The authors’ account of the results: “This paper presents the continuation of Part I [Semin. Anal. 1985/86, 55-91 (1986; Zbl 0631.35023)] on the asymptotic behaviour of solutions of “simulation” elliptic boundary value problems in a cone. Boundary value problems with variable coefficients are investigated herewith, the results of part I being essentially used. Boundary value problem operators are considered as “small” perturbations of the model ones and our main purpose is to find out minimal conditions on the coefficients under which the main results of part I are valid.
Two-weight “coercive” \(L_ p\) and \(C^ \alpha\) estimates for the solutions of the perturbed problem are obtained \((\S2)\). An estimate of \(L_ p\)-means of the solution derivatives is established and an example shows its exactness. Theorems on asymptotics of the perturbed problem solutions near the cone vertex and at infinity are proved \((\S4)\).
Within the theorems on estimates of \(L_ p\)-means and asymptotics, coefficients of the perturbing operator are subdued to unimprovable, in a sense, integral “Dini-type” conditions, their form depending on lengths of Jordan chains of an assisting spectral problem. We note in conclusion that general corollaries on the asymptotics of the solutions and on Noetherianess of boundary value problems in regions with conical boundary points may be easily obtained from the theorems, proved in \(\S2-4\).”.