184 research outputs found
Nodal sets of magnetic Schroedinger operators of Aharonov-Bohm type and energy minimizing partitions
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.Comment: 32 page
Liouville theorems and -dimensional symmetry for solutions of an elliptic system modelling phase separation
We consider solutions of the competitive elliptic system 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
, increasing in , such that if
is a solution and then . This
means that the number of components of the solution imposes an increasing
in minimal growth on the solution itself. If , the expression of
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 we can
prove the -dimensional symmetry of the solutions satisfying suitable
assumptions, extending known results which are available for . The proofs
rest upon a blow-down analysis and on some monotonicity formulae.Comment: 27 page
Solutions to nonlinear Schr\"odinger equations with singular electromagnetic potential and critical exponent
We investigate existence and qualitative behaviour of solutions to nonlinear
Schr\"odinger equations with critical exponent and singular electromagnetic
potentials. We are concerned with magnetic vector potentials which are
homogeneous of degree -1, including the Aharonov-Bohm class. In particular, by
variational arguments we prove a result of multiplicity of solutions
distinguished by symmetry properties
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