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I have been asked the following question, and I have to admit that I have no idea about the answer.

Assume that $f \colon (a,b) \to \mathbb{R}$ is a function. Assume also that there exists a function $F\colon \mathbb{R}\times \mathbb{R} \times (a,b) \to \mathbb{R}$ such that $$F(\epsilon_1,\epsilon_2,x) = F\left(\sqrt{\epsilon_1^2+\epsilon_2^2},x\right),$$ $F(0,0,x)=f(x)$ for every $x \in (a,b)$ and $F \in W^{2,2}((0,1)\times (0,1) \times (a,b))$. What can be said about the regularity of $f$?

In other words, we "lift" $f$ to a function with cylindrical symmetry, so that the "lifted" function lies in a Sobolev space. We then look at $f$ as the restriction of $F$ to the cartesian axis corresponding to $\epsilon_1=\epsilon_2=0$. Can we say that $f$ inherits some regularity property from $F$?

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  • $\begingroup$ I think there is some form of typo in your equation since $F$ has three arguments on the left hand side and two in the right hand one. Probably you mean something like $$F(\epsilon_1, \epsilon_2, x) = F\big(\sqrt{\epsilon_1^2 + \epsilon_2^2}, 0, x \big).$$ In any case if $f \in W^{2,2}(B_0(1) \times (a,b))$ it is over the critical exponent for the Sobolev embedding theorem ($k = 3/2$) and by Morrey's inequality $f$ will be Hölder continuous. $\endgroup$
    – Adrián González Pérez
    Commented Jan 24, 2018 at 12:59

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