In this talk, we address the localization of general nonlocal functionals of double-integral type with fractional dependence on the state variable, inspired by peridynamics. Localization is carried out as the interaction horizon among particles tends to zero. As a main result, we obtain an explicit formulation of the local $Γ$-limit, also covering the vectorial case. Applications of this result to nonlinear elasticity and the p-Laplacian eigenvalue problem will be discussed.

Given a weight $w$ on the unit circle, consider the orthogonal polynomials on the unit circle generated by $w$. Steklov famously conjectured that if $w$ is bounded below, then the polynomials all ought to be uniformly bounded above. While false, this conjecture begs the follow-up question: under what regularity conditions on $w$ are the polynomials uniformly bounded in $L^p(w)$ for some $p\gt 2$? Building upon a preliminary answer given by Nazarov for when $w$ is bounded above and below, we provide a positive answer when $w$ is an $A_2$ weight. This is joint work with Alexander Aptekarev and Sergey Denisov.

The Malmquist-Takenaka (MT) system is a complete orthonormal system in $H^2(T)$ generated by an arbitrary sequence of points in the unit disk that do not approach the boundary very fast. The nth point of the sequence is responsible for multiplying the nth and subsequent terms of the system by a Möbius transform taking the point to 0. One can recover the classical trigonometric system, its perturbations or conformal transformations, as particular examples of the MT system. However, for many interesting choices of the generating sequence, the MT system is less understood. We prove almost everywhere convergence of the MT series for three different classes of generating sequences.

The Q-curvature equation, a fourth-order elliptic partial differential equation with a critical exponent, is a prominent class of conformal equations, largely due to its connection with a natural concept of curvature. In light of the significant advances in the existence theory for the Q-curvature equation, in parallel with the Yamabe problem, this talk discusses the existence theory in both the compact and non-compact cases. We will also provide several interesting constructions based on techniques such as gluing and Lyapunov–Schmidt reduction, which shed light on the solution set of this equation.

In the last decades, there has been fascinating progress in the variational theory for the area functional – that is, the codimension 1 volume – using tools from PDEs and Geometric Measure Theory, and in connection with the problem of finding prescribed mean curvature (PMC) hypersurfaces.

In this talk, we describe some recent contributions from joint work with Jared Marx-Kuo (Rice University) in which we construct infinitely many PMCs for a large class of prescribing functions in a compact Riemannian manifold containing a strictly stable minimal hypersurface.