We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we first characterize all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators on (a, b), ∞ ≤ a < b ≤ ∞, and then apply this result to obtain families of optimal Hardy inequalities for general linear second-order elliptic operators in higher dimensions. As an application, we prove a new Rellich inequality.2000 Mathematics Subject Classification. Primary 35B09; Secondary 35J08, 35J20, 47J20, 49J40.

The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain [Formula: see text] that satisfy an oblique boundary condition on a portion of [Formula: see text]. First, we study weak solutions for the degenerate mixed boundary value problem [Formula: see text] where [Formula: see text] is a bounded Lipschitz domain, [Formula: see text] is a relatively open portion of [Formula: see text], and [Formula: see text] is an oblique (Robin) boundary operator defined on [Formula: see text] in a weak sense. In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a minimal positive Green function. Then we establish a criticality theory for positive weak solutions of the operator [Formula: see text] in a general domain [Formula: see text] with no boundary condition on [Formula: see text] and no growth condition at infinity. The paper extends results obtained by Pinchover and Saadon for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of [Formula: see text], and the boundary [Formula: see text] are assumed.

The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain Ω ⊆ R n that satisfy an oblique boundary condition on a portion of ∂Ω. First, we study the degenerate mixed boundary value problemwhere Ω is a bounded Lipschitz domain, ∂Ω Rob is a relatively open portion of ∂Ω, ∂Ω Dir is a closed set of ∂Ω, and B is an oblique (Robin) boundary operator defined on ∂Ω Rob .In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a positive minimal Green function. Then we establish a criticality theory for positive weak solutions of the operator (P, B) in a general domain Ω with no boundary condition on ∂Ω Dir and no growth condition at infinity. The paper generalizes and extends results obtained by Pinchover and Saadon (2002) for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of (P, B), and the boundary ∂Ω Rob are assumed.

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We construct new optimal L p Hardy-type inequalities for elliptic Schrödingertype operators with a potential term.