20 problems

Pointwise positivity conjecture. The estimates

Let $\Omega$ be a bounded domain, strictly starshaped with respect to $0\in\Omega$, and suppose the condition … . Consider the problem with $u=v=0$ on $\partial\Omega$. Lane–Emden…

Comparison-principle conjecture. Under these assumptions, the comparison principle of the paper's system comparison theorem extends to solutions in $\mathcal{A}_g^k$. This conjectu…

Generalized convex-growth existence and uniqueness conjecture. Under these assumptions, the system admits a unique solution $u\in\mathcal{A}_g^k$. This conjecture proposes extendin…

Wang–Wei classification conjecture. If $n\leq 7$, then $u$ is one-dimensional: either $u\equiv \pm1$, or

Let $u$ solve the divergence-form elliptic equation … in $B(0,1)$, with … and … Let $Z(u)$ be the nodal set, $S(u)$ the singular set, and $\beta_u$ the frequency quantity used in t…

The uniqueness conjecture. Then $u=u_3$. The inequalities need not be uniform in $z$.

BMS conjecture. Assume that $(M,g)$ is geodesically complete. Then $(M,g)$ is $L^{2}$-positivity preserving.

Let $B_R\subset\mathbb{R}^N$ be the ball of radius $R$, let $2^*=\frac{2N}{N-2}$, and let $(u_k)_{k\geq 0}$ be radial solutions of the critical Lin–Ni–Takagi problem with exponent…

Asymptotic classification conjecture. If $p\in(1,2_m^*-1)$, then

Hang–Yang conjecture. Every such solution $u_\epsilon$ is constant.

Let $f\in C_0^\infty(\mathbb{R})$ satisfy $f\ge0$ and $f(s)=0$ for all $s\le0$. Let $u$ solve … and suppose that the blow-down limit … exists and has connected positive phase…

Let $f\in C_0^\infty(\mathbb{R})$ satisfy $f\ge0$, and let $u$ solve … Assume that $u$ has finitely many critical points and that $u(x)>0$ implies $x_1>0$. Half-space rigidity conj…

Let $f\in C_0^\infty(\mathbb{R})$ satisfy $f\ge0$. Let $u$ solve … and suppose there is a homeomorphism $\Phi:\mathbb{R}^n\to\mathbb{R}^n$ and a one-dimensional solution $U$ of the…

Let $u$ be a classical solution to the one-phase free-boundary problem in the punctured ball $\mathbb{B}_1\setminus\mathbb{B}_\delta$, with contractible positive and zero phases. F…

Set … Let $u$ be a classical solution to the one-phase free-boundary problem in $\mathbb{B}_1\setminus\mathcal{C}_\delta$, with contractible positive and zero phases. Flat-implies-…

Let a strict supersolution be understood in the sense of the Perron process for the one-phase free-boundary problem, and consider solutions obtained as infima of such strict supers…

Liouville-type conjecture. Every weak solution $u\in\mathcal D^{1,p}_0({\mathbb R}_+^N)$ of this equation is identically zero:

Nonvanishing imaginary-part conjecture. Either $\operatorname{Im}(f_z)$ is constant, or

Let $\{\phi_k\}_{k=1}^{\infty}$ be eigenfunctions of the Dirichlet-to-Neumann operator, with corresponding eigenvalues $\lambda_k$, and let $v_{\phi_k}$ denote the $\gamma$-harmoni…