Linear perturbations of the operator div (Q1920779)

The main aim of the article is to study solvability conditions for the boundary value problem \[ Lu= \text{div }u+ (\sigma,u)= f,\quad x\in\omega,\quad u=\varphi,\quad x\in\partial\omega,\tag{1} \] where \(\omega\subset\mathbb{R}^m\) is a bounded domain, \(u(x)\) and \(\sigma(x)\) are \(m\)-dimensional vector-functions, and \((\cdot,\cdot)\) stands for the scalar product in \(\mathbb{R}^m\). In contrast to the results of [Sib. Mat. Zh. 27, No. 5(159), 140-154 (1986; Zbl 0625.35011); Engl. translation: Sib. Math. J. 27, 744-756 (1986)], the coefficient \(\sigma(x)\) is only assumed to be summable to a sufficiently large power and \(\partial\omega\) is assumed to be a surface of class \(C^{0,1}\). Section 1 of the article collects auxiliary information about the operator div. Section 2 is devoted to studying conditions for existence of nonzero solutions to the problem \[ L^*p\equiv -\nabla p+\sigma p=0,\quad p\in W^1_1(\omega). \] In Section 3 we study the solvability of problem (1). Problem (1) is redued to a compact perturbation of the identity operator in the scale of the spaces \(L_s\), \(s\in[s_0,\infty]\), where \(s_0\) is some limit parameter. We find a criterion, in terms of the function \(\sigma\), under whose validity problem (1) is solvable for all \(f\) and \(\varphi\) in the corresponding function spaces; also, we find conditions on \(f\) and \(\varphi\) for the solvability of the problem in case \(\sigma\) does not satisfy this criterion.