Quantitative and Qualitative properties of solutions of partial differential equations

Louisiana State University, Baton Rouge LA

Investigators

Abstract

The proposal is concerned with the analysis and applications of nonlinear partial differential equations. The model problems in this proposal arise from the study of various nonlinear phenomena and other scientific disciplines, including condensed matter physics, elasticity, inverse problem, electrodynamics, quantum mechanics, fluid mechanics, mathematics biology, differential geometry, etc. The focus of the proposal research is the investigation of the quantitative and qualitative properties of solutions for partial differential equations. Providing quantitative and qualitative information for the solutions is fundamental and essential in the study of partial differential equations, which lies in the core of mathematical analysis. It is often the case that the most effective and economical way in scientific research is to explore properties of solutions and then to develop algorithm in accordance. Besides being very useful in applied science, the investigation of various kinds of structures and properties of solutions for various types of equations absolutely leads to new theories in mathematics. The proposed projects include quantitative uniqueness, eigenfunction and eigenvalue estimates, as well as Liouville-type theorems. Techniques and ideas from analysis area, such as elliptic estimates and Fourier analysis, will be combined and applied into this project. The proposed research should enhance the understanding of classical and Steklov eigenvalue problems, semilinear and higher order elliptic equations, wave equations, fractional Laplacians, fully nonlinear equations, etc. Further research will be devoted to the study of quantitative uniqueness of parabolic differential equations and other important equations from mathematical physics. Another related direction is the study of phase separations phenomenon in Bose-Einstein condensate. Emphasis will be placed on the two components Gross-Pitaevskki system. An important part of proposed research is on eigenfunction and eigenvalue estimates. Techniques and insights in the various areas cross-fertilize each other in a fruitful way in this area. The topics consist of measure of nodal sets (zero level sets), asymptotic behavior of eigenvalues, Lebesgue norm estimates, as well as doubling estimates of Steklov eigenfunctions and classical eigenfunctions. Much effort will be made towards Yau's conjecture asserting that the size of nodal sets is comparable to its frequency. The principal investigator will also continue the previous investigation on Liouville-type theorems on nonexistence of solutions for fractional Laplacian equations and fully nonlinear partial differential equations.