Dirac方程解的估计及其在一个椭圆-抛物问题中的应用
Estimates for solutions of Dirac equations and an application to a geometric elliptic-parabolic problem
摘要：
我们开发的解决方案的估计，并得出各种局部边值问题的狄拉克方程，改善文献中已知的所有相关结果的存在性和唯一性结果。利用这些估计，我们得到了具有这种边界条件的Dirac方程解的存在唯一性和正则性定理。我们也将这些估计应用到一个新的非线性椭圆抛物问题，黎曼自旋流形上的狄拉克调和热流。这个问题的动机是超对称非线性$σ$模型，并结合了一个谐波热流型方程与狄拉克方程，非线性地依赖于流量。
英文摘要：
We develop estimates for the solutions and derive existence and uniqueness results of various local boundary value problems for Dirac equations that improve all relevant results known in the literature. With these estimates at hand, we derive a general existence, uniqueness and regularity theorem for solutions of Dirac equations with such boundary conditions. We also apply these estimates to a new nonlinear elliptic-parabolic problem, the Dirac-harmonic heat flow on Riemannian spin manifolds. This problem is motivated by the supersymmetric nonlinear $σ$-model and combines a harmonic heat flow type equation with a Dirac equation that depends nonlinearly on the flow.
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