1,721,053 research outputs found
Inequalities between dirichlet and neumann eigenvalues of the polyharmonic operators
Full text linkWe prove that μmk +m < Λmk , where μmk (Λmk ) are the eigenvalues of (-Δ)m on Ω ⊂ Rd, d ≥ 2, with Neumann (Dirichlet) boundary conditions
ON TRACE THEOREMS FOR SOBOLEV SPACES
Full text linkWe survey a few trace theorems for Sobolev spaces on N-dimensional Euclidean domains. We include known results on linear subspaces, in particular hyperspaces, and smooth boundaries, as well as less known results for Lipschitz boundaries, including Besov's Theorem and other characterizations of traces on planar domains, polygons in particular, in the spirit of the work of P. Grisvard. Finally, we present a recent approach, originally developed by G. Auchmuty in the case of the Sobolev space H-1(Omega) on a Lipschitz domain Omega, and which we have further developed for the trace spaces of H-k(Omega), k >= 2, by using Fourier expansions associated with the eigenfunctions of new multi-parameter polyharmonic Steklov problems
Conformal upper bounds for the eigenvalues of the p-Laplacian
Full text linkIn this note we present upper bounds for the variational eigenvalues of the (Formula presented.) -Laplacian on smooth domains of complete (Formula presented.) -dimensional Riemannian manifolds and Neumann boundary conditions, and on compact (boundaryless) Riemannian manifolds. In particular, we provide upper bounds in the conformal class of a given metric for (Formula presented.), and upper bounds for all (Formula presented.) when we fix a metric. To do so, we use a metric approach for the construction of suitable test functions for the variational characterization of the eigenvalues. The upper bounds agree with the well-known asymptotic estimate of the eigenvalues due to Friedlander. We also present upper bounds for the variational eigenvalues on hypersurfaces bounding smooth domains in a Riemannian manifold in terms of the isoperimetric ratio
Isoparametric foliations and the Pompeiu property
Full text linkA bounded domain ω in a Riemannian manifold M is said to have the Pompeiu property if the only continuous function which integrates to zero on ω and on all its congruent images is the zero function. In some respects, the Pompeiu property can be viewed as an overdetermined problem, given its relation with the Schiffer problem. It is well-known that every Euclidean ball fails to have the Pompeiu property while spherical balls have the property for almost all radii (Ungar's Freak theorem). In the present paper we discuss the Pompeiu property when M is compact and admits an isoparametric foliation. In particular, we identify precise conditions on the spectrum of the Laplacian on M under which the level domains of an isoparametric function fail to have the Pompeiu property. Specific calculations are carried out when the ambient manifold is the round sphere, and some consequences are derived. Moreover, a detailed discussion of Ungar's Freak theorem and its generalizations is also carried out
On the explicit representation of the trace space H32 and of the solutions to biharmonic dirichlet problems on lipschitz domains via multi-parameter Steklov problems
Full text linkWe consider the problem of describing the traces of functions in H2(Ω) on the boundary of a Lipschitz domain Ω of RN, N≥ 2. We provide a definition of those spaces, in particular of H32(∂Ω), by means of Fourier series associated with the eigenfunctions of new multi-parameter biharmonic Steklov problems which we introduce with this specific purpose. These definitions coincide with the classical ones when the domain is smooth. Our spaces allow to represent in series the solutions to the biharmonic Dirichlet problem. Moreover, a few spectral properties of the multi-parameter biharmonic Steklov problems are considered, as well as explicit examples. Our approach is similar to that developed by G. Auchmuty for the space H1(Ω) , based on the classical second order Steklov problem
On the critical points of Steklov eigenfunctions
No full textWe consider the critical points of Steklov eigenfunctions on a compact, smooth n-dimensional Riemannian manifold M with boundary ∂M. For generic metrics on M we establish an identity which relates the sum of the indexes of a Steklov eigenfunction, the sum of the indexes of its restriction to ∂M, and the Euler characteristic of M. In dimension 2 this identity gives a precise count of the interior critical points of a Steklov eigenfunction in terms of the Euler characteristic of M and of the number of sign changes of u on ∂M. In the case of the second Steklov eigenfunction on a genus 0 surface, the identity holds for any metric. As a by-product of the main result, we show that for generic metrics on M Steklov eigenfunctions are Morse functions in M
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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