Discrete versions of continuous isoperimetric problems

Discrete isoperimetric variational problems which model single and double integral isoperimetric problems are formulated and some multiplier rules are derived. For quardratic functionals, the Euler-Lagrange equation is linear and can be analyzed bydeterminant methods. difference equations methods. or numerically bythealgebraic eigenvalue problem. A specific example is given wherethe eigenfuntions and eigenvalue of this discrete problem converge. as the step size goes to zero, to the eigenfunctionsand eigenvalues of the corresponding, continuous problem Recent developments in algebraic geometry offer the hopeof using Groebner (=Gröbner or Grobner) basis methods to numerically solve some systems of equationsgenerated by Lagrange multiplier methods. These methods mayapply to nonlinear systems of equations whenever they can be reduced to systems of polynomial equations. The Groebner basis algorithm of Buchberger is an extension of Gaussian elimination to polynomial systems.