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Search Results: 1 - 10 of 333 matches for " Benedetta Noris "
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Existence and bounds of positive solutions for a nonlinear Schroedinger system
Benedetta Noris,Miguel Ramos
Mathematics , 2009,
Abstract: We prove that, for any real $\lambda$, the system $-\Delta u +\lambda u = u^3-\beta uv^2$, $ -\Delta v+\lambda v =v^3-\beta vu^2$, $ u,v\in H^1_0(\Omega),$ where $\Omega$ is a bounded smooth domain of $R^3$, admits a bounded family of positive solutions $(u_{\beta}, v_{\beta})$ as $\beta \to +\infty$. An upper bound on the number of nodal sets of the weak limits of difference $u_{\beta}-v_{\beta}$ is also provided. Moreover, for any sufficiently large fixed value of $\beta >0$ the system admits infinitely many positive solutions.
Positive constrained minimizers for supercritical problems in the ball
Massimo Grossi,Benedetta Noris
Mathematics , 2010,
Abstract: We provide a sufficient condition for the existence of a positive solution to $-\Delta u+V(|x|) u=u^p$ in $B_1$, when p is large enough. Here $B_1$ is the unit ball of $R^n$, n greater or equal to 2, and we deal both with Neumann and Dirichlet homogeneous boundary conditions. The solution turns to be a constrained minimum of the associated energy functional. As an application we show that, in case $V(|x|)$ is smooth, nonnegative and not identically zero, and p is sufficiently large, the Neumann problem always admits a solution.
Nodal sets of magnetic Schroedinger operators of Aharonov-Bohm type and energy minimizing partitions
Benedetta Noris,Susanna Terracini
Mathematics , 2009,
Abstract: In this paper we consider a stationary Schroedinger operator in the plane, in presence of a magnetic field of Aharonov-Bohm type with semi-integer circulation. We analyze the nodal regions for a class of solutions such that the nodal set consists of regular arcs, connecting the singular points with the boundary. In case of one magnetic pole, which is free to move in the domain, the nodal lines may cluster dissecting the domain in three parts. Our main result states that the magnetic energy is critical (with respect to the magnetic pole) if and only if such a configuration occurs. Moreover the nodal regions form a minimal 3-partition of the domain (with respect to the real energy associated to the equation), the configuration is unique and depends continuously on the data. The analysis performed is related to the notion of spectral minimal partition introduced in [20]. As it concerns eigenfunctions, we similarly show that critical points of the Rayleigh quotient correspond to multiple clustering of the nodal lines.
A remark on natural constraints in variational methods and an application to superlinear Schr?dinger systems
Benedetta Noris,Gianmaria Verzini
Mathematics , 2011,
Abstract: For a regular functional J defined on a Hilbert space X, we consider the set N of points x of X such that the projection of the gradient of J at x onto a closed linear subspace V(x) of X vanishes. We study sufficient conditions for a constrained critical point of J restricted to N to be a free critical point of J, providing a unified approach to different natural constraints known in the literature, such as the Birkhoff-Hestenes natural isoperimetric conditions and the Nehari manifold. As an application, we prove multiplicity of solutions to a class of superlinear Schr\"odinger systems on singularly perturbed domains.
On the eigenvalues of Aharonov-Bohm operators with varying poles: pole approaching the boundary of the domain
Benedetta Noris,Manon Nys,Susanna Terracini
Mathematics , 2014,
Abstract: We continue the analysis started in [Noris,Terracini,Indiana Univ Math J,2010] and [Bonnaillie-No\"el,Noris,Nys,Terracini,Analysis & PDE,2014], concerning the behavior of the eigenvalues of a magnetic Schr\"odinger 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.
Thomas-Fermi approximation for coexisting two component Bose-Einstein condensates and nonexistence of vortices for small rotation
Amandine Aftalion,Benedetta Noris,Christos Sourdis
Mathematics , 2014, DOI: 10.1007/s00220-014-2281-9
Abstract: We study minimizers of a Gross--Pitaevskii energy describing a two-component Bose-Einstein condensate confined in a radially symmetric harmonic trap and set into rotation. We consider the case of coexistence of the components in the Thomas-Fermi regime, where a small parameter $\ep$ conveys a singular perturbation. The minimizer of the energy without rotation is determined as the positive solution of a system of coupled PDE's for which we show uniqueness. The limiting problem for $\ep =0$ has degenerate and irregular behavior at specific radii, where the gradient blows up. By means of a perturbation argument, we obtain precise estimates for the convergence of the minimizer to this limiting profile, as $\ep$ tends to 0. For low rotation, based on these estimates, we can show that the ground states remain real valued and do not have vortices, even in the region of small density.
Existence and orbital stability of the ground states with prescribed mass for the L^2-critical and supercritical NLS on bounded domains
Benedetta Noris,Hugo Tavares,Gianmaria Verzini
Mathematics , 2013,
Abstract: We study solutions of a semilinear elliptic equation with prescribed mass and Dirichlet homogeneous boundary conditions in the unitary ball. Such problem arises in the search of solitary wave solutions for nonlinear Schr\"odinger equations (NLS) with Sobolev subcritical power nonlinearity on bounded domains. Necessary and sufficient conditions are provided for the existence of such solutions. Moreover, we show that standing waves associated to least energy solutions are always orbitally stable when the nonlinearity is L^2-critical and subcritical, while they are almost always stable in the L^2-supercritical regime. The proofs are obtained in connection with the study of a variational problem with two constraints, of independent interest: to maximize the L^{p+1}-norm among functions having prescribed L^2 and H^1_0-norm.
Stable solitary waves with prescribed $L^2$-mass for the cubic Schr?dinger system with trapping potentials
Benedetta Noris,Hugo Tavares,Gianmaria Verzini
Mathematics , 2014,
Abstract: For the cubic Schr\"odinger system with trapping potentials in $\mathbb{R}^N$, $N\leq3$, or in bounded domains, we investigate the existence and the orbital stability of standing waves having components with prescribed $L^2$-mass. We provide a variational characterization of such solutions, which gives information on the stability through of a condition of Grillakis-Shatah-Strauss type. As an application, we show existence of conditionally orbitally stable solitary waves when: a) the masses are small, for almost every scattering lengths, and b) in the defocusing, weakly interacting case, for any masses.
Increasing radial solutions for Neumann problems without growth restrictions
Denis Bonheure,Benedetta Noris,Tobias Weth
Mathematics , 2011, DOI: 10.1016/j.anihpc.2012.02.002
Abstract: We study the existence of positive increasing radial solutions for superlinear Neumann problems in the ball. We do not impose any growth condition on the nonlinearity at infinity and our assumptions allow for interactions with the spectrum. In our approach we use both topological and variational arguments, and we overcome the lack of compactness by considering the cone of nonnegative, nondecreasing radial functions of H^1.
Sub and supersolutions, invariant cones and multiplicity results for p-Laplace equations
Maria-Magdalena Boureanu,Benedetta Noris,Susanna Terracini
Mathematics , 2012,
Abstract: For a class of quasilinear elliptic equations involving the p-Laplace operator, we develop an abstract critical point theory in the presence of sub-supersolutions. Our approach is based upon the proof of the invariance under the gradient flow of enlarged cones in the $W^{1,p}_0$ topology. With this, we prove abstract existence and multiplicity theorems in the presence of variously ordered pairs of sub-supersolutions. As an application, we provide a four solutions theorem, one of the solutions being sign-changing.
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