4. METHOD OF SOLUTION
Consider a domain G(X,Y) with the boundary DG, where a function U(X,Y) has to be calculated, which minimzes the quadratic functional over G with the integrand:
F(U) = F1*UXX2 + 2*F2*UX*UY + F3*UYY2 + F4*U2 + 2*F5*U
The Fi are functions of X and Y, which have to satisfy the conditions:
F1,F3 > 0; F2 >=0 and F1*F3 - F2*F2 > 0
On the boundary DGE, essential boundary conditions
have to be prescribed. On the rest of the boundary DG-DGE, the solution fulfills the natural boundary conditions
(F1*UX+F2*UY)*NX+(F2*UX+F3*UY)*NY = 0
automatically. NX and NY are the direction-cosines of the outer normal of the boundary.
The solution is equivalent to the solution of the partial differential equation
DX(F1*UX+F2*UY) + DY(F2*UX+F3*UY) = F4*U + F5
under the boundary conditions described above.