Related papers: A solution to Newton's least resistance problem is…

We consider the Dirichlet problem u_t &= \Delta u + f(x, u, \nabla u)+ h(x, t),& \qquad &(x, t) \in \Omega \times (0, \infty), u &= 0, & \qquad &(x, t) \in \partial\Omega \times (0, \infty), on a bounded domain $\Omega \subset…

We deal with existence and uniqueness of nonnegative solutions to \begin{equation*} \left\{ \begin{array}{l} -\Delta u = f(x) \text{ in }\Omega, \frac{\partial u}{\partial \nu} + \lambda(x) u = \frac{g(x)}{u^\eta} \text{ on }…

In the previous work [Interfaces Free Bound., 19, 351-369, 2017], de Queiroz and Shahgholian investigated the regularity of the solution to the obstacle problem with singular logarithmic forcing term \begin{equation*} -\Delta u = \log u \,…

In this paper we deal with parabolic problems whose simplest model is $$ \begin{cases} u'- \Delta_{p} u + B\frac{|\nabla u|^p}{u} = 0 & \text{in} (0,T) \times \Omega,\newline u(0,x)= u_0 (x) &\text{in}\ \Omega, \newline u(t,x)=0 &\text{on}\…

We consider the problem of minimizing the Lagrangian $\int [F(\nabla u)+f\,u]$ among functions on $\Omega\subset\mathbb{R}^N$ with given boundary datum $\varphi$. We prove Lipschitz regularity up to the boundary for solutions of this…

We investigate the uniqueness of symmetric weak solutions to the stationary Navier-Stokes equation in a two-dimensional exterior domain $\Omega$. It is known that, under suitable symmetry condition on the domain and the data, the problem…

We analyze nonnegative solutions of the nonlinear elliptic problem $\Delta u=\frac{\lambda f(x)}{u^2}+P$, where $\lambda>0$ and $P\geq0$, on a bounded domain $\Omega$ of $\mathbb{R}^N$ ($N\geq 1$) with a Dirichlet boundary condition. This…

We consider Newton's problem of minimal resistance, in particular we address the problem arising in the limit if the height goes to infinity. We establish existence of solutions and lack radial symmetry of solutions. Moreover, we show that…

We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain $\Omega\subset \mathbb{R}^{N}$, within a suitable class of sign-changing weights. Denoting with $u$…

While there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less in known about critical points of the corresponding energy. Saddle…

In this article we consider a special type of degenerate elliptic partial differential equations of second order in convex domains that satisfy the interior sphere condition. We show that any positive viscosity solution $u$ of $-|\nabla…

In this paper we prove a higher differentiability result for the solutions to a class of obstacle problems in the form \begin{equation*} \label{obst-def0} \min\left\{\int_\Omega F(x,Dw) dx : w\in \mathcal{K}_{\psi}(\Omega)\right\}…

We focus here on the analysis of the regularity or singularity of solutions $\Om_{0}$ to shape optimization problems among convex planar sets, namely: $$ J(\Om_{0})=\min\{J(\Om),\ \Om\ \textrm{convex},\ \Omega\in\mathcal S_{ad}\}, $$ where…

We noisily observe solutions of an ordinary differential equation $\dot u = f(u)$ at given times, where $u$ lives in a $d$-dimensional state space. The model function $f$ is unknown and belongs to a H\"older-type smoothness class with…

We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c l c} -\Delta u & = &|u|^{q-2}u & \textrm{in }\Omega, \\ u_n & = & 0 & \textrm{on }\partial\Omega,\end{array}\right. \end {equation*} where $\Omega \subset…

We consider the optimization problem of minimizing $\int_{\Omega}G(|\nabla u|)+\lambda \chi_{\{u>0\}} dx$ in the class of functions $W^{1,G}(\Omega)$ with $u-\phi_0\in W_0^{1,G}(\Omega)$, for a given $\phi_0\geq 0$ and bounded.…

We are concerned with singular elliptic equations of the form $-\Delta u= p(x)(g(u)+ f(u)+|\nabla u|^a)$ in $\RR^N$ ($N\geq 3$), where $p$ is a positive weight and $0< a <1$. Under the hypothesis that $f$ is a nondecreasing function with…

We initiate the study of the finiteness condition $\int_{\Omega}u(x)^{-\beta}\,dx\leq C(\Omega,\beta)<+\infty$ where $\Omega\subseteq{\mathbb{R}}^n$ is an open set and $u$ is the solution of the Saint Venant problem $\Delta u=-1$ in…

Newton's problem of minimal resistance is one of the first problems of optimal control: it was proposed, and its solution given, by Isaac Newton in his masterful Principia Mathematica, in 1686. The problem consists of determining, in…

In this paper, we study the critical points of stable solutions for the following $p$-laplacian equation \begin{equation*} \begin{cases} -div\big(|\nabla u|^{p-2}\nabla u\big)=f(u)&in\ \Om,\\ u>0&in\ \Om,\\ u=0&on\ \partial\Om, \end{cases}…