Along this note, solution surfaces of (23), that is extremals
of [W.sub.[PHI]](S), will be called Willmore-like surfaces.
In order to characterize the primal form of the first-order condition, namely, F'([bar.x], [bar.u]; [eta], v) = 0 for all admissible directions, in terms of the weak maximum principle we introduce the notions of extremals
and the weak-normality for the problem (L[Q.sup.[sigma]).
Tarantello, The role of Onofri type inequalities in the symmetry properties of extremals
for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions, Ann.
On the other hand, the most important concepts can be appropriately modified; for example, the Hilbert-Weierstrass extremality theory together with the Hamilton-Jacobi equations [18-21] since the Poincare-Cartan forms [??] make "absolute sense" along the extremals
[s.sub.*](x) admissible for (14) might be an ordinary (nongeneralized) function.
Moreover, the solutions of the implicit homogeneous PDE system of first order (2.8) are extremals
for the Lagrangian
We note that the proofs in [4,5] are based on an approximation argument involving truncations, which does not allow to characterize the extremals
Any members of this subset have a legitimate claim to being considered extremal
(and therefore interesting).
In order to show the structure of extremals
it is useful to recall the Poisson bracket.
Theorem 4.4 (Existence and Nonexistence of extremals
OF THE FUNCTIONALS REPRESENTED BY PATH-INDEPENDENT CURVILINEAR INTEGRALS
Considerable progress has been made to determine necessary and sufficient conditions that any extremal
for the variational functional with fractional calculus must satisfy in recent years.