一类双调和映照型偏微分方程组正则性研究

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Research on the Regularity of a Class of Biharmonic Map-Type Partial Differential Equation Systems

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Abstract:

Biharmonic mappings are an important class of geometric mappings, but the partial differential equations that are satisfied are very complex, making their regularity study difficult. In order to study this class of problems, in this note we consider a class of biharmonic map-type fourth order elliptic partial differential equation system

Δ^{2} u = Q_{1} (x, u, \nabla u, \nabla^{2} u) + d i v Q_{2} (x, u, \nabla u, \nabla^{2} u) in B_{1},

$\Delta^{2}u=Q_{1}(x,u,\nabla u,\nabla^{2}u)+{\rm div}\,\boldsymbol{Q}_{2}(x,u,\nabla u,\nabla^{2}u)\qquad\text{in }B_{1},$

where $B_1=\{x\in\mathbb{R}^{n}:|x|<1\}$ with $n\ge4$ , and $Q_{1},{\rm Q_{2}}$ satisfy critical growth conditions with respect to $\nabla u$ and $\nabla^2 u$ . Then, under suitable smallness assumption, this note proves that the solutions of this system of equations all have Hölder regularity, thus generalising related results in the literature. This result helps to deepen the understanding of the structure of biharmonic mappings and the research on the regularity theory.

Key words: biharmonic mappings, elliptic systems with critical nonlinearity, H?lder regularity, reverse Poincaré inequality, decay estimate

CLC Number:

O175.25

Anqi Liu,Ting Yu,Changlin Xiang. Research on the Regularity of a Class of Biharmonic Map-Type Partial Differential Equation Systems[J].Acta mathematica scientia,Series A, 2025, 45(2): 408-417.

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