论文标题
具有对数电势的随机allen-cahn方程
Stochastic Allen-Cahn Equation with Logarithmic Potential
论文作者
论文摘要
在同质的诺伊曼边界条件下，我们证明了具有对数电位和对数电势和乘法性的Wiener噪声的随机allen-cahn方程的解决方案的存在和独特性。通过涉及非线性Yosida正则化的近似方程，以变异意义获得了溶液的存在。假定噪声在物理相关域的极端位置消失，并满足合适的Lipschitz-continition属性，从而证明了近似溶液的均匀估计。然后进行对极限的段落，并验证对解决方案的初始基准的连续依赖性。在另外的假设下，证明了分析上强的解决方案的存在。最后，鉴于研究与随机allen-cahn方程相关的最佳控制问题的研究，得出了对数电位的衍生物的估计。
We prove existence and uniqueness of a solution for the stochastic Allen-Cahn equation with logarithmic potential and multiplicative Wiener noise, under homogeneous Neumann boundary condition. The existence of a solution is obtained in the variational sense by means of an approximated equation involving a Yosida regularization of the nonlinearity. The noise is assumed to vanish at the extremal points of the physical relevant domain and to satisfy a suitable Lipschitz-continuity property, allowing to prove uniform estimates of the approximated solution. The passage to the limit is then carried out and continuous dependence on the initial datum of the solution is verified. Under an additional assumption, the existence of an analytically strong solution is proved. Finally, estimates for the derivatives of the logarithmic potential are derived, in view of the study of an optimal control problem associated to the stochastic Allen-Cahn equation.
