We study the distributional d'Alembertian of a signed Lorentz distance function to an achronal set in a metric measure spacetime obeying the timelike measure contraction property. We show precise representation formulas and comparison estimates (both upper and lower bounds). Under a condition we call "infinitesimally strict concavity" (known for infinitesimally Minkowskian structures and established here for Finsler spacetimes), we prove the associated distribution is a signed measure certifying the integration by parts formula. This treatment of the d'Alembertian expands upon its recent nonlinear yet elliptic interpretation; even in the smooth case, our formulas seem to pioneer its exact shape across the timelike cut locus. In the second part of our work, we present several applications of these insights. First, we show the equivalence of the timelike curvature-dimension condition with a Bochner-type inequality. Second, we set up synthetic mean curvature (as well as barriers for CMC sets) exactly. Third, we prove synthetic volume and area estimates of Heintze-Karcher-type, which enable us to show several synthetic volume singularity theorems.
The paper is devoted to a standing open problem: the existence of normalized solutions to a mass supercritical nonlinear Schrödinger system with general potentials {[ -Δ u_1+λ _1u_1+V_1(x) u_1=μ _1u_1^3+β u_1u_2^2,; -Δ u_2+λ _2u_2+V_2(x) u_2=μ _2u_2^3+β u_1^2u_2,; ]. in Ω , having prescribed masses ∫ _Ω |u_1|^2dx=α _1 and ∫ _Ω |u_2|^2dx=α _2, where Ω is either the entire space ℝ^3 or a bounded smooth star-shaped domain of ℝ^3 . Due to the presence of the potential V_i(x), i=1,2, the approach based on the Pohozaev identity to deal with the autonomous case (i.e., V_i(x)≡ 0, i=1,2 ) is not available. We develop an approximation method, which does not essentially depend on the Pohozaev identity, to obtain the existence of normalized solutions to the cooperative nonlinear Schrödinger system ( β >0 ) under mild conditions on V_i(x) . We believe that our approach could be adopted to treat other systems with potentials. We further investigate asymptotic behaviors of the normalized solution when the coupling parameter β tends to infinity, which is new even for the autonomous case.
In this paper, we present a rigorous mathematical analysis of topological defects in the Georgi–Machacek model, which is a well-motivated extension of the Standard Model Higgs sector. We focus specifically on the existence and properties of symmetric vortex solutions, which represent a class of topologically non-trivial field configurations. By employing a direct minimization approach, we establish the existence of smooth, finite-energy vortex solutions satisfying physically relevant boundary conditions. A comprehensive analysis of the solutions is conducted, proving their boundedness, monotonicity, and precise asymptotic behavior both at the origin and at infinity. Furthermore, we demonstrate the uniqueness of these solutions when the parameter satisfies a suitable lower bound. Our work provides a solid mathematical foundation for the physical understanding of topological defects in extended Higgs sectors and reveals new structural properties specific to the Georgi–Machacek model. The analytical framework developed here is potentially applicable to the study of other topological solitons in quantum field theories with non-minimal Higgs sectors.
It is shown that 3 disjoint sets with equal Gaussian volumes that partition ℝ^n with nearly minimum total Gaussian surface area must be close to adjacent 120 degree sectors, when n≥ 2 . These same results hold for any number m≤ n+1 of sets partitioning ℝ^n , conditional on the solution of a finite-dimensional optimization problem (similar to the endpoint case of the Plurality is Stablest Problem, or the Propeller Conjecture of Khot and Naor). When m>3 , the minimal Gaussian surface area is achieved by the cones over a regular simplex. We therefore strengthen the Milman–Neeman Gaussian multi bubble theorem to a “stability” statement. Consequently, we obtain the first known dimension-independent bounds for the Plurality is Stablest Conjecture for three candidates for a small amount of noise (and for m>3 candidates, conditional on the solution of a finite-dimensional optimization problem). In particular, we classify all stable local minima of the Gaussian surface area of m sets. We focus exclusively on volume-preserving variations of the sets, avoiding the use of matrix-valued partial differential inequalities. Lastly, we remove the convexity assumption from our previous result on the minimum Gaussian surface area of a symmetric set of fixed Gaussian volume.
A novel approach based on a minimax-flow-degree scheme is employed to prove the existence of infinitely many multi-peak solutions to the p-Laplace equation -Δ _p u+V(x)u^p-1=u^q-1, u>0 in ℝ^n, where 1
