WEIGHTED INEQUALITIES AND APPLICATIONS TO DEGENERATE ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

The basic objective of this thesis is to prove interior and boundary continuity for solutions of degenerate elliptic partial differential equations of divergence type. To accomplish this a number of weighted inequalities are proven, including a characterization of weights for several Sobolev inequalities. A much larger class of degeneracies is considered than previously dealt with in the literature, and results are established which until now have been known only in the strongly elliptic case.;(VBAR)A(VBAR) (LESSTHEQ) (mu)(x)(VBAR)(DEL)u(VBAR)('p-1) + a(,1)(x)u('p-1) + a(,2)(x),;A (.) (DEL)u (GREATERTHEQ) (lamda)(x)(VBAR)(DEL)u(VBAR)('p) - c(,1)(x)u('p) - c(,2)(x),;(VBAR)B(VBAR) (LESSTHEQ) b(,0)(lamda)(x)(VBAR)(DEL)u(VBAR)('p) + b(,1)(x)(VBAR)(DEL)u(VBAR)('p-1) + b(,2)(x) u('p-1) + b(,3)(x).;Two approaches are taken to the problem of establishing continuity. The first uses a Harnack inequality and the second, Morrey spaces. The first applies to equations of the form div A = B, where A, B satisfy the growth conditions.;A Harnack inequality is proven for weights (mu), (lamda) satisfying certain capacitary conditions. Interior continuity follows immediately from this, and a Wiener criterion is established for continuity at the boundary. This generalizes work of Edmunds and Peletier, Gariepy and Ziemer, Murthy and Stampacchia, Trudinger, and Kruzkov.;A theory of weighted Morrey spaces is developed which establishes continuity estimates for a wide class of weighted Sobolev spaces W(,(omega))('1,p) with p>n, n the spatial dimension. This is in turn applied to prove the continuity of solutions of systems of the form div A(,i) = B(,i), i = 1 , ... , m, where A(,i) and B(,i) satisfy growth conditions similar to the above with p > n-(epsilon). In the nondegenerate case this is due to Widman and, in more general form, to N. Meyers and A. Elcrat.;The basic tools developed in this thesis to accomplish the above are: (1) A characterization of the weights (omega), (nu), (lamda), in terms of capacities for the inequalities ((INT) u('q) d(omega))('1/q) (LESSTHEQ) c((INT) (VBAR)(DEL)u(VBAR)('p) d(nu))('1/p) and ((INT) (VBAR)u - (INT) u d(lamda)(VBAR)('q) d(omega))('1/q) (LESSTHEQ) c((INT) (VBAR)(DEL)u(VBAR)('p) d(nu))('1/p), 1(LESSTHEQ)p(LESSTHEQ)q<(INFIN). The characterization for the second inequality, even with (omega)=(nu)=(lamda)=Lebesgue measure, leads to a new result in that it characterizes the domains (OMEGA) (L-HOOK EQ) (//R)('n) which support such an inequality. This has been done by C. Amick for the case 1