C^{1, α} Interior Regularity For P-Harmonic Type Systems Under The Subcritical Growth And It's Three-spheres Theorem

In my dissertation, I mainly study some interior regularity prob-lems of weak solutions for P-harmonic type degenerate elliptic systems under the subcritical growth in non-homogeneous items and Hadamard's three-spheres theroems for a class of elliptic operators (p-harmonic operator and Pucci opera-tor). The paper is made up of four parts as follows:In the first chapter, we briefly introduce the general development of the regu-larity of second-order elliptic equation (systems) in high dimension and the back-ground of considering these problems and their developments in this paper.In the second chapter, we briefly use Morrey space and some imbedding the-ory of Campanato space and H(o|¨)lder continuous space,through establishing the comparing relations of P-harmonic type systems and the corresponding systems with non-homogeneous items,based on relations application of Campanato semi-norm and H(o|¨)lder continuous semi-norm of derivative Du about operator V(Du) = |Du|^(p-2)/2 Du got by Duzaar-Mingione recently, we get the interior H(o|¨)lder continuity of weak solutions and it's derivatives with some exponent for P-Harmonic type systems under the subcritical growth. Comparing to P-harmonic mapping,our hy-pothesis is the best with respect to the growth exponent of lower order items .In the third chapter, we will introduce the study status of three-circles and three-spheres theorems and establish Hadamard's three-spheres theorem for P-harmonic equations, and show that the maximal value function M(r)=max(?)u(x) is a convex function to r^-(n-p)/(p-1) and logr respectively in the cases of 1 < p < n and p = n.In the forth chapter, we will consider Hadamard's three- spheres theorems of classic sub-solutions for another elliptic equations-Pucci extremal equations, as P_λ,Λ⁺[D²u]≥0 and P_λ,Λ^-[D²u]≥0．In the following four cases:1≤Λ/λn-1),Λ/λ=n-1(orΛ=λand n=2),Λ/λ≥1 and n=3,Λ/λ>1and n=2,we show that the maximal value function M(r)=max(?) isa convex function with respect to r^{(1-n)λ/Λ+1},logr,r^{(1-n)Λ/λ+1} and r^1-Λ/λ,respectively.