Lp-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions

作者：Guo, Chang-Yu; Wang, Changyou; Xiang, Chang-Lin^*

来源：Calculus of Variations and Partial Differential Equations, 2023, 62(1): 31.

DOI：10.1007/s00526-022-02373-7

• BIHARMONIC MAPS
• HARMONIC MAPS
• CONSERVATION-LAWS
• WEAK SOLUTIONS
• OPERATOR
• SPACES

We establish an optimal L-p-regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions n >= 5: @@@ Delta(2)u = Delta(D. del u) + div(E . del u) + (Delta Omega + G) . del u + f in B-n, @@@ where Omega is an element of W-1,W-2 (B-n, so(m)) is antisymmetric and f is an element of L-p (B-n), and D, E, Omega, G satisfy the growth condition (GC-4), under the smallness condition of a critical scale invariant norm of del(u) and del(2)u. This system was brought into lights from the study of regularity of (stationary) biharmonic maps between manifolds by Lamm-Riviere, Struwe, and Wang. In particular, our results improve Struwe's Holder regularity theorem to any Holder exponent alpha is an element of (0, 1) when f 0, and have applications to both approximate biharmonic maps and heat flow of biharmonic maps. As a by-product of our techniques, we also partially extend the L-p-regularity theory of approximate harmonic maps by Moser to Riviere-Struwe's second order elliptic systems with antisymmetric potentials under the growth condition (GC-2) in all dimensions n >= 2 when p >= n/2, which partially confirms an interesting expectation by Sharp.

• 单位
  山东大学