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 equation
  Assume 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 super-exponential Minkowski-curvature 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)