Hamilton Bueno

Solutions of the Cheeger problem via torsion functions

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 .

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

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

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 with dependence on the gradient

We prove a result of existence of positive solutions for the p-Laplacian problem −Δpu = ω(x)f(u, |∇u|) with Dirichlet boundary condition in a bounded domain , where ω is a weight function. As in previous results by the authors, and in contrast with the hypotheses usually made, no asymptotic behaviour is assumed on f, but simple geometric assumptions in a neighbourhood 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 at both the origin and +∞, which is a remarkable fact. We apply our method to the p-growth problem −Δpu = λu(x)q−1(1 + |∇u(x)|p) (1 < q < p) in Ω with Dirichlet boundary conditions and give examples of super-linear nonlinearities which are also handled by our method.

Stationary Solutions of a Model for the Growth of Tumors and a Connection between the Nonnecrotic and Necrotic Phases

Results of existence of stationary solutions are proved for a problem modeling the growth of a spheroid tumor in absence of inhibitor agents, for both the nonnecrotic and necrotic cases. The results obtained for the nonnecrotic case are used to prove the existence of stationary solutions for the necrotic case, thus clarifying the connection between both cases. Some bounds for the inner and external radii of the necrotic tumor are given. We also discuss the critical nutrient concentration that determines the necrotic phase.

Torsion Functions and the Cheeger Problem: A Fractional Approach

Abstract Let Ω be a Lipschitz bounded domain of ℝ N

Existence and multiplicity of positive solutions for the p-Laplacian with nonlocal coefficient

{\mathbb{R}^{N}}

Remarks about a fractional Choquard equation: Ground state, regularity and polynomial decay

, N ≥ 2

A quasilinear problem with fast growing gradient

{N\geq 2}

On the p-torsion functions of an annulus

. The fractional Cheeger constant h s ⁢ ( Ω )