A few days before, I was in need to learn real quick some basic results about Holder estimates for Poisson's Equation $\;\Delta u =f\;$ so I started to read from Gilbarg & Trudinger's book chapter 4.

I want to prove the following:

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My attempt:

To begin with, using Th. 4.8 I have:

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so if the right member of (4.19) is bounded then $\;{\vert u \vert}^*_{2,\alpha;\Omega}\;$ is also bounded $\;(\#)\;$.

I'm pretty sure 4.7 follows by Arzela-Ascoli Theorem but I miss some key points.To be more specific:

  1. Why does $\;(\#)\;$ imply $\;u\;$ is uniform bounded? Does $\;{\vert \cdot \vert}^*_{2,\alpha;\Omega}\;$ envolve the uniform norm of $\;u\;$? I don't really understand the definition (which I'll leave down below)
  2. Why does $\;(\#)\;$ imply $\;u\;$ is equicontinuous? I know if a function satisfy Holder condition i.e. $\;\vert f(x)-f(y) \vert \le M {\vert x-y \vert}^{\alpha}\;$ then it's equicontinuous. My problem here is that by $\;(\#)\;$ I believe only the second derivative of $\;u\;$ satisfy the Holder condition so only this is equicontinuous. Neither $\;u\;$ nor its first derivative are Holder continuous.

All the above questions might be a consequence of the confusion I have about the definitions of the norms in this book.Here is the definition of $\;{\vert \cdot \vert}^*_{2,\alpha;\Omega}\;$:

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I apologize in advance for this long post but I wanted to be as specific as possible. Any help would be valuable.



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