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How should I go proving that (say) there are no estimates on the hessian of $u$ of the form $\|\nabla^2u\|_{\infty} \leq C \|f\|_{\infty}$? (Here one can consider $f$ continuous on the closure of the ball of radius 2 and trying to control $u$ on a smaller ball for example, using the control of $f$ on the bigger ball.) Any help (reference, proof,...) would be greatly appreciated... Edit: $\Delta u=f$ here sorry...

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    $\begingroup$ How are $u$ and $f$ related? $\endgroup$
    – Deane
    Commented Jul 9 at 23:47

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Using polar coordinates, try the function $r^2\log r\cos2\theta$. It's Laplacian should be $4\cos2\theta$ which is bounded, but it's Hessian is not.

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