $u\in L^1_{loc}(\Omega)$ has weak derivative $\partial_i u ...
1
vote
0answers
25 views

Question about Hubbard's analytic definition of quasiconformality. Aren't weak derivatives only defined up to a set of measure zero?

I'm a bit confused about something that appears in the fourth chapter of Hubbard's Teichmüller Theory text. In his statement of Weyl's lemma, he says that if $f$ is a distribution whose weak/...
1
vote
0answers
26 views

Weak form of cahn hilliard equation

I am trying to write the weak formulation of Cahn-hilliard equation as under. $\frac{\partial \phi}{\partial t} = \nabla .\{\phi(1-\phi) [\phi(1-\phi)\nabla \mu - \nabla (A(\phi)q)]\}$ $\frac{\partial ...
0
votes
0answers
46 views

Strong and weak derivatives

Let $\Omega \subset \Bbb{R}^n$ be a connected open subset with $\partial \Omega$ Lipschitz. If a function $u$ is infinitely weakly differentiable within $\Omega$ wrt Lebesgue measure $\lambda$, i.e. ...
0
votes
0answers
13 views

If $g_n\in C^\infty_c((0,\infty)\times (0,1))\to\mathbf{1}_{(-\infty,T]}(t) e^{-x}$ strongly in $L^1$, what happens to $\partial_tg_n, \partial_xg_n$?

Let $g_n \in C^\infty_c((0,T)\times (0,1))$. Suppose that the sequence $g_n$ converges strongly in $L^1$ to $\mathbf{1}_{(-\infty,T]}(t) e^{-x}$. What do $\partial_t g_n$ and $\partial_x g_n$ converge ...
3
votes
2answers
63 views

Find a weak solution of the ODE

Find a weak solution to the following ODE: $u' + u = H_0(x)$ where $H_0(x) = \begin{cases} 0 & x < 0 \\ 1 & x \geq 0 \end{cases}$ My professor advised that we try to guess the solution and ...
3
votes
1answer
47 views

Find weak derivative of sign-like function

Let $f:\mathbb{R}^2\to \mathbb{R}$ be a function defined as follows. $$f(x,y)=\begin{cases} 1, &\text{if } x>y\\ -1,&\text{if } x<y. \end{cases}$$ To compute the weak derivative $f_x$ of ...
0
votes
0answers
18 views

Convergence of Galerkin methods with shape functions satisfying boundary conditions

I have seen multiple references here in Math Stack Exchange and online to solution of PDEs using functions that satisfy a given set of boundary conditions. However, the convergence of those, is rarely ...
0
votes
1answer
39 views

Check if a function is in H2

I want to check if the Function $u(x)=|x-1|$ is in $H^{2}(\Omega) , \Omega = (0,2) $ My idea: I want to check if the function has a weak derivative. But that would only proof $L^{1}$ what should I do ...
2
votes
0answers
39 views

Showing that $u(x,y)=xy\ln|\ln\sqrt{x^2+y^2}|$ belongs to $ H^2$, but not to $C^2$.

I want to show that that $u(x,y)=xy\ln|\ln\sqrt{x^2+y^2}|$ is $C^1(\Omega)$ and $H^2(\Omega)$, but not $C^2(\Omega)$, where $\Omega=B_{\frac{1}{2}}(0)$ is the ball centered at $0$ and with radius $1/2$...
1
vote
0answers
51 views

Weak derivative for sobolev spaces in manifolds and vector bundles

In these lecture notes https://www3.nd.edu/~lnicolae/Lectures.pdf I read how one can define sobolev spaces between a riemannian manifold and a vector bundle over it. First I don't understand the ...
1
vote
1answer
39 views

Let $\Omega\subset \mathbb{R}^n$ be open and bounded. Is it true that $C^{0,1}(\Omega) \subset W^{1,\infty}(\Omega)$?

Let $\Omega\subset \mathbb{R}^n$ be open and bounded. If $\partial \Omega \subset C^1$, the result follows from Theorem 4 on page 279 of "Partial Differential Equations" by Evans. Without ...
0
votes
1answer
23 views

Approximate $f\in \text{Lip}_K(\Omega)$ with $f\geq 0,f=0$ on $\partial \Omega$ by $f_n\in C_c(\Omega)\cap \text{Lip}_K(\Omega)$ with $f_n\geq 0$.

Let $\Omega\subset \mathbb{R}^n$ be open and bounded. For $0<K\leq \infty$, $\text{Lip}_K(\Omega)$ denotes the set of lipschitz-continuous functions on $\Omega$ with lipschitzconstant less than or ...
2
votes
1answer
66 views

Why do we write $H_0^1(\Omega) \cap H^2$ instead of only $H^2_0(\Omega)$?

I've seen that when we deal with Poisson equation with homogeneous boundary conditions, let's say in 2D with a convex domain $\Omega$, we write that the regularity of $u$ is $H^2 \cap H_0^1(\Omega)$. ...
1
vote
0answers
20 views

Explicit bound for $C^{2,\alpha}$ elliptic theory

Let $\Omega$ be an open, bounded with $C^2$ boundary (or smooth as we want). A result about elliptic regularity is given as follows. If $\Omega_0\subset\subset \Omega$ and $u$ is a weak solution of $...
0
votes
0answers
42 views

Finite element method with two different Dirichlet boundary conditions

I have the problem like this $$ -\Delta u = f \ \ \text{on}\ \Omega \\ u = g_1 \ \ \text{on} \ \partial \Omega_1 \\ u = g_2 \ \ \text{on} \ \partial \Omega_2 $$ If we choose $$ V_1 = \{ \nu_1 \in ...
2
votes
0answers
39 views

If $u\in L^2((0,T),H_0^1(\Omega))$ satisfies $u(0)=u_0$ in $H^{-1}(\Omega)$, do we necessarily have $\left.u_0\right|_{\partial\Omega}=0$?

Let $\Omega\subseteq\mathbb R^d$ be bounded and open, $V:=H_0^1(\Omega)$, $H:=L^2(\Omega)$, $u_0\in C(\overline\Omega)$, $T\in(0,\infty]$, $I:=(0,T)$ and $u\in\mathcal L^2(I,V)$ admit a weak ...
3
votes
1answer
114 views

Difference between weak and distributional derivatives

I'm studing weak and distributional derivatives and solutions and I have a few questions about it. From my understanding, one defines a weak derivative of $u \in L^{1}_{loc}(\Omega)$ such that $$ \...
3
votes
0answers
56 views

Compact embedding $H^1_{0,rad}(B(0,1),|\log(1/|x|)|)$ in $L^1(B(0,1))$

I was studying the paper "On Trudinger–Moser type inequalities with logarithmic weights" (by Martha Calanchi and Bernhard Ruf) when on page 1987 there was a sequence $(u_n)$ converging ...
3
votes
0answers
68 views

Generalized second derivative of a concave and piecewise $C^2$ function

It is mentioned in page 20 of this paper that if $f: \mathbb{R}_+ \to \mathbb R$ is a concave and piecewise $C^2$ function, then the generalized second derivative of $f$ is a signed measure $\mu_f$ ...
1
vote
1answer
44 views

An continuity property in $W^{1,1}$

How can we prove that for any $\phi_n\to \phi$ in $W^{1,1}(\Omega)$ then there exist always: $$\lim\limits_{n\to\infty} \int_{\Omega}|\nabla\phi_n(x)| dx = \int_{\Omega} |\nabla\phi(x)|\ dx$$ ? The $\...
0
votes
1answer
37 views

Sobolev space of maps with higher dimensional target?

Consider the space $W^{2,2}(\Omega)$ where $\Omega = [0,1]^3$. This space contains continuous functions of the form $f:\Omega \to \mathbb{R}$. I am curious, can a Sobolev space be constructed by maps ...
-1
votes
1answer
58 views

Can I extend a weak derivative to the whole $\mathbb R$?

Let $\rho(x,t)>0$ be a Holder-continuous function on $\mathbb R^d\times(0,\infty)$. I know that: there exists (in weak sense) $\partial_t\rho\in L^1_\text{loc}(\mathbb R^d\times(0,\infty))$; there ...
0
votes
1answer
132 views

Show that D'Alembert solution gives distributional solution to the wave-equation

I am given the following question: Let $\phi, \psi\in C^0(\mathbb{R})$ and $$u(x,t)=\frac{1}{2}(\phi(x+t)+\phi(x-t)+\frac{1}{2}\int_{x-t}^{x+t}\psi(y)dy$$ Show that $u$ is a solution to the wave-...
3
votes
1answer
88 views

integration by parts in Sobolev space $W^{1,1}(\mathbb R^d)$

Let $f,g\in W^{1,1}(\mathbb R^d)$ such that: $$ \int_{\mathbb R^d}|\nabla f|\,|g|\,<\infty\,,\quad \int_{\mathbb R^d}|f|\,|\nabla g| <\infty \,.$$ Can I say that: $$ \int_{\mathbb R^d} \nabla\!f\...
3
votes
0answers
74 views

Inclusion of Sobolev spaces (from a proof by Bogachev - Krylov - Röckner - Shaposhnikov)

In a book by Bogachev-Krylov-Rockner-Shaposhnikov, I found the following statement that concludes a proof but I do not understand. I underlined in red the critical parts. $\rho(\cdot,t)>0$ is a ...
0
votes
1answer
50 views

Continuity of the $L^2$-norm with respect to a parameter

Let $\psi:\mathbb R^d\times (0,\infty)\to\mathbb (0,\infty)$ be a Holder-continuous function such that $$ \psi(\cdot,t)\in W^{1,2}(\mathbb R^d)$$ for almost every $t>0$. Using the continuity of $\...
1
vote
0answers
18 views

Stay $\|f\| \in W^{1,1}(0,T;\mathbb{R})$ provided $f \in W^{1,1}(0,T;H)$

Supose that $f \in W^{1,1}(0,T; H)$ where $H$ is a Hilbert space. Can we conclude that $\|f\| \in W^{1,1}(0,T;\mathbb{R})$? Since $f \in L^1(0,T;H)$ its clear that $\|f\|:(0,T) \to \mathbb{R} \in L^1(...
1
vote
0answers
55 views

Do convergence in $L^2$ and boundedness of gradients in $L^2$ imply convergence in $W^{1,2}$?

Let $s\in C(\mathbb R^d)$ and $s_n\in C^\infty(\mathbb R^d)$ be positive functions, both in $L^2(\mathbb R^d)$ for every $n\in\mathbb N$. Suppose that $$ s_n \,\to\,s \ \textrm{ in }L^2(\mathbb R^d)\ \...
2
votes
0answers
74 views

Is the "Dirichlet Laplacian" an extension of $\Delta$ on $C^2(Ω)$ or only on $\{u\in C(\overline Ω)\cap C^2(Ω):\left.u\right|_{\partial\Omega}=0\}$?

Let $\Omega\subseteq\mathbb R^d$ be bounded and open, $V:=H_0^1(\Omega)$ and $H:=L^2(\Omega)$. We know that there is a nondecreasing sequence $(\lambda_n)_{n\in\mathbb N}\subseteq(0,\infty)$ with $\...
0
votes
1answer
89 views

Sufficient conditions for derviative of this product to be in $L^2$.

Let $I=(l_0,l_1)$, $y \in H^2(I)$ and $\rho(x) \in L^\infty(I)$ with $\rho(x)>c\geq 0$. Define $D_\rho(x)=\rho(x)\chi_{(a_0,b_0)}$ with $l_0\leq a_0<b_0\leq l_1$. Now, For any $\phi \in C_c^\...
1
vote
1answer
46 views

Write $D^{\alpha} f$ as a convolution

I am trying to figure out an analysis problem related to Fourier transform and Young's convolution inequality. Here is the problem statement: Let $f \in L^1(\mathbb{R}^n) \cap L^p(\mathbb{R}^n) \cap C^...
0
votes
0answers
26 views

Non negative distributional derivative imply a.e. monotonicity

Suppose $f\in L^1_{loc}(\mathbb{R})$ and that $f'\le 0$ in the sense of distribution, i.e. $\forall \varphi \in C^\infty_C(\mathbb{R})$ $\varphi\ge0$ we have $\int_\mathbb{R}f\varphi'\ge 0$. How can I ...
5
votes
1answer
85 views

Gluing two functions from Sobolev spaces.

I am studying Galdi's Introduction to the mathematical theory of Navier-Stokes equations and somewhere in a proof, he "glued" two functions in $W^{1, 2}(\Omega_1)$ and $W^{1, 2}(\Omega_2)$ ...
0
votes
0answers
56 views

Determine $p$ such that $u(x,y)=1-\max\{|x|,|y|\}\in W^{1,p}((-1,1)\times(-1,1))$

Let $\Omega$ be the open square $(-1,1)^2\subset\mathbb{R}^2$, $u\in L^1(\Omega)$ such that for each $(x,y)\in\Omega$, $u(x,y)=1-\max\{|x|,|y|\}$. Determine the weak gradient of $u$ and find $p$ such ...
1
vote
1answer
64 views

Determine $a$ and $p$ such that $u(x)=|x|^{-a}\in W^{1,p}(B_1(0))$

Let $\Omega=B_1(0)=\{x\in\mathbb{R}^N:|x|<1\}$ and let $u\in L^1(\Omega)$ such that $u(x)=|x|^{-a}$, with $0<a<N$. Determine $\nabla u$ as distributional derivative. Then determine $a$ and $p$...
1
vote
0answers
24 views

Show that $|x|^{\mu} \in H^{2}(B_1(0))$ where $0<\mu<1$

In order to prove that $|x|^{\mu} \in H^{2}(B_1(0))=W^{2,2}(B_1(0))$, I tried to show that the Sobolev norm is bounded, i.e. $||~|x|^{\mu}~||_{2,2,B_1(0)}<\infty$, where $B_1(0)$ is the open unit ...
1
vote
0answers
36 views

Every function $f \in L^2(\Omega)$ admits a weak derivative.

I have write a little argument that seems to show that every function $f \in L^2(\Omega)$ admits a weak derivative: Let $f \in L^2(\Omega)$. We define the functional on $H^1(\Omega) = W^{1, 2}(\Omega)$...
0
votes
0answers
24 views

Weak form of boundary value problem with Neumann condition using divergence theorem

I have a silly question for the weak form of pde. Let's say that I have a pde in the form of $$-u_{xx}+u =0,$$ for the weak form I multiply the pde with test function $v$ so I have $$-u_{xx}v+uv =0$$ ...
0
votes
1answer
43 views

Is the maximum of two weak subsolutions still a weak subsolution?

We say $u\in\operatorname{W}^{1,2}(\Omega)$ is the weak subsolution to the elliptic PDE \begin{equation}\label{1} -D_i(a_{ij}D_j u)=0\quad \text{in}\,\, \Omega\,, \end{equation} if there holds $$\int_\...
2
votes
1answer
25 views

Let $(a,b)$ and $f\in L_{\text{loc}}^1(a,b)$. For $x_0\in (a,b)$, $F(t)=\int_{x_0}^{t}f(s)ds$. Prove that, $DF=f$ (towards theoretical distribution).

Let $(a,b)$ and $f\in L_{\text{loc}}^1(a,b)$. For $x_0\in (a,b)$, consider $$ F(t)=\int_{x_0}^{t}f(s)ds. $$ Prove that, $DF=f$ (towards theoretical distribution). I thought of the following: Let $\...
0
votes
0answers
20 views

A question on a.e. zero weak derivative

Suppose that $\varOmega$ is a bounded connected domain with smooth boundary. If $u$ has weak derivative, and $Du=0$ a.e. in $\varOmega$, then $u\equiv$constant a.e. in $\varOmega$. I wanna know how to ...
2
votes
1answer
26 views

Weak differentiability in first $n$ coordinates

Assume we have $f \in W^{1, \infty}(\mathbb{R}^n \times \mathbb{R})$. Can we then deduce that the mapping $x \mapsto f(x, y)$ for almost every $y \in \mathbb{R}$ is weakly differentiable? Intuitively, ...
1
vote
1answer
56 views

definition of weak formulation

the definition of weak formulation on wiki is: Let $V$ a Banach space. We want to find the solution $u \in V$ of the equation: $$Au=f$$ where $A:V \rightarrow V'$ and $f \in V'$. This is equivalent to ...
1
vote
0answers
34 views

Condition under which the Clarke's subdifferential is locally Lipschitzian

Given a locally Lipschitz continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. The Clarke's subdifferential set is the set given by $$ \partial f (x) = {\text{cl co}}\left\lbrace \lim_{k\in\...
1
vote
0answers
38 views

How to prove that a given function $\varphi : \mathbb{R}^2 \to \mathbb{R}$ is weakly differentiable?

For $0<r<R$, we define $\varphi : \mathbb{R}^2 \to \mathbb{R}$ by $$\varphi(w):=\left\{ \begin{array}{ll} 1, & \lvert w\rvert \leq r \\ \frac{log \lvert w\...
0
votes
0answers
32 views

A weak notion of uniform continuity of Clarke's subdifferential

Given a locally Lipschitz continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. The Clarke's subdifferential set is the set given by $$ \partial f (x) = {\text{cl co}}\left\lbrace \lim_{k\in\...
1
vote
1answer
246 views

weak variational formulation of Poisson equation with Dirichlet boundary conditions

I have given the Poisson equation with Dirichlet boundary conditions \begin{cases} -\Delta u & = f & \text{in} &\Omega \\ \quad u & = g & \text{on} & \partial\Omega \end{cases}...

1
2 3 4 5
12