Related papers: Density estimates for degenerate double-well poten…
We provide density estimates for level sets of minimizers of the energy $\frac{1}{2} \int_{\Omega}\int_{\Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dxdy+\int_{\Omega}\int_{\mathbb{R}^n\setminus\Omega}…
In this paper we provide density estimates for a class of functions which includes all the minimizers of the energy $\mathcal{E}_s^p(u,\Omega):=(1-s)\left(\frac{1}{2}\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\,dx\,dy…
We consider a class of Allen-Cahn equations associated with Ginzburg-Landau energies involving degenerate double-well potentials that vanish of order $m$ at the minima \begin{equation} J(v,\Omega)=\int_{\Omega}\Big\{|\nabla…
In models of phase coexistence, the precise form of the double-well potential is of central importance, yet it cannot be derived from first principles. In this paper, we investigate an inverse problem: starting from a prescribed transition…
In this short remark on a previous paper \cite{SZ25}, we continue the study of Allen-Cahn equations associated with Ginzburg-Landau energies \begin{equation*} J(v,\Omega)=\int_{\Omega}\Big\{F(\nabla v,v,x)+W(v,x)\Big\}dx, \end{equation*}…
We consider degenerate nonautonomous energies $$ \int_\Omega f(x, Dv)\, dx, $$ for vector-valued functions $v \in W^{1,1}(\Omega, \mathbb{R}^N)$, where the integrand $f(x,P)$ satisfies growth and weak uniform quasiconvexity assumption…
The treatment of degenerate states within Kohn-Sham density functional theory (KS-DFT) is a problem of longstanding interest. We propose a solution to this mapping from the interacting degenerate system to that of the noninteracting fermion…
We discuss "Banach SN spaces", which include Hilbert spaces, negative Hilbert spaces, and the product of any real Banach space with its dual. We introduce "L-positive" sets, which generalize monotone multifunctions from a Banach space into…
We consider a functional obtained by adding a trace term to the Allen-Cahn phase segregation model and we prove some density estimates for the level sets of the interfaces. We treat in a unified way also the cases of possible degeneracy and…
A bivariate perspective on Kohn-Sham density functional theory is proposed, treating potential and density as simultaneous independent variables, and used to make fruitful connection between Lieb's rigorous foundational framework and…
In three previous papers, we discussed quasidense multifunctions from a Banach space into its dual, or, equivalently, quasidense subsets of the product of a Banach space and its dual. In this paper, we survey (without proofs) some of the…
Forty-five years after the point de d\'epart [1] of density functional theory, its applications in chemistry and the study of electronic structures keep steadily growing. However, the precise form of the energy functional in terms of the…
We establish interior regularity and optimal growth estimates for sign-changing minimizers of the $p-$singular or $p-$degenerate quasilinear Alt--Phillips functional throughout the full range of $1<p<\infty$ and of the nonlinearity power…
We refine the asymptotic estimates for minimizers of a class of nonlocal energy functionals of the form \[ \frac{1}{4} \iint_{\R^{2n} \setminus (\R^n \setminus \Omega)^2} \snr{u(x) - u(y)}^2 K(x - y) \,dx\,dy + \int_\Omega W(u(x)) \,dx, \]…
We discuss some results related to a phase transition model in which the potential energy induced by a double-well function is balanced by a fractional elastic energy. In particular, we present asymptotic results (such as…
We describe how density-functional theory, well-known for its many uses in ab initio calculations of electronic structure, can be used to study the ground state of inhomogeneous model Hamiltonians. The basic ideas and concepts are discussed…
The recently proposed universal relations between the moments of the polydispersity distributions of a phase-separated weakly polydisperse system are analyzed in detail using the numerical results obtained by solving a simple density…
Localization of a particle in the wells of an asymmetric double-well (DW) potential is investigated here. Information entropy-based uncertainty measures, such as Shannon entropy, Fisher information, Onicescu energy, etc., and phasespace…
We study diffuse phase interfaces under asymmetric double-well potential energies with degenerate minima and demonstrate that the limiting sharp profile, for small interface energy cost, on a finite space interval is in general not…
We derive a generalized gradient approximation to the exchange energy to be used in density functional theory calculations of two-dimensional systems. This class of approximations has a long and successful history, but it has not yet been…