Holder Continuity of the Higher Derivatives of Solutions to Elliptic Systems
of Nonlinear PDE of Arbitrary Order

A. P. Kopylov


Consider the system


L_j (x; f₁(x), ......f_m(x); ......∂^p1f_χ (x), .....; ...., ∂^p2f_χ (x), ....; ...∂^plf_χ (x), ...) = 0           (1) 
j = 0, 1, 2, .....k,           

where x = (x₁, x₂, . . ., x_n)   Π Rⁿ ;

p_ν = (p_ν1, p_ν2, ......p_νn)  is a multi-index of order |p_ν| = Σ_s=1ⁿ  p_νs    with ν = 0, 1, ......l; 

∂^p1f_χ = [(∂₁)^pν1 ° (∂₁)^pν2 ° ........  ° (∂₁)^pνn)]f_χ     (∂_s = ∂/∂x_s)

the partial derivative of the function f_χ (=∂₁^po1 f_χ)  χ = 1, 2, ..., m, corresponding to p_ν  and (1) includes the symbols of all such partial derivatives of each function  f_χ   χ = 1, 2, ..., m    up to l-th order. This is an l-th-order system of k PDE in m unknown real functions 

 f_χ   χ = 1, 2, ..., m     of n real variables. Here L_j are real functions of class C¹:

L_j =  L_j ( x; ...., vp0,χ ...., vp1,χ ...., vpl,χ ....),      (2)

( x; ...., vp0,χ ...., vp1,χ ...., vpl,χ ....) = y Î Y, Y is a domain in R^Nl ,

N_l =n + m Σ_ν=1^l  n_ν  ,   n_ν = (n+ν-1)!/ν!(n-1)!  ν = 0, 1, 2, ....... l.

A solution f: U ® R^m (U is a domain in Rⁿ) of class C^l(U, R^m) to (1) is called elliplic if

for all z Î Rⁿ \ {0} and y = (x; .... ∂^p0 f_χ(x), .....∂^p1f_χ (x), .....; ....;...., ∂^plf_χ (x), ....), x  ÎU

Theorem. Let the functions L_j in (1), (2) belong to the class Cⁱ(Y, R^k) and f: U ® R^m (U Ì Rⁿ) be an elliptic C^l-solution to (1). If O < a < 1 and E is a compact subset of U, then there exisis a number C_a,E ³ 0 such that

|∂^plf_χ (x') - ∂^plf_χ (x'')| £ C_a,E |x' - x''|^a 

x',x" Î E, |p^l| = l, χ = 1, 2, ..., m.


This assertion is a generalization of Nirenberg's and Morrey's well-known results.