\( \newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\norm}[1]{\|{#1}\|} \)

The questions below emerged during the course of my research. Some concern longstanding problems, others are more recent, and the remainder were posed by me.

1. Given a bounded open set $\Omega \subset \mathbb{R}^n$, it is a classical result that the eigenvalues $\lambda \in \mathbb{R}$ satisfying \begin{equation*} \left\{ \begin{array}{rcll} -\Delta u &=& \lambda u & \qquad \text{in } \Omega, \\ u &=& 0 & \qquad \text{on } \partial\Omega \end{array}\right. \end{equation*} are positive and discrete, denoted by $$0 < \lambda_1(\Omega) \leq \lambda_2(\Omega) \leq \cdots$$ A central question in spectral theory is to study the optimization problem \begin{equation}\label{lak} \min \{\lambda_k(\Omega)\,;\,\Omega \in \mathcal{A}\}, \end{equation} where $\mathcal{A}$ is a prescribed class of admissible sets. When $k=1$ and $\mathcal{A} = \{\Omega \,;\, |\Omega| = 1\}$, the celebrated Faber--Krahn inequality shows that the unit ball minimizes \eqref{lak}. The analogous result for higher eigenvalues remains largely unsolved. For instance, the following problem is still open:
   
   Suppose $\Omega \subseteq \mathbb{R}^2$ and let \[ \mathcal{A} = \{\Omega \,;\, \Omega \text{ is a polygon with $S$ sides and } |\Omega|=1\}. \] For each $k \in \mathbb{N}$ and $S\ge 3$, determine the optimal shape for problem \eqref{lak} or show that there isn't one.
2. (Suggested by X. Cabre) A regular solution $u$ to $$\begin{equation*} \left\{ \begin{array}{rcll} -\Delta_p u &=& f(u) & \qquad \text{in } \Omega, \\ u &>& 0 & \qquad \text{in } \Omega, \\ u &=& 0 & \qquad \text{on } \partial\Omega \end{array}\right. \end{equation*}$$ is said to be stable if the second variation of the associated energy functional at $u$ is nonnegative, that is, $$\begin{equation*} \int_{\Omega}\left\{\abs{\nabla u}^{p-2}\abs{\nabla\xi}^2 +(p-2)\abs{\nabla u}^{p-4}\left(\nabla u\cdot\nabla\xi\right)^2\right\}\,dx - \int_\Omega f'(u)\xi^2\,dx \,\geq\, 0 \end{equation*}$$ for every $\xi \in \mathcal{T}_u$, where $\mathcal{T}_u$ is the space of admissible test functions defined by $$\begin{equation*} \mathcal{T}_u := \begin{cases} W^{1,2}_{\sigma,0}(\Omega), & \text{if } p \geq 2, \\[6pt] \{\xi\in W^{1,2}_0(\Omega) : \, \norm{\nabla \xi}_{L_\sigma^2(\Omega)} < \infty \}, & \text{if } p \in (1,2). \end{cases} \end{equation*}$$ The following question remains open when $p \neq 2$:
   
   Let $u$ be a nonnegative stable solution in the unit ball $B_1$, and let $f$ be nonnegative, nondecreasing, and convex. Show that there exists a constant $C>0$ such that \[ \norm{u}_{L^1(B_1)} \leq C \norm{u_r}_{L^1(B_1)}, \] where $u_r$ denotes the radial derivative of $u$, i.e., $u_r(x) = \tfrac{x}{\abs{x}} \cdot \nabla u(x)$.
3. This is one of the most interesting unsolved problems in elliptic PDEs, despite its simple formulation. Consider the equation $$\begin{equation}\label{cpp} -\Delta_p u = f(x) \end{equation}$$ in the weak sense, where $f \in L^\infty$. The $C^{p'}$-conjecture states the following:
   
   Weak solutions of \eqref{cpp} are locally $C^{1,\frac{1}{p-1}}$-regular.
   
   It is already known that $u \in C^{1,\alpha}_{\mathrm{loc}}$ for some small $\alpha > 0$ , the conjecture states that the optimal exponent is $\frac{1}{p-1}$. The conjecture has been solved in the plane, but remains widely open in higher dimensions. A new proof in the planar case has recently been obtained by myself.
4. Systems of elliptic equations arise naturally in a wide range of mathematical and physical contexts, including reaction-diffusion processes, population dynamics, nonlinear elasticity, and phase separation phenomena. Unlike single-equation models, systems often exhibit a richer structure due to the interaction of multiple components, which leads to delicate analytical challenges. For instance, consider the system \begin{equation*} \begin{cases} -\Delta u = \dfrac{f(x) v^{\theta}}{u^{\theta}} \qquad & \text{in } \Omega,\\[6pt] -\dfrac{\Delta v}{(1+v)^{\gamma}} = g(x) u^{\alpha} \qquad & \text{in } \Omega,\\[6pt] u>0, \, v>0 & \text{in } \Omega,\\[6pt] u=0, \, v=0 & \text{on } \partial \Omega, \end{cases} \end{equation*} The principal feature of this system is the interplay between degenerate coercivity and the singular nonlinearity, which may significantly influence the existence and uniqueness of bounded solutions. This motivates the following natural question
   
   Does the system admit bounded solutions? If so, are they unique?
   
   A Related problem has been recently analyzed in by myself here.
5. This long-standing problem concerns the optimal regularity of viscosity solutions to the PDE \[ u_i u_j u_{ij} = 0. \] One can easily check that the function \[ u(x,y) = x^{\tfrac{4}{3}} + y^{\tfrac{4}{3}} \] satisfies the equation above (in the viscosity sense) in the plane. It is then natural to ask whether or not the exponent $\tfrac{4}{3}$ is sharp. More precisely, the following problem remains unsolved:
   
   The optimal regularity of infinity harmonic functions is $C^{1,\frac{1}{3}}$.
   
   The best result available to date, due to Evans and Smart, establishes everywhere differentiability. Recent advances and private communications suggest that the problem may be approachable using a priori smooth estimates.