On the Uniqueness Classes of Solutions of Boundary Value Problems for Third-Order Equations of the Pseudo-Elliptic Type

Abdukomil Risbekovich Khashimov,Dana Smetanová

doi:10.3390/axioms9030080

Abdukomil Risbekovich Khashimov, Dana Smetanová

Open Access

https://doi.org/10.3390/axioms9030080

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Journal: Axioms	Publication Date: Jul 16, 2020
Citations: 1	License type: CC BY 4.0

Affiliation: Institute of Technology and Business

Abstract

The paper is devoted to solutions of the third order pseudo-elliptic type equations. An energy estimates for solutions of the equations considering transformation’s character of the body form were established by using of an analog of the Saint-Venant principle. In consequence of this estimate, the uniqueness theorems were obtained for solutions of the first boundary value problem for third order equations in unlimited domains. The energy estimates are illustrated on two examples.

Highlights

The uniqueness theorem for the first boundary value problem of the planar theory of elasticity in unlimited domains and Pharagmen–Lindelöf type theorems were obtained as a corollary of the energetic estimate
An analog of the Saint-Venant principle, uniqueness theorems in unlimited domains, and Pharagmen–Lindelöf type theorems in the theory of elasticity were derived for the system
The analogy of the Saint-Venant principle is established for the generalized solution of the third order pseudoelliptical type equation

Summary

Introduction

Barré de Saint-Venant studied the planar theory of elasticity His principle is expressed as a prior estimate for a solution of a biharmonic equation satisfying homogeneous boundary conditions of the first boundary value problem in the part of the domain boundary (c.f., [1,2]). The uniqueness theorem for the first boundary value problem of the planar theory of elasticity in unlimited domains and Pharagmen–Lindelöf type theorems were obtained as a corollary of the energetic estimate. Boundary value problems have applications in fluid dynamics, astrophysics, hydrodynamic, hydromagnetic stability, astronomy, beam and long wave theory, induction motors, engineering, and applied physics. An analog of the Saint-Venant principle, uniqueness theorems in unlimited domains, and Pharagmen–Lindelöf type theorems in the theory of elasticity were derived for the system.

Notations and Formulation of the Problem

Energy Inequalities

Conclusions