On the Variational Symmetries of P.D.E's Incorporating Boundary Value Constraints

A crucial interface between Optimization theory, Trace theory, and Lie Symmetry theory is brought to the fore in this paper, as an enhancement of standard known results established in Partial Differential Equation (P.D.E) analysis. In particular, the incorporation of compatible Boundary Value constraints in the re-assessment of admitted variational symmetries is a key process in the development of the discussions and results. Classical variational formulation techniques for a few Boundary Value Problems are considered. Ramifications of standard and modified admitted variational symmetries in simplification of PDEs constitute further relevant crucial details addressed appreciably.