This page contains a list of open problems. If you have the solution, write us to get your prize!

An old pendulum conjecture.Prove or disprove the following proposition.
For every continuous function \(q(t)\) such that \(\int_0^Tq(t)\,\mathrm{d}t=0\) there is a constant \(\delta_q>0\) such that, for every \(c\in[\delta_q,\delta_q]\), the equation \[x''+\sin x=c+q(t) \] has at least one \(T\)periodic solution.Prize: A sailing boat trip in the Gulf of Trieste (offered by A. Fonda, but only if you are not seasick...) 
Improving the Jacobowitz–Hartman Theorem.Prove that there exists a solution of the periodic problem \[ \begin{cases} \, x''+f(t,x)=0,\\ \, x(0)=x(T),\quad x'(0)=x'(T), \end{cases} \] assuming \(f\colon[0,T]\times\mathbb{R}\to\mathbb{R}\) to be a continuous function such that \[ \lim_{x\to\infty}\frac{f(t,x)}{x}=+\infty, \quad \text{uniformly in \(t\in[0,T]\).} \]Prize: a bottle of good Portuguese Ginjinha (offered by A. Fonda)

A regularity result for BV heteroclinics/homoclinics of a prescribed mean curvature equationAssume that \(f \colon [0, 1] \to \mathbb{R}\) is continuous and \(f(0)=0=f(1)\). Consider the equation \[\left(\frac{v'}{\sqrt{1+(v')^2}}\right)' + f(v) = 0, \quad v=v(z), \; z \in \mathbb{R},\] to be settled in the space \( BV_{loc}(\mathbb{R}) \). Prove that (or disprove that, or characterize the assumptions under which) all its \( BV_{loc}(\mathbb{R})\) heteroclinic solutions connecting \(0\) and \(1\) (or all its homoclinics to \(0\) or \(1\), when existing) belong to \(SBV_{loc}(\mathbb{R})\) (that is, their Cantor part vanishes) and possess a finite number of jump discontinuities.Prize: a bottle of original Ratafià liquor from Piemonte and Raveo’s S (offered by M. Garrione and E. Sovrano)

A uniqueness result for the equation \(u'' + a(t) u^\gamma = 0\).Let \(a\colon [0,T] \to \mathbb{R} \) be a stepwise function with a single change of sign. Let \(\gamma\in \mathbb{R} \setminus\{1,0,1\}\) and assume that \(\gamma \cdot \int_0^T a(t)\,\mathrm{d}t < 0\). Consider the Neumann and the periodic boundary value problems associated with \[ u'' + a(t) u^\gamma = 0. \] Uniqueness results for positive solutions are available for \( \gamma\in\mathopen{]}\infty,3\mathclose{]}\cup \mathopen{]}0,1\mathclose{[} \cup \mathopen{]}1,+\infty\mathclose{[} \) (see [Boscaggin, Feltrin, Zanolin, Open Math. 2021] and the references therein). The case \( \gamma \in \mathopen{]}3,1\mathclose{[} \cup \mathopen{]}1,0\mathclose{[} \) is open.Prize: 2 bottles of red wine from Piemonte (offered by A. Boscaggin and G. Feltrin)

A uniqueness result for a superexponential Minkowskicurvature equation.Let \(a\colon [0,T] \to \mathbb{R} \) be a stepwise function with a single change of sign and such that \(\int_0^T a(t)\,\mathrm{d}t < 0\). In [Boscaggin, Feltrin, Zanolin, preprint 2020], the authors prove the existence of at least one positive solution of the Neumann and periodic problems associated with the equation \[ \left( \frac{u'}{\sqrt{1(u')^2}}\right)' + a(t) \, (e^{u^p} 1) = 0, \quad p>1. \] Numerical evidence of uniqueness is provided, see also [Boscaggin, Feltrin, Zanolin, Open Math. 2021]. A proof of the uniqueness does not exist.Prize: 2 bottles of wine from Collio (FVG) and an autographed copy of this book ☺ (offered by A. Boscaggin and G. Feltrin)