**1. Introduction**

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., [3–5] 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 [8], 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 [9] and for elliptic equations of higher order in the papers [2,6,7,10–14]. The Saint-Venant principle for a cylindrical body was studied in [15].

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 [18].

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 [21].

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

$$L\_0 l u + L\_1 u + M u = f(x, y, t) \tag{1}$$

where

$$M u = u\_t + a^k(\mathbf{x})u\_{\mathbf{x}\_k} + a\_0(\mathbf{x})u\_\prime \quad L\_1 u = b^{ij}(\mathbf{x})u\_{\mathbf{x}\_i \mathbf{x}\_j} + b^i(\mathbf{x})u\_{\mathbf{x}\_i \mathbf{x}\_j}$$

$$L\_o u = u\_t - a^{ij}(\mathbf{x})u\_{\mathbf{x}\_i \mathbf{x}\_j} + a^i(\mathbf{x})u\_{\mathbf{x}\_i} + a\_0(\mathbf{x})u\_\prime$$

$$M u = c^{pq}(\mathbf{x})u\_{y\_py\_q} + c^p(\mathbf{x})u\_{y\_p} + c\_0(\mathbf{x})u\_\prime$$

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 *Lo*, *M* are uniformly elliptic, i.e.,

$$a^{ij} = a^{ji}, \quad \lambda\_0 |\xi|^2 \le a^{ij} \xi\_i \xi\_j \le \lambda\_1 |\xi|^2, \quad \text{for all} \quad (\mathbf{x}, y, t) \in \mathcal{Q} \cup \partial \mathcal{Q}, \quad \text{for all} \quad \xi \in \mathbb{R}^{n+m+1}$$

$$c^{pq} = c^{qp}, \quad \mu\_0 |\xi|^2 \le a^{ij} \xi\_i \xi\_j \le \mu\_1 |\xi|^2, \quad \text{for all} \quad (\mathbf{x}, y, t) \in \mathcal{Q} \cup \partial \mathcal{Q}, \quad \text{for all} \quad \xi \in \mathbb{R}^{n+m+1}. \tag{2}$$

Let *G* = *D* × Ω and *<sup>ν</sup>*(*x*)=(*<sup>ν</sup>x*1 , ... , *<sup>ν</sup>xn* , *<sup>ν</sup>y*1 , ... , *<sup>ν</sup>ym* , *<sup>ν</sup>t*) is a vector of the inner normal of *Q* in the point (*<sup>x</sup>*, *y*, *t*).

We break up the bound of *Q*. Denote

$$\begin{aligned} \sigma\_0 &= \{ (\mathbf{x}, \mathbf{y}, t) \in \partial G \times (0, T) : \mathfrak{a}^k \boldsymbol{\nu}\_k = 0 \}, \\\\ \sigma\_1 &= \{ (\mathbf{x}, \mathbf{y}, t) \in \partial G \times (0, T) : \mathfrak{a}^k \boldsymbol{\nu}\_k > 0 \}, \\\\ \sigma\_2 &= \{ (\mathbf{x}, \mathbf{y}, t) \in \partial G \times (0, T) : \mathfrak{a}^k \boldsymbol{\nu}\_k < 0 \}, \end{aligned}$$

Consider in *Q* the boundary value problem

$$L\_0 l u + L\_1 u + M u = f(x, y, t),$$

$$u|\_{\\\\\partial Q} = 0, \quad \mathfrak{a}^k u\_{x\_k}|\_{\\\partial \mathcal{D}} = 0. \tag{3}$$

Define the operator *d*:

$$du = (b^{ij} + a^k a^{ij}\_{x\_k} - a\_0 a^{ij} + a^{ij}\_t) u\_{\mathbf{x}; \mathbf{x}\_j} + (b^i + a\_0 a^i - a^i a^k\_{x\_k} + a^i a\_0 - a^i\_t) u\_{\mathbf{x}\_i} + (a\_{0\_i} - a\_0 a\_0) u \equiv 0$$

$$d^{ij} u\_{\mathbf{x}\_i \mathbf{x}\_j} + d^i u\_{\mathbf{x}\_i} + du.$$

Assume that the condition

$$d^{ij} = d^{ji}, \quad \gamma\_0 |\underline{\xi}|^2 \le d^{ij} \underline{\chi}\_i \underline{\chi}\_j \le \gamma\_1 |\underline{\xi}|^2, \quad \text{for all} \quad (\mathbf{x}, \underline{y}, t) \in Q \cup \partial Q, \quad \text{for all} \quad \underline{\zeta} \in \mathbb{R}^{n+m+1} \tag{4}$$

holds. Let

$$Q\_{\mathsf{T}} = Q \cap \{ (x, y, t) : 0 < y\_1 < \mathsf{T} \}, \quad \partial G\_{\mathsf{T}} = \partial G \cap \{ y : 0 < y\_1 < \mathsf{T} \},$$

$$\sigma\_{0, \mathsf{T}} = \{ (x.y.t) \in \partial G\_{\mathsf{T}} \times (0, T) : a^k \nu\_k = 0 \},$$

$$\sigma\_{1, \mathsf{T}} = \{ (x, y, t) \in \partial G\_{\mathsf{T}} \times (0, T) : a^k \nu\_k > 0 \},$$

$$\sigma\_{2, \mathsf{T}} = \{ (x, y, t) \in \partial G\_{\mathsf{T}} \times (0, T) : a^k \nu\_k < 0 \}.$$

For some *h* > 0, define

$$\sigma\_{2,h,\mathsf{r}} = \{ (\mathsf{x}, y, t) \in \sigma\_{2,\mathsf{r}} : \rho((\mathsf{x}, y, t), \partial \sigma\_{2,\mathsf{r}}) > h \}, \quad \sigma\_{2,\mathsf{r}}^h = \sigma\_{2,\mathsf{r}} \nmid \sigma\_{2,h,\mathsf{r}}.$$

Let *<sup>E</sup>*(*Qτ*) be a set of functions *υ* ∈ *C*<sup>2</sup> *Qτ* such that *υ* = 0 in *∂Gτ* × (0, *T*) and *<sup>α</sup>kυxk* = 0 on *<sup>σ</sup>*0,*τ* ∪ *<sup>σ</sup>*1,*τ* ∪ *<sup>σ</sup>h*2,*τ* for some *h* > 0.

We denote as *<sup>H</sup>*(*Qτ*) the Hilbert space obtained by closing *<sup>E</sup>*(*Qτ*) with respect to the norm

$$\|u\|\|\_{H(Q\_{\mathbb{T}})} = \left\{ \int\_{\mathcal{Q}\_{\mathbb{T}}} \left( d\_1^{ij} u\_{\mathcal{X}\_l} u\_{\mathcal{X}\_j} + u\_{\mathcal{Y}p} u\_{\mathcal{Y}q} + u\_t^2 + u^2 \right) dx \, dy \, dt - \int\_{\mathcal{Q}\_{\mathbb{T}}} a^k \nu\_k d^{ij} u\_{\mathcal{X}\_l} u\_{\mathcal{X}\_j} ds \right\}^{\frac{1}{2}},$$

where

$$d\_1^{ij} = -\frac{1}{2}a^j a\_{\chi\_j}^{ij} - \frac{1}{2}a\_t^{ij} + a^j a^i + d^{ij} - \frac{1}{2\lambda\_0} a^{ij}\gamma$$

*dij*1 = *dji*1 , *β*0|*ξ*|<sup>2</sup> ≤ *dij*1 *ξiξj* ≤ *β*1|*ξ*|2, for all (*<sup>x</sup>*, *y*, *t*) ∈ *Q* ∪ *∂Q*, for all *ξ* ∈ R*n*+*m*<sup>+</sup>1.

Now consider bilinear form

$$a(\boldsymbol{u}, \boldsymbol{v}) = \int\_{Q\_{\boldsymbol{v}}} \left[ a^k d^j u\_{\boldsymbol{x}\_i} \boldsymbol{v}\_{\boldsymbol{x}\_j \boldsymbol{x}\_k} + a^{ij} u\_{\boldsymbol{x}\_i} \boldsymbol{v}\_{\boldsymbol{x}\_j \boldsymbol{t}} + \left( a^k a^{ij}\_{\boldsymbol{x}\_j} - a^i a^k \right) \boldsymbol{u}\_{\boldsymbol{x}\_i} \boldsymbol{v}\_{\boldsymbol{x}\_j} + \right.$$

$$d^{ij} \boldsymbol{u}\_{\boldsymbol{x}\_i} \boldsymbol{v}\_{\boldsymbol{x}\_j} + \left( d^i - d^{ij}\_{\boldsymbol{x}\_j} \right) \boldsymbol{u} \boldsymbol{v}\_{\boldsymbol{x}\_i} + \left( a^{ij}\_{\boldsymbol{x}\_i} + a^i + a^i \right) \boldsymbol{u}\_{\boldsymbol{x}\_i} \boldsymbol{v}\_{\boldsymbol{t}} + c^{pq} \boldsymbol{u}\_{\boldsymbol{y}\_p} \boldsymbol{v}\_{\boldsymbol{y}\_q} + \left( c^p - c^{pq}\_{\boldsymbol{y}\_q} \right) \boldsymbol{u} \boldsymbol{v}\_{\boldsymbol{y}\_p} + \boldsymbol{v}\_{\boldsymbol{y}\_q} \boldsymbol{u}\_{\boldsymbol{y}\_q} \boldsymbol{v}\_{\boldsymbol{x}\_i} \in \boldsymbol{I} + \left( \boldsymbol{u}\_{\boldsymbol{x}\_i} - \boldsymbol{c}\_{\boldsymbol{y}\_q} \boldsymbol{u}\_{\boldsymbol{y}\_q} + d^i + d^i\_{\boldsymbol{x}\_i \boldsymbol{x}\_j} \right) \boldsymbol{u} \boldsymbol{v}\_{\boldsymbol{x}\_i} \boldsymbol{d} \boldsymbol{x} \right.$$

**Definition 1.** *If <sup>u</sup>*(*<sup>x</sup>*, *y*, *t*) ∈ *<sup>H</sup>*(*Qτ*) *for any τ* < +∞ *and*

$$a(\mu, \upsilon) = \int\_{Q\_{\tau}} f \upsilon \, d\mathbf{x} \, dy \, dt \tag{5}$$

*for an arbitrary function υ* ∈ *<sup>E</sup>*(*Qτ*), *<sup>υ</sup>*|*<sup>S</sup>τ* = 0 *where Sτ* = *Q* ∩ {(*<sup>x</sup>*, *y*, *t*) : *y*1 = *<sup>τ</sup>*}, *then the function <sup>u</sup>*(*<sup>x</sup>*, *y*, *t*) *is said to be a generalized solution of the problem (1),(3) in the domain Q*.

#### **3. Energy Inequalities**

**Theorem 1.** (Analog of the Saint-Venant principle)

*Let* −1 ≤ *aijxi* + *ai* + *a*0 ≤ 0; *θ* ≡ *d*0 − 12 *dijxixj* + 12 *dixi* − 12 *cpqypyq* + 12 *cpyp* − *c*0 ≤ *θ*0 < 0, *for all* (*<sup>x</sup>*, *y*, *t*) ∈ *Q* ∪ *∂Q*.

*If <sup>u</sup>*(*<sup>x</sup>*, *y*, *t*)*is generalized solution of the problem (1), (3) and f*(*<sup>x</sup>*, *y*, *t*) = 0 *at y*1 ≤ *τ*2, *then for any τ*1 *such that* 0 ≤ *τ*1 ≤ *τ*2, *takes place*

$$\int\_{Q\_{\mathbb{F}\_1}} E(u) dx \, dy \, dt \le \Phi^{-1}(\mathfrak{r}\_1, \mathfrak{r}\_2) \int\_{Q\_{\mathbb{F}\_2}} E(u) dx \, dy \, dt \tag{6}$$

*where <sup>E</sup>*(*u*) = *dijuxiuxj* + *cpquypuyq* + *u*2*t* − *θu*2. *Here* <sup>Φ</sup>(*<sup>τ</sup>*, *<sup>τ</sup>*2) *is a solution of the problem*

$$\Phi' = -\mu(\tau)\Phi,\quad \tau\_1 \le \tau \le \tau\_2.$$

$$\Phi(\tau\_2, \tau\_2) = 1,$$

*μ*(*τ*) *is an arbitrary continuous function such that*

$$0 < \mu(\mathbf{r}) \le \inf\_{N} \left\{ \int\_{\tilde{S}\_{\mathbf{r}}} E(\upsilon) dx \, dy' \, dt \left| \int\_{\tilde{S}\_{\mathbf{r}}} P(\upsilon) dx \, dy' \, dt \right|^{-1} \right\},\tag{8}$$

$$y' = (y\_2, y\_3, \dots, y\_m),$$

$$P(\upsilon) = -c^{p1}\upsilon\upsilon\_{y\_p} + \frac{1}{2}\left(c^1 - c\_{y\_q}^{1\eta}\right)\upsilon^2,\tag{9}$$

*N is the set of continuously differentiable functions in the neighborhood of Sτ which are equal to zero in Sτ* ∩ (*∂Gτ* × (0, *<sup>T</sup>*)).

**Proof.** Assume in (5) *υ* = *um*(Ψ(*y*1) − 1) where <sup>Ψ</sup>(*y*1) = <sup>Φ</sup>(*<sup>τ</sup>*1, *<sup>τ</sup>*2) if 0 ≤ *y*1 ≤ *τ*1, <sup>Ψ</sup>(*y*1) = <sup>Φ</sup>(*y*1, *<sup>τ</sup>*2) if *τ*1 ≤ *y*1 ≤ *τ*2, and <sup>Ψ</sup>(*y*1) = 1 if *τ*2 ≤ *y*1.

$$
\mu\_m \in E(Q\_\mathbb{\tau}), \quad \|\mu\_m - \mathfrak{u}\|\_{H(Q\_\mathbb{\tau})} \to 0, \quad \mathfrak{u} \in H(Q).
$$

Then

$$a(\mu - \mu\_m + \mu\_{m\prime}\mu\_m(\Psi - 1)) = 0 \text{ in } Q\_{\tau\_2}\text{-}\mathbb{1}$$

Therefore

$$a(u\_{\mathfrak{m}\_1} u\_{\mathfrak{m}}(\Psi - 1)) = \delta\_{\mathfrak{m}} \text{ in } \mathcal{Q}\_{\mathfrak{T}\_2} \tag{10}$$

where *δm* = −*<sup>a</sup>*(*<sup>u</sup>* − *um*, *um*(<sup>Ψ</sup> − <sup>1</sup>)).

> It is obvious that *δm* → 0 at *m* → +<sup>∞</sup>. Integrating by parts (10), we have

$$\int\_{Q\_{r\_2}} E(u\_{\mathfrak{m}}) (\Psi - 1) dx \, dy \, dt \le \int\_{Q\_{r\_2}} P(u\_{\mathfrak{m}}) \Psi' dx \, dy \, dt + \delta\_{\mathfrak{m}}.$$

Hence

$$\int\_{Q\_2} \mathbb{E}(u\_m)(\Psi - 1) \mathrm{d}x \, dy \, dt \le \int\_{Q\_2} \mathbb{P}(u\_m) \mu \Psi \mathrm{d}x \, dy \, dt + \delta\_m. \tag{11}$$

The estimation (6) follows from (8) and (11) at *m* → +<sup>∞</sup>.

Now we will estimate *μ*(*y*1) in case when *Sτ* can be included to the (*n* + *m*)-dimensional parallelepiped which smallest edge is equal to *<sup>λ</sup>*(*τ*). Suppose that

$$\max\_{S\_{\mathbb{T}}} \left\{ \left( \frac{1}{2} c^1 - c\_{y\_q}^{1q} \right), 0 \right\} = \gamma(\tau), \quad \max\_{S\_{\mathbb{T}}} c\_{p1} = \beta(\tau).$$

Applying the Friedreich and Cauchy–Bunyakovsky inequalities, we have from (9)

$$\left| \int\_{\mathbb{S}\_{\mathbb{T}}} P(\boldsymbol{\upsilon}) d\boldsymbol{x} \, d\boldsymbol{y}' \, dt \right| \leq \left| \int\_{\mathbb{S}\_{\mathbb{T}}} \boldsymbol{\upsilon}^{p1} \boldsymbol{\upsilon} \boldsymbol{\upsilon}\_{\mathcal{Y}\_{p}} d\boldsymbol{x} \, d\boldsymbol{y}' \, dt \right| + \left| \int\_{\mathbb{S}\_{\mathbb{T}}} \frac{1}{2} \left( \boldsymbol{\varepsilon}^{1} - \boldsymbol{\varepsilon}\_{\mathcal{Y}\_{q}}^{1q} \right) \boldsymbol{\upsilon}^{2} d\boldsymbol{x} \, d\boldsymbol{y}' \, dt \right| \leq$$

$$
\beta(\tau) \left[ \int\_{\mathcal{S}\_{\tau}} v^2 dx \, dy' \, dt \right]^{\frac{1}{2}} \left[ \int\_{\mathcal{S}\_{\tau}} v\_{y\_{\mathcal{T}}}^2 dx \, dy' \, dt \right]^{\frac{1}{2}} + \gamma(\tau) \int\_{\mathcal{S}\_{\tau}} v^2 dx \, dy' \, dt \le 0
$$

$$
\left( \frac{\beta(\tau) \lambda(\tau)}{\pi \gamma\_0} + \frac{\gamma(\tau) \lambda^2(\tau)}{\pi^2 \gamma\_0} \right) \int\_{\mathcal{S}\_{\tau}} E(v) dx \, dy' \, dt.
$$

Therefore we can set

$$
\mu(\tau) = \pi^2 \gamma\_0 \left( \pi \beta(\tau) \lambda(\tau) + \lambda^2(\tau) \gamma(\tau) \right)^{-1}.
$$

If *c*1 − <sup>2</sup>*c*<sup>1</sup>*qyq* ≤ 0 in *S<sup>τ</sup>*, then *γ*(*τ*) = 0. Consequently

$$
\mu(\tau) = \frac{\pi \gamma \rho}{\beta(\tau) \lambda(\tau)}.\tag{12}
$$
