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i%\ dxtdx, £ i 3x< be defined on an open set ü c R. We consider solutions u=u(x) e C(£ï)n C(ù) of Lu^.0 which attain their maximum value M at P, a characteristic boundary point of £1. PROPAGATION SET. (See [3] and [1].) Let the diffusion vector field oc(X) be the kth column vector of the nxn matrix a(x), where a=tf, and let the drift vector field f}(x) be defined by Pi=bi—^j^1{aij)Xi9 i = l , • • • , n. Assume that a, a, • • • , a, j8 e C^), where # ç P n is open and Ü£ 2?. For P e ôQ9 the propagation set S(P9 CÏ) is generated by segments of trajectories in CI of vector fields Xop+X1aL +X2cn +' • • + ŵ<*> ^o=0, where the scalar functions AA.=Afc(̂ :) e C , &=0, 1, • • • , n. S(P9 Q) is the closure of S(P9 Q) in D. CURVATURE CONDITION. Let P eô£l be a characteristic boundary point, that is, va(P)v~09 where v is the unit inner normal to ÔQ, at P. Then m(P)=0 for each k9 A:=l, • • • , n. Let n * be the plane of oc*(P) and r through P. In this plane, the cross-section of ôQ, and the projection of the trajectory of a through P are curves which are perpendicular to v at P. Let the curvatures of these curves at P be rk9 for the sectional curvature of ôQ9 and ak9 for the 'shadow curvature' [4] of the trajectory of a. Finally, define the 'excess curvature' pk to be the difference rfc—ak. LEMMA. / J V J J U P*|a|=*v-2£-i*l*l *' ( r / ^ ^ ^ ö r 6 ö ^ r those kfor which a*(P)^0.)

A boundary maximum principle for degenerate elliptic-parabolic inequalities, for characteristic boundary points