Related papers: Area minimizing currents mod $2Q$: linear regulari…
In this paper, we consider an area minimizing integral $m$-current $T$ within a submanifold $\Sigma$ of $\mathbb{R}^{m+n}$, taking a boundary $\Gamma$ with arbitrary multiplicity $Q \in \mathbb{N} \setminus \{0\}$, where $\Gamma$ and…
In this paper, we propose new linearly convergent second-order methods for minimizing convex quartic polynomials. This framework is applied for designing optimization schemes, which can solve general convex problems satisfying a new…
We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative,…
The paper studies a particular class of analytic solutions for the Generalized Ohm's Law, approached by means of the so called formal powers of the Pseudoanalytic Function Theory. The reader will find a description of the electrical current…
For any real $k\geq 2$ and large prime $q$, we prove a lower bound on the $2k$-th moment of the Dirichlet character sum \begin{equation*} \frac{1}{\phi(q)} \sum_{\substack{\chi \text{ mod }q\\ \chi\neq \chi_0}} \Big| \sum_{n\leq x}…
We study an unbounded operator arising naturally after linearizing the system modelling the motion of a rigid body in a viscous incompressible fluid. We show that this operator is $\mathcal{R}$ sectorial in $L^q$ for every $q\in…
In this paper, we first establish the regularity theorem for suitable weak solutions to the Ericksen-Leslie system in dimensions two. Building on such a regularity, we then establish the existence of a global weak solution to the…
We consider the "thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in $\mathbb R^{n+1}_+$ plus the area of the positivity set of that function in $\mathbb R^n$. We establish full…
In 2000, Ambrosio and Kirchheim, with the paper "Currents in metric spaces", settled the foundations of a theory of currents on metric spaces and used it to pose and solve Plateau's problem in a wide class of Banach spaces. Following an…
The regularity theory of the Campanato space $\mathcal{L}^{(q,\lambda)}_k(\Omega)$ has found many applications within the regularity theory of solutions to various geometric variational problems. Here we extend this theory from…
The aim of this paper is to show two applications of metric currents to complex analysis. After recalling the basic definitions, we give a detailed proof of the comparison theorem between metric currents and classical ones on a manifold. In…
A natural construction of the logarithmic extension of the M(2,p) minimal models is presented, which generalises our previous model [0708.0802] of percolation (p=3). Its key aspect is the replacement of the minimal model irreducible modules…
In this paper, we introduce notions about local regularity of the super-potential and prove equidistribution of positive closed $(p, p)$-currents of $\mathbb{P}^k$ whose super-potentials are bounded near an invariant analytic subset, for…
We show weighted non-autonomous $L^q(L^p)$ maximal regularity for families of complex second-order systems in divergence form under a mixed regularity condition in space and time. To be more precise, we let $p,q \in (1,\infty)$ and we…
In this paper, we consider multi-valued graphs with a prescribed real analytic interface that minimize the Dirichlet energy. Such objects arise as a linearized model of area minimizing currents with real analytic boundaries and our main…
In the first part of this doctoral thesis we develop a regularity theory for a polyconvex functional in compressible elasticity. In the second part, we will concentrate on uniqueness questions in various situations of finite elasticity.…
The aim of these notes is to give a complete self-contained account of the current state of the art in the regularity for planar minimizers and critical points of the Mumford-Shah functional.
We establish general sufficient conditions for exact (and global) regularity in the $\bar\partial$-Neumann problem on $(p,q)$-forms, $0 \leq p \leq n$ and $1\leq q \leq n$, on a pseudoconvex domain $\Omega$ with smooth boundary $b\Omega$ in…
Assuming the generalized Lindel\"of hypothesis for Dirichlet $L$-functions, we establish that the least prime $p\equiv a\pmod{q}$ satisfies $p\ll_{\varepsilon} q^{2+\varepsilon}$. This achieves a bound that nearly matches the classical…
We demonstrate our simple strategy for renormalization with QED at one-loop level, basing on an elaboration of the effective field theory philosophy. No artificial regularization or deformation of the original theory is introduced here and…