Variational Problems of Certain Functionals

Abstract

We get the existence and regularity of minimizers of certain functionals, which may be degenerate and have non-polynomial growth. Applying these results we can find exponentially harmonic maps in every homotopy class of maps from a connected Riemannian C4-manifold into [Formula: see text], where [Formula: see text] is a compact Riemannian C4-manifold.

On Eells-Sampson’s existence theorem for harmonic maps via exponentially harmonic maps

“…Also, for n ≥ 2, Naito [8] showed that an E-minimizer u : Ω → R n , where Ω ⊆ R m is a bounded domain, is smooth in the interior of Ω. Thereafter, Duc [1] at last showed the following strongest regularity theorem for E-minimizers. (2) As mentioned in [1,Section 3], the Hölder norm du C α of the gradient of an exponentially harmonic map u is estimated by a constant depending only on (M, g), (N, h), E(u), and the Lipschitz constant du L ∞ .…”

“…Thereafter, Duc [1] at last showed the following strongest regularity theorem for E-minimizers. (2) As mentioned in [1,Section 3], the Hölder norm du C α of the gradient of an exponentially harmonic map u is estimated by a constant depending only on (M, g), (N, h), E(u), and the Lipschitz constant du L ∞ . Therefore, in order to verify Theorem 1.1, it suffices to show that du ε L ∞ is uniformly bounded as ε → 0.…”

“…Also, for n ≥ 2, Naito [8] showed that an E-minimizer u : Ω → R n , where Ω ⊆ R m is a bounded domain, is smooth in the interior of Ω. Thereafter, Duc [1] at last showed the following strongest regularity theorem for E-minimizers.…”