Vol. 22, No. 1, 1967

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An integral inequality with applications to the Dirichlet problem

James Calvert

Vol. 22 (1967), No. 1, 19–29

An existence theorem for the elliptic equation Δuqu = f can be based on minimization of the Dirichlet integral D(u,u) = |∇u|2 + q|u|2 dx. 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.

Mathematical Subject Classification
Primary: 35.04
Received: 3 October 1966
Published: 1 July 1967
James Calvert