A well-known theorem of Evans and Krylov states that in an equation of the form $F(D^2 u)=g$, provided that the right hand side and $u$ both have Lipschitz gradient, and that $F$ is concave or convex and of class $C^2$ on the space of symmetric matrices, and uniformly elliptic at $u$, the solution is of class $C^{2,\alpha}$ for some $\alpha$.

The proof in Gilbarg-Trudinger uses differentiaion of the equation twice, and therefore $L^\infty$ second derivatives of $g$ are needed. My question is about minimal regularity known to be necessary for the function $g$. Does one really need Lipscitz gradients for $g$ or it can be relaxed?


1 Answer 1


The equation $F(D^2u) = g$ for $g \in C^{\alpha}$ should have a $C^{2,\alpha}$ estimate by perturbation theory. See for instance Caffarelli-Cabre, Ch. 8. The idea is that the constant-coefficient equation $F(D^2w) = f(x_0)$ has a $C^{2,\alpha}$ estimate (Evans-Krylov), so by using the quadratic approximation to $w$, ABP, rescaling and iteration we can produce a paraboloid approximating $u$ to order $O(|x-x_0|^{2+\alpha})$. (Since we use ABP I think that some Holder condition on the growth of the $L^n$ norm of $g$ should suffice).

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    $\begingroup$ This is correct. Also, it is not necessary to assume $u$ to have Lipschitz gradient or $F \in C^2$. I think that the Holder condition on the growth of the $L^n$ norm of $g$ that you suggest is equivalent to $g \in C^\alpha$ by Campanato's theorem. $\endgroup$ Jun 5, 2013 at 16:18
  • $\begingroup$ Thank Luis; Can one remove the condition on Lipschitz gradeint? How? Could you provide a reference? $\endgroup$
    – S.A.A
    Jun 6, 2013 at 0:06
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    $\begingroup$ Check Theorem 6.6 in the book of Caffarelli and Cabre for the version with zero right hand side. The case of $g \in C^\alpha$ follows from Theorem 8.1 in the same book after you decipher those hypothesis. $\endgroup$ Jun 6, 2013 at 22:34

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