On the Boundary Point Principle for Elliptic Equations in the Plane

1. Let D be an open connected subset of E (n*z2) and denote by £a(D) the class of second order uniformly elliptic operators of the form L= 23?»/-x a>ijd/dXidxj with coefficients defined in D and satisfying there the condition 2?*/-x ##£*& = # 23?» x ??> f° r some constant a in the range 0<a^l/n, and the normalization £ X i # n = = l . I t is well known [ l ] — [4] that such differential operators enjoy the following strong minimum and boundary point principles: A nonconstant, twice differentiate function u(x), satisfying Lu^O in D, cannot attain a local minimum in D. Moreover if u attains a local minimum at a boundary point x° where dD has the inner sphere property, and if v is a unit vector directed internally to the sphere, then