# 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

## Related Questions

### Rings whose Frobenius is flat

0  Asked on December 30, 2020 by anon1432

### $ell^1$-norm of eigenvectors of Erdős-Renyi Graphs

1  Asked on December 30, 2020 by stefan-steinerberger

### The problems of global asymptotic freeness

1  Asked on December 29, 2020 by iliyo

### What OEIS sequence is this?

1  Asked on December 29, 2020 by a-z

### Reference for the rectifiablity of the boundary hypersurface of convex open set

1  Asked on December 28, 2020

### Tannakian group of Galois representations coming from geometry

0  Asked on December 27, 2020 by smn

### Why there is no 3-category or tricategory of bicategories?

1  Asked on December 27, 2020 by lolman

### Analytically controlling sizes in modular arithmetic to demonstrate Dirichlet pigeonhole application

1  Asked on December 27, 2020

### “Weakly” nuclear operators (terminology)

0  Asked on December 26, 2020 by pea

### holomorphy in infinite dimensions (holomorphic families of operators)

2  Asked on December 26, 2020 by andr-henriques

### Max weighted matching where edge weight depends on the matching

2  Asked on December 25, 2020

### Faltings’ height theorem for isogenies over finite fields

0  Asked on December 21, 2020 by asvin

### Probability of positivity of rational solutions to a diophantine system?

0  Asked on December 21, 2020 by vs

### On sup over boundaries of Sobolev functions

0  Asked on December 20, 2020 by yongmin-park

### On weaker forms of the abc conjecture from the theory of Hölder and logarithmic means

1  Asked on December 18, 2020 by user142929

### Set of points with a unique closest point in a compact set

2  Asked on December 18, 2020 by piotr-hajlasz

### Complex $2$ manifold $M$ with a given real 2 dimensional submanifold which meets all complex limit cycles of foliations of $M$

0  Asked on December 17, 2020 by ali-taghavi

### Rigorous multivariate differentiation of integral with moving boundaries (Leibniz integral rule)

1  Asked on December 17, 2020 by amir-sagiv

### Closed form of the sum $sum_{rge2}frac{zeta(r)}{r^2}$

0  Asked on December 15, 2020 by epic_math