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Let p ∈ (1,∞). The question of existence of a curve in ℝ₊² starting at (0,0) and such that at every point (x,y) of this curve, the $l^{p}$ -distance of the points (x,y) and (0,0) is equal to the Euclidean length of the arc of this curve between these points is considered. This problem reduces to a nonlinear differential equation. The existence and uniqueness of solutions is proved and nonelementary explicit solutions are given.

A Differential Geometric Characterization of the First Painlevé Transcendent.

N. Kamran, W.F. Shadwick (1987/1988)

Mathematische Annalen

A differential inclusion : the case of an isotropic set

Gisella Croce (2005)

ESAIM: Control, Optimisation and Calculus of Variations

In this article we are interested in the following problem: to find a map $u : Ω \to ℝ^{2}$ that satisfies $\{\begin{matrix} D u \in E & a.e. in Ω \\ u (x) = ϕ (x) & x \in \partial Ω \end{matrix}$ where $Ω$ is an open set of $ℝ^{2}$ and $E$ is a compact isotropic set of $ℝ^{2 \times 2}$ . We will show an existence theorem under suitable hypotheses on $ϕ$ .

A differential inclusion: the case of an isotropic set

Gisella Croce (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this article we are interested in the following problem: to find a map $u : Ω \to ℝ^{2}$ that satisfies $\{\begin{matrix} D u \in E & a.e. in Ω \\ u (x) = ϕ (x) & x \in \partial Ω \end{matrix}$ where Ω is an open set of $ℝ^{2}$ and E is a compact isotropic set of $ℝ^{2 \times 2}$ . We will show an existence theorem under suitable hypotheses on φ.

A Differential Inequality for the Distance Function in Normed Linear Spaces.

Ray Redheffer, Wolfgang Walter (1974)

Mathematische Annalen

A differential Puiseux theorem in generalized series fields of finite rank

Mickaël Matusinski (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

We study differential equations $F (y, ..., y^{(n)}) = 0$ where $F$ is a formal series in $y, y^{'}, ..., y^{(n)}$ with coefficients in some field of generalized power series $𝕂_{r}$ with finite rank $r \in ℕ^{*}$ . Our purpose is to express the support $Supp y_{0}$ , i.e. the set of exponents, of the elements $y_{0} \in 𝕂_{r}$ that are solutions, in terms of the supports of the coefficients of the equation, namely $Supp F$ .

A diffusion reaction system modelling spatial patterns

Jäger, Willi (1982)

Equadiff 5

A direct approach to the Weiss conjecture for bounded analytic semigroups

Bounit Hamid, Driouich Adberrahim, El-Mennaoui Omar (2010)

Czechoslovak Mathematical Journal

We give a new proof of the Weiss conjecture for analytic semigroups. Our approach does not make any recourse to the bounded $H^{\infty}$ -calculus and is based on elementary analysis.

A direct proof of a theorem by Kolmogorov in hamiltonian systems

L. Chierchia, C. Falcolini (1994)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

A Dirichlet Type Result for Ordinary Differential Operators.

W.N. Everitt, M. Giertz (1973)

Mathematische Annalen

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