Weak Solutions of Boundary Value Problems in Nontube Domains for Fourth-Order Equations of Composite Type

In the paper existence and uniqueness of weak solutions of boundary value problems in nontube domains for fourth-order equations of composite type are proved by methods of functional analysis.


Introduction
In the paper the existence and uniqueness of weak solution of boundary value problem in nontube domain for fourth-order equation of composite type are proved by methods of functional analysis. The main part of the operator is a composition of the wave operator and the operator of Laplace type. The analysis of such equations is caused by the need to develop further the fundamental theory of well-posed problems for linear partial differential equations, since such theory for the second-order equations of hyperbolic and elliptic types has been already developed. The interest to equations of composite type from a practical point of view is caused by a wide class of applications of in many fields of scientific knowledge and industry.
Boundary value problems for equations of composite type ∂ ∂x Δu = 0, ∂ 2 ∂x∂y Δu = 0, (1.1) equations are investigated. We don't know any papers devoted to the equation we consider in this paper. The purpose of this paper is to extend the theorems of existence and uniqueness of weak solutions of boundary value problems for the fourth-order equations of composite type in tube domains to nontube domains. A tube domain is a domain with the following property: the generator of the surface being part of the boundary of the domain and called lateral is parallel to the axis of x 0 . We call nontube a domain which doesn't have this property. In the case of tube domains this problem was considered in the paper [20], the proof of existence and uniqueness of weak solutions of other problems in tube domains for this equation can be found in the papers [15,20]. The papers [14] and [21] are devoted to proving existence and uniqueness of strong solutions of boundary value problems in tube domains for fourth-order equation of composite type. The equation of composite type of more common form is considered in the paper [22].
We denote by C l (Q) a set of continuously differentiable functions up to the order l in the closure Q of the domain Q, where l is nonnegative integer. In equation (2.1) a (α) (x) are given functions and a (α) (x) ∈ C 2 (Q).
Boundary ∂Q of the domain Q consists of three parts: . . , ν n (x)) is the outward with respect to the domain Q unit normal on the hypersurface ∂Q at a point x ∈ ∂Q, |ν | 2 = ν 2 1 + . . . + ν 2 n , δ 1 > 0. The tangent to the hypersurface S 2 at a point x ∈ S 2 vector τ (x) = (1, τ 1 (x), . . . , τ n (x)) satisfies the conditions where δ 2 is a sufficiently small positive number. For equation (2.1) we formulate the following boundary conditions: is derivative in the direction of p i from vector field P i , which is not tangent to S i , i = 1, 2. If boundary conditions (2.3)-(2.5) are nonhomogeneous, then they are reduced to the homogeneous ones by extending their right parts to the domain Q by functions from suitable spaces and replacing the desired function [23].

Definition of Weak Solution
We consider problem (2.1)-(2.5) and define some functional spaces in which a weak solution is defined. Weak solution is defined by some equality of the corresponding functionals. To achieve this, along with problem (2.1)-(2.5) we consider an adjoint boundary value problem: is derivative in the direction of p 3 from vector field P 3 , which is not We define domains of definitions for the operators L and L as follows: It is easy to check that for any functions u ∈ D(L) and v ∈ D(L ), where (·,·) L2(Q) is value of scalar product in the space L 2 (Q) of square integrable in Q functions. We denote by H l (Q) Hilbert space elements of which u ∈ L 2 (Q) and their weak derivatives D α u, |α| l, belong to L 2 (Q). A scalar product in  and v ∈ H 1 0 (Q), which is extension by continuity of the bilinear form (w, v) L2(Q) , where w ∈ L 2 (Q), v ∈ H 1 0 (Q).
As far as the set , u ∈ D(L).

7)
u ∈ H 3 0 (Q), is linear continuous functional in the dense set H 3 0 (Q) of the space H 1 0 (Q) in topology induced from the space H 1 0 (Q). Then this functional allows continuous extension to space H 1 0 (Q). Consequently, a unique element L v ∈ H −1 0 exists, such that Ψ (u, v) = u, L v for v ∈ D(L ) and any u ∈ H 1 0 (Q), where u, w is canonical bilinear form for u ∈ H 1 0 (Q) and w ∈ H −1 0 , which is extension by continuity of the bilinear form (u, w) L2(Q) , where u ∈ H 1 0 (Q) and w ∈ L 2 (Q). As far as the set H 3 0 (Q) is dense in the space H 1 0 (Q), then , v ∈ D(L ).

Existence and Uniqueness of the Weak Solution
Existence of the weak solution is proved on the basis of (3.8), if corresponding a priori estimations for the operators L and L are obtained. Here we take into account that the spaces H 2 0 (Q) and 0 H 2 (Q) are reflexive.

Theorem 1. The inequalities
are valid, where c i are some positive constants independent of u and v.
Proof. We prove that inequality (4.1) is correct. We suppose at first u ∈ H 2 0 (Q) ∩ H 3 (Q). It follows from (3.5) that in order that the functional v → Φ(u, v) is continuous over the space H 1 0 (Q), the equality ∂ 2 u ∂p 2 1 | S1 = 0 must be satisfied, i.e. actually u ∈ H 3 0 (Q), if u ∈ D(L). In this case the value of the functional Φ(u, v) can be represented in the form The functional Φ(u, v) written in form (4.3) is extended by continuity to all space H 1 0 (Q). We suppose in (4.
We represent the main part of the expression under the integral sign in the last relation in the divergent form: where We apply the Ostrogradskii formula to I 1 and get I 1 = 0 by virtue of (2.3)-(2.5). With the help of the inequality We integrate over domain Q (τ ) = {x ∈ Q | 0 < x 0 < τ < T }, T = sup x∈Q x 0 , the identity where c 9 is sufficiently large positive constant: and inequality (4.7) takes the form v(τ ) 1 ε τ 0 v(t) dt + εc 9 u H 2 0 (Q) . We add it to inequality (4.6), choose ε such that the inequality c 7 − εc 9 > 0 is satisfied and apply the Gronwall inequality: The right part of inequality (4.8) is independent of τ , therefore in its left part one can pass to the least upper bound with respect to τ . We get the following inequality: Taking into account estimate sup 0<τ <T u 2 is proved in case u ∈ H 2 0 (Q) ∩ H 3 (Q). Now let u ∈ D(L). Then J k u ∈ H 2 0 (Q) ∩ H 3 (Q), where J k is mollifier with variable step [3,16,17]. It is proved above that . (4.9) We represent the functional Φ(J k u, v) in the following form: Φ(J k u, v) = Φ(u, J * k v) + K 1 (u, v; k) + K 2 (u, v; k), Proof. Uniqueness of weak solution of problem (2.1)-(2.5) follows from Theorem 1. The operator L is closed, therefore for proving existence of weak solution of problem (2.1)-(2.5) and, thus, for finishing the proof of the theorem it remains to prove that R(L) = H −1 0 . It is enough to show the density of the elements Lu, where u ∈ H 3 0 (Q), is in the space H −1 0 . Let v ∈ 0 H 3 (Q) be such that for any function u from the indicated class Lu, v = 0. As u ∈ H 3 0 (Q) and v ∈ 0 H 3 (Q), then by the virtue of (3.8) u, L v = 0. Consequently, it follows from (4.2) that v = 0 in H 3 (Q).