Regularity of Exponentially Harmonic Functions

Existence and uniqueness of weak solutions for a non-uniformly parabolic equation

“…The corresponding elliptic case, which originated from the exponential harmonic mappings has been studied in [21,12,17], especially the regularity theory. Naito [21] proved existence, uniqueness and C α regularity of the minimizer.…”

“…Naito [21] proved existence, uniqueness and C α regularity of the minimizer. Duc and Eells [12], Lieberman [17] respectively proved the C ∞ or C 1,α regularity of the minimizer. Lieberman [19] proved the interior C 1,α -estimate for the parabolic counterpart.…”

On a class of non-uniformly elliptic equations

“…It has been studied in [10,16,17], especially for the regularity theory. Naito [17] proved the existence, uniqueness and C α regularity of the minimizer.…”

“…Naito [17] proved the existence, uniqueness and C α regularity of the minimizer. Duc and Eells [10], Lieberman [16] proved the C ∞ and C 1,α regularity of the minimizer, respectively. Besides, Siepe [21] studied the Lipschitz regularity of the minimizers of the more general functional This condition implies that Φ(ξ) would be controlled by a polynomial of |ξ|.…”

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

“…However, the faster the growth of a functional, the higher the regularity of its minima that we can expect. Indeed, in the case of N = R, Duc and Eells [2] showed that an E-minimizer u : (M, g) → R of the Dirichlet problem is smooth in the interior of M , where (M, g) is a compact Riemannian manifold with boundary, and Lieberman [7] showed the global regularity T. OMORI for u : Ω → R, where Ω ⊆ R m is an open subset. 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.…”