[Paper Review] On the Boundary Value Problem with the Operator in Boundary Conditions for the Operator-Differential Equation of the Third Order

А. Р. Алиев, Sevindj F. Babayeva|arXiv (Cornell University)|Jul 25, 2011

Spectral Theory in Mathematical Physics4 references3 citations

TL;DR

This paper investigates the regular solvability of a third-order operator-differential equation on the semi-axis with a boundary condition perturbed by a bounded linear operator K. Using functional analysis in Sobolev-type spaces and spectral theory of self-adjoint operators, the authors establish sufficient conditions—particularly involving the norm of perturbation operators and a critical inequality—ensuring unique, stable solutions in the weighted space $ W_2^3(R_+; H; A) $.

ABSTRACT

In this paper the boundary value problem for one class of the operator-differential equations of the third order on a semi-axis, where one of the boundary conditions is perturbed by some linear operator is researched. There are received sufficient conditions on the operator coefficients of the considered boundary value problem, providing its correct and univalent resolvability in Sobolev type space.

Motivation & Objective

• To address the lack of research on third-order operator-differential equations with operator-valued boundary conditions, especially in the context of visco-elastic fluid dynamics.
• To extend the theory of boundary value problems for odd-order differential equations with operator coefficients in Hilbert spaces.
• To analyze the impact of a bounded linear operator K on the second boundary condition, which modifies the standard initial condition $ u'(0) = 0 $.
• To establish sufficient conditions for the unique, stable (regular) solvability of the problem in the space $ W_2^3(R_+; H; A) $.
• To generalize existing results for second-order equations to the third-order case, particularly under perturbed boundary conditions.

Proposed method

• Formulates the boundary value problem for a third-order operator-differential equation on the semi-axis $ R_+ $, with $ u'''(t) - A^3u(t) + \sum_{j=1}^3 A_j u^{(3-j)}(t) = f(t) $, where $ A $ is a self-adjoint, positive-definite operator.
• Defines the solution space $ W_2^3(R_+; H; A) $ as the completion of functions with square-integrable third derivative and $ A^3u $, equipped with a norm combining both terms.
• Introduces a modified solution space $ \mathop{W_{2;K}^{3}}\limits^{o}(R_+; H; A) $, where the second boundary condition is $ u'(0) = Ku $, with $ K \in L(W_2^3(R_+; H; A), H_{3/2}) $.
• Applies the theory of intermediate derivatives and trace theorems to control the behavior of $ u $, $ u' $, and $ u'' $ at $ t = 0 $, ensuring continuity into fractional-order Hilbert scales.
• Uses the operator decomposition $ P = P_0 + P_1 $, where $ P_0 = d^3/dt^3 - A^3 $ and $ P_1 = \sum A_j u^{(3-j)} $, and proves $ P_0 $ is an isomorphism from $ \mathop{W_{2;K}^{3}}\limits^{o} $ to $ L_2(R_+; H) $.
• Applies the method of small perturbations: shows that $ P = P_0(I + P_0^{-1}P_1) $ is invertible if the spectral-type condition $ \alpha(\kappa) < 1 $ holds, with $ \alpha(\kappa) $ depending on operator norms and $ \kappa = \|K\| $.

Experimental results

Research questions

• RQ1 Under what conditions is the third-order operator-differential equation $ u''' - A^3u + \sum A_j u^{(3-j)} = f $ regularly solvable on $ R_+ $ with a perturbed boundary condition $ u'(0) = Ku $?
• RQ2 How does the presence of a bounded linear operator $ K $ in the boundary condition affect the solvability and stability of the solution in the Sobolev-type space $ W_2^3(R_+; H; A) $?
• RQ3 What are the sufficient conditions on the coefficients $ A_j $, expressed via $ B_j = A_j A^{-j} $, that guarantee the invertibility of the full operator $ P = P_0 + P_1 $?
• RQ4 Can the theory of intermediate derivatives and trace theorems in scale of Hilbert spaces be used to derive uniform estimates for the solution in terms of the data $ f $?
• RQ5 How do the results generalize known second-order results (e.g., from Gasymov, Mirzoev, Yakubov) to the third-order case with operator-valued boundary conditions?

Key findings

• The operator $ P_0 = d^3/dt^3 - A^3 $ is an isomorphism from $ \mathop{W_{2;K}^{3}}\limits^{o}(R_+; H; A) $ onto $ L_2(R_+; H) $, ensuring well-posedness of the unperturbed problem.
• The solution satisfies the estimate $ \|u\|_{W_2^3(R_+; H; A)} \leq C \|f\|_{L_2(R_+; H)} $, with $ C $ depending on $ \kappa = \|K\| $, $ \kappa < 1 $, and the norms of $ B_j = A_j A^{-j} $.
• A sufficient condition for regular solvability is $ \alpha(\kappa) = \sum_{j=0}^{2} C_j(\kappa) \|B_{3-j}\|_{H\to H} < 1 $, where $ C_j(\kappa) $ are explicit constants derived from trace and interpolation inequalities.
• For $ K = 0 $, the condition reduces to $ \alpha(0) = \frac{2^{1/3}}{3^{1/2}} (\|B_1\| + \|B_2\|) + \|B_3\| < 1 $, recovering known results from earlier works.
• The paper provides explicit estimates for intermediate derivatives (e.g., $ \|A^{3-j} u^{(j)}\|_{L_2} $) in terms of the data, which are of independent interest in interpolation theory.
• The theory is valid in the scale of Hilbert spaces $ H_\gamma $ generated by the self-adjoint operator $ A $, allowing for a general functional-analytic framework applicable to various physical models.