One-dimensional $L_p$-Hardy-type inequalities for special weight functions and their applications

Abstract

We establish one-dimensional $L_p$-Hardy inequalities with additional terms and use them for justifying their multidimensional analogues in convex domains with finite volumes. We obtain variational inequalities with power-law weights being generalizations of the corresponding inequalities presented earlier in papers by M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev and J. Tidblom. We formulate and prove inequalities valid for arbitrary domains, and then we simplify them substantially for the class of convex domains. The constants in the additional terms in these spatial inequalities depend on the volume or on the diameter of the domain. As a corollary of the obtained results we get estimates for the first eigenvalue of the $p$-Laplacian subject to the Dirichlet boundary conditions.