Title: Linear non-divergence elliptic equations in a bounded, infinitely winding planar domain Show Chinese title

Title: 有界且无限缠绕的平面区域中的线性非散度椭圆方程

Abstract: We study the second order elliptic equations of non-divergence form in a planar domain with complicated geometry. In this case the domain winds around a fixed circle infinitely many times and converges to it when the rotating angle goes to infinity. For the homogeneous equation and the homogeneous Dirichlet boundary condition, in the case of bounded drifts, we prove that the maximum of the solution on the cross-section corresponding to a given rotating angle either grows or decays exponentially as the angle goes to infinity. Results for the oscillation and its asymptotic estimates are also obtained for inhomogeneous Dirichlet data. If the drift is unbounded but does not grow to infinity too fast, then the above maximum also goes to either zero or infinity. For the inhomogeneous equation, we obtain the estimates in the case of bounded forcing functions. Moreover, we establish the uniqueness of the solution and its continuous dependence on the boundary data and the forcing function. Show Chinese abstract

Abstract: 我们研究平面区域中非散度形式的二阶椭圆方程。在这种情况下，区域围绕一个固定圆无限次缠绕，并且当旋转角趋于无穷时收敛于该圆。对于齐次方程和齐次狄利克雷边界条件，在漂移有界的情况下，我们证明了对应于给定旋转角的横截面上解的最大值随着角度趋于无穷时要么指数增长要么指数衰减。对于非齐次狄利克雷数据，也得到了振荡及其渐近估计的结果。如果漂移是无界的但不会太快趋于无穷，那么上述最大值也会趋于零或无穷。对于非齐次方程，我们在有界强迫函数的情况下得到了估计。此外，我们建立了解的唯一性以及其对边界数据和强迫函数的连续依赖性。