Workshop onNonlinear PDE and
Elliptic Regularity

Abstract

We present a logarithmic Sobolev inequality for closed submanifolds of Riemannian manifolds with nonnegative curvature. The proof draws on ideas from the Alexandrov–Bakelman–Pucci maximum principle and the classical Heintze–Karcher estimates for the volumes of tubular neighborhoods.

Abstract

We study maximal hypersurfaces with multiple light cones in Lorentz–Minkowski space by considering weak solutions of the mean curvature equation with multiple Dirac masses. Our approach is based on an approximation argument.

Abstract

We study some local and global properties of solutions of equation$$-\Delta u=e^u+m|\nabla u|^q,$$in a punctured domain$\Omega\setminus\{0\}$, or in an exterior domain of$\mathbb R^N$($N\ge2$), where$m$is a positive parameter and$q>1$. We give a priori estimates and a precise description of the local behaviour of solutions near the singularity or near$\infty$, and prove the existence of solutions with the behaviours previously described. There is a strong competition between the gradient and the exponential source terms. The behaviour changes drastically according$q$is smaller or larger than$2$. Many results are obtained by introducing various dynamical systems associated to the equation.

Institut Denis Poisson–CNRS UMR 7013, Université de Tours, France

Abstract

For$0<a<1$, we study the exterior-ball obstacle problem posed by Fanghua Lin in his lectures (Example 1.5,Lectures on Elliptic Free Boundary Problems):$$\min\left\{\int_{B_1}|Du|^2\,\mathrm{d}x:\quad \begin{aligned} &u\in W^{1,2}(B_1;\mathbb R^n),\\ &u=x\ \text{on }\partial B_1,\quad |u|\ge a\ \text{a.e. in }B_1 \end{aligned} \right\}.$$We prove that the canonical radial map is the unique global minimizer in dimensions$3\le n\le6$. Together with the result of Figalli, Guerra, Kim, and Shahgholian for$n\ge7$, this gives a complete solution in every dimension$n\ge3$. In particular, the coincidence set is a concentric ball, confirming Lin’s proposed geometric picture and resolving the remaining cases of the vortex minimality problem. The proof combines singularity analysis and a sharp whole-space energy estimate with a monotonicity principle in the obstacle radius$a$and an analysis of the equality cases. This is joint work with Jiahuan Li, Yilu Liu, and Xi-nan Ma.

Abstract

The Lane–Emden system is an important model for Hamiltonian elliptic systems. We consider the zero Dirichlet problem in a half-space for arbitrary exponents$p,q\geq 1$. We prove that every nonnegative classical solution that is nondecreasing in the normal direction is necessarily trivial. No bound of any kind is imposed at infinity. This result extends the known scalar monotonicity theorem.

Abstract

We study some properties of solutions of$$u_{tt}+\Delta_\sigma u+\lambda u+\epsilon|u|^{q-1}u=0$$in the cylinder$\mathbb R\times S^{N-1}$in the range$q>1$,$\lambda\in\mathbb R$and$\epsilon=\pm1$. We recall the origin of the problem and study too particularly the equation in$\mathbb R\times S^1$pointing out the role of the set ofstationary solutionsi.e. the solutions of$$\omega''+\lambda\omega+\epsilon|\omega|^{q-1}\omega=0\quad\text{on }S^1,$$and theelliptic waves solutions, i.e. solutions$u(t,\theta)=\omega_\alpha(\alpha t+\theta)$where$$(1+\alpha^2)\omega_\alpha''+\lambda\omega_\alpha +\epsilon|\omega_\alpha|^{q-1}\omega_\alpha=0\quad\text{on }S^1.$$

Abstract

We study the Sobolev critical semilinear elliptic equation in the half-space with an isolated boundary singularity and zero Dirichlet boundary condition. We address two open questions: the existence of Delaunay-type log-periodic solutions and the asymptotic classification of singular solutions. We establish a global branch of positive log-periodic solutions, construct the corresponding concentrating family, and prove its local uniqueness. Consequently, the expected stationary asymptotic classification fails, and no universal critical scale-invariant upper bound holds throughout the half-space, in sharp contrast with the classical theory of interior singularities. This is joint work with Huayang Wang.

Abstract

We discuss Aleksandrov–Bakelman–Pucci (ABP) type estimates for hypoelliptic kinetic equations with rough coefficients. The proof is based on an adjoint argument and avoids the convex geometric gradient map used in classical ABP theory.

Abstract

We prove smooth solvability of the Dirichlet problem for quaternionic Monge–Ampère equations on smooth relatively compact domains in$\mathbb H^m/\Gamma$, where$m\ge2$and$\Gamma$is a discrete additive subgroup: for a smooth positive density, the existence of a smooth strictly plurisubharmonic subsolution guarantees the existence and uniqueness of the smooth admissible solution. The technique of the proof is a simple adaptation from the Ph.D. Thesis of Jingyong Zhu [25] in 2017, supervised by Xinan Ma, which is a joint work of Ma and Zhu. The claim [OpenAI, Theorem A.1] by OpenAI on Sogge’s local smoothing conjecture is a special case of our theorem. This is joint work with Jingyong Zhu.

Related reference: X.-N. Ma and J. Zhu,arXiv:2611.03214(2026). Reference [25] follows the numbering in this preprint.