LAPLACIANS AND SOBOLEV GRADIENTS

In [9] Weyl discusses the problem of determining when a vector field i s the gradient of some function. He introduces a method of orthogonal projections to solve this problem for all square integrable (but not n ecessarily differentiable) vector fields. In the construction of Sobol ev gradients for problems in nonlinear partial differential equations [6] a family of problems of a somewhat similar nature arises. Using an idea of Beurling and Deny [2],[3] we give a generalization of the Lax -Milgram Theorem [5] which unifies a wide class of Sobolev gradient co nstructions.