1,721,030 research outputs found
Sharp asymptotic estimates for eigenvalues of Aharonov-Bohm operators with varying poles
No full textWe 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
Perturbed eigenvalues of polyharmonic operators in domains with small holes
Full text linkWe study singular perturbations of eigenvalues of the polyharmonic operator on bounded domains under removal of small interior compact sets. We consider both homogeneous Dirichlet and Navier conditions on the external boundary, while we impose homogeneous Dirichlet conditions on the boundary of the removed set. To this aim, we develop a notion of capacity which is suitable for our higher-order context, and which permits to obtain a description of the asymptotic behaviour of perturbed simple eigenvalues in terms of a capacity of the removed set, in dependence of the respective normalized eigenfunction. Then, in the particular case of a subset which is scaling to a point, we apply a blow-up analysis to detect the precise convergence rate, which turns out to depend on the order of vanishing of the eigenfunction. In this respect, an important role is played by Hardy-Rellich inequalities in order to identify the appropriate functional space containing the limiting profile. Remarkably, for the biharmonic operator this turns out to be the same, regardless of the boundary conditions prescribed on the exterior boundary
On the leading term of the eigenvalue variation for Aharonov-Bohm operators with a moving pole
No full textWe study the behavior of certain eigenvalues for magnetic Aharonov-Bohm operators with half-integer circulation and Dirichlet boundary conditions in a planar domain. We analyze the leading term in the Taylor expansion of the eigenvalue function as the pole moves in the interior of the domain, proving that it is a harmonic homogeneous polynomial and determining its exact coefficients
Almgren-type monotonicity methods for the classification of behavior at corners of solutions to semilinear elliptic equations
No full textA monotonicity approach to the study of the asymptotic behaviour 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
Unique continuation and classification of blow-up profiles for elliptic systems with Neumann boundary coupling and applications to higher order fractional equations
No full textIn this paper we develop a monotonicity formula for elliptic systems with Neumann boundary coupling, proving unique continuation and classification of blow-up profiles. As an application, we obtain strong unique continuation for some fourth order equations and higher order fractional problems
Unique continuation principles for a higher order fractional Laplace equation
Full text linkIn this paper we prove the strong unique continuation principle and the unique continuation from sets of positive measure for solutions of a higher order fractional Laplace equation in an open domain. Our proofs are based on the Caffarelli-Silvestre (2007 Commun. PDE 32 1245-60) extension method combined with an Almgren type monotonicity formula. The corresponding extended problem is formulated as a system of two second order equations with singular or degenerate weights in a half-space, for which asymptotic estimates are derived by a blow-up analysis
On the sharp effect of attaching a thin handle on the spectral rate of convergence
No full textConsider 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 condition, the case of resonant domains features polynomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions. © 2013 Elsevier Inc
Singularity of eigenfunctions at the junction of shrinking tubes, Part II
No full textIn continuation with [17], 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 [17]; most of all, we provide the exact asymptotic behavior of the implicit normalization for solutions given in [17] and thus describe the (N - 1)-order singularity developed at a junction of the tube (where N is the space dimension). © 2014 Elsevier Inc
Eigenvalue variation under moving mixed Dirichlet-Neumann boundary conditions and applications
Full text linkWe deal with the sharp asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet-Neumann boundary conditions. In case of simple eigenvalues, we compute explicitly the constant appearing in front of the expansion's leading term. This allows inferring some remarkable consequences for Aharonov-Bohm eigenvalues when the singular part of the operator has two coalescing poles
On the behavior at collisions of solutions to Schrodinger equations with many-particle and cylindrical potentials
No full textThe asymptotic behavior of solutions to Schr ̈odinger equations
with singular homogeneous potentials is investigated. Through an Almgren
type monotonicity formula and separation of variables, we describe the exact
asymptotics near the singularity of solutions to at most critical semilinear elliptic
equations with cylindrical and quantum multi-body singular potentials.
Furthermore, by an iterative Brezis-Kato procedure, pointwise upper estimate
are derived
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