Coercive Estimates and Separability for the Fourth-Order Operator in Weight Space

Abstract

The paper investigates the coercive properties of a fourth-order operator in the weight space . This paper presents a detailed investigation into the coercive properties and separability conditions of a fourth-order differential operator within a weighted Hilbert space framework. The motivation stems from the critical role such operators play in mathematical physics and engineering, particularly in modeling elastic structures and complex boundary value problems. While the theory of second-order and biharmonic operators is well developed, there remains a need to deepen the understanding of higher-order operators, especially in non-uniform or weighted settings. We begin by defining the analytical setting, including the class of admissible weight functions and functional spaces, and formulate the problem through a general fourth-order elliptic equation. Building upon existing approaches in the theory of coercivity and separability, we derive new sufficient conditions under which the operator admits a coercive inequality. These inequalities are essential in demonstrating well-posedness and are shown to lead directly to separability, which ensures the operator’s spectral decomposition and solution uniqueness. The core result of the paper is a theorem establishing that, under certain regularity and boundedness conditions on the weight functions, the considered operator is separable in the weighted space. The proof employs a combination of integration by parts, approximation techniques, and delicate functional estimates that rely on properties of the weight function and the structure of the differential operator. This work generalizes prior results on coercive solvability from second-order settings to fourth-order frameworks and introduces a systematic method to verify separability using inequalities derived from the operator's coercive structure. As such, the results have broader implications for the study of nonlinear and non-divergent elliptic operators in weighted domains.