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We consider the minimization of an energy functional given by the sum of a crystalline perimeter and a nonlocal interaction of Riesz type, under volume constraint. We show that, in the small mass regime, if the Wulff shape of the…
We present a design strategy for selecting the effective polarizability distribution for a metasurface aperture needed to form a desired radiation pattern. A metasurface aperture consists of an array of subwavelength metamaterial elements,…
The model introduced in [Kholmatov-Piovano 2020] in the framework of the theory on Stress-Driven Rearrangement Instabilities (SDRI) [Asaro-Tiller 1972; Grinfeld 1993} for the morphology of crystalline materials under stress is considered.…
This paper presents analysis of important issues associated with the design of refractive laser beam shaping systems. The concept of singular radius is introduced along with solutions to minimize its adverse effect on shaper performance. In…
We study the regularity of stable solutions to the problem $$ \left\{ \begin{array}{rcll} (-\Delta)^s u &=& f(u) & \text{in} \quad B_1\,, u &\equiv&0 & \text{in} \quad \mathbb R^n\setminus B_1\,, \end{array} \right. $$ where $s\in(0,1)$.…
We introduce a compact yet highly expressive design space for shellular metamaterials. By employing only a few dozen degrees of freedom, this design space represents geometries ranging from simple planar configurations to complex triply…
Recently, the use of intelligent reflecting surface (IRS) has gained considerable attention in wireless communications. By intelligently adjusting the passive reflection angle, IRS is able to assist the base station (BS) to extend the…
We extend the celebrated theorem of Kellogg for conformal diffeomorphisms to the minimizers of Dirichlet energy. Namely we prove that a diffeomorphic minimiser of Dirichlet energy of Sobolev mappings between doubly connected Riemanian…
Low-rank optimization problems with sparse simplex constraints involve variables that must satisfy nonnegativity, sparsity, and sum-to-1 conditions, making their optimization particularly challenging due to the interplay between low-rank…
In the paper, we prove the existence of radial solutions to \begin{equation}\notag%\label{main-eq-abstarct} %\begin{aligned} -\Delta_p u+({\rm sgn}(p-s)+V(x))|u|^{p-2}u+\lambda |u|^{s-2}u=|u|^{q-2}u\qquad\text{in}\,\R^N \\…
The paper is concerned with the sparse approximation of functions having hybrid regularity borrowed from the theory of solutions to electronic Schr\"odinger equations due to Yserentant [43]. We use hyperbolic wavelets to introduce…
For open radial sets $\Omega\subset \mathbb{R}^N$, $N\geq 2$ we consider the nonlinear problem \[ (P)\quad Iu=f(|x|,u) \quad\text{in $\Omega$,}\quad u\equiv 0\quad \text{on $\mathbb{R}^N\setminus \Omega$ and }\lim_{|x|\to\infty} u(x)=0, \]…
We consider a variant of Gamow's liquid drop model with an anisotropic surface energy. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface…
In this work we consider an energy subcritical semi-linear wave equation ($3 < p < 5$) \[ \partial_t^2 u - \Delta u = \phi(x) |u|^{p-1} u, \qquad (x,t) \in {\mathbb R}^3 \times {\mathbb R} \] with initial data $(u,u_t)|_{t=0} = (u_0,u_1)\in…
In this paper, we present a theoretical analysis of separating images consisting of pointlike and $C^{ \beta}$-curvelike structures, where $\beta \in (1,2] $. Our approach is based on $l_1$-minimization, in which the sparsity of the desired…
We study local approximation properties in hierarchical spline spaces through a twofold approach. First, we design and analyze a robust adaptive refinement algorithm to construct locally graded meshes. Second, we establish rigorous…
We characterize the solution to the Newton minimal resistance problem in a class of radial q-concave profiles. We also give the corresponding result for one-dimensional profiles. Moreover, we provide a numerical optimization algorithm for…
The paper deals with two fundamental types of trigonometric polynomials and splines on uniform grids, which allow us to construct interpolation approximations that depend linearly on the values of the interpolated function. Fundamental on…
We present a method for approximating the solution of a parameterized, nonlinear dynamical system using an affine combination of solutions computed at other points in the input parameter space. The coefficients of the affine combination are…
We consider weak non-negative solutions to the critical $p$-Laplace equation in $\mathbb{R}^N$, $-\Delta_p u =u^{p^*-1}$ in the singular case $1<p<2$. We prove that if the nonlinearity is locally Lipschitz continuous, namely $p^*\geqslant2$…