7 problems

Radial characterization conjecture. The solution $u$ is the minimal radial entire solution about some $x_*\in\mathbb{R}^N$ if and only if

Let $\rho$ be an integrable, nonnegative radially symmetric density on $\mathbb{R}^d$ with $\int\rho=N$, and let $\rho_{\rm rad}(r)=\omega_d r^{d-1}\rho_0(r)$ when…

Consider the compressible Navier–Stokes equations referred to as, and backward self-similar solutions with radial symmetry. Nonexistence conjecture. There are no backward self-simi…

In the atomic case, let $V(x)=-Z/|x|$, and consider a minimizer of the variational problem $I_V(m)$, if one exists. Lieb's radial symmetry conjecture. Any such minimizer must be ra…

Consider the critical nonhomogeneous porous medium equation in dimension $N=2$ with general, not necessarily radially symmetric, initial data. The preceding discussion establishes…

Radial blowup profile conjecture. This class of radially symmetric decreasing solutions, including a Dirac mass at the origin, describes well the local behavior at the blowup time…

Let $p>2$ and let $u_p=f_p(r)e^{i\theta}$ be the unique minimizer of the $p$-Ginzburg–Landau energy within the class of radially symmetric degree-one maps, where … The minimizer sa…