An existence theorem for the
elliptic equation Δu−qu = f can be based on minimization of the Dirichlet integral
D(u,u) =∫|∇u|2+ q|u|2dx. The usual assumption that q(x) ≧ 0 is relaxed in this
paper.
Actually the paper deals directly with the general second order formally
self-adjoint elliptic differential equation ∑i,kDt(aikDku) + qu = f where q(x) is
positive and “not too large” in a sense which will be made precise later. The
technique consists in showing that the quadratic form whose Euler-Lagrange
equation is the P.D.E. above is positive for a sufficiently large class of
functions.
