17 problems

Necessity conjecture. This condition is not only sufficient but also necessary: the Flamant solution should not apply for models with quadratic growth in other asymptotic regimes.

Solvability conjecture. For a fixed external body force $\boldsymbol{b}$ and all sufficiently small $\delta$, solvability of the linearized equilibrium equations implies solvabilit…

Anisotropic surface-term conjecture. Adding the Ciarlet–Nečas condition to the nonlinear Griffith functional should produce an additional anisotropic surface term, possibly dependi…

Regular-class attainment conjecture. The neo-Hookean energy $E$ attains its infimum in $\mathcal{A}_s^r$. The relaxed energy attains its infimum in a larger class, and the original…

Let $\Omega$, $\widetilde{\Omega}$ and $\boldsymbol{b}$ be axisymmetric, and let … where $\mathcal{A}_s$ consists of axisymmetric maps…

Conjecture on composition operators and weak quasiconformal mappings. The language of composition operators and its geometric representations as weak quasiconformal mappings should…

Let (\text{\cursive{\begin{scriptsize}\,b\,\end{scriptsize}}}^b_\varepsilon)_\varepsilon denote the sequence of scaled transverse gradients in the membrane component, and let…

Let $M^-_3(\lambda,\infty)$ denote the lower cubic supremum at infinite cutoff, and let $\nu_1^{A,-}\approx0.1923$. Lower cubic-supremum conjecture. The numerical investigations su…

Let $M_3(\lambda,c_0)$ be the cubic supremum, and let $c_1(\lambda)$ be a threshold such that $M_3(\lambda,c_0)=M_3(\lambda,c_1)$ for all $c_0\geq c_1(\lambda)$. For…

Let $G$ be the energy function and let $M_2(\lambda,c_0)$ denote the corresponding supremum, with parameters $\lambda>0$ and $c_0>0$. Quadratic-supremum conjecture. For all…

Let $Q=[-1,1]^2\subset\mathbb{R}^2$, and consider shear maps of the form … where $\sigma$ belongs to the specified subclass of $W^{1,2}(Q;\mathbb{R})$ and satisfies the uniform Lip…

For $n\in\mathbb{N}$, let $\lambda_1,\lambda_2,\ldots,\lambda_n,\mu_1,\mu_2,\ldots,\mu_n\in\mathbb{R}_+$, and let … be the elementary symmetric polynomials. The sum of squared loga…

Let $U$ be the right stretch tensor and let $K_2^2=\|\operatorname{dev}_n\log U\|^2$. Global TSTS-M$^+$ nonexistence conjecture. For $n=2,3$, there is no elastic energy of the form…

Rank-one convexity region conjecture. For $n=2,3$, the energy $F\mapsto\frac{\mu}{k}e^{k\|\operatorname{dev}_n\log U\|^2}$ with $k>\frac{3}{16}$ is rank-one convex on a set contain…

For $F\in\mathbb{R}^{n\times n}$, define the exponentiated Hencky energy by … where $U$ is the right stretch tensor and $\mu,\kappa,k,\widehat{k}$ are parameters. Planar polyconvex…

For an isotropic elastic energy, let TSTS-M$^+$ denote the strict monotonicity of the Cauchy stress tensor with respect to the left Hencky strain, in the sense that the correspondi…

Let $W_{\rm eH}$ denote the exponentiated Hencky-logarithmic strain energy, and let E$^+$-TSS denote the strict tension–tension stress inequalities. The exponentiated Hencky energy…