A Study Of The Monotonicity Of The Region For The Principal Eigenvalue Of The Laplace Operator

Laplace operator is a representative operator,and it is also one of the important mathematical models used to solve problems in mathematical physics and other fields.The eigenvalues of Laplace operator are arranged from small to large,and the principal eigenvalue is the first non-zero eigenvalue in these eigenvalues.The principal eigenvalue is of great significance studying the steady state of the equation,that is,the large time behavior of the solution of the equation.Because the size of the principal eigenvalue of Laplace operator changes with the change of region,the paper mainly studies the regional monotonicity of the principal eigenvalue of Laplace operator under the condition of Neumann boundary.Through multiple instances,the solution and proof of the principal eigenvalue of Laplace operator are discussed in different regions of coordinate system,polar coordinate system and spatial coordinate systemand,and the monotonic properties of the principal eigenvalue are also discussed.A conclusion that the region monotonicity of the principal eigenvalue of Laplace operator is not certain under Neumann boundary condition is confirmed.In order to solve the problem,the commonly used equations in partial differential equations,such as Bessel equation,are introduced in the paper.The basic calculation method of solving the equation is the method of separating variables.The full text is divided into four chapters:The first chapter introduces the research background,significance and current situation of Laplace operator eigenvalue,and gives the main research results.The second chapter gives the basic theory used in the proof process.The third chapter studies the principal eigenvalue of the laplace operator under Neumann boundary condition in two-dimensional space.Using the method of separating variables,the two-dimensional equation is reduced to one-dimensional,the principal eigenvalue are found and proved.The monotonic properties of the principal eigenvalue in rectangular,circular,sector and sector ring regions are given by positive and negative derivatives of the principal eigenvalue.The fourth chapter studies the principal eigenvalue of Laplace operator under Neumann boundary condition in three-dimensional space,and gives the monotone properties of the principal eigenvalue in the region defined by sphere and area defined by concentric spheres.