平面域上p-Laplace算子最小特征值的数值研究
Numerical investigation of the smallest eigenvalues of the p-Laplace operator on planar domains
AI总结：
使用约束下降法和约束山口算法数值研究平面域上p-Laplace算子的两个最小特征值随p的变化，并探讨第二特征函数的对称性变化。
摘要：
考虑在具有零Dirichlet边界条件的平面域上，p>1的p-Laplace算子的特征值问题。在Sobolev空间设置中，使用约束下降法和约束山口算法数值研究两个最小特征值对p的依赖性。对p在1.1到10之间的值进行计算。还数值研究了第二特征函数的对称性。对于圆盘，关于将圆盘分成两半的节线的奇对称性在所有考虑的p值下保持不变，而对于矩形和三角形，对称性随p变化。基于数值证据，这种情况下的对称性变化发生在某个依赖于区域的p_0值处。
英文摘要：
The eigenvalue problem for the p-Laplace operator with p>1 on planar domains with the zero Dirichlet boundary condition is considered. The Constrained Descent Method and the Constrained Mountain Pass Algorithm are used in the Sobolev space setting to numerically investigate the dependence of the two smallest eigenvalues on p. Computations are conducted for values of p between 1.1 and 10. Symmetry properties of the second eigenfunction are also examined numerically. While for the disk an odd symmetry about the nodal line dividing the disk in halves is maintained for all the considered values of p, for rectangles and triangles symmetry changes as p varies. Based on the numerical evidence the change of symmetry in this case occurs at a certain value p_0 which depends on the domain.
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