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.
In the 19th century, A.J.C. 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]). Many recent recent results are inspired by Saint-Venant principle (c.f., [3,4,5] and many others).
The energetic estimates were received first in [6,7]. These estimates do not take into account character of transformation of the body form at moving off from those part of the bound where exterior forces are applied. In the paper , a proof of the Saint-Venant principle in the planar theory of elasticity was obtained by different way. The energetic estimate was gained in the connection considered character of transformation of the body form. The uniqueness theorem for the first boundary value problem of the planar theory of elasticity in unlimited domains and also Pharagmen–Lindelöf type theorems were obtained as a corollary of the energetic estimate. The proofs of the Pharagmen–Lindelöf type theorems were done for equations of the theory of elasticity in  and for elliptic equations of higher order in the papers [2,6,7,10,11,12,13,14]. The Saint-Venant principle for a cylindrical body was studied in .
Boundary value problems have applications in fluid dynamics, astrophysics, hydrodynamic, hydromagnetic stability, astronomy, beam and long wave theory, induction motors, engineering, and applied physics. Boundary value problems of higher order is studied in papers [16,17]. An overview of some results on the class of functions with subharmonic behaviour and their invariance properties under conformal and quasiconformal mappings is presented in .
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 of equations in the case of space with boundary conditions of the first boundary value problem (c.f., [19,20]). Similar results were obtained for the mixed problems in .
We shall note else work [12,22], which by means of principle Saint-Venant’s is studied asymptotic characteristic of the solutions of the third order equations of the composite type and dynamic systems.
Boundary value problems have applications in fluid dynamics, astrophysics, hydrodynamic, hydromagnetic stability, astronomy, beam and long wave theory, induction motors, engineering, and applied physics.
2. Notations and Formulation of the Problem
Consider in the unlimited domain Q the equation
We suppose here and later on that the summation is carried out by repeating indexes, all coefficients in (1) and their derivatives are bounded and measurable in any finite subdomain of the domain Q. Furthermore, we suppose that boundary of Q is smooth or piecewise-smooth. We assume that the operators are uniformly elliptic, i.e.,
Let and is a vector of the inner normal of Q in the point
We break up the bound of Q. Denote
Consider in Q the boundary value problem
Define the operator d:
Assume that the condition
For some define
Let be a set of functions such that in and on for some
We denote as the Hilbert space obtained by closing with respect to the norm
Now consider bilinear form
If for any and
for an arbitrary function where then the function is said to be a generalized solution of the problem (1),(3) in the domain
3. Energy Inequalities
(Analog of the Saint-Venant principle)
Let for all
If is generalized solution of the problem (1), (3) and at then for any such that takes place
Here is a solution of the problem
is an arbitrary continuous function such that
N is the set of continuously differentiable functions in the neighborhood of which are equal to zero in
Assume in (5) where if if and if
It is obvious that at Integrating by parts (10), we have
The estimation (6) follows from (8) and (11) at ☐
Now we will estimate in case when can be included to the -dimensional parallelepiped which smallest edge is equal to Suppose that
Applying the Friedreich and Cauchy–Bunyakovsky inequalities, we have from (9)
Therefore we can set
If in then Consequently
Let as the domain Q lies inside the rotation body i.e., We have from (15)
In this case, from the inequality (6) we have
Consider an example of Q for which
It is clear that if the domain Q is narrowing at If then and this case includes domains lying in the band with the width If then Q can be extended respectively at For this kind of domains, we can assume
Then the estimate (6) is valid for considered domains if
As a corollary of the Saint-Venant principle, we have the uniqueness theorem for the problem (1), (3) in unlimited domain Q for classes of functions increasing in infinity depending from
Let in Q and conditions of theorem 1 hold. If is a generalized solution of the problem (1), (3) in Q and for a sequence at and some
where at then in
We have from (6) considering (13)
at Hence in
Further for any fixed we have
Hence, in Since was chosen arbitrary, in ☐
In the present paper, the analogy of the Saint-Venant principle is established for the generalized solution of the third order pseudoelliptical type equation. Furthermore, uniqueness theorems are obtained for solutions of the first boundary value problem in classes of functions increasing in infinity depending on the geometric characteristics of the domain were is bounded domain. Boundary value problems for the third order pseudoelliptical type equations in bounded domains were considered in .
The main goal of our research on these problems consists of the following parts:
Establish energy estimates (analogous to the Saint-Venant’s principle) that allow us to determine the widest class of uniqueness of solutions to the problem depending on the geometric characteristics of the domain.
Construction of the solution of the problem under study on an unbounded domain in classes of functions growing at infinity.
Establish estimates for solutions of the problem and its derivatives at infinitely remote boundary points.
The first part of our research on these problems is given in this paper. The remaining two parts will be studied in the future, which will be performed on the basis of this paper. Therefore, the results of this article are necessary and relevant for further qualitative research to solve third-order equations in the vicinity of irregular boundary points.
Conceptualization, methodology, validation, formal analysis, investigation A.R.K.; validation, formal analysis, D.S. All authors have read and agreed to the published version of the manuscript.
The project is funded by the Institute of Technology and Business in České Budějovice, grant numbers: IGS 8210-004/2020 and IGS 8210-017/2020.
Conflicts of Interest
The authors declare no conflict of interest.
The following abbreviations are used in this manuscript:
Abdukomil Risbekovich Khashimov
Barré de Saint-Venant, A.J.C. De la torsion des prismes. Mem. Divers Savants Acad. Sci. Paris1855, 14, 233–560. [Google Scholar]
Gurtin, M.E. The Linear Theory of Elasticity. In Handbuch der Physik; Springer: Berlin, Germany, 1972. [Google Scholar]
Marin, M.; Oechsner, A.; Craciun, E.M. A generalization of the Saint-Venant’s principle for an elastic body with dipolar structure. Contin. Mech. Thermodyn.2020, 32, 269–278. [Google Scholar] [CrossRef]
Roohi, M.; Soleymani, K.; Salimi, M.; Heidari, M. Numerical evaluation of the general flow hydraulics and estimation of the river plain by solving the Saint-Venant equation. Model. Earth Syst. Environ.2020, 6, 645–658. [Google Scholar] [CrossRef]
Xiao, B.; Sun, Z.X.; Shi, S.X.; Gao, C.; Lu, Q.C. On random elastic constraint conditions of Levinson beam model considering the violation of Saint-Venant’s principle in dynamic. Eur. Phys. J. Plus2020, 135, 168. [Google Scholar] [CrossRef]
Flavin, J.N. On Knowels version of Saint-Venant’s principle two dimensional elastostatics. Arch. Ration. Mech. Anal.1974, 53, 366–375. [Google Scholar] [CrossRef]
Knowels, J.K. On Saint-Venant’s principle in the two-dimensional linear theory of elasticity. Arch. Ration. Mech. Anal.1966, 21, 1–22. [Google Scholar] [CrossRef]
Oleinik, O.A.; Yosifian, G.A. On Saint-Venant’s principle in the planar theory of elasticity. Dokl. AN SSSR1978, 239, 530–533. [Google Scholar]
Worowich, I.I. Formulation of boundary value problems of the theory of elasticity at infinite energy integral and basis properties of homogeneous solutions. In Mechanics of Deformable Bodies and Constructions; Mashinistroenie: Moscow, Russia, 1975; pp. 112–118. [Google Scholar]
Galaktionov, V.A.; Shishkov, A.E. Saint-Venant’s principle in blow-up for higher-order quasilinear parabolic equations. Proc. Sec. A Math. R. Soc. Edinb.2003, 133, 1075–1119. [Google Scholar] [CrossRef]
Galaktionov, V.A.; Shishkov, A.E. Structure of boundary blow-up for higher order quasilinear parabolic PDE. Proc. Roy. Soc. Lond. A2004, 460, 3299–3325. [Google Scholar] [CrossRef]
Khashimov, A.R. Estimation of derived any order of the solutions of the boundary value problems for equation of the third order of the composite type on infinity. Uzb. Math. J.2016, 2, 140–148. [Google Scholar]
Kozhanov, A.I. Boundary Value Problems for Equations of Mathematical Physics of the Odd Order; Nauka: Novosibirsk, Russia, 1990; 149p. [Google Scholar]
Landis, E.M. On behavior of solutions of elliptic higher order equations in unlimited domains. Trudy Mosk. Mat. Ob.1974, 31, 35–38. [Google Scholar]
Fabiano, N.; Nikolić, N.; Shanmugam, T.; Radenović, S.; Čitaković, N. Tenth order boundary value problem solution existence by fixed point theorem. J. Inequal. Appl.2020, 166, 1–11. [Google Scholar] [CrossRef]
Shanmugam, T.; Muthiah, M.; Radenović, S. Existence of Positive Solution for the Eighth-Order Boundary Value Problem Using Classical Version of Leray—Schauder Alternative Fixed Point Theorem. Axioms2019, 8, 129. [Google Scholar] [CrossRef]
Todorčević, V. Subharmonic behavior and quasiconformal mappings. Anal. Math. Phys.2019, 9, 1211–1225. [Google Scholar] [CrossRef]
Oleinik, O.A.; Radkevich, E.V. Analyticity and Liouville and Pharagmen–Lindelöf type theorems for general elliptic systems of differential equations. Mate. Sb.1974, 95, 130–145. [Google Scholar]
Oleinik, O.A.; Yosifian, G.A. On singularities at the boundary points and uniqueness theorems for solutions of the first boundary value problem of elasticity. Comm. Partial. Differ. Equ.1977, 2, 937–969. [Google Scholar] [CrossRef]
Oleinik, O.A.; Yosifian, G.A. Saint-Venant’s principle for the mixed problem of the theory of elasticity and its applications. Dokl. AN SSSR1977, 233, 824–827. [Google Scholar]
Berdichevsky, V.; Foster, D.J. On Saint-Venant’s principle in the dynamics of elastic beams. Int. J. Solids Struct.2003, 40, 3293–3310. [Google Scholar] [CrossRef]