Related papers: Some optimization problems for nonlinear elastic m…

An adaptive direct collocation method is developed for solving optimal control problems constrained by parabolic partial differential equations. The partial differential equation is first reformulated in a variational setting, where the…

In this series of papers, we present a motion planning framework for planning comfortable and customizable motion of nonholonomic mobile robots such as intelligent wheelchairs and autonomous cars. In Part I, we presented the mathematical…

We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…

In this paper we consider positive supersolutions of the nonlinear elliptic equation \[- \Delta u = \rho(x) f(u)|\nabla u|^p, \qquad \hfill \mbox{ in } \Omega,\] where $0\le p<1$, $ \Omega$ is an arbitrary domain (bounded or unbounded) in $…

The aim of this paper is to study the singular solutions to fractional elliptic equations with absorption $$ \left\{\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+|u|^{p-1}u=0,\quad & \rm{in}\quad\Omega\setminus\{0\},\\[2mm]…

Submodular optimization finds applications in machine learning and data mining. In this paper, we study the problem of maximizing functions of the form $h = f-c$, where $f$ is a monotone, non-negative, weakly submodular set function and $c$…

We prove the monotonicity of positive solutions to the problem $-\Delta u = f(u)$ in $\mathbb{R}^N_+ := \{(x',x_N)\in\mathbb{R}^N \mid x_N>0 \}$ under zero Dirichlet boundary condition with a possible singular nonlinearity $f$. In some…

We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\Delta u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in…

In this work, we obtain an existence of nontrivial solutions to a minimization problem involving a fractional Hardy-Sobolev type inequality in the case of inner singularity. Precisely, for $\lambda>0$ we analyze the attainability of the…

This document introduces a strategy to solve linear optimization problems. The strategy is based on the bounding condition each constraint produces on each one of the problem's dimension. The solution of a linear optimization problem is…

We establish a new $W^{1,2\frac{n-1}{n-2}}$ estimate for the extremal solution of $-\Delta u=\lambda f(u)$ in a smooth bounded domain $\Omega$ of $\mathbb{R}^n$, which is convex, for arbitrary positive and increasing nonlinearities $f\in…

We study equilibrium configurations for the Euler-Plateau energy with elastic modulus, which couples an energy functional of Euler-Plateau type with a total curvature term often present in models for the free energy of biomembranes. It is…

Progresses in additive manufacturing technologies allow the realization of finely graded microstructured materials with tunable mechanical properties. This paves the way to a wealth of innovative applications, calling for the combined…

We study the existence and uniqueness of the positive solutions of the problem (P): $\partial_tu-\Delta u+u^q=0$ ($q>1$) in $\Omega\times (0,\infty)$, $u=\infty$ on $\partial\Omega\times (0,\infty)$ and $u(.,0)\in L^1(\Omega)$, when…

We investigate the existence and the multiplicity of solutions of the problem $$ \begin{cases} -\Delta_p u-\Delta_q u = g(x, u)\quad & \mbox{in } \Omega,\\ \displaystyle{u=0} & \mbox{on } \partial\Omega, \end{cases} $$ where $\Omega$ is a…

Let $\Omega\subseteq \mathbb{R}^N$ a bounded open set, $N\geq 2$, and let $p>1$; we prove existence of a renormalized solution for parabolic problems whose model is $$ \begin{cases} u_{t}-\Delta_{p} u=\mu & \text{in}\…

Topological optimization finds a material density distribution minimizing a functional of the solution of a partial differential equation (PDE), subject to a set of constraints (typically, a bound on the volume or mass of the material).…

Non-linear performance measures are widely used for the evaluation of learning algorithms. For example, $F$-measure is a commonly used performance measure for classification problems in machine learning and information retrieval community.…

This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:\mathbb{R}^+ \mapsto \mathbb{R}^+$: $$\begin{cases} L[u]=f(u) &\qquad in \{u>0\}\\ u \geq 0 &\qquad in\, \Omega\\ u=g…

We study the elliptic inclusion given in the following divergence form \begin{align*} & -\mathrm{div}\, A(x,\nabla u) \ni f\quad \mathrm{in}\quad \Omega, & u=0\quad \mathrm{on}\quad \partial \Omega. \end{align*} As we assume that $f\in…