arXiv subjects

Grey Ercole

Publications and source records attributed to Grey Ercole.

25 records · Page 2Linked to original sources

arXiv2012

Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift

In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition for arbitrary bounded domains $Ω\subset R^{N}$. This method, which has a direct functional analysis approach, does not approximate the eigenvalues of the Laplacian as those of a finite linear operator. It is based on the uniform convergence away from nodal surfaces and can produce a simple and fast algorithm for computing the eigenvalues with minimal computational requirements, instead of using the ubiquitous Rayleigh quotient of finite linear algebra. Also, an alternative expression for the Rayleigh quotient in the associated infinite dimensional Sobolev space which avoids the integration of gradients is introduced and shown to be more efficient. The method can also be used in order to produce the spectral decomposition of any given function $u\in L^{2}(Ω)$.

math.SP↗

arXiv2012

Absolute continuity of the best Sobolev constant of a bounded domain

Let $λ_{q}:=\inf{\Vert\nabla u\Vert_{L^{p}(Ω)}^{p}/\Vertu\Vert_{L^{q}(Ω)}^{p}:u\in W_{0}^{1,p}(Ω)\setminus{0}} $, where $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N},$ $1<p<N$ and $1\leq q\leq p^{\star}% :=\frac{Np}{N-p}.$ We prove that the function $q\mapstoλ_{q}$ is absolutely continuous in the closed interval $[1,p^{\star}].$

math.AP↗

arXiv2011

Solutions of the Cheeger problem via torsion functions

The Cheeger problem for a bounded domain $Ω\subset\mathbb{R}^{N}$, $N>1$ consists in minimizing the quotients $|\partial E|/|E|$ among all smooth subdomains $E\subsetΩ$ and the Cheeger constant $h(Ω)$ is the minimum of these quotients. Let $ϕ_{p}\in C^{1,α}(\barΩ)$ be the $p$-torsion function, that is, the solution of torsional creep problem $-Δ_{p}ϕ_{p}=1$ in $Ω$, $ϕ_{p}=0$ on $\partialΩ$, where $Δ_{p}u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator, $p>1$. The paper emphasizes the connection between these problems. We prove that $\lim_{p\rightarrow1^{+}}(\|ϕ_{p}\|_{L^{\infty}(Ω)})^{1-p}=h(Ω)=\lim_{p\rightarrow1^{+}}(\|ϕ_{p}\|_{L^{1}(Ω)})^{1-p}$. Moreover, we deduce the relation $\lim_{p\to1^{+}}\|ϕ_{p}\|_{L^{1}(Ω)}\geq C_{N}\lim_{p\to1^{+}}\|ϕ_{p}\|_{L^{\infty}(Ω)}$ where $C_{N}$ is a constant depending only of $N$ and $h(Ω)$, explicitely given in the paper. An eigenfunction $u\in BV(Ω)\cap L^{\infty}(Ω)$ of the Dirichlet 1-Laplacian is obtained as the strong $L^{1}$ limit, as $p\rightarrow1^{+}$, of a subsequence of the family $\{ϕ_{p}/\|ϕ_{p}\|_{L^{1}(Ω)}\}_{p>1}$. Almost all $t$-level sets $E_{t}$ of $u$ are Cheeger sets and our estimates of $u$ on the Cheeger set $|E_{0}|$ yield $|B_{1}|h(B_{1})^{N}\leq |E_{0}|h(Ω)^{N},$ where $B_{1}$ is the unit ball in $\mathbb{R}^{N}$. For $Ω$ convex we obtain $u=|E_{0}|^{-1}χ_{E_{0}}$.

math.AP↗

arXiv2011

Computing the first eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions

We introduce an iterative method for computing the first eigenpair $(λ_{p},e_{p})$ for the $p$-Laplacian operator with homogeneous Dirichlet data as the limit of $(μ_{q,}u_{q}) $ as $q\rightarrow p^{-}$, where $u_{q}$ is the positive solution of the sublinear Lane-Emden equation $-Δ_{p}u_{q}=μ_{q}u_{q}^{q-1}$ with same boundary data. The method is shown to work for any smooth, bounded domain. Solutions to the Lane-Emden problem are obtained through inverse iteration of a super-solution which is derived from the solution to the torsional creep problem. Convergence of $u_{q}$ to $e_{p}$ is in the $C^{1}$-norm and the rate of convergence of $μ_{q}$ to $λ_{p}$ is at least $O(p-q)$. Numerical evidence is presented.

math.AP↗

arXiv2010

Positive Solutions for the p-Laplacian with Dependence on the Gradient

We prove a result of existence of positive solutions of the Dirichlet problem for $-Δ_p u=\mathrm{w}(x)f(u,\nabla u)$ in a bounded domain $Ω\subset\mathbb{R}^N$, where $Δ_p$ is the $p$-Laplacian and $\mathrm{w}$ is a weight function. As in previous results by the authors, and in contrast with the hypotheses usually made, no asymptotic behavior is assumed on $f$, but simple geometric assumptions on a neighborhood of the first eigenvalue of the $p$-Laplacian operator. We start by solving the problem in a radial domain by applying the Schauder Fixed Point Theorem and this result is used to construct an ordered pair of sub- and super-solution, also valid for nonlinearities which are super-linear both at the origin and at $+\infty$. We apply our method to the Dirichlet problem $-Δ_pu = λu(x)^{q-1}(1+|\nabla u(x)|^p)$ in $Ω$ and give examples of super-linear nonlinearities which are also handled by our method.

math.AP↗

arXiv2010

Computing the $\sin_{p}$ function via the inverse power method

In this paper, we discuss a new iterative method for computing $\sin_{p}$. This function was introduced by Lindqvist in connection with the unidimensional nonlinear Dirichlet eigenvalue problem for the $p$-Laplacian. The iterative technique was inspired by the inverse power method in finite dimensional linear algebra and is competitive with other methods available in the literature.

math.CA↗

arXiv2010

A quasilinear problem in two parameters depending on the gradient

The existence of positive solutions is considered for the Dirichlet problem \[ \left\{ \begin{array} [c]{rcll}% -Δ_{p}u & = & λω_{1}(x)\left\vert u\right\vert ^{q-2}% u+βω_{2}(x)\left\vert u\right\vert ^{a-1}u|\nabla u|^{b} & \text{in }Ω\\ u & = & 0 & \text{on }\partialΩ, \end{array} \right. \] where $λ$ and $β$ are positive parameters, $a$ and $b$ are positive constants satisfying $a+b\leq p-1$, $ω_{1}(x)$ and $ω_{2}(x)$ are nonnegative weights and $1<q\leq p$. The homogeneous case $q=p$ is handled by making $q\rightarrow p^{-}$ in the sublinear case $1<q<p,$ which is based on the sub- and super-solution method. The core of the proof of this problem is then generalized to the Dirichlet problem $-Δ_{p}u=f(x,u,\nabla u)$ in $Ω$, where $f$ is a nonnegative, continuous function satisfying simple, geometrical hypotheses. This approach might be considered as a unification of arguments dispersed in various papers, with the advantage of handling also nonlinearities that depend on the gradient, even in the $p$-growth case. It is then applied to the problem $-Δ_{p}u=λω(x)u^{q-1}\left( 1+|\nabla u|^{p}\right) $ with Dirichlet boundary conditions in the domain $Ω$.

math.AP↗