CORRECT RESTRICTION OF NONLINEAR OPERATORS OF S.L. SOBOLEV

Abstract

In this work, we first consider the solvability of a linear model boundary value problem for a Sobolev-type differential expression and prove the equivalence of two types of boundary value problems for it. Based on this, we investigate the solvability of nonlinear Sobolev-type differential operators that arise in the dynamics of stratified media. The analysis employs the theory of wellposed operators in Banach spaces, particularly those that can be represented as operator products. Further, two main theorems are formulated and proved: the first theorem establishes the unique solvability of a nonlinear Sobolev-type differential operator in a cylindrical domain; the second therorem generalizes the result of Theorem 1 and considers a Bitsadze–Samarsky-type problem that relates the boundary data to the values of the sought-after function on a smooth surface located inside the cylindrical domain. The study shows that the application of the theory of correct operator restrictions is effective for analyzing complex nonlinear problems and can be extended to more general geometric and physical models.