Application of a Variant of Mountain Pass Theorem in Modeling Real Phenomena

Abstract

Mountain Pass Theorem (MPT) is an important result in variational methods with multiple applications in partial differential equations involved in mathematical physics. Starting from a variant of MPT, a new result concerning the existence of the solution for certain mathematical physics problems involving p-Laplacian and p-pseudo-Laplacian has been obtained. Based on the main theorem, the existence, possibly the uniqueness, and characterization of solutions for models such as nonlinear elastic membrane, glacier sliding, and pseudo torsion problem have been obtained. The novelty of the work consists of the formulation of the central result under weaker conditions requested by the chosen variant of MPT, the proof of this statement, and its application in solving above mentioned problems. While the expressions of such Dirichlet and/or von Neumann problems were already completed, this proposed solving method suggests some specific numerical methods to construct the appropriate solution. A general goal of this paper is the extension of the applicative pallet of this way to construct the solutions encountered in modeling real processes developed within new emerging technologies.

Solutions for Some Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 1

“…Solving such problems via results obtained with Ekeland variational principle and other generalized variational principles has been the goal of other works of the author where some Dirichlet or Newmann problems have been studied as in [12,14,16] with the use of a perturbed variational principle, in [17] with variational procedures, as also in [18]. Mountain pass theorem variants and applications involved in modeling real phenomena are performed by the author in [19], while other applied variational methods are capitalized on by the author in [20,21], and several variational principles, together with generalizations and variants, have been compared and analyzed in the monograph [22].…”

Solutions for Some Specific Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 2

“…Finally, the last frame, which originated in a perturbed variational principle, was provided to intervene in the study of a generalized Helle-Show flow of a power-law fluid, and also for the pseudo torsion problem. For some of the problems derived from real phenomena modeling presented in this work, the author proposed different solving methods to those reported in the paper [21]. Starting from theoretical results based on a namely version of Mountain Pass Theorem, the study [21] aimed to gain a better understanding of the passage from the high abstract frame to characterize the solution, with designing new models for special physical phenomena being the final goal.…”

“…For some of the problems derived from real phenomena modeling presented in this work, the author proposed different solving methods to those reported in the paper [21]. Starting from theoretical results based on a namely version of Mountain Pass Theorem, the study [21] aimed to gain a better understanding of the passage from the high abstract frame to characterize the solution, with designing new models for special physical phenomena being the final goal. Some other applications for problems involving the p-Laplacian having the theoretical background in a perturbed variational principle have been presented by the author in [22].…”