1,721,006 research outputs found
The nodal set of solutions to some elliptic problems: Singular nonlinearities
Full text linkThis paper deals with solutions to the equation-Delta u = lambda(+ )(u(+))(q-1) - lambda(-)(u(-))(q-1 )in B-1where lambda(+), lambda(-) > 0, q is an element of (0, 1), B-1 = B-1(0) is the unit ball in R-N, N >= 2, and u(+) := maxu, 0, u(-) := max-u, 0 are the positive and the negative part of u, respectively. We extend to this class of singular equations the results recently obtained in [25] for sublinear and discontinuous equations, 1 <= q < 2, namely: (a) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most N - 2 (locally finite when N = 2). As an intermediate step, we establish the regularity of a class of not necessarily minimal solutions.The proofs are based on a priori bounds, monotonicity formulae for a 2-parameter family of Weiss-type functionals, blow-up arguments, and the classification of homogeneous solutions. (C) 2019 Elsevier Masson SAS. All rights reserved
On a Neumann problem involving two critical Sobolev exponents: remarks on geometrical and topological aspects
No full textSingularity 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
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
Regularity of the optimal sets for some spectral functionals
Full text linkIn this paper we study the regularity of the optimal sets for the shape optimization problem min{λ1(Ω)+⋯+λk(Ω) : Ω⊂Rd open, |Ω|=1}, where λ1(·) , ... , λk(·) denote the eigenvalues of the Dirichlet Laplacian and | · | the d-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer Ωk∗ is composed of a relatively open regular part which is locally a graph of a C∞ function and a closed singular part, which is empty if d< d∗, contains at most a finite number of isolated points if d= d∗ and has Hausdorff dimension smaller than (d- d∗) if d> d∗, where the natural number d∗∈ [ 5 , 7 ] is the smallest dimension at which minimizing one-phase free boundaries admit singularities. To achieve our goal, as an auxiliary result, we shall extend for the first time the known regularity theory for the one-phase free boundary problem to the vector-valued case
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
A Note on Local Asymptotics of Solutions to Singular Elliptic Equations via Monotonicity Methods
No full textThis paper concerns the asymptotic behavior of solutions and their
gradients to linear and nonlinear elliptic equations with singular coefficients of
fuchsian type
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