A new proof of compactness in G(S)BD

We prove a compactness result in GBD {\operatorname{GBD}} which also provides a new proof of the compactness theorem in GSBD {\operatorname{GSBD}} , due to Chambolle and Crismale. Our proof is based on a Fréchet–Kolmogorov compactness criterion and does not rely on Korn or Poincaré–Korn inequalities.

Sutured ECH is a natural invariant

We show that sutured embedded contact homology is a natural invariant of sutured contact 3 3 -manifolds which can potentially detect some of the topology of the space of contact structures on a 3 3 -manifold with boundary. The appendix, by C. H. Taubes, proves a compactness result for the completion of a sutured contact 3 3 -manifold in the context of Seiberg–Witten Floer homology, which enables us to complete the proof of naturality.

Blow-up solutions with minimal mass for nonlinear Schrödinger equation with variable potential

This paper studies the mass-critical variable coefficient nonlinear Schrödinger equation. We first get the existence of the ground state by solving a minimization problem. Then we prove a compactness result by the variational characterization of the ground state solutions. In addition, we construct the blow-up solutions at the minimal mass threshold and further prove the uniqueness result on the minimal mass blow-up solutions which are pseudo-conformal transformation of the ground states.

Action convergence of operators and graphs

We present a new approach to graph limit theory that unifies and generalizes the two most well-developed directions, namely dense graph limits (even the more general $L^p$ limits) and Benjamini–Schramm limits (even in the stronger local-global setting). We illustrate by examples that this new framework provides a rich limit theory with natural limit objects for graphs of intermediate density. Moreover, it provides a limit theory for bounded operators (called P-operators) of the form $L^\infty (\Omega )\to L^1(\Omega )$ for probability spaces $\Omega $ . We introduce a metric to compare P-operators (for example, finite matrices) even if they act on different spaces. We prove a compactness result, which implies that, in appropriate norms, limits of uniformly bounded P-operators can again be represented by P-operators. We show that limits of operators, representing graphs, are self-adjoint, positivity-preserving P-operators called graphops. Graphons, $L^p$ graphons, and graphings (known from graph limit theory) are special examples of graphops. We describe a new point of view on random matrix theory using our operator limit framework.

On a quasilinear elliptic problem involving the 1-biharmonic operator and a Strauss type compactness result

In this paper we prove the compactness of the embeddings of the space of radially symmetric functions of BL(ℝN) into some Lebesgue spaces. In order to do so we prove a regularity result for solutions of the Poisson equation with measure data in ℝN, as well as a version of the Radial Lemma of Strauss to the space BL(ℝN). An application is presented involving a quasilinear elliptic problem of higher-order, where variational methods are used to find the solutions.