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Raul Fernandes Horta

Publications and source records attributed to Raul Fernandes Horta.

5 recordsLinked to original sources
arXiv2026

On the relations between fundamental frequency and torsional rigidity in the case of anisotropic energies

We consider variational energies of the form \[E_H(u)=\frac12\int_ΩH^2(\nabla u)\,dx\] defined on the Sobolev space $H^1_0(Ω)$, where $H$ is a general seminorm. Our primary objective is to investigate optimization problems associated with the first eigenvalue $λ_H(Ω)$ and the torsional rigidity $T_H(Ω)$ induced by the seminorm $H$. In particular, we focus on functionals of the type \[F_{q,Ω}(H)=λ_H(Ω)\,T_H^q(Ω),\] where $q>0$ is a fixed real parameter. The optimization is performed with respect to the control $H$; we analyze both minimization and maximization problems for $F_{q,Ω}(H)$, as $H$ ranges over a suitable class of seminorms.

math.OC↗
arXiv2025

On the characterization of the Dirichlet and Fucik spectra of the one-dimensional anisotropic p-Laplace operator

The paper is concerned with the Dirichlet spectrum $Λ^{a,b}_p(0,L)$ of the anisotropic $p$-Laplace operator $- Δ^{a,b}_{p}$ on an interval $(0,L)$ where \[ Δ^{a,b}_p u:= \left(a^{p}[(u')^{+}]^{p-1}-b^{p}[(u')^{-}]^{p-1}\right)', \ \ a, b > 0. \] The set $Λ^{a,b}_p(0,L)$ and the respective eigenfunctions are completely characterized for $a \neq b$ in terms of the corresponding ones within the isotropic context. As an interesting application, we derive a new optimal Poincaré inequality that is stronger than the classical counterpart. The leading ideas are based on glue arguments of conveniently modified eigenfunctions and maximum type principles. More generally, our approach allows to characterize the Fu\v cík spectrum $Σ^{a,b}_p(0,L)$ of $- Δ^{a,b}_{p}$ on $(0,L)$ and mainly the corresponding solutions. All results are novelty even for the nonlinear operator $Δ^{a,b}_2$.

math.SP↗
arXiv2024

Sharp isoanisotropic estimates for fundamental frequencies of membranes and connections with shapes

The underlying motivation of the present work lies on a cornerstone question in spectral optimization that consists of determining sharp lower and upper uniform estimates for fundamental frequencies of a set of uniformly elliptic operators on a fixed membrane. We solve completely the problem in the plane for the general class of anisotropic operators in divergence form generated by arbitrary norms, which also includes the computation of optimal constants and the characterization of corresponding anisotropic extremizers (if they exist). Our approach is based on an isoanisotropic optimization formulation which, in turn, demands to be addressed within the broader environment of nonnegative, convex and 1-homogeneous anisotropies. A fine and detailed analysis of least energy levels associated to anisotropies with maximum degeneracy leads to a central connection between shapes and fundamental frequencies of rather degenerate elliptic operators. Such a linking also permits to establish that the supremum of anisotropic fundamental frequencies over all fixed-area membranes is infinite for any nonzero anisotropy. As a by-product, the well-known maximization conjecture for fundamental frequencies of the p-Laplace operator is proved for any p other than 2.

math.AP↗
arXiv2024

Optimal anisotropies for p-Laplace type operators in the plane

Sharp lower and upper uniform estimates are obtained for fundamental frequencies of $p$-Laplace type operators generated by quadratic forms. Optimal constants are exhibited, rigidity of the upper estimate is proved, anisotropic attainability of the lower estimate is derived as well as characterization of anisotropic extremizers for circular and rectangular membranes. Sharp quantitative anisotropic inequalities associated with lower constants are also established, providing as a by-product information on anisotropic stability. When the uniform ellipticity condition is relaxed, we show that the optimal lower constant remains positive, while anisotropic extremizers no longer exist. Our sharp lower estimate can be viewed as an isoanisotropic counterpart of the Faber-Krahn isoperimetric inequality in the plane.

math.AP↗