1,721,021 research outputs found
Absence of Critical Mass Densities for a Vibrating Membrane
No full textWe study the dependence of the eigenvalues of a N-dimensional vibrating membrane upon variation of the mass density. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically on the mass density and that such functions have no critical points with constant mass constraint. In particular, the elementary symmetric functions of the eigenvalues, hence all simple eigenvalues, have no local maxima or minima on the set of those mass densities with a prescribed total mass
Representation of the eigenvalues of the p-Laplacian by means of a surface integral
No full textWe prove a natural generalization to the p-Laplacian of the celebrated Rellich identity
Monotonicity, continuity and differentiability results for the Lp Hardy constant
No full textWe consider the L^p Hardy inequality involving the distance to the boundary for a domain in the
n-dimensional Euclidean space. We study the dependence on p of the corresponding best constant and we prove monotonicity, continuity and differentiability results. The focus is on non-convex domains in which case such a constant is in general not explicitly known
Extension and embedding theorems for Campanato spaces on C0,γ domains
Full text linkWe consider Campanato spaces with exponents, on domains of class (0,)in the N-dimensional Euclidean space endowed with a natural anisotropic metricdepending on. We discuss several results including the appropriate Campanato's embedding theorem and we prove that functions of those spaces canbe extended to the whole of the Euclidean space without deterioration of the exponents ,
A maximum principle in spectral optimization problems for elliptic operators subject to mass density perturbations
No full textWe consider eigenvalue problems for general elliptic operators of arbitrary order subject to homogeneous boundary conditions on open subsets of the Euclidean N-dimensional space. We prove stability results for the dependence of the eigenvalues upon variation of the mass density and we prove a maximum principle for extremum problems related to mass density perturbations which preserve the total mass
On the sharpness of a certain spectral stability estimate for the Dirchlet Laplacian
No full textWe consider a spectral stability estimate by Burenkov and Lamberti concerning the variation of the eigenvalues of second order uniformly elliptic operators on variable open sets in the N-dimensional euclidean space, and we prove that it is sharp for any dimension N. This is done by studying the eigenvalue problem for the Dirichlet Laplacian on special open sets inscribed in suitable spherical cones
Neumann to Steklov eigenvalues: Asymptotic and monotonicity results
Full text linkWe consider the Steklov eigenvalues of the Laplace operator as limiting Neumann
eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss
the asymptotic behaviour of the Neumann eigenvalues and find explicit formulae for
their derivatives in the limiting problem. We deduce that the Neumann eigenvalues
have a monotone behaviour in the limit and that Steklov eigenvalues locally minimize
the Neumann eigenvalues
Spectral stability of the p-Laplacian
No full textWe study the dependence of the eigenvalues of the p-Laplacian upon domain perturbation. We prove Lipschitz-type estimates for the deviation of the eigenvalues following a domain perturbation. Such estimates are expressed in terms of suitable measures of vicinity between open sets, such as the 'atlas distance' and the 'lower Hausdorff-Pompeiu deviation'. In the case of open sets with Holder continuous boundaries, our results improve a result known for the first eigenvalue
Shape sensitivity analysis of the Hardy constant
No full textWe consider the Hardy constant associated with a domain in the nn-dimensional Euclidean space and we study its variation upon perturbation of the domain. We prove a Fréchet differentiability result and establish a Hadamard-type formula for the corresponding derivatives. We also prove a stability result for the minimizers of the Hardy quotient. Finally, we prove stability estimates in terms of the Lebesgue measure of the symmetric difference of domains
On a classical spectral optimization problem in linear elasticity
No full textWe consider a classical shape optimization problem for the eigenvalues of elliptic operators with homogeneous boundary conditions on domains in the N-dimensional Euclidean space. We survey recent results concerning the analytic dependence of the elementary symmetric functions of the eigenvalues upon domain perturbation and the role of balls as critical points of such functions subject to volume constraint. Our discussion concerns Dirichlet and buckling-type problems for polyharmonic operators, the Neumann and the intermediate problems for the biharmonic operator, the Lame' and the Reissner-Mindlin systems
- …
