15,343 research outputs found
Quasilinear eigenvalues
In this work, we review and extend some well known results for the
eigenvalues of the Dirichlet Laplace operator to a more general class of
monotone quasilinear elliptic operators. As an application we obtain some
homogenization results for nonlinear eigenvalues.Comment: 23 pages, Rev. UMA, to appea
The p-Laplace equation in domains with multiple crack section via pencil operators
The p-Laplace equation
\n \cdot (|\n u|^n \n u)=0 \whereA n>0, in a bounded domain \O \subset
\re^2, with inhomogeneous Dirichlet conditions on the smooth boundary \p \O
is considered. In addition, there is a finite collection of curves
\Gamma = \Gamma_1\cup...\cup\Gamma_m \subset \O, \quad \{on which we assume
homogeneous Dirichlet boundary conditions} \quad u=0, modeling a multiple
crack formation, focusing at the origin 0 \in \O. This makes the above
quasilinear elliptic problem overdetermined. Possible types of the behaviour of
solution at the tip 0 of such admissible multiple cracks, being a
"singularity" point, are described, on the basis of blow-up scaling techniques
and a "nonlinear eigenvalue problem". Typical types of admissible cracks are
shown to be governed by nodal sets of a countable family of nonlinear
eigenfunctions, which are obtained via branching from harmonic polynomials that
occur for . Using a combination of analytic and numerical methods,
saddle-node bifurcations in are shown to occur for those nonlinear
eigenvalues/eigenfunctions.Comment: arXiv admin note: substantial text overlap with arXiv:1310.065
Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application
We develop further the theory of symmetrization of fractional Laplacian
operators contained in recent works of two of the authors. The theory leads to
optimal estimates in the form of concentration comparison inequalities for both
elliptic and parabolic equations. In this paper we extend the theory for the
so-called \emph{restricted} fractional Laplacian defined on a bounded domain
of with zero Dirichlet conditions outside of .
As an application, we derive an original proof of the corresponding fractional
Faber-Krahn inequality. We also provide a more classical variational proof of
the inequality.Comment: arXiv admin note: substantial text overlap with arXiv:1303.297
Ground Energy of the Magnetic Laplacian in Polyhedral Bodies
The asymptotic behavior of the first eigenvalues of magnetic Laplacian
operators with large magnetic fields and Neumann realization in polyhedral
domains is characterized by a hierarchy of model problems. We investigate
properties of the model problems (continuity, semi-continuity, existence of
generalized eigenfunctions). We prove estimates for the remainders of our
asymptotic formula. Lower bounds are obtained with the help of a classical IMS
partition based on adequate coverings of the polyhedral domain, whereas upper
bounds are established by a novel construction of quasimodes, qualified as
sitting or sliding according to spectral properties of local model problems.Comment: 59 page
The model magnetic Laplacian on wedges
We study a model Schr\"odinger operator with constan tmagnetic field on an
infinite wedge with natural boundary conditions. This problem is related to the
semiclassical magnetic Laplacian on 3d domains with edges. We show that the
ground energy is lower than the one coming from the regular part of the wedge
and is continuous with respect to the geometry. We provide an upper bound for
the ground energy on wedges of small opening. Numerical computations enlighten
the theoretical approach
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