# Tagged Questions

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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### Why are these functions Locally Integrable?

Apparently the following are locally integrable: (a) $x^r\in L_1(0,\infty)$, $\forall r\in\mathbb{R}$. But this couldn't be right because $\frac{1}{x}\notin L_\text{loc}(\mathbb{R})$ (since ...
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### How to handle a diffusion equation with a variable diffusion coefficient with finite elements?

Based on the notes here I created a finite elements solver for the stationary heat equation (Poission's equation) $$-u''(x) = f(x)$$ However I would like to solve the stationary heat equation that ...
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### Exercice 4 chapter 7 Evans book

I want to solve the following problem: Suppose $H$ is a Hilbert space, if $u_{n}\rightharpoonup u$ in $L^{2}(0,T;H)$ and $u'_{n}\rightharpoonup v$ in $L^{2}(0,T;H^{'})$. I want to proof the $v=u'$. ...
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### Weak differentiability and diffeomorphisms

Let $U,V\subset\mathbb{R}^n$ be open sets and assume the existence of a $\mathcal{C}^1$-diffeomorphism $\phi:U\rightarrow V$. Let $u\in W^{1,p}(U)$, $1\leq p\leq\infty$, and define ...
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### Derivative is a Radon measure if time increment bounded

Here, $u$ belongs to some $L^p$ space. Can someone tell me what result the author uses to say that $u_t$ is a Radon measure?
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### Weak convergence and weak convergence of time derivatives

I am working in $H^1(S^1)$, the space of absolutely continuous $2\pi$-periodic functions $\mathbb R\to\mathbb R^{2n}$ wih square integrable derivwtives. I have a sequence $z_j$ (for the record, it ...
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### Weak derivative of modulus

I am stuck on an introductory problem on Sobolev spaces. Any help would be appreciated! Suppose $\Omega\subseteq\mathbb{R}^n$ is bounded and open and $u:\Omega\rightarrow\mathbb{R}$ has an ...
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### Distributional derivative of $2$-variable indicator function

Let us say we have the indicator function $\chi_{\{|x|\leq 1\}}$ in $\mathbb{R}^2$. How can I write out the weak derivative of this indicator function? Is it $\delta_{|x|=1}$? Or it should be vector ...
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### Heaviside function and weak solutions

If a solution to a PDE has a solution f(x) in which f(x) is a Heaviside function, independent on its argument, can I say the solution is unstable, therefore being weak?
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### A question on vanishing viscosity in Evans PDE

In Evans book on PDE (2nd edition) he uses the vanishing viscosity method to prove existence of a weak solution to the following linear hyperbolic system ...