A STUDY OF SINGULARITIES AND PHRAGMEN-LINDELOF THEOREMS FOR CERTAIN CLASSES OF NONLINEAR ELLIPTIC SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS
Abstract
Theorems of Phragmen-Lindelof type for the following classes of elliptic partial differential inequalities in an arbitrary unbounded domain (OMEGA)(L-HOOK)(//R)('n), n (GREATERTHEQ) 2, are obtained: (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where a(,ij),b(,i) are measurable functions such that the coefficients a(,ij) define an uniformly elliptic matrix and b(,i) (ELEM)L('(INFIN))((OMEGA)); (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) with a(,ij),b(,i) (ELEM)L(,loc)('n)((OMEGA)) and such that a(,ij)(x) are uniformly elliptic and Holder continuous in a neighborhood of infinity, and b(,i) = O((VBAR)(VBAR)x(VBAR)(VBAR)('-1)) as x (--->) (INFIN); and (A-3) div(<(DEL)u,(DEL)u>('(p-2)/2)(DEL)u) (GREATERTHEQ) f(u), p > 1. Always above f is a positive, nondecreasing and locally Lipschitz function and u is assumed to be in the corresponding natural weak-Sobolev spaces. Namely, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) in (A-1) and (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) or (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) in (A-2) or (A-3) respectively. As a consequence of these Phragmen-Lindelof type of results we obtain nonexistence and Liouville type of theorems. We also deal with the general problem of removable singularities. Namely, we obtain theorems about removable singularities for the equations (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) and (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where a(,ij),b(,i) are C('0,1)((OMEGA)) in (A-4) and a(,ij),b(,i) (ELEM)C('o,(mu))((OMEGA)), 0 < (mu) (LESSTHEQ) 1, in (A-5) and (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) and for x-uniformly in (OMEGA) we have (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) with C a positive constant, (OMEGA) = {x (ELEM) (//R)('n): (VBAR)(VBAR)x(VBAR)(VBAR) < 1} n > 2 and g (GREATERTHEQ) n/(n-2). Our results relate the power q, the ellipticity constants and the Hausdorff dimension of the set to be removed.
Degree
Ph.D.
Subject Area
Mathematics
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