A STUDY OF DEGENERATE ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

Abstract

ABSTRACT In this thesis, two types of second order elliptic partial differential equations will be studied. The first type is the following equation A(க,) +2B(க,0) தர + C(க,ப) ர -0 for a function u of Sobolev class W2,2(n2). Here A, B and C are measurable functions on with A > 0,C > 0 and AC - B2 > 0 a.e. Our main result will be that u is of class C1(n2) provided that locally integrable on n2. The second equation we will study is the non-homogeneous p-harmonic equation (A+C)2 AC - B2 is div |Vu|P-2 Vu = div f for a function u E WI,P(R"), where f e W1,9(9") with - += pq =1. Our main result is the following: THEOREM. Let u be a non-homogeneous p-harmonic function on R" of class W1P(R"), where 1 g € W1,2(8", 82") and the following uniform estimate holds Among other applications, this theorem will be used to establish higher integrability of Vu.