159 problems

Equilateral triangle torsional-rigidity conjecture. The equilateral triangle of unit width minimizes the torsional rigidity among shapes having unit minimal width.

Choksi–Peletier conjecture. There exists a unique threshold $\varepsilon^*>0$ such that, for $\varepsilon\leq\varepsilon^*$, the unit ball is the unique minimizer of $G$ under the…

Meissner's conjecture. Meissner's tetrahedra minimize volume among all three-dimensional convex bodies of constant width $d$.

Let $\Omega$ be a simply connected domain representing the cross-section of a beam, with prescribed area. Its torsional rigidity $T(\Omega)$ is defined by the associated Rayleigh q…

Let $\Omega\subset\mathbb{R}^2$ be a domain of fixed area, and let $\operatorname{l}_{1,\alpha}(\Omega)$ denote its first Robin eigenvalue for $\alpha<0$. For sufficiently small…

Let $\Omega$ range over bounded Lipschitz domains of fixed volume, and let $R^\Omega_\alpha$ denote the Robin Laplacian on $\Omega$ with fixed parameter $\alpha<0$. Reverse Faber–K…

Among planar simply connected domains with fixed Lebesgue measure, let $\mu_1(\Omega)$ denote the first nonzero eigenvalue of the classical Neumann Laplacian. Kornhauser–Stakgold c…

Let $\Omega$ be a bounded $C^{2,\epsilon}$ domain in $\mathbb{R}^{N+1}$, with $\epsilon\in(0,1)$ and connected boundary. Let $u$ be a nontrivial solution of the overdetermined eige…

Let $\Omega\subset\mathbb{R}^N$ be a domain with prescribed volume $V(\Omega)=m$, and let the load satisfy $\rho\in L^\infty(\Omega,[-1,1])$. Denote by…

Let $\Omega\subset\mathbb{R}^2$ be an open set of finite measure, and let $\lambda_3(\Omega)$ denote the third strictly positive Dirichlet eigenvalue of the Laplacian on $\Omega$,…

Let $\Omega$ be a domain, let $A$ be the prescribed parameter, and let $D=D_{\alpha,A}$ be an optimal configuration. Say that $D$ has the same symmetries as $\Omega$ when every sym…

Consider an annulus of thickness $\tau$, and write $\delta=A/|\Omega|$ for the prescribed area fraction. Let $D=D_{\alpha,A}$ be an optimal configuration, and let symmetry breaking…

Let $\Omega_h$ be a dumbbell domain, and let $D=D_{\alpha,A}$ be an optimal configuration. Dumbbell symmetry conjecture. For every $\alpha>0$, there is $\rho_1(\alpha,h)>0$ such th…

Let $\Omega_h$ be a dumbbell domain, let $D=D_{\alpha,A}$ be an optimal configuration, and let $\psi_{\Omega}$ be the function whose maxima determine the relevant localization regi…

Let $D$ be an optimal configuration in a planar domain, and let $\partial D$ denote its free boundary. Two-dimensional free-boundary regularity conjecture. In dimension two, the fr…

Let $\Omega$ be a convex domain, and let $D=D_{\alpha,A}$ be an optimal configuration in $\Omega$. Write $D^c$ for its complement and let $\overline{\alpha}_\Omega(A)$ denote the t…

Let $\Omega\subseteq\mathbb{R}^2$ range over planar domains, and let $\lambda_1(\Omega)\leq\lambda_2(\Omega)\leq\lambda_3(\Omega)\leq\cdots$ be the Dirichlet Laplacian eigenvalues.…

Let $\Omega$ be a planar domain, and let $\lambda_1(\Omega)<\lambda_2(\Omega)\leq\lambda_3(\Omega)\leq\cdots$ denote the Dirichlet Laplacian eigenvalues. A local maximum is underst…

Let $B$ denote the ball in $\mathbb{R}^3$, and let $lambda_1(Omega)$ be the first Maxwell eigenvalue of a domain $Omega$. Consider domains subject to either a fixed-volume constrai…

Let $B$ denote the ball in $\mathbb{R}^3$, and let the first three Maxwell eigenvalues of a domain $Omega$ be $lambda_1(Omega),lambda_2(Omega),lambda_3(Omega)$. For $j=1,2,3$, let…

Let $\Omega$ be a domain in $\mathbb{R}^n$ with prescribed volume, let $\beta<0$ be the Robin parameter, and let $\operatorname{l}_{1}(\Omega,\beta)$ denote the first Robin eigenva…

Let $\Omega\subset\mathbb{R}^d$ be a domain of volume $|\Omega|$, let $B_R$ be a ball with $|B_R|=|\Omega|$, and let $\lambda_1^\alpha(\Omega)$ denote the lowest eigenvalue of the…

Let $B$ be a ball and let $d4b\Omega\subset\mathbb{R}^N$ be a domain. Denote by $d706_{1,p}(\Omega)$ the first eigenvalue of the Dirichlet $p$-Laplacian on $d4b\Omega$, and by…

Small-mass rigidity conjecture. There exists a universal constant $q_0>0$ such that, for $q<q_0$, balls are rigid minimizers for Problem $$ .

Let $\beta>0$ be a Robin parameter and let $\beta_k$ denote a threshold depending on the eigenvalue index $k$. An optimizer is a domain attaining the minimum of the $k$-th Robin ei…