New estimates for the Hardy constants of multipolar Schrodinger operators

In this paper, we study the optimization problem mu* (Omega) := inf(u is an element of D 1,2 (Omega)) integral(Omega) vertical bar del u vertical bar(2) dx / integral(Omega) Vu(2) dx in a suitable functional space D-1,D-2 (Omega). Here, V is the multi-singular potential given by V := Sigma(1 <= i<j <= n) vertical bar a(i) - a(j)vertical bar(2) / vertical bar x - a(i)vertical bar(2)vertical bar x - a(j)vertical bar(2) and all the singular poles a(1),..., a(n), n >= 2, arise either in the interior or at the boundary of a smooth open domain Omega subset of R-N, with N >= 3 or N >= 2, respectively. For a bounded domain Omega containing all the singularities in the interior, we prove that mu* (Omega) > mu* (R-N) when n >= 3 and mu* (Omega) = mu* (R-N) when n = 2 (it is known from [C. Cazacu and E. Zuazua, Improved multipolar hardy inequalities, in Studies in Phase Space Analysis with Application to PDEs, Progress in Nonlinear Differential Equations and Their Applications, Vol. 84 ( Birkhauser, New York, 2013), pp. 37-52] that mu* (R-N) = (N - 2)(2) / n(2)). In the situation when all the poles are located on the boundary, we show that mu* (Omega) = N-2/n(2) if Omega is either a ball, the exterior of a ball or a half-space. Our results do not depend on the distances between the poles. In addition, in the case of boundary singularities we obtain that mu* (Omega) is attained in H-0(1) (Omega) when Omega is a ball and n >= 3. Besides, mu* (Omega) is attained in D-1,D-2 (Omega) when Omega is the exterior of a ball with N >= 3 and n >= 3 whereas in the case of a half-space mu* (Omega) is attained in D-1,D-2 (Omega) when n >= 3. We also analyze the critical constants in the so-called weak Hardy inequality which characterizes the range of mu's ensuring the existence of a lower bound for the spectrum of the Schrodinger operator - Delta - mu V. In the context of both interior and boundary singularities, we show that the critical constants in the weak Hardy inequality are (N - 2)(2) / (4n - 4) and N-2 / (4n - 4), respectively.