A Generalized Palais-Smale Condition in the Fr\'{e}chet space setting

The Palais-Smale condition was introduced by Palais and Smale in the mid-sixties and applied to an extension of Morse theory to infinite dimensional Hilbert spaces. Later this condition was extended by Palais for the more general case of real functions over Banach-Finsler manifolds in order to obtain Lusternik-Schnirelman theory in this setting. Despite the importance of Fr\'{e}chet spaces, critical point theories have not been developed yet in these spaces.Our aim in this paper is to extend the Palais-Smale condition to the cases of $C^1$-functionals on Fr\'{e}chet spaces and Fr\'{e}chet-Finsler manifolds of class $C^1$. The difficulty in the Fr\'{e}chet setting is the lack of a general solvability theory for differential equations. This restricts us to adapt the deformation results (which are essential tools to locate critical points) as they appear as solutions of Cauchy problems. However, Ekeland proved the result, today is known as Ekleand’s variational principle, concerning the existence of almost-minimums for a wide class of real functions on complete metric spaces. This principle can be used to obtain minimizing Palais-Smale sequences. We use this principle along with the introduced conditions to obtain some customary results concerning the existence of minima in the Fr\'{e}chet setting.Recently it has been developed the projective limit techniques to overcome problems (such as solvability theory for differential equations) with Fr\'{e}chet spaces. The idea of this approach is to represent a Fr\'{e}chet space as the projective limit of Banach spaces. This approach provides solutions for a wide class of differential equations and every Fr\'{e}chet space and therefore can be used to obtain deformation results. This method would be the proper framework for further development of critical point theory in the Fr\'{e}chet setting.

Solvability Theory and Iteration Method for One Self-Adjoint Polynomial Matrix Equation

The solvability theory of an important self-adjoint polynomial matrix equation is presented, including the boundary of its Hermitian positive definite (HPD) solution and some sufficient conditions under which the (unique or maximal) HPD solution exists. The algebraic perturbation analysis is also given with respect to the perturbation of coefficient matrices. An efficient general iterative algorithm for the maximal or unique HPD solution is designed and tested by numerical experiments.

Comparison of Sharp Interface Dendritic Growth Simulation With Microscopic Solvability Theory

We present and validate a numerical technique for computing dendritic growth of crystals from pure melts. The solidification process is computed in the diffusion-driven limit. The mixed Eulerian-Lagrangian framework treats the immersed phase boundary as a sharp solid-fluid interface and a conservative finite volume formulation allows boundary conditions at the moving surface to be exactly applied. The case of discontinuous material properties is also computed. The results from our calculations are compared with two-dimensional microscopic solvability theory. It is shown that the method predicts dendrite tip details in good agreement with solvability theory. The ability of the method to treat the front as a sharp entity and therefore to respect discontinuous material property variation at the solid-liquid interface is also shown to produce results in agreement with solvability and with other sharp interface simulations.