99 research outputs found
Almgren-type monotonicity methods for the classification of behavior at corners of solutions to semilinear elliptic equations
A monotonicity approach to the study of the asymptotic behavior near corners
of solutions to semilinear elliptic equations in domains with a conical
boundary point is discussed. The presence of logarithms in the first term of
the asymptotic expansion is excluded for boundary profiles sufficiently close
to straight conical surfaces
Sharp asymptotic estimates for eigenvalues of Aharonov-Bohm operators with varying poles
We investigate the behavior of eigenvalues for a magnetic Aharonov-Bohm
operator with half-integer circulation and Dirichlet boundary conditions in a
planar domain. We provide sharp asymptotics for eigenvalues as the pole is
moving in the interior of the domain, approaching a zero of an eigenfunction of
the limiting problem along a nodal line. As a consequence, we verify
theoretically some conjectures arising from numerical evidences in preexisting
literature. The proof relies on an Almgren-type monotonicity argument for
magnetic operators together with a sharp blow-up analysis
Global minimizers of coexistence for competing species
A class of variational models describing ecological systems of k species
competing for the same resources is investigated. The occurrence of coexistence
in minimal energy solutions is discussed and positive results are proven for
suitably differentiated internal dynamics
Classification of local asymptotics for solutions to heat equations with inverse-square potentials
Asymptotic behavior of solutions to heat equations with spatially singular
inverse-square potentials is studied. By combining a parabolic Almgren type
monotonicity formula with blow-up methods, we evaluate the exact behavior near
the singularity of solutions to linear and subcritical semilinear parabolic
equations with Hardy type potentials. As a remarkable byproduct, a unique
continuation property is obtained
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