arXiv subjects

Hugo Tavares

Publications and source records attributed to Hugo Tavares.

At least 37 records · Page 2Linked to original sources

arXiv2021

On the least-energy solutions of the pure Neumann Lane-Emden equation

We study the pure Neumann Lane-Emden problem in a bounded domain \[ -Δu = |u|^{p-1} u \text{ in }Ω, \qquad \partial_νu=0 \text{ on }\partial Ω, \] in the subcritical, critical, and supercritical regimes. We show existence and convergence of least-energy (nodal) solutions (l.e.n.s.). In particular, we prove that l.e.n.s. converge to a l.e.n.s. of a problem with sign nonlinearity as $p\searrow 0$; to a l.e.n.s. of the critical problem as $p\nearrow 2^*$ (in particular, pure Neumann problems exhibit no blowup phenomena at the critical Sobolev exponent $2^*$); and we show that the limit as $p\to 1$ depends on the domain. Our proofs rely on different variational characterizations of solutions including a dual approach and a nonlinear eigenvalue problem. Finally, we also provide a qualitative analysis of l.e.n.s., including symmetry, symmetry-breaking, and monotonicity results for radial solutions.

math.AP↗

arXiv2020

Nodal Solutions for sublinear-type problems with Dirichlet boundary conditions

We consider nonlinear second order elliptic problems of the type \[ -Δu=f(u) \text{ in } Ω, \qquad u=0 \text{ on } \partial Ω, \] where $Ω$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=|s|^{p-1}s$ with $0 1$ and $λ>λ_2(Ω)$ (the second Dirichlet eigenvalue of the Laplacian). We prove the existence of a least energy nodal (i.e. sign changing) solution, and of a nodal solution of mountain-pass type. We then give explicit examples of domains where the associated levels do not coincide. For the case where $Ω$ is a ball or annulus and $f$ is of class $C^1$, we prove instead that the levels coincide, and that least energy nodal solutions are nonradial but axially symmetric functions. Finally, we provide stronger results for the Allen-Cahn type nonlinearities in case $Ω$ is either a ball or a square. In particular we give a complete description of the solution set for $λ\sim λ_2(Ω)$, computing the Morse index of the solutions.

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arXiv2020

Regularity of all minimizers of a class of spectral partition problems

We study a rather broad class of optimal partition problems with respect to monotone and coercive functional costs that involve the Dirichlet eigenvalues of the partitions. We show a sharp regularity result for the entire set of minimizers for a natural relaxed version of the original problem, together with the regularity of eigenfunctions and a universal free boundary condition. Among others, our result covers the cases of the following functional costs \[ (ω_1, \dots, ω_m) \mapsto \sum_{i=1}^{m} \left( \sum_{j=1}^{k_i} λ_{j}(ω_i)^{p_i}\right)^{1/p_i}, \quad \prod_{i=1}^{m} \left( \prod_{j=1}^{k_i} λ_{j}(ω_i)\right), \quad \prod_{i=1}^{m} \left( \sum_{j=1}^{k_i} λ_{j}(ω_i)\right) \] where $(ω_1, \dots, ω_m)$ are the sets of the partition and $λ_{j}(ω_i)$ is the $j$-th Laplace eigenvalue of the set $ω_i$ with zero Dirichlet boundary conditions.

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arXiv2019

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

In this paper we investigate the existence of solutions to the following Schrödinger system in the critical case \begin{equation*} -Δu_{i}+λ_{i}u_{i}=u_{i}\sum_{j = 1}^{d}β_{ij}u_{j}^{2} \text{ in } Ω, \quad u_i=0 \text{ on } \partial Ω, \qquad i=1,...,d. \end{equation*} Here, $Ω\subset \mathbb{R}^{4}$ is a smooth bounded domain, $d\geq 2$, $-λ_{1}(Ω)<λ_{i}<0$ and $β_{ii}>0$ for every $i$, $β_{ij}=β_{ji}$ for $i\neq j$, where $λ_{1}(Ω)$ is the first eigenvalue of $-Δ$ with Dirichlet boundary conditions. Under the assumption that the components are divided into $m$ groups, and that $β_{ij}\geq 0$ (cooperation) whenever components $i$ and $j$ belong to the same group, while $β_{ij}<0$ or $β_{ij}$ is positive and small (competition or weak cooperation) for components $i$ and $j$ belonging to different groups, we establish the existence of nonnegative solutions with $m$ nontrivial components, as well as classification results. Moreover, under additional assumptions on $β_{ij}$, we establish existence of least energy positive solutions in the case of mixed cooperation and competition. The proof is done by induction on the number of groups, and requires new estimates comparing energy levels of the system with those of appropriate sub-systems. In the case $Ω=\mathbb{R}^4$ and $λ_1=\ldots=λ_m=0$, we present new nonexistence results. This paper can be seen as the counterpart of [Soave-Tavares, J. Differential Equations 261 (2016), 505-537] in the critical case, while extending and improving some results from [Chen-Zou, Arch. Ration. Mech. Anal. 205 (2012), 515--551], [Guo-Luo-Zou, Nonlinearity 31 (2018), 314--339].

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arXiv2018

A fountain of positive Bubbles on a Coron's Problem for a Competitive Weakly Coupled Gradient System

We consider the following critical elliptic system: \begin{equation*} \begin{cases} -Δu_i=μ_i u_i^{3}+βu_i^{ } \sum\limits_{j\neq i} u_j^{2} \quad \hbox{in}\ Ω_\varepsilon \\ u_i=0 \hbox{ on } \partialΩ_\varepsilon , \qquad u_i>0 \hbox{ in } Ω_\varepsilon \end{cases}\qquad i=1,\ldots, m, \end{equation*} in a domain $Ω_\varepsilon \subset \mathbb{R}^4$ with a small shrinking hole $B_\varepsilon(ξ_0)$. For $μ_i>0$, $β<0$, and $\varepsilon>0$ small, we prove the existence of a non-synchronized solution which looks like a fountain of positive bubbles, i.e. each component $u_i$ exhibits a towering blow-up around $ξ_0$ as $\varepsilon \to 0$. The proof is based on the Ljapunov-Schmidt reduction method, and the velocity of concentration of each layer within a given tower is chosen in such a way that the interaction between bubbles of different components balance the interaction of the first bubble of each component with the boundary of the domain, and in addition is dominant when compared with the interaction of two consecutive bubbles of the same component.

math.AP↗

arXiv2018

Bose fluids and positive solutions to weakly coupled systems with critical growth in dimension two

We prove, using variational methods, the existence in dimension two of positive vector ground states solutions for the Bose-Einstein type systems \begin{equation} \begin{cases} -Δu+λ_1u=μ_1u(e^{u^2}-1)+βv\left(e^{uv}-1\right) \text{ in } Ω, &\\ -Δv+λ_2v=μ_2v(e^{v^2}-1)+βu\left(e^{uv}-1\right)\text{ in } Ω, &\\ u,v\in H^1_0(Ω) \end{cases} \end{equation} where $Ω$ is a bounded smooth domain, $λ_1,λ_2>-Λ_1$ (the first eigenvalue of $(-Δ,H^1_0(Ω))$, $μ_1,μ_2>0$ and $β$ is either positive (small or large) or negative (small). The nonlinear interaction between two Bose fluids is assumed to be of critical exponential type in the sense of J. Moser. For `small' solutions the system is asymptotically equivalent to the corresponding one in higher dimensions with power-like nonlinearities.

math.AP↗

arXiv2018

Normalized solutions for Nonlinear Schrödinger systems on bounded domains

We analyze $L^2$-normalized solutions of nonlinear Schrödinger systems of Gross-Pitaevskii type, on bounded domains, with homogeneous Dirichlet boundary conditions. We provide sufficient conditions for the existence of orbitally stable standing waves. Such waves correspond to global minimizers of the associated energy in the $L^2$-subcritical and critical cases, and to local ones in the $L^2$-supercritical case. Notably, our study includes also the Sobolev-critical case.

math.AP↗

arXiv2018

Sharp concentration estimates near criticality for radial sign-changing solutions of Dirichlet and Neumann problems

We consider radial solutions of the slightly subcritical problem $-Δu_\varepsilon = |u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon$ either on $\mathbb R^n$ ($n\geq 3$) or in a ball $B$ satisfying Dirichlet or Neumann boundary conditions. In particular, we provide sharp rates and constants describing the asymptotic behavior (as $\varepsilon\to 0$) of all local minima and maxima of $u_\varepsilon$ as well as its derivative at roots. Our proof is done by induction and uses energy estimates, blow-up/normalization techniques, a radial pointwise Pohozaev identity, and some ODE arguments. As corollaries, we complement a known asymptotic approximation of the Dirichlet nodal solution in terms of a tower of bubbles and present a similar formula for the Neumann problem.

math.AP↗

arXiv2018

On a coupled system of a Ginzburg-Landau equation with a quasilinear conservation law

We study the Cauchy problem for a coupled system of a complex Ginzburg-Landau equation with a quasilinear conservation law $$ \left\{\begin{array}{rlll} e^{-iθ}u_t&=&u_{xx}-|u|^2u-αg(v)u& v_t+(f(v))_x&=&α(g'(v)|u|^2)_x& \end{array}\right. \qquad x\in\mathbb{R},\, t \geq 0, $$ which can describe the interaction between a laser beam and a fluid flow (see [Aranson, Kramer, Rev. Med. Phys. 74 (2002)]). We prove the existence of a local in time strong solution for the associated Cauchy problem and, for a certain class of flux functions, the existence of global weak solutions. Furthermore we prove the existence of standing waves of the form $(u(t,x),v(t,x))=(U(x),V(x))$ in several cases.

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arXiv2018

Least energy nodal solutions of Hamiltonian elliptic systems with Neumann boundary conditions

We study existence, regularity, and qualitative properties of solutions to the system \[ -Δu = |v|^{q-1} v\quad \text{ in }Ω,\qquad -Δv = |u|^{p-1} u\quad \text{ in }Ω,\qquad \partial_νu=\partial_νv=0\quad \text{ on }\partialΩ, \] with $Ω\subset \mathbb R^N$ bounded; in this setting, all nontrivial solutions are sign changing. Our proofs use a variational formulation in dual spaces, considering sublinear $pq< 1$ and superlinear $pq>1$ problems in the subcritical regime. In balls and annuli we show that least energy solutions (l.e.s.) are foliated Schwarz symmetric and, due to a symmetry-breaking phenomenon, l.e.s. are not radial functions; a key element in the proof is a new $L^t$-norm-preserving transformation, which combines a suitable flipping with a decreasing rearrangement. This combination allows us to treat annular domains, sign-changing functions, and Neumann problems, which are non-standard settings to use rearrangements and symmetrizations. In particular, we show that our transformation diminishes the (dual) energy and, as a consequence, radial l.e.s. are strictly monotone. We also study unique continuation properties and simplicity of zeros. Our theorems also apply to the scalar associated model, where our approach provides new results as well as alternative proofs of known facts.

math.AP↗

arXiv2017

Variational problems with long-range interaction

We consider a class of variational problems for densities that repel each other at distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional \[ D(\mathbf{u}) = \sum_{i=1}^k \int_Ω |\nabla u_i|^2 \quad \text{or} \quad R(\mathbf{u}) = \sum_{i=1}^k \frac{\int_Ω |\nabla u_i|^2}{\int_Ω u_i^2} \] minimized in the class of $H^1(Ω,\mathbb{R}^k)$ functions attaining some boundary conditions on $\partial Ω$, and subjected to the constraint \[ \mathrm{dist} (\{u_i > 0\}, \{u_j > 0\}) \ge 1 \qquad \forall i \neq j. \] For these problems, we investigate the optimal regularity of the solutions, prove a free-boundary condition, and derive some preliminary results characterizing the free boundary $\partial \{\sum_{i=1}^k u_i > 0\}$.

math.AP↗

arXiv2016

Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping

We study regularity issues for systems of elliptic equations of the type \[ -Δu_i=f_{i,β}(x)-β\sum_{j\neq i} a_{ij} u_i |u_i|^{p-1}|u_j|^{p+1} \] set in domains $Ω\subset \mathbb{R}^N$, for $N \geq 1$. The paper is devoted to the derivation of $\mathcal{C}^{0,α}$ estimates that are uniform in the competition parameter $β> 0$, as well as to the regularity of the limiting free-boundary problem obtained for $β\to + \infty$. The main novelty of the problem under consideration resides in the non-trivial grouping of the densities: in particular, we assume that the interaction parameters $a_{ij}$ are only non-negative, and thus may vanish for specific couples $(i,j)$. As a main consequence, in the limit $β\to +\infty$, densities do not segregate pairwise in general, but are grouped in classes which, in turn, form a mutually disjoint partition. Moreover, with respect to the literature, we consider more general forcing terms, sign-changing solutions, and an arbitrary $p>0$. In addition, we present a regularity theory of the emerging free-boundary, defined by the interface among different segregated groups. These equations are very common in the study of Bose-Einstein condensates and are of key importance for the analysis of optimal partition problems related to high order eigenvalues.

math.AP↗

arXiv2016

Paths to uniqueness of critical points and applications to partial differential equations

We prove a unified and general criterion for the uniqueness of critical points of a functional in the presence of constraints such as positivity, boundedness, or fixed mass. Our method relies on convexity properties along suitable paths and significantly generalizes well-known uniqueness theorems. Due to the flexibility in the construction of the paths, our approach does not depend on the convexity of the domain and can be used to prove uniqueness in subsets, even if it does not hold globally. The results apply to all critical points and not only to minimizers, thus they provide uniqueness of solutions to the corresponding Euler-Lagrange equations. For functionals emerging from elliptic problems, the assumptions of our abstract theorems follow from maximum principles, decay properties, and novel general inequalities. To illustrate our method we present a unified proof of known results, as well as new theorems for mean-curvature type operators, fractional Laplacians, Hamiltonian systems, Schrödinger equations, and Gross-Pitaevski systems.

math.AP↗

arXiv2016

Spiked solutions for Schrödinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions

In this paper we deal with the nonlinear Schrödinger system \[ -Δu_i =μ_i u_i^3 + βu_i \sum_{j\neq i} u_j^2 + λ_i u_i, \qquad u_1,\ldots, u_m\in H^1_0(Ω) \] in dimension 4, a problem with critical Sobolev exponent. In the competitive case ($β<0$ fixed or $β\to -\infty$) or in the weakly cooperative case ($β\geq 0$ small), we construct, under suitable assumptions on the Robin function associated to the domain $Ω$, families of positive solutions which blowup and concentrate at different points as $λ_1,\ldots, λ_m\to 0$. This problem can be seen as a generalization for systems of a Brezis-Nirenberg type problem.

math.AP↗

arXiv2016

Dividing the circle

There are known constructions for some regular polygons, usually inscribed in a circle, but not for all polygons - the Gauss-Wantzel Theorem states precisely which ones can be constructed. The constructions differ greatly from one polygon to the other. There are, however, general processes for determining the side of the $n$-gon (approximately, but sometimes with great precision), which we describe in this paper. We present a joint mathematical analysis of the so-called Bion and Tempier approximation methods, comparing the errors and trying to explain why these constructions would work at all.

math.HO↗

arXiv2015

Extremality conditions and regularity of solutions to optimal partition problems involving Laplacian eigenvalues

Let $Ω\subset \mathbb{R}^N$ be an open bounded domain and $m\in \mathbb{N}$. Given $k_1,\ldots,k_m\in \mathbb{N}$, we consider a wide class of optimal partition problems involving Dirichlet eigenvalues of elliptic operators, of the following form \[ \inf\left\{F(λ_{k_1}(ω_1),\ldots, λ_{k_m}(ω_m)):\ (ω_1,\ldots, ω_m)\in \mathcal{P}_m(Ω)\right\}, \] where $λ_{k_i}(ω_i)$ denotes the $k_i$--th eigenvalue of $(-Δ,H^1_0(ω_i))$ counting multiplicities, and $\mathcal{P}_m(Ω)$ is the set of all open partitions of $Ω$, namely \[ \mathcal{P}_m(Ω)=\left\{(ω_1,\ldots,ω_m):\ ω_i\subset Ω\text{ open},\ ω_i\cap ω_j=\emptyset\ \forall i\neq j\right\}. \] While existence of a quasi-open optimal partition $(ω_1,\ldots, ω_m)$ follows from a general result by Bucur, Buttazzo and Henrot [Adv. Math. Sci. Appl. 8, 1998], the aim of this paper is to associate with such minimal partitions and their eigenfunctions some suitable extremality conditions and to exploit them, proving as well the Lipschitz continuity of some eigenfunctions, and regularity of the partition in the sense that the free boundary $\cup_{i=1}^m \partial ω_i\cap Ω$ is, up to a residual set, locally a $C^{1,α}$ hypersurface. This last result extend the ones in the paper by Caffarelli and Lin [J. Sci. Comput. 31, 2007] to the case of higher eigenvalues.

math.AP↗

arXiv2015

Semitrivial vs. fully nontrivial ground states in cooperative cubic Schrödinger systems with $d\ge3$ equations

In this work we consider the weakly coupled Schrödinger cubic system \[ \begin{cases} \displaystyle -Δu_i+λ_i u_i= μ_i u_i^{3}+ u_i\sum_{j\neq i}b_{ij} u_j^2 \\ u_i\in H^1(\mathbb{R}^N;\mathbb{R}), \quad i=1,\ldots, d, \end{cases} \] where $1\leq N\leq 3$, $λ_i,μ_i >0$ and $b_{ij}=b_{ji}>0$ for $i\neq j$. This system admits semitrivial solutions, that is solutions $\mathbf{u}=(u_1,\ldots, u_d)$ with null components. We provide optimal qualitative conditions on the parameters $λ_i,μ_i$ and $b_{ij}$ under which the ground state solutions have all components nontrivial, or, conversely, are semitrivial. This question had been clarified only in the $d=2$ equations case. For $d\geq 3$ equations, prior to the present paper, only very restrictive results were known, namely when the above system was a small perturbation of the super-symmetrical case $λ_i\equiv λ$ and $b_{ij}\equiv b$. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the $d=2$ case.

math.AP↗

arXiv2015

Ground States for a nonlinear Schrödinger system with sublinear coupling terms

We study the existence of ground states for the coupled Schrödinger system \begin{equation} \left\{\begin{array}{lll} \displaystyle -Δu_i+λ_i u_i= μ_i |u_i|^{2q-2}u_i+\sum_{j\neq i}b_{ij} |u_j|^q|u_i|^{q-2}u_i \\ u_i\in H^1(\mathbb{R}^n), \quad i=1,\ldots, d, \end{array}\right. \end{equation} $n\geq 1$, for $λ_i,μ_i >0$, $b_{ij}=b_{ji}>0$ (the so-called "symmetric attractive case") and $1<q<n/(n-2)^+$. We prove the existence of a nonnegative ground state $(u_1^*,\ldots,u_d^*)$ with $u_i^*$ radially decreasing. Moreover we show that, for $1<q<2$, such ground states are positive in all dimensions and for all values of the parameters.

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