Grey Ercole

Computing the First Eigenpair of the p-Laplacian via Inverse Iteration of Sublinear Supersolutions

We introduce an iterative method for computing the first eigenpair (λp,ep) for the p-Laplacian operator with homogeneous Dirichlet data as the limit of (μq,uq) as q→p−, where uq is the positive solution of the sublinear Lane-Emden equation

Absolute continuity of the best Sobolev constant

-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}

Solutions of the Cheeger problem via torsion functions

with the 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 uq to ep is in the C1-norm and the rate of convergence of μq to λp is at least O(p−q). Numerical evidence is presented.

Asymptotic behaviour of quasi-stationary solutions of a nonlinear problem modelling the growth of tumours

Abstract Let λ q ≔ inf { ‖ ∇ u ‖ L p ( Ω ) p / ‖ u ‖ L q ( Ω ) p : u ∈ W 0 1 , p ( Ω ) ∖ { 0 } } be the best Sobolev constant of the immersion W 0 1 , p ( Ω ) ↪ L q ( Ω ) , where Ω is a bounded and smooth domain of R N , 1 p N and 1 ≤ q ≤ p ⋆ ≔ N p N − p . We prove that the function q ∈ [ 1 , p ⋆ ] ⟼ λ q is absolutely continuous.

On the resonant Lane–Emden problem for the p-Laplacian

Abstract The Cheeger problem for a bounded domain Ω ⊂ R N , N > 1 consists in minimizing the quotients | ∂ E | / | E | among all smooth subdomains E ⊂ Ω and the Cheeger constant h ( Ω ) is the minimum of these quotients. Let ϕ p ∈ C 1 , α ( Ω ¯ ) be the p -torsion function, that is, the solution of torsional creep problem − Δ p ϕ p = 1 in Ω , ϕ p = 0 on ∂ Ω , where Δ p u : = div ( | ∇ u | p − 2 ∇ u ) is the p -Laplacian operator, p > 1 . The paper emphasizes the connection between these problems. We prove that lim p → 1 + ( ‖ ϕ p ‖ L ∞ ( Ω ) ) 1 − p = h ( Ω ) = lim p → 1 + ( ‖ ϕ p ‖ L 1 ( Ω ) ) 1 − p . Moreover, we deduce the relation lim p → 1 + ‖ ϕ p ‖ L 1 ( Ω ) ⩾ C N lim p → 1 + ‖ ϕ p ‖ L ∞ ( Ω ) where C N is a constant depending only of N and h ( Ω ) , explicitely given in the paper. An eigenfunction u ∈ BV ( Ω ) ∩ L ∞ ( Ω ) of the Dirichlet 1-Laplacian is obtained as the strong L 1 limit, as p → 1 + , 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 ⩽ | E 0 | h ( Ω ) N , where B 1 is the unit ball in R N . For Ω convex we obtain u = | E 0 | − 1 χ E 0 .

Computing the sinP Function via the inverse power method.

We study the global stability of quasi-steady solutions for a simple mathematical model describing the growth of a spherical vascularized tumour consisting only of living cells. By assuming the rates of proliferation and absorption to be increasing nonlinear functions of the nutrient concentration, we establish the existence of a non-trivial steady solution and conditions for the existence and uniqueness of a quasi-steady solution for each initial configuration. Also, we prove that all these quasi-steady solutions converge uniformly to a non-trivial steady solution. The quasi-steady approach is justified by the smallness of the parameter that measures the ratio between the timescales for the diffusion of nutrients and growth of the tumour.

Positive Solutions for the p-Laplacian and Bounds for its First Eigenvalue

We study the positive solutions of the Lane–Emden problem -Δpu = λp|u|q-2u in Ω, u = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded and smooth domain, N ≥ 2, λp is the first eigenvalue of the p-Laplacian operator Δp, p > 1, and q is close to p. We prove that any family of positive solutions of this problem converges in to the function θpep when q → p, where ep is the positive and L∞-normalized first eigenfunction of the p-Laplacian and . A consequence of this result is that the best constant of the immersion is differentiable at q = p. Previous results on the asymptotic behavior (as q → p) of the positive solutions of the nonresonant Lane–Emden problem (i.e. with λp replaced by a positive λ ≠ λp) are also generalized to the space and to arbitrary families of these solutions. Moreover, if uλ,q denotes a solution of the nonresonant problem for an arbitrarily fixed λ > 0, we show how to obtain the first eigenpair of the p-Laplacian as the limit in , when q → p, of a suitable scaling of the pair (λ, uλ,q). For computational purposes the advantage of this approach is that λ does not need to be close to λp. Finally, an explicit estimate involving L∞- and L1-norms of uλ,q is also derived using set level techniques. It is applied to any ground state family {vq} in order to produce an explicit upper bound for ‖vq‖∞ which is valid for q ∈ [1, p + ϵ] where .

Asymptotics for the best Sobolev constants and their extremal functions

Abstract In this paper, we discuss a new iterative method for computing sinp. 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. Keywords: p-Laplacian, eigenvalues, eigenfunctions, sinp, inverse power method.

Positive solutions for the p-Laplacian with dependence on the gradient

Abstract We prove a result of existence and localization of positive solutions of the Dirich- let problem for -Δpu = w(x)f(u) in a bounded domain Ω, where Δp is the p-Laplacian, w is a weight function and the nonlinearity f(u) satisfies certain local bounds. As in previous results in radially symmetric domains by two of the authors, and in contrast with the hypotheses usually made, no asymptotic behavior is assumed on f. A positive solution is obtained by applying the Schauder Fixed Point Theorem and such approach allows us to construct ordered sub- and super-solutions for the problem, thus producing an iterative method to obtain a positive solution. Our result leads not only to asymptotic conditions on the nonlinearity which provides the existence of a sequence of positive solutions for the problem with arbitrarily large sup norm, but also to an estimate of the first eigenvalue λp(Ω, w) of the p-Laplacian operator with weight w. For w ≡ 1, we compare our lower bound for λp(Ω, 1) with that obtained by means of the Cheeger constant h(Ω). We give a characterization of this constant in terms of the solution of the torsional creep problem -Δpɸp = 1 in Ω with Dirichlet boundary data, which offers a good approximation of the first eigenvalue of the p-Laplacian for p near 1.

Positive solutions for the p -Laplacian in annuli

Let p > 1 and let be a bounded and smooth domain of R N , N ≥ 2. It is well known that the infimum λq() := inf n k∇uk p : u ∈ W 1,p 0 () and kuk q = 1 o 0 () and also in C()). Moreover, we prove that any minimizer up ofp() satisfies −�pup = up(xp)�p()δxp , where δxp is the Dirac delta distribution concentrated at the only point xp satisfying |up(xp)| = kupk ∞ = 1. In the second part of the paper we prove that limp→∞ �p() 1 p = 1 kρk1 where ρ denotes the distance function to the boundary ∂. We also prove that there exist pn → ∞, x∗ ∈ and u∞ ∈ W 1,∞ 0 () such that: ρ(x∗) = kρk ∞ , xpn → x∗, upn → u∞ uniformly in , 0 < u∞ ≤ ρ kρk1 in and