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Manifolds with geodesics which minimize length globally
Does the limit $\lim_{(x,y)\to (0,0)} \frac {x^3y^2}{x^4+y^6}$ exist

Let $\Omega$ be a bounded domain. I write $L^2$ instead of $L^2(\Omega)$ etc.

Let $u \in L^2(0,T;H^1)$ with weak derivative $u’ \in L^2(0,T;H^{-1})$.

Consider

$$f(u(x,t)) = \begin{cases}

-1 &: u(x,t) \in (-\infty, -1)\\

u &: u(x,t) \in (-1, 1)\\

1 &: u(x,t) \in (1, \infty).

\end{cases}$$

- Definition of weak time derivative
- $L^{2}(0,T; L^{2}(\Omega))=L^{2}(\times\Omega)$?
- Weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$
- Is $L^2(0,\infty;L^2(\Omega)) = L^2((0,\infty)\times \Omega)$?

We have $f(u) \in L^2(0,T;L^2)$. Does it make sense to talk of a weak derivative of $f(u)$? We could naively write

$$(f(u))’ = \begin{cases}

0 &: u(x,t) \in (-\infty, -1)\\

u’ &: u(x,t) \in (-1, 1)\\

0 &: u(x,t) \in (1, \infty).

\end{cases}$$

and then write this as $(f(u))’ = \chi_{\{u \in (-1,1)\}}u’$ but I can’t make sense of this.

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- Compact inclusion in $L^p$
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- How to prove that an operator is compact?
- Rainwater theorem, convergence of nets, initial topology
- Equivalent Definitions of the Operator Norm
- The kernel of a continuous linear operator is a closed subspace?
- Is there any handwavy argument that shows that $\int_{-\infty}^{\infty} e^{-ikx} dk = 2\pi \delta(x)$?
- Can every closed subspace be realized as kernel of a bounded linear operator from a Banach space to itself?

To begin with, a function

$$

f(\xi)\overset{\rm def}{=}

\begin{cases}

-1,\quad \xi<-1,\\

\xi,\quad \xi\in [-1,1],\\

1,\quad \xi>1,

\end{cases}

$$

is Lipschitz, and hence by theorem 2.1.11 from the textbook

*“Weakly differentiable functions”* by W.P. Ziemer,

given any $u\in L^2(0,T;H^1)$, for Sobolev weak derivatives,

holds the chain rule

$$

\partial_{x_k} f(u)=\partial_{x_k} (f\circ u)=f'(u)\cdot\partial_{x_k} u

$$

implying that a superposition $f(u)=f\circ u\in L^2(0,T;H^1)$, while

a superposition $f'(u)=f’\circ u\in L^{\infty}(0,T;H^1)$. Indicated

in the post just $f(u)\in L^2(0,T;L^2)$ is not enough for a distributional

derivative $\partial_t$ to be correctly defined, which requires a rather

more delicate approach than Sobolev weak derivatives $\partial_{x_k}$.

Namely, the chain rule

$$

\partial_t f(u)=\partial_{x_k} (f\circ u)=f'(u)\cdot\partial_t u

$$

holds with the *R.H.S.* being a correctly defined product of a distribution

$\partial_t u \in L^2(0,T;H^{-1})$ by a function $f'(u)=f’\circ u\in

L^{\infty}(0,T;H^1)$, which correctly defines a distribution $\partial_t

f(u)\in L^2(0,T;H^{-1})$, with distributions understood here as functional-valued

Bochner measurable functions $(0,T)\to H^{-1}\overset{\rm def}{=}

\bigl(H_0^1(\Omega)\bigr)^{\ast}$.

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- finding the limit $\lim\limits_{x \to \infty }(\frac{1}{e}(1+\frac{1}{x})^x)^x$
- Solving $\lim\limits_{x\to0} \frac{x – \sin(x)}{x^2}$ without L'Hospital's Rule.
- How can I prove it to be a well-defined binary operation?
- Proof that the real numbers are countable: Help with why this is wrong
- Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V.
- Determing the number of possible March Madness brackets