arXiv subjects

Alessandra De Luca

Publications and source records attributed to Alessandra De Luca.

10 recordsLinked to original sources

arXiv2026

Edge asymptotics for weighted Robin equations on cylinders with applications to spectral fractional Laplacians

For a class of weighted degenerate or singular problems on a cylinder, Almgren-type monotonicity methods are employed to derive asymptotic estimates of solutions near the edge between the base and the lateral surface, where a Robin boundary condition is imposed. As a relevant application, local asymptotics and unique continuation from the boundary are obtained for the Robin and Neumann spectral fractional Laplacians. The local decay rates of the solutions are the same for both the Neumann and the Robin problems and are explicitly quantized by a weighted spherical spectral problem with some symmetry.

math.AP↗

arXiv2025

On the subcritical Lane-Emden equation on Riemannian models with polynomial volume growth

We focus on the problems of existence and non-existence of positive solutions for the Sobolev-subcritical Lane-Emden equation on certain Riemannian manifolds (mainly models) with asymptotically negative curvature, which, from the viewpoint of the volume growth of geodesic balls, can be regarded as intermediate settings between the Euclidean and the hyperbolic spaces. A number of interesting phenomena arise: the subcritical regime naturally divides into three further ranges, characterized by existence phenomena (slightly subcritical), non-existence phenomena (strongly subcritical), and by a mixed behavior where existence and non-existence strongly depend on additional assumptions on the manifold (intermediate). In the intermediate regime, we further show that the radial homogeneous Dirichlet problem in geodesics balls may admit multiple positive solutions, thereby revealing substantial differences with respect to both the Euclidean and the hyperbolic settings.

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arXiv2025

Stability properties of solutions to convection-reaction equations with nonlinear diffusion

In this paper we study a convection-reaction-diffusion equation of the form \begin{equation*} u_t=\varepsilon(h(u)u_x)_x-f(u)_x+f'(u), \quad t>0, \end{equation*} with a nonlinear diffusion in a bounded interval of the real line. In particular, we first focus our attention on the existence of stationary solutions with at most one zero inside the interval, studying their behavior with respect to the viscosity coefficient $\varepsilon>0$ and their stability/instability properties. Then, we investigate the large time behavior of the solutions for finite times and the asymptotic regime. We also show numerically that, for a particular class of initial data, the so-called metastable behavior occurs, meaning that the time-dependent solution persists for an exponentially long (with respect to $\varepsilon$) time in a transition non-stable phase, before converging to a stable configuration.

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arXiv2024

A note on geometric assumptions for unique continuation from the edge of a crack

The present paper aims at representing an improvement of the result in [2], where a strong unique continuation property and a description of the local behaviour around the edge of a crack for solutions to an elliptic problem are established, by relaxing the star-shapedness condition on the complement of the crack. More specifically, this assumption will be dropped off by applying a suitable diffeomorphism which straightens the boundary of the crack, before performing the approximation procedure developed in [2] in order to derive a suitable monotonicity formula. This will yield the appearence of a matrix in the equation, which shall be handled appropriately: for this we will take a hint from [4].

math.AP↗

arXiv2024

Unique continuation from conical boundary points for fractional equations

We provide fine asymptotics of solutions of fractional elliptic equations at boundary points where the domain is locally conical; that is, corner type singularities appear. Our method relies on a suitable smoothing of the corner singularity and an approximation scheme, which allow us to provide a Pohozaev type inequality. Then, the asymptotics of solutions at the conical point follow by an Almgren type monotonicity formula, blow-up analysis and Fourier decomposition on eigenspaces of a spherical eigenvalue problem. A strong unique continuation principle follows as a corollary.

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arXiv2023

Layered patterns in reaction-diffusion models with Perona-Malik diffusions

In this paper we deal with a reaction-diffusion equation in a bounded interval of the real line with a nonlinear diffusion of Perona-Malik's type and a balanced bistable reaction term. Under very general assumptions, we study the persistence of layered solutions, showing that it strongly depends on the behavior of the reaction term close to the stable equilibria $\pm1$, described by a parameter $θ>1$. If $θ\in(1,2)$, we prove existence of steady states oscillating (and touching) $\pm1$, called $compactons$, while in the case $θ=2$ we prove the presence of $metastable$ $solutions$, namely solutions with a transition layer structure which is maintained for an exponentially long time. Finally, for $θ>2$, solutions with an unstable transition layer structure persist only for an algebraically long time.

math.AP↗

arXiv2023

Strong unique continuation from the boundary for the spectral fractional Laplacian

We investigate unique continuation properties and asymptotic behaviour at boundary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian. An extension procedure leads us to study a degenerate or singular equation on a cylinder, with a homogeneous Dirichlet boundary condition on the lateral surface and a non homogeneous Neumann condition on the basis. For the extended problem, by an Almgren-type monotonicity formula and a blow-up analysis, we classify the local asymptotic profiles at the edge where the transition between boundary conditions occurs. Passing to traces, an analogous blow-up result and its consequent strong unique continuation property is deduced for the nonlocal fractional equation.

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arXiv2022

Nonlocal capillarity for anisotropic kernels

We study a nonlocal capillarity problem with interaction kernels that are possibly anisotropic and not necessarily invariant under scaling. In particular, the lack of scale invariance will be modeled via two different fractional exponents $s_1, s_2\in (0,1)$ which take into account the possibility that the container and the environment present different features with respect to particle interactions. We determine a nonlocal Young's law for the contact angle and discuss the unique solvability of the corresponding equation in terms of the interaction kernels and of the relative adhesion coefficient.

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arXiv2022

Strong unique continuation and local asymptotics at the boundary for fractional elliptic equations

We study local asymptotics of solutions to fractional elliptic equations at boundary points, under some outer homogeneous Dirichlet boundary condition. Our analysis is based on a blow-up procedure which involves some Almgren type monotonicity formulae and provides a classification of all possible homogeneity degrees of limiting entire profiles. As a consequence, we establish a strong unique continuation principle from boundary points.

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arXiv2020

Unique continuation from the edge of a crack

In this work we develop an Almgren type monotonicity formula for a class of elliptic equations in a domain with a crack, in the presence of potentials satisfying either a negligibility condition with respect to the inverse-square weight or some suitable integrability properties. The study of the Almgren frequency function around a point on the edge of the crack, where the domain is highly non-smooth, requires the use of an approximation argument, based on the construction of a sequence of regular sets which approximate the cracked domain. Once a finite limit of the Almgren frequency is shown to exist, a blow-up analysis for scaled solutions allows us to prove asymptotic expansions and strong unique continuation from the edge of the crack.

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