## DEG1 CHALLENGES

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

• 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)