arXiv subjects

Susanna Terracini

Publications and source records attributed to Susanna Terracini.

At least 55 records · Page 3Linked to original sources

arXiv2018

The nodal set of solutions to some elliptic problems: sublinear equations, and unstable two-phase membrane problem

We are concerned with the nodal set of solutions to equations of the form \begin{equation*} -Δu = λ_+ \left(u^+\right)^{q-1} - λ_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $λ_+,λ_- > 0$, $q \in [1,2)$, $B_1=B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N \ge 2$, and $u^+:= \max\{u,0\}$, $u^-:= \max\{-u,0\}$ are the positive and the negative part of $u$, respectively. This class includes, the \emph{unstable two-phase membrane problem} ($q=1$), as well as \emph{sublinear} equations for $1<q<2$. We prove the following main results: (a) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most $N-2$ (locally finite when $N=2$); (d) a partial stratification theorem. Ultimately, the main features of the nodal set are strictly related with those of the solutions to linear (or superlinear) equations, with two remarkable differences. First of all, the admissible vanishing orders can not exceed the critical value $2/(2-q)$. At threshold, we find a multiplicity of homogeneous solutions, yielding the \emph{non-validity} of any estimate of the $(N-1)$-dimensional measure of the nodal set of a solution in terms of the vanishing order. The proofs are based on monotonicity formulæ\ for a $2$-parameter family of Weiss-type functionals, blow-up arguments, and the classification of homogenous solutions.

math.AP↗

arXiv2017

On the Asymptotic Growth of Positive Solutions to a Nonlocal Elliptic Blow-up System Involving Strong Competition

For a competition-diffusion blow-up system involving the fractional Laplacian of the form \begin{equation*}\label{syst1} -(-Δ)^su=uv^2,\quad-(-Δ)^sv=vu^2,\quad u,v>0 \ \mathrm{in} \ \mathbb{R}^N, \end{equation*} whith $s\in(0,1)$, we prove that the maximal asymptotic growth rate for its entire solutions is $2s$. Moreover, since we are able to construct symmetric solutions to the problem, when $N=2$ with prescribed growth arbitrarily close to the critical one, we can conclude that the asymptotic bound found is optimal. Finally, we prove existence of genuinely higher dimensional solutions, when $N\geq 3$.

math.AP↗

arXiv2017

Regularity of the optimal sets for some spectral functionals

In this paper we study the regularity of the optimal sets for the shape optimization problem \[ \min\Big\{λ_1(Ω)+\dots+λ_k(Ω)\ :\ Ω\subset\mathbb{R}^d,\ \text{open}\ ,\ |Ω|=1\Big\}, \] where $λ_1(\cdot),\dots,λ_k(\cdot)$ denote the eigenvalues of the Dirichlet Laplacian and $|\cdot|$ the $d$-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer $Ω_k^*$ is composed of a relatively open regular part which is locally a graph of a $C^{1,α}$ function and a closed singular part, which is empty if $d d^*$, where the natural number $d^*\in[5,7]$ is the smallest dimension at which minimizing one-phase free boundaries admit singularities. To achieve our goal, as an auxiliary result, we shall extend for the first time the known regularity theory for the one-phase free boundary problem to the vector-valued case.

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

Scattering parabolic solutions for the spatial N-centre problem

For the $N$-centre problem in the three dimensional space, $$ \ddot x = -\sum_{i=1}^{N} \frac{m_i \,(x-c_i)}{\vert x - c_i \vert^{α+2}}, \qquad x \in \mathbb{R}^3 \setminus \{c_1,\ldots,c_N\}, $$ where $N \geq 2$, $m_i > 0$ and $α\in [1,2)$, we prove the existence of entire parabolic trajectories having prescribed asymptotic directions. The proof relies on a variational argument of min-max type. Morse index estimates and regularization techniques are used in order to rule out the possible occurrence of collisions.

math.CA↗

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

Multi-layer radial solutions for a supercritical neumann problem

In this paper we study the Neumann problem\begin{equation*}\begin{cases}-Δu+u=u^p \& \text{ in }B\_1 \\u \textgreater{} 0, \& \text{ in }B\_1 \\\partial\_νu=0 \& \text{ on } \partial B\_1,\end{cases}\end{equation*}and we show the existence of multiple-layer radial solutions as $p\rightarrow+\infty$.

math.AP↗

arXiv2015

Liouville theorems and $1$-dimensional symmetry for solutions of an elliptic system modelling phase separation

We consider solutions of the competitive elliptic system \[ \left\{ \begin{array}{ll} -Δu_i = - \sum_{j \neq i} u_i u_j^2 & \text{in $\mathbb{R}^N$} \\ u_i >0 & \text{in $\mathbb{R}^N$} \end{array}\right. \qquad i=1,\dots,k. \] We are concerned with the classification of entire solutions, according with their growth rate. The prototype of our main results is the following: there exists a function $δ=δ(k,N) \in \mathbb{N}$, increasing in $k$, such that if $(u_1,\dots,u_k)$ is a solution and \[ u_1(x)+\cdots+u_k(x) \le C(1+|x|^d) \qquad \text{for every $x \in \mathbb{R}^N$}, \] then $d \ge δ$. This means that the number of components $k$ of the solution imposes an increasing in $k$ minimal growth on the solution itself. If $N=2$, the expression of $δ$ is explicit and optimal, while in higher dimension it can be characterized in terms of an optimal partition problem. We discuss the sharpness of our results and, as a further step, for every $N \ge 2$ we can prove the $1$-dimensional symmetry of the solutions satisfying suitable assumptions, extending known results which are available for $k=2$. The proofs rest upon a blow-down analysis and on some monotonicity formulae.

math.AP↗

arXiv2014

On the eigenvalues of Aharonov-Bohm operators with varying poles: pole approaching the boundary of the domain

We continue the analysis started in [Noris,Terracini,Indiana Univ Math J,2010] and [Bonnaillie-Noël,Noris,Nys,Terracini,Analysis & PDE,2014], concerning the behavior of the eigenvalues of a magnetic Schrödinger operator of Aharonov-Bohm type with half-integer circulation. We consider a planar domain with Dirichlet boundary conditions and we concentrate on the case when the singular pole approaches the boundary of the domain. The $k$-th magnetic eigenvalue converges to the $k$-th eigenvalue of the Laplacian. We can predict both the rate of convergence and whether the convergence happens from above or from below, in relation with the number of nodal lines of the $k$-th eigenfunction of the Laplacian. The proof relies on a detailed asymptotic analysis of the eigenfunctions, based on an Almgren-type frequency formula for magnetic eigenfunctions and on the blow-up technique.

math.AP↗

arXiv2014

Avoiding collisions under topological constraints in variational problems coming from celestial mechanics

In a singular potential setting, we generalize a method which allows to show that minimizers under topological constraints of the action functional (or of the Maupertuis') are collision-free. This methods applies to $3$-dimensional problems of celestial mechanics exhibiting a particular cylindrical symmetry, as well as to planar problems of $N$-centre type, where it gives optimal results.

math.DS↗

arXiv2014

Positive harmonic functions in union of chambers

We characterize the set of positive harmonic functions with Dirichlet boundary conditions in unbounded domains which are union of several different chambers. We analyze the asymptotic behavior of the solutions in connection with the changes in the domain's geometry. Finally we classify all (possibly sign-changing) infinite energy solutions having minimal frequency at the infinite ends of the domain.

math.AP↗

arXiv2013

On the eigenvalues of Aharonov-Bohm operators with varying poles

We consider a magnetic operator of Aharonov-Bohm type with Dirichlet boundary conditions in a planar domain. We analyse the behavior of its eigenvalues as the singular pole moves in the domain. For any value of the circulation we prove that the k-th magnetic eigenvalue converges to the k-th eigenvalue of the Laplacian as the pole approaches the boundary. We show that the magnetic eigenvalues depend in a smooth way on the position of the pole, as long as they remain simple. In case of half-integer circulation, we show that the rate of convergence depends on the number of nodal lines of the corresponding magnetic eigenfunction. In addition, we provide several numerical simulations both on the circular sector and on the square, which find a perfect theoretical justification within our main results, together with the ones in [Bonnaillie-Noël, V., Helffer, B., Exp. Math. 20 (2011), no. 3, 304-322; MR2836255 (2012i:35274)].

math.AP↗

arXiv2013

On the sharp effect of attaching a thin handle on the spectral rate of convergence

Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichlet Laplacian is continuous with respect to the channel section parameter. This in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation. Under an explicit nondegeneracy condition, we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain. As an application, we prove that, again under an explicit nondegeneracy con- dition, the case of resonant domains features polinomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions.

math.AP↗

arXiv2013

Singularity of eigenfunctions at the junction of shrinking tubes. Part I

Consider two domains connected by a thin tube: it can be shown that, generically, the mass of a given eigenfunction of the Dirichlet Laplacian concentrates in only one of them. The restriction to the other domain, when suitably normalized, develops a singularity at the junction of the tube, as the channel section tends to zero. Our main result states that, under a nondegeneracy condition, the normalized limiting profile has a singularity of order N-1, where N is the space dimension. We give a precise description of the asymptotic behavior of eigenfunctions at the singular junction, which provides us with some important information about its sign near the tunnel entrance. More precisely, the solution is shown to be one-sign in a neighborhood of the singular junction. In other words, we prove that the nodal set does not enter inside the channel.

math.AP↗

arXiv2013

Uniform Hölder regularity with small exponent in competition-fractional diffusion systems

For a class of competition-diffusion nonlinear systems involving fractional powers of the Laplacian, including as a special case the fractional Gross-Pitaevskii system, we prove that uniform boundedness implies Hölder boundedness for sufficiently small positive exponents, uniformly as the interspecific competition parameter diverges. This implies strong convergence of the family of solutions as the segregation of their support occurs.

math.AP↗

arXiv2013

Singularity of eigenfunctions at the junction of shrinking tubes, Part II

In continuation with the paper arXiv:1202.4414, we investigate the asymptotic behavior of weighted eigenfunctions in two half-spaces connected by a thin tube. We provide several improvements about some convergences stated in arXiv:1202.4414; most of all, we provide the exact asymptotic behavior of the implicit normalization for solutions given in arXiv:1202.4414 and thus describe the (N-1)-order singularity developed at a junction of the tube (where N is the space dimension).

math.AP↗

arXiv2012

Uniform Holder bounds for strongly competing systems involving the square root of the laplacian

For a class of competition-diffusion nonlinear systems involving the square root of the Laplacian, including the fractional Gross-Pitaevskii system, we prove that uniform boundedness implies Holder boundedness for every exponent less than 1/2, uniformly as the interspecific competition parameter diverges. Moreover we prove that the limiting profile is Holder continuous of exponent 1/2. This system arises, for instance, in the relativistic Hartree-Fock approximation theory for mixtures of Bose-Einstein condensates in different hyperfine states.

math.AP↗