Reference to a Classical Regularity Theorem

MathOverflow Asked by artful_dodger on January 3, 2022


I need a reference to the following result:

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


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

for some $0<beta<1$, 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 $0<alpha<1$.


$$B_1^+ = {x = (x’,x_{n+1}) in mathbb{R}^{n+1} : |x| < 1, , , x_{n+1} > 0}$$
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 $alpha$.

One Answer

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

Add your own answers!

Related Questions

What OEIS sequence is this?

1  Asked on December 29, 2020 by a-z


Ask a Question

Get help from others!

© 2023 All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP, SolveDir