A function $f\colon \mathbb R^{n\times N}\to\mathbb R$ is called rank-one convex if $$f(\lambda x+(1-\lambda)y)\leq\lambda f(x)+(1-\lambda)f(y),$$ for all $\lambda\in[0,1]$ and $x,y\in\mathbb R^{n\times N}$ such that $\mathrm{rank}(x-y)=1.$ A locally bounded and Borel measurable function $f\colon\mathbb R^{n\times N}\to\mathbb R$ is calles quasiconvex if for any $\xi\in\mathbb R^{n\times N},$ any $\varphi\in C_0^1([0,1]^n,\mathbb R^N)$ there holds: $$\int_{[0,1]^n} f(\xi+\nabla\varphi(x))dx\geq f(\xi).$$ It is know that quasiconvexity implies rank-one convexity. Sverak proved in 1992 that rank-one convexity does not imply quasicomnvexity in the case $n\geq 2$ and $N\geq 3.$ The problem whether those two are the same in the case $N=2$ and $n\geq 2$ is open (this is probably the central question of the calculus of variations and materials science). ...

Let us consider 2D Euler equations in a bounded domain or on a torus, with a linear damping and a random external force: \begin{equation*} \partial_t u + \langle u,\nabla\rangle u+\gamma u+\nabla p=\eta(t,x), \quad \mathrm{div} u=0. \end{equation*} Here $u=(u_1,u_2)$ and $p$ are unknown velocity field and pressure, $\gamma>0$ is a parameter, and $\eta$ is a random force such that the equation in question generates a Markov process. For instance, $\eta$ can be a process white in time and regular in space (see Section 2.2 in [1]) or a piecewise independent random process (see Section 1 in [2]). Recall that a measure on the phase space is said to be stationary if it does not change under the evolution (see Section 1.3 in [1]). ...

Problem. Do there exist homogeneous degree $m\ge 2$ polynomial $p$-harmonic functions (i.e. solutions of $\,\,\mathrm{div} |\nabla u|^{p-2}\nabla u=0\,$), $x\in \mathbb{R}^n$ for some $p>1$ and $p\ne 2$?

The problem comes back to [1]. It is known that the answer is negative in the following cases:

$\bullet$ $n=2$ and any $m\ge 2$ [1]; ...

Fix $u \in C^1(\Omega)$ with $|\nabla u| >0$ in $\Omega$. Let $\tilde{u} \in C^1(\Omega)$ be another function such that $u|_{\partial \Omega}=\tilde{u}|_{\partial \Omega}$, and $$ \nabla u \cdot J=|J||\nabla u| \ \ \hbox{and}\ \ \nabla \tilde{u} \cdot \tilde{J}=|\tilde{J}||\nabla \tilde{u}| \ \ $$ for some $J,\tilde{J}\in L^{\infty}(\Omega)$. Suppose $\parallel u-\tilde{u}\parallel_{L^1} \leq C_1 \parallel |J| -|\tilde{J}| \parallel_{L^{\infty}}$ and $\parallel J-\tilde{J} \parallel _{L^1}\leq C_2 \parallel |J| -|\tilde{J}| \parallel_{L^{\infty}}$, for some $C_1, C_2>0$. Is $\parallel \nabla u- \nabla\tilde{u}\parallel_{L^1} \leq C \parallel |J| -|\tilde{J}| \parallel_{L^{\infty}}$, for some $C>0$ independent of $\tilde{u}$? if not, provide a counterexample. One may assume $J, \tilde{J}$ are divergence-free, if necessary. ...

Edit from 17/01/2022. The first open question described below is partially solved in the recent preprint Lamboley-Novruzi-Pierre : Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains (https://arxiv.org/abs/2109.10669) More precisely, it is proven that the free boundary is locally a polygonal line. It remains to show that the free boundary is "globally polygonal". ...

Definitions Suppose $A = U \cap \{ x \mathop : \boldsymbol \Theta^m ( \| V \|, x ) \geq 1 \}$, $X$ is the set of $x \in U$ such that $\mathrm{Tan}^m ( \| V \|, x )$ is an $m$ dimensional plane, $Y$ is the vectorspace of symmetric endomorphisms of $\mathbf R^n$, the tangent plane map $\tau : X \to Y$ is defined by requiring that $\tau (x) \in Y$ is the canonical orthogonal projection of $\mathbf R^n$ onto $\mathrm{Tan}^m ( \| V \|, x )$ for $x \in X$, $\Theta = \boldsymbol \Theta^m ( \| V \|, \cdot )$, and the trace $T : \mathrm{Hom} ( \mathbf R^n, Y) \to \mathbf R^n$ satisfies $T ( g ) = \sum_{i=1}^n g(u_i)(u_i)$ whenever $g \in \mathrm{Hom} ( \mathbf R^n, Y )$ and $u_1, \ldots, u_n$ form an orthonormal basis of $\mathbf R^n$. ...

Definition Whenever $a \in \mathbf R^n$, $0 < r < \infty$, $\mathbf B(a,r) \subset U$, and $T \in \mathbf G(n,m)$, we define \begin{align*} \phi_q (a,r,T) & = r^{-m/q} \big ( {\textstyle\int_{\mathbf B(a,r) \times \mathbf G(n,m)}} | S_\natural - T_\natural |^q \, \mathrm d V \, (x,S) \big)^{1/q}\ \text{for $1 \leq q < \infty$}, \\ \phi_\infty (a,r,T) & = \inf \big \{ t \mathop : V ( ( \mathbf B(a,r) \times \mathbf G (n,m) ) \cap \{ (x,S) \mathop : | S_\natural - T_\natural | > t \} ) = 0 \big \}. \end{align*} This quantity is usually referred to as the tilt-excess with exponent $q$ of the varifold $V$ in $\mathbf B(a,r)$ with respect to $T$. ...