Related papers: A constrained minimization problem related to two …
We study the minimizer of the electrostatic Born--Infeld energy \begin{equation*} \int_{\mathbb{R}^n}1-\sqrt{1-|D v|^2}\ dx-\int_{\mathbb{R}^n}\rho v\ dx, \end{equation*} which vanishes at infinity. We show that the minimizer $u$ is…
We study the 'bad science matrix problem': among all matrices $A\in\mathbb{R}^{n\times n}$ whose rows have unit $\ell_2$-norm, determine the maximum of $\beta(A)=\frac{1}{2^n}\sum_{x\in\{\pm1\}^n}\|Ax\|_\infty$. Steinerberger [1]…
A double-well energy expressed as a minimum of two quadratic functions, called phase energies, is studied with taking into account the minimization of the corresponding integral functional. Such integral, as being not sequentially weakly…
As a continuation of [14], we study new pattern formations of ground states $(u_1,u_2)$ for two-component Bose-Einstein condensates (BEC) with homogeneous trapping potentials in $R^2$, where the intraspecies interaction $(-a,-b)$ and the…
In this work a linearly constrained minimization of a positive semidefinite quadratic functional is examined. Our results are concerning infinite dimensional real Hilbert spaces, with a singular positive operator related to the functional,…
We study conditions of reduction of multidimensional wave equations - a system of d'Alembert and Hamilton equations. Necessary conditions for compatibility of such reduction conditions are proved. Possible types of the reduced equations and…
This paper is concerned with the following fractional Schr\"{o}dinger equations involving critical exponents: \begin{eqnarray*} (-\Delta)^{\alpha}u+V(x)u=k(x)f(u)+\lambda|u|^{2_{\alpha}^{*}-2}u\quad\quad \mbox{in}\ \mathbb{R}^{N},…
We study the problem of the existence, uniqueness and stability of solutions of reflected stochastic differential equations (SDEs) with a minimality condition depending on the law of the solution (and not on the paths). We require that some…
In this paper we study the existence of minimizers for interaction energies with the presence of external potentials. We consider a class of subharmonic interaction potentials, which include the Riesz potentials $|{\bf…
In this paper, we study the existence of minimizers for a class of constrained minimization problems derived from the Schr\"{o}dinger-Poisson equations: $$-\Delta u+V(x)u+(|x|^{-1}*u^2)u-|u|^\frac{4}{3}u=\lambda u,~~x\in\R^3$$ on the…
We establish a $\Gamma$-convergence result for $h\to 0$ of a thin nonlinearly elastic 3D-plate of thickness $h>0$ which is assumed to be glued to a support region in the 2D-plane $x_3=0$ over the $h$-2D-neighborhood of a given closed set…
We focus on the minimization of the least square loss function either under a $k$-sparse constraint or with a sparse penalty term. Based on recent results, we reformulate the $\ell_0$ pseudo-norm exactly as a convex minimization problem by…
We investigate existence, uniqueness and asymptotic behavior of minimizers of a family of non-local energy functionals of the type $$ \frac{1}{4}\iint_{\mathbb{R}^{2n}\setminus (\mathbb{R}^n \setminus \Omega)^2}|u(x)-u(y)|^2 K(x-y) \,dx dy…
We study the one-dimensional symmetry of solutions to the nonlinear Stokes equation $$ \begin{cases} -\Delta u+\nabla W(u)=\nabla p&\text{in }\mathbb{R}^d,\\ \nabla\cdot u=0&\text{in }\mathbb{R}^d, \end{cases} $$ which are periodic in the…
We present a new perturbation theory for quantum mechanical energy eigenstates when the potential equals the sum of two localized, but not necessarily weak potentials $V_{1}(\vec{r})$ and $V_{2}(\vec{r})$, with the distance $L$ between the…
In this paper, we study that the nearly critical nonlocal problem \begin{equation*} \left\lbrace \begin{aligned} &-\Delta u=(|x|^{-{(n-2)}}\ast u^{p-\epsilon})u^{p-1-\epsilon} \quad \mbox{in}\quad \Omega, &u>0\quad…
Analytic and approximate solutions for the energy eigenvalues generated by a confined softcore Coulomb potentials of the form a/(r+\beta) in d>1 dimensions are constructed. The confinement is effected by linear and harmonic-oscillator…
In this paper, we consider the existence of solutions for the linearly coupled Choquard system with potentials \begin{align*} \left\{\begin{aligned} &-\Delta u+\lambda_1 u+V_1(x)u=\mu_1(I_{\alpha}\star|u|^p)|u|^{p-2}u+\beta(x) v,\\ &-\Delta…
We consider the following coupled fractional Schr\"{o}dinger system: \begin{equation*} \left\{ \begin{aligned} &(-\Delta)^su+\lambda_1u=\mu_1|u|^{2p-2}u+\beta|v|^p|u|^{p-2}u\\ &(-\Delta)^sv+\lambda_2v=\mu_2|v|^{2p-2}v+\beta|u|^p|v|^{p-2}v\\…
A common optimization problem is the minimization of a symmetric positive definite quadratic form $< x,Tx >$ under linear constrains. The solution to this problem may be given using the Moore-Penrose inverse matrix. In this work we extend…