DSpace Repository :: Browsing by Author "Ercole, Grey"

Browsing by Author "Ercole, Grey"

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An optimal pointwise Morrey-Sobolev inequality.
(2020) Ercole, Grey; Pereira, Gilberto de Assis
Let Ω be a bounded, smooth domain of RN , N ≥ 1. For each p > N we study the optimal function s = sp in the pointwise inequality |v(x)| ≤ s(x) ∇vLp(Ω) , ∀ (x, v) ∈ Ω × W1,p 0 (Ω). We show that sp ∈ C0,1−(N/p) 0 (Ω) and that sp converges pointwise to the distance function to the boundary, as p → ∞. Moreover, we prove that if Ω is convex, then sp is concave and has a unique maximum point.
Asymptotic behavior of extremals for fractional Sobolev inequalities associated with singular problems.
(2019) Ercole, Grey; Pereira, Gilberto de Assis; Sanchis, Remy de Paiva
Let be a smooth, bounded domain of RN , ω be a positive, L1-normalized function, and 0 < s < 1 < p. We study the asymptotic behavior, as p → ∞, of the pair p p, u p, where p is the best constant C in the Sobolev-type inequality C exp (log |u| p)ωdx ≤ [u] p s,p ∀ u ∈ Ws,p 0 () and u p is the positive, suitably normalized extremal function corresponding to p. We show that the limit pairs are closely related to the problem of minimizing the quotient |u|s / exp (log |u|)ωdx , where |u|s denotes the s-Hölder seminorm of a function u ∈ C0,s 0 ().
Asymptotic behavior of the p-torsion functions as p goes to 1.
(2016) Bueno, Hamilton; Ercole, Grey; Macedo, Shirley da Silva
Let Ω be a Lipschitz bounded domain of RN, N ≥ 2, and let up ∈ W1,p 0 (Ω) denote the p-torsion function of Ω, p > 1. It is observed that the value 1 for the Cheeger constant h(Ω) is threshold with respect to the asymptotic behavior of up, as p → 1+, in the following sense: when h(Ω) > 1, one has limp→1+ up L∞(Ω) = 0, and when h(Ω) < 1, one has limp→1+ up L∞(Ω) = ∞. In the case h(Ω) = 1, it is proved that lim supp→1+ up L∞(Ω) < ∞. For a radial annulus Ωa,b, with inner radius a and outer radius b, it is proved that limp→1+ up L∞(Ωa,b) = 0 when h(Ωa,b) = 1.
Asymptotic behaviour as p → ∞ of least energy solutions of a (p, q(p))-Laplacian problem.
(2019) Alves, Claudianor Oliveira; Ercole, Grey; Pereira, Gilberto de Assis
We study the asymptotic behaviour, as p → ∞, of the least energy solutions of the problem −(Δp + Δq(p))u = λp|u(xu)| p−2u(xu)δxu in Ω u = 0 on ∂Ω, where xu is the (unique) maximum point of |u|, δxu is the Dirac delta distribution supported at xu, limp→∞ q(p) p = Q ∈ (0, 1) if N 0 is such that min ∇u∞ u∞ : 0 ≡ u ∈ W1,∞(Ω) ∩ C0(Ω) limp→∞(λp) 1/p < ∞.
Computing the best constant in the Sobolev inequality for a ball.
(2017) Ercole, Grey; Espírito Santo, Júlio César do; Martins, Eder Marinho
Let B1 be the unit ball of R N , N ≥ 2, and let p ? = N p/(N − p) if 1 < p < N and p ? = ∞ if p ≥ N. For each q ∈ [1, p? ) let wq ∈ W1,p 0 (B1) be the positive function such that kwqkLq(B1) = 1 and λq(B1) := min ( k∇uk p Lp(B1) kuk p Lq(B1) : 0 6≡ u ∈ W1,p 0 (B1) ) = k∇wqk p Lp(B1) . In this paper we develop an iterative method for obtaining the pair (λq(B1), wq), starting from w1. Since w1 is explicitly known, the method is computationally practical, as our numerical tests show. 2010 Mathematics Subject Classification. 34L16; 35J25; 65N25 Keywords: Best Sobolev constant; extremal functions; inverse iteration method; p-Laplacian.
Computing the first eigenpair of the p-Laplacian in annuli.
(2015) Ercole, Grey; Espírito Santo, Júlio César do; Martins, Eder Marinho
We propose a method for computing the first eigenpair of the Dirichlet p-Laplacian, p > 1, in the annulus Ωa,b = {x ∈ RN : a < |x| < b}, N > 1. For each t ∈ (a, b), we use an inverse iteration method to solve two radial eigenvalue problems: one in the annulus Ωa,t, with the corresponding eigenvalue λ−(t) and boundary conditions u(a) = 0 = u (t); and the other in the annulus Ωt,b, with the corresponding eigenvalue λ+(t) and boundary conditions u (t) = 0 = u(b). Next, we adjust the parameter t using a matching procedure to make λ−(t) coincide with λ+(t), thereby obtaining the first eigenvalue λp. Hence, by a simple splicing argument, we obtain the positive, L∞-normalized, radial first eigenfunction up. The matching parameter is the maximum point ρ of up. In order to apply this method, we derive estimates for λ−(t) and λ+(t), and we prove that these functions are monotone and (locally Lipschitz) continuous. Moreover, we derive upper and lower estimates for the maximum point ρ, which we use in the matching procedure, and we also present a direct proof that up converges to the L∞-normalized distance function to the boundary as p → ∞. We also present some numerical results obtained using this method.
Computing the first eigenvalue of the p-Laplacian via the inverse power method.
(2009) Biezuner, Rodney Josué; Ercole, Grey; Martins, Eder Marinho
In this paper, we discuss a new method for computing the first Dirichlet eigenvalue of the p-Laplacian inspired by the inverse power method in finite dimensional linear algebra. The iterative technique is independent of the particular method used in solving the p-Laplacian equation and therefore can be made as efficient as the latter. The method is validated theoretically for any ball in Rn if p >1 and for any bounded domain in the particular case p = 2. For p >2 the method is validated numerically for the square.
Computing the sinP Function via the inverse power method.
(2010) Biezuner, Rodney Josué; Ercole, Grey; Martins, Eder Marinho
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.
Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift.
(2012) Biezuner, Rodney Josué; Ercole, Grey; Giacchini, Breno Loureiro; Martins, Eder Marinho
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 X _ RN. 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 2 L2ðXÞ.
Existence and multiplicity of positive solutions for the p-Laplacian with nonlocal coefficient.
(2008) Bueno, H.; Ercole, Grey; Ferreira, Wenderson Marques; Santos, Antônio Zumpano Pereira
We consider the Dirichlet problem with nonlocal coefficient given by −a(Ω|u|q dx)_pu = w(x)f (u) in a bounded, smooth domain Ω ⊂ Rn (n _ 2), where _p is the p-Laplacian, w is a weight function and the nonlinearity f (u) satisfies certain local bounds. In contrast with the hypotheses usually made, no asymptotic behavior is assumed on f . We assume that the nonlocal coefficient a(_Ω|u|q dx) (q _ 1) is defined by a continuous and nondecreasing function a : [0,∞)→[0,∞) satisfying a(t) > 0 for t > 0 and a(0) _ 0. A positive solution is obtained by applying the Schauder Fixed Point Theorem. The case a(t) = tγ/q (0 < γ < p − 1) will be considered as an example where asymptotic conditions on the nonlinearity provide the existence of a sequence of positive solutions for the problem with arbitrarily large sup norm.
Fractional Sobolev inequalities associated with singular problems.
(2018) Ercole, Grey; Pereira, Gilberto de Assis
In this paper we study Sobolev-type inequalities associated with singular problems for the fractional p-Laplacian operator in a bounded domain of RN , N ≥ 2.
On a singular minimizing problem.
(2018) Ercole, Grey; Pereira, Gilberto de Assis
For each q ∈ (0, 1) let λq(Ω) := inf k∇vk p Lp(Ω) : v ∈ W1,p 0 (Ω) and Z Ω |v| q dx = 1, where p > 1 and Ω is a bounded and smooth domain of R N , N ≥ 2. We first show that 0 < μ(Ω) := lim q→0+λq(Ω)|Ω| p q < ∞, where |Ω| = R Ω dx. Then, we prove that μ(Ω) = min (k∇vk p Lp(Ω) : v ∈ W1,p 0 (Ω) and lim q→0+ 1 |Ω| Z Ω |v| q dx 1 q = 1) and that μ(Ω) is reached by a function u ∈ W1,p 0 (Ω), which is positive in Ω, belongs to C 0,α(Ω), for some α ∈ (0, 1), and satisfies − div(|∇u| p−2 ∇u) = μ(Ω)|Ω| −1 u −1 in Ω, and Z Ω log udx = 0. We also show that μ(Ω)−1 is the best constant C in the following log-Sobolev type inequality exp 1 |Ω| Z Ω log |v| p dx ≤ C k∇vk p Lp(Ω) , v ∈ W1,p 0 (Ω) and that this inequality becomes an equality if, and only if, v is a scalar multiple of u and C = μ(Ω)−1.
Torsion functions and the Cheeger problem : a fractional approach.
(2016) Bueno, Hamilton Prado; Ercole, Grey; Macedo, Shirley da Silva; Pereira, Gilberto A.
Let Ω be a Lipschitz bounded domain of ℝN, N ≥ 2. The fractional Cheeger constant hs(Ω), 0 < s < 1, is defined by hs(Ω) = inf E⊂Ω Ps(E) |E| , where Ps(E) = ∫ ℝN ∫ ℝN |χE(x) − χE(y)| |x − y| N+s dx dy, with χE denoting the characteristic function of the smooth subdomain E. The main purpose of this paper is to show that lim p→1 + |ϕ s p | 1−p L∞(Ω) = hs(Ω) = lim p→1 + |ϕ s p | 1−p L 1(Ω) , where ϕ s p is the fractional (s, p)-torsion function of Ω, that is, the solution of the Dirichlet problem for the fractional p-Laplacian: −(∆) s p u = 1 in Ω, u = 0 in ℝN \ Ω. For this, we derive suitable bounds for the first eigenvalue λ s 1,p (Ω) of the fractional p-Laplacian operator in terms of ϕ s p . We also show that ϕ s p minimizes the (s, p)-Gagliardo seminorm in ℝN, among the functions normalized by the L 1 -norm.

Browsing by Author "Ercole, Grey"

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