Explore the vast range of titles available.

We deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.

Let Ω be a cone in with n ≥  2. For every fixed we find the best constant in the Rellich inequality for . We also estimate the best constant for the same inequality on . Moreover we show improved Rellich inequalities with remainder terms involving logarithmic weights on cone-like domains.

Given a constant \\[k>1\\] and a real-valued function K on the hyperbolic plane \\[{\\mathbb {H}}^2\\], we study the problem of finding, for any \\[\\varepsilon \\approx 0\\], a closed and embedded curve \\[u^\\varepsilon \\] in \\[{\\mathbb {H}}^2\\] having geodesic curvature \\[k+\\varepsilon K(u^\\varepsilon )\\] at each point.

We give a unified approach to strong maximum principles for a large class of nonlocal operators of order s ∈ (0, 1) that includes the Dirichlet, the Neumann Restricted (or Regional) and the Neumann Semirestricted Laplacians.

Let k:C→Rk:\\mathbb {C}\\to \\mathbb {R} be a smooth given function. A kk-loop is a closed curve uu in C\\mathbb {C} having prescribed curvature k(p)k(p) at every point p∈up\\in u. We use variational methods to provide sufficient conditions for the existence of kk-loops. Then we show that a breaking symmetry phenomenon may produce multiple kk-loops, in particular when kk is radially symmetric and somewhere increasing. If k>0k>0 is radially symmetric and non-increasing, we prove that any embedded kk-loop is a circle; that is, round circles are the only convex loops in C\\mathbb {C} whose curvature is a non-increasing function of the Euclidean distance from a fixed point. The result is sharp, as there exist radially increasing curvatures k>0k>0 which have embedded kk-loops that are not circles.

We are interested in variational problems involving weights that are singular at a point of the boundary of the domain. More precisely, we study a linear variational problem related to the Poincaré inequality and to the Hardy inequality for maps in H01(Ω), where Ω is a bounded domain in ℝN, N ≥ 2, with 0 ∈ ∂Ω. In particular, we give sufficient and necessary conditions so that the best constant is achieved.