# Reference to a Classical Regularity Theorem

MathOverflow Asked by artful_dodger on January 3, 2022

(Edited)

## I need a reference to the following result:

If $$u in H^2(B_1^+) cap {rm Lip}(B_1^+)$$ satisfies
$$begin{cases} {rm div}(F(x,u,nabla u)) = F_0(x,u,nabla u) quad & {rm in} B_1^+ \ u = 0 & {rm on} B_1′ end{cases}$$

where

$$F in C^{1,beta}(B_1^+timesmathbb{R}timesmathbb{R}^{n+1};mathbb{R}^{n+1}), quad F_0 in C^{0,beta}(B_1^+timesmathbb{R}timesmathbb{R}^{n+1};mathbb{R})$$

for some $$0, and

$$langle D_p F(x,u,p) xi,xi rangle ge lambda(M) |xi|^2$$

for some $$0 < lambda(M) < + infty$$, for every $$x in overline{B_1^+}$$, $$u in mathbb{R}$$, and $$|p| le M$$,

then $$u in C^{2,alpha}(overline{B_{1/2}^+})$$ for some $$0.

## Notations:

$$B_1^+ = {x = (x’,x_{n+1}) in mathbb{R}^{n+1} : |x| < 1, , , x_{n+1} > 0}$$
is the half-ball and
$$B_1′ = {x = (x’,0) in mathbb{R}^{n+1} : |x’| < 1}$$
is the flat part of its boundary.
Also, we have $$n ge 1$$.
$$H^2$$ denotes the Sobolev Space of functions with second order weak derivatives in $$L^2$$ and $${rm Lip}$$ is the space of Lipschitz-continuous funcions, whilst $$C^{k,alpha}$$ is the space of functions whose $$k$$-th order classical derivatives are Hölder-continuous of exponent $$alpha$$.

The discussion from Section 13.1 in the book of Gilbarg and Trudinger shows that $$u in C^{1,,alpha}left(B_{3/4}^+right)$$. From here one can apply Schauder estimates for linear equations. For example, one can pass the divergence on the left hand side and view $$u$$ as a solution to a non-divergence form linear equation with Hölder continuous coefficients (namely $$F^i_j(nabla u)$$, in the case that $$F$$ depends only on $$nabla u$$). For the relevant linear theory, see e.g. Section 5.5 from the book of Giaquinta and Martinazzi here.

Answered by Connor Mooney on January 3, 2022

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