Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations

Abstract

This is a collection of open problems concerning various areas in function theory, functional analysis, theory of linear and nonlinear partial differential equations.

Uniqueness and Bifurcation Branches for Planar Steady Navier–Stokes Equations Under Navier Boundary Conditions

“…There exists no multiplicity result valid in any situation, nor any detailed description of how the bifurcation from uniqueness might occur; see however [26]. Therefore, a complete comprehension of these phenomena is a challenging task, see [21,Problem 67].…”

Sharp Uncertainty Principle inequality for solenoidal fields

“…for unconstrained fields [17,19]. This result was further refined in the series of the recent papers [14,10,8,11]; in particular, the removal of the axisymmetry condition on the test solenoidal fields was achieved without changing the best constant from Costin-Maz'ya's one, which also includes a solution to Maz'ya's another open problem [20,Section 9.4]. Moreover, the second-order version of Hardy-Leray inequality, called the Rellich-Leray inequality, was found in [15,16,9,12] with the new best constant for curl-free or solenoidal fields.…”

The sharp second order Caffareli-Kohn-Nirenberg inequality and stability estimates for the sharp second order uncertainty principle

“…In fact, the inequality (1.3) without the sharp constant can be derived from (1.1) and Cauchy-Schwartz inequality (see the comment in the introduction in [13]). The inequality (1.3) is motivated by an open question of Maz'ya [35] on finding the sharp constant in (1.1) when we replace u by a divergence-free vector field U. In particular, the inequality (1.3) answers affirmatively the question of Maz'ya in the case n = 2.…”