Searcharxiv⌕ Search

arXiv subjects

J. Fernandez Bonder

Publications and source records attributed to J. Fernandez Bonder.

9 recordsLinked to original sources
arXiv2023

Existence of Eigenvalues for Anisotropic and Fractional Anisotropic Problems via Ljusternik-Schnirelmann Theory

In this work, our interest lies in proving the existence of critical values of the following Rayleigh-type quotients $$Q_{\mathbf p}(u) = \frac{\|\nabla u\|_{\mathbf p}}{\|u\|_{\mathbf p}},\quad\text{and}\quad Q_{\mathbf s,\mathbf p}(u) = \frac{[u]_{\mathbf s,\mathbf p}}{\|u\|_{\mathbf p}}, $$ where $\mathbf p = (p_1,\dots,p_n)$, $\mathbf s=(s_1,\dots,s_n)$ and $$ \|\nabla u\|_{\mathbf p} = \sum_{i=1}^n \|u_{x_i}\|_{p_i} $$ is an anisotropic Sobolev norm, $[u]_{\mathbf s,\mathbf p}$ is a fractional version of the same anisotropic norm, and $\|u\|_{\mathbf p}$ is an anisotropic Lebesgue norm. Using the Ljusternik-Schnirelmann theory, we prove the existence of a sequence of critical values and we also find an associated Euler-Lagrange equation for critical points. Additionally, we analyze the connection between the fractional critical values and its local counterparts.

math.AP↗
arXiv2017

Some nonlocal optimal design problems

In this paper we study two optimal design problems associated to fractional Sobolev spaces $W^{s,p}(Ω)$. Then we find a relationship between these two problems and finally we investigate the convergence when $s\uparrow 1$.

math.AP↗
arXiv2017

An optimization problem for the first eigenvalue of the $p-$fractional laplacian

In this paper we analyze an eigenvalue problem related to the nonlocal $p-$laplace operator plus a potential. After reviewing some elementary properties of the first eigenvalue of these operators (existence, positivity of associated eigenfunctions, simplicity and isolation) we investigate the dependance of the first eigenvalue on the potential function and establish the existence of some {\em optimal} potentials in some admissible classes.

math.AP↗
arXiv2009

Precise asymptotic of eigenvalues of resonant quasilinear systems

In this work we study the sequence of variational eigenvalues of a system of resonant type involving $p-$ and $q-$laplacians on $Ω\subset \R^N$, with a coupling term depending on two parameters $α$ and $β$ satisfying $α/p + β/q = 1$. We show that the order of growth of the $k^{th}$ eigenvalue depends on $α+β$, $\lam_k = O(k^{\frac{α+β}{N}})$.

math.AP↗
arXiv2009

Refined asymptotics for eigenvalues on domains of infinite measure

In this work we study the asymptotic distribution of eigenvalues in one-dimensional open sets. The method of proof is rather elementary, based on the Dirichlet lattice points problem, which enable us to consider sets with infinite measure. Also, we derive some estimates for the the spectral counting function of the Laplace operator on unbounded two-dimensional domains.

math.AP↗
arXiv2008

Some optimization problems for nonlinear elastic membranes

In this paper we study some optimization problems for nonlinear elastic membranes. More precisely, we consider the problem of optimizing the cost functional $\J(u)=\int_{\partialΩ} f(x) u \rd \H^{N-1}$ over some admissible class of loads $f$ where $u$ is the (unique) solution to the problem $-Δ_p u + |u|^{p-2}u = 0$ in $Ω$ with $|\nabla u|^{p-2}u_ν= f$ on $\partial Ω$.

math.AP↗
arXiv2006

Estimates for the Sobolev trace constant with critical exponent and applications

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality $S\|u\|^p_{L^{p_*}(\partialΩ) \hookrightarrow \|u\|^p_{W^{1,p}(Ω)}$ that are independent of $Ω$. This estimates generalized those of [3] for general $p$. Here $p_* := p(N-1)/(N-p)$ is the critical exponent for the immersion and $N$ is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of [16]. Finally, we study an optimal design problem with critical exponent.

math.AP↗