arXiv subjects

Marta Calanchi

Publications and source records attributed to Marta Calanchi.

8 recordsLinked to original sources

arXiv2026

Some remarks on patterns for semilinear Neumann problems

We study semilinear elliptic equations \begin{equation*} \begin{cases} -\Delta u = f(u) & \text{in } \Omega, \\ \partial_\nu u = 0 & \text{on } \partial\Omega, \end{cases} \end{equation*} with homogeneous Neumann boundary conditions in bounded domains. A classical result by Casten-Holland and Matano shows that stable nonconstant solutions cannot exist in convex domains, although unstable spatial patterns may still occur. In this paper we investigate rigidity properties of classical solutions without imposing stability assumptions and aim to identify structural conditions on the nonlinearity ensuring that all solutions are constant. We prove that every classical solution of the Neumann problem is constant provided the nonlinearity satisfies a suitable `monotonicity' condition, which includes the cases where the nonlinearity has a fixed sign or changes sign in a controlled way around one of its zeros. This yields a rigidity result depending solely on the structure of the nonlinearity and does not require convexity assumptions on the domain. We also discuss the sharpness of our assumptions by constructing examples of nonlinearities for which nonconstant solutions exist. In particular, inspired by the approach of Lin-Ni-Takagi, we consider exponential-type nonlinearities in dimension $N=2$, and show that when a parameter crosses a critical threshold, the associated Neumann problem admits nontrivial and nonconstant solutions for sufficiently small diffusion.

math.AP↗

arXiv2025

Low regularity results for degenerate Poisson problems

In this paper we study the Poisson problem, \[ \begin{cases} -{\rm div}(d^\beta\nabla u)=f&{\rm in}\ \Omega\\ u=0&{\rm on}\ \partial\Omega, \end{cases} \] where $\Omega\subset\mathbb R^N$, $N\ge2$ is a smooth bounded domain, $f$ is a continuous function, $\beta< 1$, and $d(x)=dist(x,\partial\Omega )$. We describe the behaviour of $u$ near $\partial\Omega$ and discuss some of its regularity properties.

math.AP↗

arXiv2024

Qualitative Properties of Solutions of Semilinear Elliptic Systems

The article explores the qualitative properties of solutions to elliptic equations and systems, focusing particularly on whether solutions retain the symmetry of their domains. According to the well-known Gidas-Ni-Nirenberg theorem, positive solutions to certain autonomous elliptic equations in radial domains are radial themselves. However, this symmetry can be broken in equations with power weight terms. The article also examines related results for systems of these weighted equations.

math.AP↗

arXiv2023

Positive eigenvectors and simple nonlinear maps

For linear operators $L, T$ and nonlinear maps $P$, we describe classes of simple maps $F = I - P T$, $F = L - P$ between Banach and Hilbert spaces, for which no point has more than two preimages. The classes encompass known examples (homeomorphisms, global folds) and the weaker, geometric, hypotheses suggest new ones. The operator $L$ may be the Laplacian with various boundary conditions, as in the original Ambrosetti-Prodi theorem, or the operators associated with the quantum harmonic oscillator, the hydrogen atom, a spectral fractional Laplacian, elliptic operators in non-divergent form. The maps $P$ include the Nemitskii map $P(u) = f(u)$ but may be non-local, even non-variational. For self-adjoint operators $L$, we employ familiar results on the nondegeneracy of the ground state. On Banach spaces, we use a variation of the Krein-Rutman theorem.

math.FA↗

arXiv2018

Global folds between Banach spaces as perturbations

Global folds between Banach spaces are obtained from a simple geometric construction: a Fredholm operator $T$ of index zero with one dimensional kernel is perturbed by a compatible nonlinear term $P$. The scheme encapsulates most of the known examples and suggests new ones. Concrete examples rely on the positivity of an eigenfunction. For the standard Nemitskii case $P(u) = f(u)$ (but $P$ might be nonlocal, non-variational), $T$ might be the Laplacian with different boundary conditions, as in the Ambrosetti-Prodi theorem, or the Schrödinger operators associated with the quantum harmonic oscillator or the Hydrogen atom, a spectral fractional Laplacian, a (nonsymmetric) Markov operator. For self-adjoint operators, we use results on the nondegeneracy of the ground state. On Banach spaces, a similar role is played by a recent extension by Zhang of the Krein-Rutman theorem.

math.AP↗

arXiv2015

Abundance of cusps and a converse to the Ambrosetti-Prodi theorem

According to the Ambrosetti-Prodi theorem, the map $F(u)= - Δu - f(u)$ between appropriate functional spaces is a global fold. Among the hypotheses, the convexity of the function $f$ is required. We show in two different ways that, under mild conditions, convexity is indeed necessary. If $f$ is not convex, there is a point with at least four preimages under $F$. More, $F$ generically admits cusps among its critical points. We present a larger class of nonlinearities $f$ for which the critical set of $F$ has cusps. The results are true for a class of boundary conditions.

math.AP↗

arXiv2015

Fibers and global geometry of functions

Since the seminal work of Ambrosetti and Prodi, the study of global folds was enriched by geometric concepts and extensions accomodating new examples. We present the advantages of considering fibers, a construction dating to Berger and Podolak's view of the original theorem. A description of folds in terms of properties of fibers gives new perspective to the usual hypotheses in the subject. The text is intended as a guide, outlining arguments and stating results which will be detailed elsewhere.

math.AP↗

arXiv2008

Multiple Solutions for a Henon-Like Equation on the Annulus

For the equation (-Δu = | |x|-2 |^αu^{p-1}), (1 < |x| < 3), we prove the existence of two solutions for (α) large, and of two additional solutions when (p) is close to the critical Sobolev exponent (2^*=2N/(N-2)). A symmetry--breaking phenomenon appears, showing that the least--energy solutions cannot be radial functions.

math.AP↗