On biharmonic maps and their generalizations
نویسندگان
چکیده
Abstract. We give a new proof of regularity of biharmonic maps from four-dimensional domains into spheres, showing first that the biharmonic map system is equivalent to a set of bilinear identities in divergence form. The method of reverse Hölder inequalities is used next to prove continuity of solutions and higher integrability of their second order derivatives. As a byproduct, we also prove that a weak limit of biharmonic maps into a sphere is again biharmonic. The proof of regularity can be adapted to biharmonic maps on the Heisenberg group, and to other functionals leading to fourth order elliptic equations with critical nonlinearities in lower order derivatives.
منابع مشابه
Stability of F-biharmonic maps
This paper studies some properties of F-biharmonic maps between Riemannian manifolds. By considering the first variation formula of the F-bienergy functional, F-biharmonicity of conformal maps are investigated. Moreover, the second variation formula for F-biharmonic maps is obtained. As an application, instability and nonexistence theorems for F-biharmonic maps are given.
متن کاملRemarks on biharmonic maps into spheres
We prove an apriori estimate in Morrey spaces for both intrinsic and extrinsic biharmonic maps into spheres. As applications, we prove an energy quantization theorem for biharmonic maps from 4-manifolds into spheres and a partial regularity for stationary intrinsic biharmonic maps into spheres. x
متن کاملEnergy identity of approximate biharmonic maps to Riemannian manifolds and its application
We consider in dimension four weakly convergent sequences of approximate biharmonic maps to a Riemannian manifold with bi-tension fields bounded in L for p > 4 3 . We prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivial biharmonic maps on R. As a corollary, we obtain an energy identity for the heat flow of biharmoni...
متن کاملRegularity and uniqueness of the heat flow of biharmonic maps
In this paper, we first establish regularity of the heat flow of biharmonic maps into the unit sphere S ⊂ R under a smallness condition of renormalized total energy. For the class of such solutions to the heat flow of biharmonic maps, we prove the properties of uniqueness, convexity of hessian energy, and unique limit at t = ∞. We establish both regularity and uniqueness for Serrin’s (p, q)-sol...
متن کاملEnergy identity for a class of approximate biharmonic maps into sphere in dimension four
We consider in dimension four weakly convergent sequences of approximate biharmonic maps into sphere with bi-tension fields bounded in L for some p > 1. We prove an energy identity that accounts for the loss of Hessian energies by the sum of Hessian energies over finitely many nontrivial biharmonic maps on R.
متن کامل