arXiv subjects

Paolo Tilli

Publications and source records attributed to Paolo Tilli.

28 records · Page 2Linked to original sources

arXiv2018

Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length

We consider the problem of maximizing the first eigenvalue of the $p$-laplacian (possibly with non-constant coefficients) over a fixed domain $Ω$, with Dirichlet conditions along $\partialΩ$ and along a supplementary set $Σ$, which is the unknown of the optimization problem. The set $Σ$, that plays the role of a supplementary stiffening rib for a membrane $Ω$, is a compact connected set (e.g. a curve or a connected system of curves) that can be placed anywhere in $\overlineΩ$, and is subject to the constraint of an upper bound $L$ to its total length (one-dimensional Hausdorff measure). This upper bound prevents $Σ$ from spreading throughout $Ω$ and makes the problem well-posed. We investigate the behavior of optimal sets $Σ_L$ as $L\to\infty$ via $Γ$-convergence, and we explicitly construct certain asymptotically optimal configurations. We also study the behavior as $p\to\infty$ with $L$ fixed, finding connections with maximum-distance problems related to the principal frequency of the $\infty$-laplacian.

math.AP↗

arXiv2017

De Giorgi's approach to hyperbolic Cauchy problems: the case of nonhomogeneous equations

In this paper we discuss an extension of some results obtained by E. Serra and P. Tilli, in [Serra&Tilli '12, Serra&Tilli '16], concerning an original conjecture by E. De Giorgi ([De Giorgi '96, De Giorgi '06]) on a purely minimization approach to the Cauchy problem for the defocusing nonlinear wave equation. Precisely, we show how to extend the techniques developed by Serra and Tilli for homogeneous hyperbolic nonlinear PDEs to the nonhomogeneous case, thus proving that the idea of De Giorgi yields in fact an effective approach to investigate general hyperbolic equations.

math.AP↗

arXiv2017

Multiple positive bound states for the subcritical NLS equation on metric graphs

We consider the Schroedinger equation with a subcritical focusing power nonlinearity on a noncompact metric graph, and prove that for every finite edge there exists a threshold value of the mass, beyond which there exists a positive bound state achieving its maximum on that edge only. This bound state is characterized as a minimizer of the energy functional associated to the NLS equation, with an additional constraint (besides the mass prescription): this requires particular care in proving that the minimizer satisfies the Euler--Lagrange equation. As a consequence, for a sufficiently large mass every finite edge of the graph hosts at least one positive bound state that, owing to its minimality property, is orbitally stable.

math.AP↗

arXiv2017

Nonlinear dynamics on branched structures and networks

Nonlinear dynamics on graphs has rapidly become a topical issue with many physical applications, ranging from nonlinear optics to Bose-Einstein condensation. Whenever in a physical experiment a ramified structure is involved, it can prove useful to approximate such a structure by a metric graph, or network. For the Schroedinger equation it turns out that the sixth power in the nonlinear term of the energy is critical in the sense that below that power the constrained energy is lower bounded irrespectively of the value of the mass (subcritical case). On the other hand, if the nonlinearity power equals six, then the lower boundedness depends on the value of the mass: below a critical mass, the constrained energy is lower bounded, beyond it, it is not. For powers larger than six the constrained energy functional is never lower bounded, so that it is meaningless to speak about ground states (supercritical case). These results are the same as in the case of the nonlinear Schrodinger equation on the real line. In fact, as regards the existence of ground states, the results for systems on graphs differ, in general, from the ones for systems on the line even in the subcritical case: in the latter case, whenever the constrained energy is lower bounded there always exist ground states (the solitons, whose shape is explicitly known), whereas for graphs the existence of a ground state is not guaranteed. For the critical case, our results show a phenomenology much richer than the analogous on the line.

math.AP↗

arXiv2016

Negative energy ground states for the $L^2$-critical NLSE on metric graphs

We investigate the existence of ground states with prescribed mass for the focusing nonlinear Schrödinger equation with $L^2$-critical power nonlinearity on noncompact quantum graphs. We prove that, unlike the case of the real line, for certain classes of graphs there exist ground states with negative energy for a whole interval of masses. A key role is played by a thorough analysis of Gagliardo-Nirenberg inequalities and on estimates of the optimal constants. Most of the techniques are new and suited to the investigation of variational problems on metric graphs.

math-ph↗

arXiv2015

Where best to place a Dirichlet condition in an anisotropic membrane?

We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with non constant coefficients, over a fixed domain $Ω$. Dirichlet conditions are imposed along $\partial Ω$ and, in addition, along a set $Σ$ of prescribed length ($1$-dimensional Hausdorff measure). We look for the best shape and position for the supplementary Dirichlet region $Σ$ in order to maximize the first eigenvalue. The limit distribution of the optimal sets, as their prescribed length tends to infinity, is characterized via $Γ$-convergence of suitable functionals defined over varifolds: the use of varifolds, as opposed to probability measures, allows one to keep track of the local orientation of the optimal sets (which comply with the anisotropy of the problem), and not just of their limit distribution.

math.OC↗

arXiv2015

Threshold phenomena and existence results for NLS ground states on graphs

We investigate the existence of ground states of prescribed mass, for the nonlinear Schroedinger energy on a noncompact metric graph G. While in some cases the topology of G may rule out or, on the contrary, guarantee the existence of ground states of any given mass, in general also some metric properties of G, and their quantitative relation with the actual value of the prescribed mass, are relevant as concerns the existence of ground states. This may give rise to interesting phase transitions from nonexistence to existence of ground states, when a certain quantity reaches a critical threshold.

math.AP↗

arXiv2014

NLS ground states on graphs

We investigate the existence of ground states for the subcritical NLS energy on metric graphs. In particular, we find out a topological assumption that guarantees the nonexistence of ground states, and give an example in which the assumption is not fulfilled and ground states actually exist. In order to obtain the result, we introduce a new rearrangement technique, adapted to the graph where it applies. Owing to such a technique, the energy level of the rearranged function is improved by conveniently mixing the symmetric and monotone rearrangement procedures.

math.AP↗

arXiv2014

Lack of ground state for NLS on bridge-type graphs

We prove the nonexistence of ground states for NLS on bridge-like graphs, i.e. graphs with two halflines and four vertices, of which two at infinity, with Kirchhoff matching conditions. By ground state we mean any minimizer of the energy functional among all functions with the same mass.

math.AP↗

arXiv2013

A minimization approach to hyperbolic Cauchy problems

Developing an original idea of De Giorgi, we introduce a new and purely variational approach to the Cauchy Problem for a wide class of defocusing hyperbolic equations. The main novel feature is that the solutions are obtained as limits of functions that minimize suitable functionals in space-time (where the initial data of the Cauchy Problem serve as prescribed boundary conditions). This opens up the way to new connections between the hyperbolic world and that of the Calculus of Variations. Also dissipative equations can be treated. Finally, we discuss several examples of equations that fit in this framework, including nonlocal equations, in particular equations with the fractional Laplacian.

math.AP↗