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Jul
15
comment Sobolev functions counterexample
Why the downvote? The answer is fine! Another way to do it (more elaborate), is to consider the function $f(x)=\operatorname {d}(x,\partial A)$ and then regularize it.
Jul
5
comment Why no trace operator in $L^2$?
Yes @tomglabst. Take a look in Lions-Magenes books: Non-Homogeneous Boundary Value Problems and Applications. For now, I do not remember the volume.
Jun
24
revised ajuda com a solução desta EDO
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Jun
16
revised Multivariable calculus chain rule for weak derivatives
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Jun
16
revised Multivariable calculus chain rule for weak derivatives
added 500 characters in body
Jun
16
revised Multivariable calculus chain rule for weak derivatives
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Jun
16
answered Multivariable calculus chain rule for weak derivatives
Jun
15
comment Showing Sobolev space $W^{1,2}$ is a Hilbert space
You are right. There is no need of Lebesgue's theorem.
Jun
15
comment The Dirichlet problem for the Laplace equation: classical solutions versus weak solution
You have that $\int_{B_r}\nabla u\nabla \phi=0$ for all $\phi\in H_0^1(B_R)$. From this, you can get enough regularity.
Jun
14
awarded  Popular Question
Jun
2
comment On a Lagrange remainder application.
Read the wikipedia article that I have cited above.
Jun
2
comment On a Lagrange remainder application.
en.wikipedia.org/wiki/…
Jun
2
comment On a Lagrange remainder application.
The correct is $c=x_0-\lambda (x_0-x)$ for some $\lambda \in (0,1)$.
May
27
comment Find a cut-off function in a ball.
In this case $f$ is Lipschitz. So it has a derivative which is defined almost everywhere. That's the meaning of $f'$.
May
27
comment Find a cut-off function in a ball.
For your information, $(\phi_\delta'\star f)=(\phi_\delta\star f')$. To answer your question, note that $f$ is a continuous cut-off function, so it remains only to regularize it, which I did by using convolution. The length $\epsilon$ is used only to guarantee some space for the regularization process.
May
27
comment Find a cut-off function in a ball.
Nice, you are almost there. For the derivative, note that if $\eta(x)=(\phi_\delta\star f)(x)$ then, $\eta'(x)=(\phi_\delta\star f')(x)$. Can you conclude now?
May
26
comment Find a cut-off function in a ball.
Yes, I will be here. Take your time and try to understand it first, then you come back here.
May
26
comment Find a cut-off function in a ball.
Well, try it first. If you not try, you will not understand it. I suggest you to draw a picture of $f$ and understand why I choose $\epsilon$ and $\delta$ as I did.
May
26
revised Find a cut-off function in a ball.
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May
26
answered Find a cut-off function in a ball.