arXiv subjects

Manuel Dias

Publications and source records attributed to Manuel Dias.

4 recordsLinked to original sources

arXiv2026

Spectral convergence of empirical integral operators with discontinuous kernels

We study the spectral behavior as the sample size $n \to +\infty$ of integral operators defined by convolution of a non-negative symmetric kernel k with respect to empirical measures $\mu_n = \frac{1}{n} \sum_{i=1}^n \delta_{X_i}$, where $\{X_i\}_{i=1}^n$ are independent uniform samples from a compact probability metric space $(\mathcal{X},d,\mu)$. Relaxing the usual positivity and continuity assumptions on k, we prove the convergence of these empirical operators to their continuous counterparts, and provide explicit convergence rates.

math.SP↗

arXiv2024

Spectral properties of symmetrized AMV operators

The symmetrized Asymptotic Mean Value Laplacian $\tilde{\Delta}$, obtained as limit of approximating operators $\tilde{\Delta}_r$, is an extension of the classical Euclidean Laplace operator to the realm of metric measure spaces. We show that, as $r \downarrow 0$, the operators $\tilde{\Delta}_r$ eventually admit isolated eigenvalues defined via min-max procedure on any compact locally Ahlfors regular metric measure space. Then we prove $L^2$ and spectral convergence of $\tilde{\Delta}_r$ to the Laplace--Beltrami operator of a compact Riemannian manifold, imposing Neumann conditions when the manifold has a non-empty boundary.

math.AP↗

arXiv2024

A note on the maximization of the first Dirichlet eigenvalue for perforated planar domains

In this work we prove that given an open bounded set $Ω\subset \mathbb{R}^2$ with a $C^2$ boundary, there exists $ε:= ε(Ω)$ small enough such that for all $0 < δ< ε$ the maximum of $\{λ_1(Ω- B_δ(x)):B_δ \subset Ω\}$ is never attained when the ball is close enough to the boundary. In particular it is not obtained when $B_δ(x)$ is touching the boundary $\partial Ω$.

math.AP↗

arXiv2023

Optimal uniform bounds for competing variational elliptic systems with variable coefficients

Let $Ω\subset \mathbb{R}^N$ be an open set. In this work we consider solutions of the following gradient elliptic system \[ -\text{div}(A(x)\nabla u_{i,β}) = f_i(x,u_{i,β}) + a(x)β|u_{i, β}|^{γ-1}u_{i, β} \mathop{\sum_{j=1}^l}_{j\neq i} |u_{j, β}|^{γ+ 1}, \] for $i=1,\ldots, l$. We work in the competitive case, namely $β<0$. Under suitable assumptions on $A$, $a$, $f_i$ and on the exponent $γ$, we prove that uniform $L^\infty$-bounds on families of positive solutions $\{u_β\}_{β<0}=\{(u_{1,β},\ldots, u_{l,β})\}_{β<0}$ imply uniform Lipschitz bounds (which are optimal). One of the main points in the proof are suitable generalizations of Almgren's and Alt-Caffarelli-Friedman's monotonicity formulas for solutions of such systems. Our work generalizes previous results, where the case $A(x)=Id$ (i.e. the operator is the Laplacian) was treated.

math.AP↗