Title: Up-to-the-Neumann-Boundary Regularity for a Free Boundary Problem in Two Dimensions
Abstract: In this talk, we will discuss the regularity properties of a free boundary problem up to a Neumann fixed boundary in two dimensions. This will extend prior work of the authors by demonstrating that the free boundary itself must be smooth and intersect with the fixed boundary at a right angle.

Title: Some uniqueness results for strongly singular problems
Abstract: We consider a semilinear problem with strongly singular nonlinearity and discuss Brezis-Oswald type uniqueness results for positive solutions. Specifically, we will present uniqueness results using comparison arguments, behavior of positive solution near the boundary and appropriate modification of test functions. This is a joint work with Francesca Faraci.

Title: Optimal mass described by a Sturm-Liouville problem with eigenparameter in the boundary condition.
Abstract: We find an optimal mass of a structure described by a Sturm-Liouville (S-L) problem with a spectral parameter in the boundary conditions. While previous work on the subject focused on a somewhat simplified model, we consider a more general S-L problem. We use the calculus of variations approach to determine a set of critical points of the corresponding functional – yet these “predesigns” themselves do not represent meaningful solutions. We additionally introduce a set of solvability conditions on the data of the S-L problem which confirm that these critical points do represent meaningful solutions we refer to as designs.

Title: Infinitely many entire solutions to the curl-curl problem with critical exponent
Abstract: We prove the existence of an unbounded sequence of solutions to the curl-curl problem with critical exponent via a variational approach and suitable group actions to recover compactness.