arXiv subjects

Camilla Chiara Polvara

Publications and source records attributed to Camilla Chiara Polvara.

6 recordsLinked to original sources

arXiv2026

Regularity results for elliptic equations on cones

We study global regularity of solutions to Dirichlet or Neumann elliptic problems in spherical sectors $S_{D,R}$ of radius $R>0$ in $\mathbb{R}^N, N\ge 2$, where $D$ is the bounded domain on the unit sphere $\mathbb{S}^{N-1}$ which spans the spherical sector. One of the main results shows that boundedness of the gradient of the solutions of Poisson equations holds whenever $λ_1(D)\ge N-1$, where $λ_1(D)$ is the first nontrivial eigenvalue of the Laplace Beltrami operator $-Δ_{\mathbb{S}^{N-1}}$ on the domain $D$ with Dirichlet or Neumann boundary conditions on $\partial D$. As an example of Maz'ya shows, the condition on the eigenvalue is sharp. For general spherical sectors and for $p$-Laplacian equations, $p>1$ we prove weighted global lipschitzianity of the solutions, as well as second order regularity.

math.AP↗

arXiv2025

Bifurcation from bubbles in nonconvex cones

We investigate the Neumann problem for the critical semilinear elliptic equation in cones. The standard bubble provides a family of radial solutions, which are known to be the only positive solutions in convex cones. For nonconvex cones, symmetry breaking may occur and the symmetry breaking is related to the first nonzero Neumann eigenvalue of the Laplace Beltrami operator on the domain $D\subset§^{N-1}$, that spans the cone. We construct a one-parameter family of domains on the sphere whose first eigenvalue crosses the threshold at which the bubble loses stability. Under the assumption that this eigenvalue is simple, we prove, via the Crandall Rabinowitz bifurcation theorem, the existence of a branch of nonradial solutions bifurcating from the standard bubble. Moreover we show that the bifurcation is global.

math.AP↗

arXiv2025

Stability for the Sobolev inequality in cones

We prove a quantitative Sobolev inequality in cones of Bianchi-Egnell type, which implies a stability property. Our result holds for any cone as long as the minimizers of the Sobolev quotient are nondegenerate, which is the case of most cones. When the minimizers are the classical bubbles we have more precise results. Finally, we show that local estimates are not enough to get the optimal constant for the quantitative Sobolev inequality.

math.AP↗

arXiv2024

On the classification of extremals of Caffarelli-Kohn-Nirenberg inequalities

We consider a family of critical elliptic equations which arise as the Euler-Lagrange equation of Caffarelli-Kohn-Nirenberg inequalities, possibly in convex cones in $\mathbb{R}^d$, with $d\geq 2$. We classify positive solutions without assuming that the solution has finite energy and when the intrinsic dimension $n \in (\frac{3}{2},5]$.

math.AP↗

arXiv2024

Classification results, rigidity theorems and semilinear PDEs on Riemannian manifolds: a P-function approach

We consider solutions to some semilinear elliptic equations on complete noncompact Riemannian manifolds and study their classification as well as the effect of their presence on the underlying manifold. When the Ricci curvature is non-negative, we prove both the classification of positive solutions to the critical equation and the rigidity for the ambient manifold. The same results are obtained when we consider solutions to the Liouville equation on Riemannian surfaces. The results are obtained via a suitable P-function whose constancy implies the classification of both the solutions and the underlying manifold. The analysis carried out on the P-function also makes it possible to classify non-negative solutions for subcritical equations on manifolds enjoying a Sobolev inequality and satisfying an integrability condition on the negative part of the Ricci curvature. Some of our results are new even in the Euclidean case.

math.AP↗

arXiv2023

Symmetry breaking and instability for semilinear elliptic equations in spherical sectors and cones

We consider semilinear elliptic equations with mixed boundary conditions in spherical sectors inside a cone. The aim of the paper is to show that a radial symmetry result of Gidas-Ni-Nirenberg type for positive solutions does not hold in general nonconvex cones. This symmetry breaking result is achieved by studying the Morse index of radial positive solutions and analyzing how it depends on the domain D on the unit sphere which spans the cone. In particular it is proved that the Neumann eigenvalues of the Laplace Beltrami operator on D play a role in computing the Morse index. A similar breaking of symmetry result is obtained for the positive solutions of the critical Neumann problem in the whole unbounded cone. In this case it is proved that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones.

math.AP↗