# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...