| In this thesis, two types of second order elliptic partial differential equations will be studied. The first type is the following equation Ax,y6 2u6x2+2B x,y62u 6x6y+Cx,y 62u6y2 =0 for a function u of Sobolev class Here A, B and C are measurable functions on Ω with A > 0, C > 0 and AC − B² > 0 a.e.; Our main result will be that u is of class C1(Ω) provided that is locally integrable on Ω.; The second equation we will study is the non-homogeneous p-harmonic equation div&vbm0;1u&vbm0;p-2 1u=divf for a function where with Our main result is the following:; THEOREM. Let u be a non-homogeneous p-harmonic function on of class where 1 < p ≤ 2. If then g∈W1,2&parl0; ℜn,ℜn&parr0; and the following uniform estimate holds Dg L2≤C p,n f2-p2p-2 Lq Df Lq. ; Among other applications, this theorem will be used to establish higher integrability of ∇u. |