Related papers: A note on Newton's problem of minimal resistance f…

Let $u$ minimize the functional $F(u) = \int_\Omega f(\nabla u(x))\, dx$ in the class of convex functions $u : \Omega \to {\mathbb R}$ satisfying $0 \le u \le M$, where $\Omega \subset {\mathbb R}^2$ is a compact convex domain with nonempty…

We consider the problem $\inf\big\{ \int\!\!\int_\Omega (1 + |\nabla u(x,y)|^2)^{-1} dx dy : \text{ the function } u : \Omega \to \mathbb{R} \text{ is concave and } 0 \le u(x,y) \le M \text{ for all } (x,y) \in \Omega =\{ (x,y): x^2 + y^2…

We consider the following problem stated in 1993 by Buttazzo and Kawohl: minimize the functional $\int\!\!\int_\Omega (1 + |\nabla u(x,y)|^2)^{-1} dx\, dy$ in the class of concave functions $u: \Omega \to [0,M]$, where $\Omega \subset…

Newton's problem of the body of minimal aerodynamic resistance is traditionally stated in the class of {\it convex} axially symmetric bodies with fixed length and width. We state and solve the minimal resistance problem in the wider class…

The aim of this paper is twofold. First, we cut off a part of a convex surface by a plane near a ridge point and characterize the limiting behavior of the surface measure in $S^2$ induced by this part of surface when the plane approaches…

Here we solve the problem posed by Comte and Lachand-Robert in (2001). Take a bounded domain \Omega in R^2 and a piecewise smooth non-positive function u : \bar{\Omega} \to R vanishing on the boundary of \Omega. Consider a flow of point…

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…

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…

A parallel flow of non-interacting point particles is incident on a body at rest. When hitting the body's surface, the particles are reflected elastically. Assume that each particle hits the body at most once (SIC condition); then the force…

This paper investigates the Newton's problem of minimal resistance for a body moving through a fluid whose density decreases exponentially with altitude. We prove the local existence and regularity of radial solutions $u(r)$ satisfying the…

The problem considered first by I. Newton (1687) consists in finding a surface of the minimal frontal resistance in a parallel flow of non-interacting point particles. The standard formulation assumes that the surface is convex with a given…

Here is one of the results obtained in this paper: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q<{{n+2}\over {n-2}}$ if $n\geq 3$ and let $\lambda_1$ be the first eigenvalue of the problem $$\cases{-\Delta…

In this short note we consider an unconventional overdetermined problem for the torsion function: let $n\geq 2$ and $\Omega$ be a bounded open set in $\mathbb{R}^n$ whose torsion function $u$ (i.e. the solution to $\Delta u=-1$ in $\Omega$,…

We present a numerical method for the solution of Newton's problem of least resistance in the class of convex functions using a convex hull approach. We observe that the numerically computed solutions possess some symmetry. Further, their…

The weak lower semicontinuity of the functional $$ F(u)=\int_{\Omega}f(x,u,\nabla u)\, dx$$ is a classical topic that was studied thoroughly. It was shown that if the function $f$ is continuous and convex in the last variable, the…

The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem ${\rm inf}{{\rm ess sup}_{x \in \Omega} f(\nabla u(x)): u \in u_0 + W^{1,\infty}_0(\Omega)}$ when the supremand $f$ is not…

Since its original formulation by Isaac Newton in 1685, the problem of determining bodies of minimal resistance moving through a fluid has been one of the classical problems in the calculus of variations. Initially posed for cylindrically…

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 study existence, structure, uniqueness and regularity of solutions of the obstacle problem \begin{equation*} \inf_{u\in BV_f(\Omega)}\int_{\mathbb{R}^n}\phi(x,Du), \end{equation*} where $BV_f(\Omega)=\{u\in BV(\Omega): u\geq \psi \text{…

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…