Regularity of polyharmonic maps in the critical dimension
The authors consider the intrinsic and extrinsic \(m\)-polyenergy functional on a domain \(\Omega\) of \(\mathbb{R}^{n}\), \(n\geq 2m\), action on functions \(u:\Omega\to N\), \(u\in W^{m,2}(\Omega,N)\), where \(N\) is a Riemannian manifold immersed in \(\mathbb{R}^K\) \[ E_m(u)=\frac{1}{2}\int_{\Omega}|D^mu|^2dx \quad \quad \tilde{E}_m(u)=\frac{1}{2}\int_{\Omega}|\nabla^{m-1}Du|^2dx, \] and study the regularity of weakly extrinsic and intrinsic \(m\)-ployharmonic maps, resp., that is, of the critical points of \(E_m\) and of \(\tilde{E}_m\), resp., for all variations \(u_t=\pi_N(u+tW)\), with \(\pi_N\) the nearest-point retraction onto \(N\). Their first main theorem states that if \(n=2m\) and \(N\) is compact, such critical points are smooth, generalizing a result of \textit{C.\ Wang} [Math. Z. 247, No.~1, 65--87 (2004; Zbl 1064.58016)] for the case \(m=2\). For the case \(n\geq 2m\), the authors establish a second main theorem that obtains smoothness of a weakly extrinsic or intrinsic \(m\)-polyharmonic map that is locally Hölder continuous. The proof of the first theorem consists in showing that \(u\) is Hölder continuous for the case of critical dimension \(n=2m\). To prove their results they use Lorentz spaces \(L^{p,q}(\Omega)\), and recall Sobolev-Lorentz embedding theorems, Poincaré-type inequalities, and derive a fundamental elliptic inequality for the \(L^{p,q}(\Omega)\)-norm of the Hessian of \(f\) on a ball of radius \(r<1\) in terms of the norms of \(\Delta f\), \(Df\) and \(f\) on the unit ball. By interpolation methods and using Hodge decompositions, the norm of higher order derivatives of \(f\) is estimated. Reformulating an orthogonal condition that extrinsically polyharmonic maps satisfy, the authors derive a new form of the Euler-Lagrangian equation, and using a Uhlenbeck's gauge theorem and arguments due to Schoen an Uhlenbeck, leads to the construction of Coulomb frames satisfying appropriate estimates, and consequent decay and growth estimates for weak polyharmonic maps \(u\in W^{m,2}\), as in the above paper of Wang. Since Hölder continuity is proved (or assumed if \(n\geq 2m\)), the proof follows as in [\textit{S.-Y. A.\ Chang, L.\ Wang} and \textit{P. C.\ Yang}, Commun.\ Pure Appl.\ Math. 52, 1099--1111 (1999; Zbl 1044.58019)] by using Nirenberg interpolation formulas, some inequalities and Schauder theory to obtain higher order estimates with corresponding higher regularity. The intrinsic case requires some extra technicalities on handling additional terms.
scientific article; zbMATH DE number 5606319