针对Heisenberg群上p-次Laplace方程,建立其弱解的Liouville型定理,证明过程主要基于Moser迭代技巧和弱解的正则性结果。In this paper, we give a Liouville type theorem for the weak solution of the p-sub-Laplacian equation on the Heisenberg group. The proof process relies on Moser iterative techniques and the regularity results of weak solutions.
本文研究了一类具有非局部项的p-Laplace方程的边界最优控制问题,通过对成本泛函的极小化序列取极限给出p-Laplace方程初边值问题最优控制函数的存在性。首先利用能量估计方法研究该问题解的存在唯一性,其次利用紧性估计和紧嵌入定理分析成本泛函极小化序列的收敛性,最后由成本泛函的弱下半连续性证明最优控制函数的存在性。In this paper, we study the boundary optimal control problem of a class of p-Laplace equations with non-local terms, and the existence of the optimal control function of the initial boundary value problem of the p-Laplace equation is given by taking the limit of the minimization sequence of the cost function. Firstly, the energy estimation method is used to study the existence uniqueness of the solution of the problem, then the tightness estimation and the tight embedding theorem are used to analyze the convergence of the cost functional minimization sequence, and finally the existence of the optimal control function is proved by the weak lower semi-continuity of the cost function.
主要介绍了一种证明弱解存在性的一种方法——变分法,变分法的基本内容是确定泛函的极值点和临界点,在一定条件下微分方程边值问题常常可以转化为变分问题来研究。首先通过给定的泛函求极值元,极值点处的方程在分部积分的意义下满足弱解定义,其次构造极小元泛函,将所求问题转化为求解相应泛函的极值元,即得方程弱解的存在性,接下来证明泛函极值元的存在性和弱解的唯一性,从而由变分方法确定该四阶定态p-Laplace方程弱解的存在性问题。This paper introduces a method to prove the existence of weak solutions—variational method. The basic content of variational method is to determine the extreme point and critical point of the functional. Under certain conditions, the boundary value problem of differential equations can often be studied by converting the variational problem. This paper first uses the given functional to find the extreme value element, and the equation at the extreme point satisfies the definition of weak solution in the sense of distribution integral. Secondly, we construct the minimal element functionals, and transform the problem into the corresponding universal extreme element, and we obtain the existence of weak solutions, and next, we prove the uniqueness of weak solutions and the existence of functional extremum elements. we finally give the existence of weak solutions for the weak solutions of the fourth-order stationary p-Laplace equation through the variational method.