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Thread: Problem in functional analysis

  1. #1
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    Problem in functional analysis

    Hello,

    One defines $\displaystyle D_i^{+h} u(x)=\frac{1}{h}(u(x+h\cdot e_i)-u(x))$.
    $\displaystyle D_i^+ u$ is defined on the maximal subregion $\displaystyle \Omega_0=\Omega_0(+h,i)$.

    Now let $\displaystyle \Omega=(-1,1)$ and $\displaystyle u\in L^1(\Omega)$.

    How to prove, that the following implication is not true in general:

    $\displaystyle \left \| D_1^+ u \right \|_1\leq c\quad \forall\ h >0 \ \ \Rightarrow\ \ u \in H^{1,1}(\Omega)$

    In a book about functional analysis I read the following lemma:
    "Let $\displaystyle 1< p\leq \infty$. If the inequality $\displaystyle \left \| D_i^{+h} u \right \|_{p;\Omega_0}\leq c$ holds for all $\displaystyle \Omega_0 \subset \subset \Omega$ and for all $\displaystyle 0< h\leq h_0(\Omega_0)$ then $\displaystyle u$ is weakly differentiable in $\displaystyle x_i$ with $\displaystyle \left \| D_i u \right \|_{p; \Omega} \leq c$."

    Here the problem is $\displaystyle p=1$, so the lemma can not be used.

    Do you have a counter example?

    Kind regards,
    Alexander
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Problem in functional analysis

    I think that you even have this in the classical sense - a function with directional derivatives at a point may not be differentiable there.
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