An $\infty$-harmonic function is a viscosity solution of $\nabla^2 u(\nabla u,\nabla u)=0$, or equivalently, an absolute minimizer of $\|\nabla u\|_{L^\infty}$. We prove a variety of new structural and regularity results in two dimensions, including: 1. $\infty$-harmonic functions in domains of $\mathbb{R}^2$ are $C^{1,1/3}$. 2. Critical points are isolated, and at each critical point, the solution has a unique quasiradial blow-up. 3. Entire solutions with polynomial growth have unique quasiradial blow-downs, and are determined by their Fourier modes at infinity. These results are consequences of a new theory relating $\infty$-harmonic functions to inverse mean curvature flow (IMCF) clusters -- which are piecewise weak solutions of IMCF with common obstacle-type boundary conditions on the interfaces (a simple example is an embedded family of cuspidal curves evolving by inverse curvature). This connection arises as the $p\to\infty$ limit of the classical duality between $p$-harmonic and $q$-harmonic functions in $\mathbb{R}^2$, where $\frac1p+\frac1q=1$.
For every $n\geq 3$, we construct a nonconstant real-valued function $U\in C^\infty(\overline{\mathbb R^n_+})$, harmonic in the upper half-space, whose complete boundary jet vanishes on a compact nowhere dense subset of $\partial\mathbb R^n_+$ of positive $(n-1)$-dimensional measure. The essential construction takes pl...
We investigate the Dirichlet problem for the variational integral $J[u] = \int_{\Omega} f(\nabla u) \, dx$ with density $f$ of linear growth satisfying appropriate ellipticity conditions. We show that the relaxed problem admits a unique solution $u$ in the space of functions of bounded variation, if the set $\Gamma_0$...
Let $\Omega\subset\mathbb R^n$ be a bounded domain, and let $f$ be a nonnegative, nondecreasing function satisfying the Keller-Osserman condition. We study boundary blow-up solutions of $\Delta u=f(u)$ in $\Omega$. Although existence is classical, uniqueness under these assumptions is known in balls but remains open ev...
We consider long-time behavior of the zero-homogeneous solutions with 3-fold symmetry to the two-dimensional Euler equation. This is the remaining case in the relaxation theory of Said, Elgindi, and Murray [Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4) \textbf{58} (2025), no.~4, 943--970], which treats $m$-fold symmetry solut...
For every pair of integers $3\leq k<n$ with $k$ odd, we construct a compact embedded rotational torus in $\mathbb{R}^{n+1}$ whose homothetic dilations satisfy the unnormalised $\sigma_k$-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has H\"older regularity $C^{1,1/k}$ and Sobolev regularity $W^...
We consider the Dirichlet problem for the mean curvature operator in Minkowski space, \[ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = \lambda u + \mu h(x,u) \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial\Omega, \] in a bounded domain $\Omega \subset \mathbb{R}^N$, where $\la...
A. Boscaggin, F. Colasuonno, Ricardo Ziegele· 1 citation· ⚡1
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
