A Semi-strong Minimum for a Multiple Integral Problem in the Calculus of Variations

and integrating over the fixed domain A. We seek sufficient conditions on a surface So which will ensure that So provides 7(5) with a relative minimum in the class of surfaces S which coincide with So on the boundary C of A. In 1917 Lichtenstein [5]C) considered the case w = 2 and by constructing a field established a sufficiency theorem for a stiong relative minimum. He supposed analyticity for the functions involved, and assumed for the second variation I2 a Jacobi condition expressed in terms of the characteristic values of a boundary value problem associated with the accessory partial differential equation. In the present paper we prove, without field theory and for the case of general «, a sufficiency theorem for a semi-strong relative minimum under much less stringent analytic requirement. We also give an estimate of the difference I(S)—I(Si). We assume for the second variation a condition of the form