arXiv subjects

Raoní Ponciano

Publications and source records attributed to Raoní Ponciano.

7 recordsLinked to original sources

arXiv2026

Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications

We prove a Pólya--Szegö inequality for functions defined on an $n$-dimensional submanifold $Σ$ of a complete noncompact Riemannian manifold with nonnegative sectional curvature. The associated rearrangement is a Schwarz rearrangement on $\mathbb R^n$, and the constant depends on the $L^n$-norm of the mean curvature of $Σ$ and an isoperimetric quantity obtained by Brendle. As applications, we derive Sobolev, Log-Sobolev, Hardy, and Gagliardo--Nirenberg inequalities on submanifolds of arbitrary codimension under a small total mean curvature assumption. In the critical Sobolev case, we obtain Moser--Trudinger inequalities on finite-volume submanifolds and exact growth inequalities on submanifolds with infinite volume. Under suitable assumptions, the Pólya--Szegö constant equals one; in this case, the critical constants in the inequalities coincide with the sharp Euclidean ones.

math.DG↗

arXiv2026

Survey on topological methods for Allen--Cahn equations and systems

We present a survey on multiplicity results for the Allen--Cahn equation and systems in the singular perturbation regime, emphasizing their geometric interpretation through $Γ$-convergence and isoperimetric theory. In the scalar case, the Allen--Cahn functional converges to perimeter, giving rise to minimal and constant-mean-curvature hypersurfaces, while vectorial Allen--Cahn systems lead to multi-phase isoperimetric clusters. The main methodological tool discussed is the photography method, a variational-topological approach based on localized approximate solutions and barycenter maps, which enables one to encode the topology of the ambient manifold into multiplicity results. We compare problems posed on closed manifolds with those on manifolds with boundary, describing the distinct geometric effects induced by Neumann and Dirichlet boundary conditions. The survey highlights both the effectiveness and the limitations of this framework, particularly in the vectorial case, where the lack of a full classification of isoperimetric clusters creates fundamental analytical challenges.

math.AP↗

arXiv2026

Radial Sobolev embeddings on spherically symmetric Riemannian manifolds

We study Sobolev spaces of radial functions on spherically symmetric Riemannian manifolds. Using geodesic polar coordinates, we give a sharp one-dimensional reduction: a radial function belongs to the Sobolev space on the manifold if and only if its radial representation lies in an associated weighted Sobolev space on an interval, with weights determined explicitly by the metric. This characterization allows us to prove optimal Sobolev-type embeddings for radial functions into weighted Lebesgue spaces on both bounded and unbounded spherically symmetric manifolds. As further consequences, we establish new radial lemmas and decay estimates that capture the precise behaviour of radial Sobolev functions near the origin and at infinity. Our results unify and extend the classical radial embeddings in Euclidean and hyperbolic spaces.

math.AP↗

arXiv2025

Sharp Sobolev and Adams-Trudinger-Moser inequalities for symmetric functions without boundary conditions on hyperbolic spaces

Embedding theorems for symmetric functions without zero boundary condition have been studied on flat Riemannian manifolds, such as the Euclidean space. However, these theorems have only been established on hyperbolic spaces for functions with zero boundary condition. In this work, we focus on sharp Sobolev and Adams-Trudinger-Moser embeddings for radial functions in hyperbolic spaces, considering both bounded and unbounded domains. One of the main features of our approach is that we do not assume boundary zero condition for symmetric functions on geodesic balls or the entire hyperbolic space. Our main results include Theorems 1.2, 1.3, and 1.4, which establish weighted Sobolev embedding theorems, and Theorems 1.5 together with 1.6, which present Adams-Trudinger-Moser type of embedding theorems. In particular, a key result is Theorem 1.1 which is a highly nontrivial comparison result between norms of the higher order covariant derivatives and higher order derivatives of the radial functions. Higher order asymptotic behavior of radial functions on hyperbolic spaces are established to prove our main theorems. This approach includes novel radial lemmata and decay properties of higher order radial Sobolev functions defined in hyperbolic space.

math.AP↗

arXiv2024

From bubbles to clusters: Multiple solutions to the Allen--Cahn system

We extend previous works on the multiplicity of solutions to the Allen-Cahn system on closed Riemannian manifolds by considering an arbitrary number of phases. Specifically, we show that on parallelizable manifolds, the number of solutions is bounded from below by topological invariants of the underlying manifold, provided the temperature parameter and volume constraint are sufficiently small. The Allen-Cahn system naturally arises in phase separation models, where solutions represent the distribution of distinct phases in a multi-component mixture. As the temperature parameter approaches zero, the system's energy approximates the multi-isoperimetric profile, leading to solutions concentrating in regions resembling isoperimetric clusters. For two or three phases, these results rely on classifying isoperimetric clusters. However, this classification remains incomplete for a larger number of phases. To address this technical issue, we employ a "volume-fixing variations" approach, enabling us to establish our results for any number of phases and small volume constraints. This offers deeper insights into phase separation phenomena on manifolds with arbitrary geometry.

math.AP↗

arXiv2023

Sharp higher order Admas' inequality with exact growth condition on weighted Sobolev spaces

This paper introduces a novel higher order Adams inequality that incorporates an exact growth condition for a class of weighted Sobolev spaces. Our rigorous proof confirms the validity of this inequality and provides insights into the optimal nature of the critical constant and the exponent within the denominator. Furthermore, we apply this inequality to study a class of ordinary differential equations (ODEs), where we successfully derive both a concept of the weak solution and a comprehensive regularity theory.

math.AP↗

arXiv2023

Sharp Sobolev and Adams-Trudinger-Moser embeddings on weighted Sobolev spaces and their applications

We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential class with a sharp constant. As applications, we prove that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions.

math.AP↗