Weighted Korn inequalities in paraboloidal domains

A weighted Korn inequality in a domain $\Omega\subset\mathbb R^n$ with paraboloidal exit $\Pi$ to infinity is obtained. Asymptotic sharpness of the inequality is achieved by using different weight factors for the longitudinal (with respect to the axis of $\Pi$) and transversal displacement vector components and by making the weight factors of the derivatives depend on the direction of differentiation. The solvability of the elasticity problem in the energy class (the closure of $C_0^\infty(\overline\Omega)^n$ in the norm generated by the elastic energy functional) is studied; the dimensions of the kernel and the cokerned of the corresponding operator depend on the exponent $s\in(-\infty,1)$ in the “grate of expansion” of the paraboloid $\Pi$.