# 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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### Finite Element Method Weak Formulation

I have a question about the weak formulation of a PDE in finite element analysis. Suppose we have the following two-dimensional PDE: $$\Delta \cdot u(x,y) = q(x,y)$$ where $q$ is given, $u$ is ...
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There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are ...
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### Does convergence in H1 imply pointwise convergence?

I'm trying to figure out if convergence in $H^1(a,b)$ implies pointwise convergence (by the way: what is the usual name of this space?). It is defined to be Hilbert space of absolutely continuous ...
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### Exercise 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 prove that $v=u'$....
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### Sobolev spaces in one-dimensional vs multidimensional

Here in Wikipedia, it is said that in the one-dimensional case, it is enough to assume that the $(k-1)$-th derivative of the function $f$, is differentiable almost everywhere and is equal almost ...
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### Finding Weak Solutions to ODEs

I'm wondering if anyone has a reference to a good set of notes on finding weak (distributional) solutions to ODEs, or has any tips or tricks. For example, $$xy^\prime=0$$ has a classical solution ...
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### Geometric Interpretation of Weak Derivative

As we know, classic derivative $f'(x)$ of a function $f(x)$ can be interpreted as the rate of change of function $f$ in each point $x.$ How about weak derivative? Since it is defined through integral ...
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### If $f_j \rightharpoonup f$ weakly in $W^{1,p}$ then $f_j \to f$ strongly in $L^p$?

Suppose $1<p<\infty$ and $\Omega$ is an open bounded set in $\mathbb R^n$ with nice boundary (say Lipschitz or even better). Let $(f_j)_j \subset W^{1,p}(\Omega)$ s.t. $f_j \rightharpoonup f$ ...
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### Differences between $C_c^\infty[0,T]$ and $C_c^\infty(0,T)$

I believe it is true that: If $f \in C_c^\infty(0,T)$, then $f(T)=f(0)=0$. $C_c^\infty(0,T) \subset C_c^\infty[0,T]$ $C^\infty(0,T) \subset C_c^\infty[0,T]$ If $f \in C_c^\infty[0,T]$, it doesn't ...
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### How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$?

How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$? Since $|u|\in L_p$, I only need to show weak derivative of $|u|$ exists and $D|u| \in L_p$. Can anyone give me some hint? Thanks!
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### Properties of weak derivatives in Sobolev spaces

In PDE Evans 2nd edition, pages 261-263, there is a theorem and its proof which concerns the four properties of weak derivatives. Unfortunately, I do not understand the fourth property, which I will ...
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### Weakly convergence in $W^{1,q}, 1<q<\infty$

Let $(x_n)$ be weakly convergent against $x$ in the Sobolev space $W^{1,q},1<q<\infty$. Now I have to show, that $(\dot{x_n})$ converges weakly against $\dot{x}$ in $L^q$. (With the point I ...
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### $u\in W^{1,1}(U) \Rightarrow u^+\in W^{1,1}(U)$ , where $U\subset\mathbb{R}^n$ open

Im trying to show that for an open set $U\subset\mathbb{R}^n$ and a function $u\in W^{1,1}(U)$, also the positive part $u^+$ is in $W^{1,1}(U)$. My idea is the following: Let $E\subseteq U$ defined ...
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### Continuous function without a weak derivative

Let $f:\Omega\to\mathbb{R}$ be a continuous function. Is it necessarily true that $f$ has a derivative in the weak sense? That is, is there some $v:\Omega\to\mathbb{R}$ such that for every test ...
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### Weak derivative: Showing a function is equal to zero a.e.

I am very beginner in the theory of weak derivative. I am trying to fix the following problem: Suppose that $f\in{L_{loc}^{1}}$ and $\int_{a}^{b}f(x)\phi({x})dx=0$ for all $\phi\in{C_{0}^{\infty}}$ ...
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### Generalized chainrule for Sobolev functions with a cut-off

let $\Omega\subset\mathbb{R}^n$ be a bounded domain and $f\in C^1(\bar\Omega \times (0,g(x)),[0,1])$ and $f(x,\cdot)$ increasing and $g(x)\in\mathbb{R}$ continuous (maybe better, Lipschitz?). I want ...
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### When the weak derivative just is the strong (or classical) derivative?

When the weak derivative just is the strong (or classical) derivative? For instance, can we prove that weak derivate $Du\in C^\alpha$(or $C^0$) implies $u\in C^{1,\alpha}$(or $C^1$).