Tarantello, On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities, Adv.
Tarantello, The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions, Ann.
His topics include the first variation, base generalizations, the homogeneous problem, variable-endpoint conditions, broken extremals
, and sufficient conditions.
Therefore the positive extremals
are presented by annual sediment yield which make <3% of the supply, the major anomalies make 36%, the small anomalies make 6-15%; the provision of negative extremals
makes >97%, the corresponding provision of large extremals
makes 97-94% and the provision of small extremals
Moreover, the extremals are given by u(x) = [alpha] sin ([[?
We note that the proofs in [4,5] are based on an approximation argument involving truncations, which does not allow to characterize the extremals.
Now the set can be plotted as a cluster of points on a plane, and more extremals
Section 3 deals with the structure of extremals
, that is, the integral curves of the Hamiltonian vector fields [Mathematical Expression Omitted] associated with H, based on the decomposition of the cotangent bundle T*G of G as the product G x L*.
In this particular case we have studied the relating problems in a various aspect and obtained interesting results such as the exact values of the best constant S = S(p,q,[alpha]) in certain cases, the existence and nonexistence of the extremals
and so on.
OF THE FUNCTIONALS REPRESENTED BY PATH-INDEPENDENT CURVILINEAR INTEGRALS
More precisely, x(*) is an extremal
for the least square type Lagrangian
If this case, each functions may be an extremal
one and such a Lagrangian is called null Lagrangian.