# Lewy方程的无穷可微解

 全部作者： 吴小庆 作者单位： 西南石油大学理学院 摘 要： 应用算子级数法求解Lewy方程，获得了局部解与整体解的解析表达式。分析Lewy定理的证明思想，认为当Lewy方程的自由项F′(t)不解析时是否有连续可微解，可以归结为复Cauchy-Riemann方程边值问题是否有连续可微解。证明了：1) 若复Cauchy-Riemann方程边值问题有无穷可微解，则Lewy方程有无穷可微解；2) 若F(t)为无穷可微函数，F(t)∈C∞(R), 且lim|t|→∞t2F(n)(t)=0, n∈N, 则复Cauchy-Riemann方程边值问题有无穷可微解；3) 存在在区间［-1，1］处处不解析的实值无穷可微函数F(t), 但Lewy方程在原点的实心邻域内存在无穷可微解。从而Lewy定理的结论不成立。并给出了Lewy方程积分形式的精确解的表达式。 关 键 词： 偏微分方程；Lewy方程；实解析；无穷可微解；紧支集 Title： The infinitely differentiable solution of Lewy equation Author： WU Xiaoqing Organization： School of Sciences, Southwest Petroleum University Abstract： This paper uses the operator series method to solve the Lewy equation, and has obtained the analytical expression of local solution and global solution. It has analyzed the demonstration of Lewy theorem and thinks that the problem whether Lewy equation has continuously differentiable solution when the free term F′(t) is non-analytical can be resolved as whether the boundary value problem of the complex Cauchy-Riemann equation has continuously differentiable solution. This paper has proved that: 1) If complex Cauchy-Riemann equation has infinitely differentiable solution, then Lewy equation has infinitely differentiable solution; 2) If F(t) is an infinitely differentiable solution F(t)∈C∞(R) and lim|t|→∞t2F(n)(t)=0, n∈N, then the boundary value problem of the complex Cauchy-Riemann equation also has infinitely differentiable solution; 3) There exists the real infinitely differentiable function F(t) which is non-analytical in the interval ［-1,1］, but there exists the infinitely differentiable solution in the solid field of the origin of Lewy equation. Then the conclusion of Lewy theorem is untenable. And this paper has given the accurate solution of the integral form of Lewy equation. Key words： partial differential equation; Lewy equation; real analytical; infinitely differential solution; compactly supported set 发表期数： 2010年1月第1期 引用格式： 吴小庆. Lewy方程的无穷可微解[J]. 中国科技论文在线精品论文，2010，3（1）：19-27. 0 评论数　0