Repositorio DSpace :: Examinando por Autor "Leonori, Tommaso"

Examinando por Autor "Leonori, Tommaso"

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Asymptotically Linear Problems and Antimaximum Principle for the Square Root of the Laplacian
(2016-03-10) Arcoya, David; Colorado, Eduardo; Leonori, Tommaso; https://orcid.org/0000-0002-7284-2413; https://orcid.org/0000-0002-1067-5752; https://orcid.org/0000-0002-0848-4463
This work deals with bifurcation of positive solutions for some asymptotically linear problems, involving the square root of the Laplacian (-Delta)(1/2). A simplified model problem is the following: {(-Delta)(1/2)u = lambda m(x)u + g(u) in Omega, u = 0 on partial derivative Omega, with Omega subset of R-N a smooth bounded domain, N >= 2, lambda > 0, m is an element of L-infinity(Omega), m(+) not equivalent to 0 and g is a continuous function which is super-linear at 0 and sub-linear at infinity. As a consequence of our bifurcation theory approach we prove some existence and multiplicity results. Finally, we also show an anti-maximum principle in the corresponding functional setting.
Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations
(American Institute of Mathematical Sciences (AIMS), 2015-12) Leonori, Tommaso; Peral, Ireneo; Primo, Ana; Soria, Fernando; https://orcid.org/0000-0002-0848-4463; https://orcid.org/0000-0003-2297-9910; https://orcid.org/0000-0003-1804-3175; https://orcid.org/0000-0001-5753-807X
In this work we consider the problems { script Lu = f in Ω, u = 0 in ℝN\Ω, and { ut + script Lu = f in QT ≡ Ω x (0,T), u(x,t) = 0 in (ℝN\Ω) x (0,T), u(x, 0) = 0 in Ω, where script L is a nonlocal differential operator and Ω is a bounded domain in ℝN, with Lipschitz boundary. The main goal of this work is to study existence, uniqueness and summability of the solution u with respect to the summability of the datum f. In the process we establish an Lp-theory, for p ≥ 1, associated to these problems and we prove some useful inequalities for the applications.
The best approximation of a given function in L2-norm by Lipschitz functions with gradient constraint
(De Gruyter, 2024-04-24) Buccheri, Stefano; Leonori, Tommaso; Rossi, Julio D.; https://orcid.org/0000-0002-0667-233X; https://orcid.org/0000-0002-0848-4463; https://orcid.org/0000-0002-5905-4412
The starting point of this paper is the study of the asymptotic behavior, as p → ∞, of the following minimization problem: min{1 p ∫ Ω |∇v|p + 1 2 ∫ (v − f)2, v ∈ W1,p(Ω)}. Ω We show that the limit problem provides the best approximation, in the L2-norm, of the datum f among all Lipschitz functions with Lipschitz constant less or equal than one. Moreover, such an approximation verifies a suitable PDE in the viscosity sense. After the analysis of the model problem above, we consider the asymptotic behavior of a related family of nonvariational equations and, finally, we also deal with some functionals involving the (N − 1)-Hausdorff measure of the jump set of the function.
Comparison principles for p-Laplace equations with lower order terms
(Springer Nature, 2016-08-06) Leonori, Tommaso; Porretta, Alessio; Riey, Giuseppe
We prove comparison principles for quasilinear elliptic equations whose simplest model is (Formula presented.), where Δpu=div(|Du|p-2Du) is the p-Laplace operator with p> 2 , λ≥ 0 , H(x, ξ) : Ω × RN→ R is a Carathéodory function and Ω ⊂ RN is a bounded domain, N≥ 2. We collect several comparison results for weak sub- and super-solutions under different setting of assumptions and with possibly different methods. A strong comparison result is also proved for more regular solutions.
Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth
(American Institute of Mathematical Sciences (AIMS), 2019-05) Leonori, Tommaso; Magliocca, Martina
In this paper we deal with uniqueness of unbounded solutions to the following problem (formula pergented) where QT = (0, T) × Ω is the parabolic cylinder, Ω is an open subset of RN, N ≥ 2, 1 < p < N, and the right hand side H(t, x, ξ): (0, T) × Ω × RN → R exhibits a superlinear growth with respect to the gradient term. © 2019 American Institute of Mathematical Sciences.
Deterministic KPZ-type equations with nonlocal “gradient terms”
(Springer Nature, 2023-12-03) Abdellaou, Boumediene; Fernández, Antonio J.; Leonori, Tommaso; Younes, Abdelbadie; https://orcid.org/0000-0002-0848-4463; https://orcid.org/0000-0001-6236-2769
The main goal of this paper is to prove existence and non-existence results for deterministic Kardar–Parisi–Zhang type equations involving non-local “gradient terms”. More precisely, let Ω ⊂ RN, N≥ 2 , be a bounded domain with boundary ∂Ω of class C2. For s∈ (0 , 1) , we consider problems of the form (Formula presented.) where q> 1 and λ> 0 are real parameters, f belongs to a suitable Lebesgue space, μ∈ L∞(Ω) and D represents a nonlocal “gradient term”. Depending on the size of λ> 0 , we derive existence and non-existence results. In particular, we solve several open problems posed in [Abdellaoui in Nonlinearity 31(4): 1260-1298 (2018), Section 6] and [Abdellaoui in Proc Roy Soc Edinburgh Sect A 150(5): 2682-2718 (2020), Section 7]. © 2022, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.
Existence of solutions for semilinear nonlocal elliptic problems via a Bolzano theorem
(Springer-Verlag, 2013-10-01) Arcoya, David; Leonori, Tommaso; Primo, Ana; https://orcid.org/0000-0002-7284-2413; https://orcid.org/0000-0002-0848-4463; https://orcid.org/0000-0003-1804-3175
In this paper we deal with the existence of positive solutions for the following nonlocal type of problems {-Δu = α/(σωg(u)dx) p f(u) in Ω u>0 in Ω u=0 on ∂ Ω where Ω is a bounded smooth domain in ℝ N (N≥1), f,g are continuous positive functions, σ>0 and pεℝ. We give sufficient conditions on the functions f and g in order to have existence of positive solutions.
Ground states of self-gravitating elastic bodies
(Springer Nature, 2014-09) Calogero, Simone; Leonori, Tommaso; https://orcid.org/0000-0003-3802-3665; https://orcid.org/0000-0002-0848-4463
The existence of static, self-gravitating elastic bodies in the non-linear theory of elasticity is established. Equilibrium configurations of self-gravitating elastic bodies close to the reference configuration have been constructed in Beig and Schmidt (Proc R Soc Lond, 109–115, 2003) using the implicit function theorem. In contrast, the steady states considered in this article correspond to deformations of the relaxed state with no size restriction and are obtained as minimizers of the energy functional of the elastic body.
Large solutions and gradient bounds for quasilinear elliptic equations
(Taylor and Francis Group, 2015-10-09) Leonori, Tommaso; Porretta, Alessio
We consider the quasilinear degenerate elliptic equation λu - Δpu + H(x,Du)= 0 in Ω where Δp is the p-Laplace operator, p > 2, λ ≥0 and Ω is a smooth open bounded subset of ℝN (N ≥ 2). Under suitable structure conditions on the function H, we prove local and global gradient bounds for the solutions. We apply these estimates to study the solvability of the Dirichlet problem, and the existence, uniqueness and asymptotic behavior of maximal solutions blowing up at the boundary. The ergodic limit for those maximal solutions is also studied and the existence and uniqueness of a so-called additive eigenvalue is proved in this context.
Large solutions to quasilinear problems involving the p-Laplacian as p diverges
(Springer Nature, 2021-01-18) Buccheri, Stefano; Leonori, Tommaso
In this paper we deal with large solutions to {u-Δpu+β|∇u|q=finΩ,u(x)=+∞on∂Ω,where Ω ⊂ RN , with N≥ 1 , is a smooth, open, connected, and bounded domain, p≥ 2 , β> 0 , p- 1 < q≤ p and f∈ C(Ω) ∩ L∞(Ω). We are interested in studying their behavior as p diverges. Our main result states that, if, in some sense, the domain Ω is large enough, such solutions converge locally uniformly to a limit function that turns out to be a large solution of a suitable limit equation (that involves the ∞-Laplacian). Otherwise, if Ω is small, we have a complete blow-up.
Local estimates for parabolic equations with nonlinear gradient terms
(Springer Nature, 2010-12-23) Leonori, Tommaso; Petitta, Francesco; https://orcid.org/0000-0002-0848-4463; https://orcid.org/0000-0001-9265-7912
In this paper we deal with local estimates for parabolic problems in ℝ with absorbing first order terms, whose model is., where T > 0, N ≥ 2, 1 < q ≤2, f (t, x) {e open} L1 (0,T;L1loc (ℝN)) and u0 {e open} L1loc(ℝn).
Nonlinear elliptic equations with Hardy potential and lower order term with natural growth
(Elsevier, 2011-07) Leonori, Tommaso; Martínez-Aparicio, Pedro J.; Primo, Ana
In this work we analyze the interaction between the Hardy potential and a lower order term to obtain the existence or nonexistence of a positive solution in elliptic problems whose model is {-Δpu=g(u)| ∇u| p+λ up-1/|x|p +f,in Ω,u>0,in Ω,u=0,on ∂Ω, where ΩℝN, N≥3, is a bounded domain containing the origin, 10, the behavior of the positive continuous function g at infinity provides the existence of a solution for such a problem.
On the comparison principle for unbounded solutions of elliptic equations with first order terms
(Elsevier, 2018-01-15) Leonori, Tommaso; Porretta, Alessio
We prove a comparison principle for unbounded weak sub/super solutions of the equation λu−div(A(x)Du)=H(x,Du) in Ω where A(x) is a bounded coercive matrix with measurable ingredients, λ≥0 and ξ↦H(x,ξ) has a super linear growth and is convex at infinity. We improve earlier results where the convexity of H(x,⋅) was required to hold globally.
Parabolic equations with natural growth approximated by nonlocal equations
(World Scientific, 2021) Leonori, Tommaso; Molino, Alexis; Segura De León, Sergio; https://orcid.org/0000-0002-0848-4463; https://orcid.org/0000-0003-2819-7282; https://orcid.org/0000-0002-8515-7108
In this paper, we study several aspects related with solutions of nonlocal problems whose prototype is {u(t) = integral N-R J (x - y)(u(y, t) u( x, t))g (u(y , t) u( x, t))dy in Omega x (0, T), u(x, 0) = u(0)(x) in Omega, where we take, as the most important instance, g(s) similar to 1 + mu/2 s/1+mu(2)s(2) with mu is an element of R as well as mu(0)is an element of L-1 (Omega), J is a smooth symmetric function with compact support and S2 is either a bounded smooth subset of R-N, with nonlocal Dirichlet boundary condition, or RN itself. The results deal with existence, uniqueness, comparison principle and asymptotic behavior. Moreover, we prove that if the kernel is resealed in a suitable way, the unique solution of the above problem converges to a solution of the deterministic Kardar Parisi Zhang equation.
Principal eigenvalue of mixed problem for the fractional Laplacian: Moving the boundary conditions
(Elsevier, 2018-07-15) Leonori, Tommaso; Medina, Maria; Peral, Ireneo; Primo, Ana; Soria, Fernando
We analyze the behavior of the eigenvalues of the following nonlocal mixed problem {(−Δ)su=λ1(D)u in Ω,u=0 in D,Nsu=0 in N. Our goal is to construct different sequences of problems by modifying the configuration of the sets D and N, and to provide sufficient and necessary conditions on the size and the location of these sets in order to obtain sequences of eigenvalues that in the limit recover the eigenvalues of the Dirichlet or Neumann problem. We will see that the nonlocality plays a crucial role here, since the sets D and N can have infinite measure, a phenomenon that does not appear in the local case (see for example [6–8]).
Quasilinear elliptic equations with singular quadratic growth terms
(World Scientific Publishing, 2011-08) Boccardo, Lucio; Leonori, Tommaso; Orsina, Luigi; Petitta, Francesco; https://orcid.org/0000-0001-9265-7912; https://orcid.org/0000-0002-0848-4463; https://orcid.org/0000-0003-3676-5526; https://orcid.org/0000-0002-8067-0121
In this paper, we deal with positive solutions for singular quasilinear problems whose model is {u=0-Δu + |∇u| 2/(1-u)γ = g in Ω on ∂Ω, where Ω is a bounded open set of ℝN, g ≥ 0 is a function in some Lebesgue space, and γ > 0. We prove both existence and nonexistence of solutions depending on the value of γ and on the size of g.
Regularity of solutions to a fractional elliptic problem with mixed Dirichlet-Neumann boundary data
(De Gruyter, 2021-10-01) Carmona, Jose; Colorado, Eduardo; Leonori, Tommaso; Ortega, Alejandro; 35996792500; 6507374993; 14521092300; 56489711800
In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the spectral fractional Laplacian.
Semilinear fractional elliptic problems with mixed Dirichlet-Neumann boundary conditions
(Elsevier, 2020-08-01) Carmona, José; Colorado, Eduardo; Leonori, Tommaso; Ortega, Alejandro; 6507374993; 14521092300; 56489711800
We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator and a concave-convex term, together with mixed Dirichlet-Neumann boundary conditions.
Strong convergence of the gradients for p-Laplacian problems as p → ∞
(Elsevier, 2021-03-01) Buccheri, Stefano; Leonori, Tommaso; Rossi, Julio D.
In this paper we prove that the gradients of solutions to the Dirichlet problem for −∆pup = f , with f > 0, converge as p → ∞ strongly in every Lq with 1 ≤ q < ∞ to the gradient of the limit function. This convergence is sharp since a simple example in 1-d shows that there is no convergence in L∞. For a nonnegative f we obtain the same strong convergence inside the support of f . The same kind of result also holds true for the eigenvalue problem for a suitable class of domains (as balls or stadiums).
A uniqueness result for a singular elliptic equation with gradient term
(Cambridge University Press, 2018-06-22) Carmona, José; Leonori, Tommaso; 35996792500; 14521092300
We prove the uniqueness of a solution for a problem whose simplest model is with k ≥ 1, 0 Lz(Ω) and Ω is a bounded domain of N, N ≥ 2. So far, uniqueness results are known for k < 1, while existence holds for any k ≥ 1 and f positive in open sets compactly embedded in a neighbourhood of the boundary. We extend the uniqueness results to the k ≥ 1 case and show, with an example, that existence does not hold if f is zero near the boundary. We even deal with the uniqueness result when f is replaced by a nonlinear term λuq with 0 < q < 1 and λ > 0.

Examinando por Autor "Leonori, Tommaso"

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