1,721,013 research outputs found
Existence and regularity of minimizers for some spectral functionals with perimeter constraint
Full text linkIn this paper we prove that the shape optimization problem {λk (Ω) : Ω ⊂ Rd, Ω open, P(Ω) = 1, |Ω| <+ ∞- has a solution for any k ∈ N and dimension d. Moreover, every solution is a bounded connected open set with boundary which is C 1,α outside a closed set of Hausdorff dimension d-8. Our results are more general and apply to spectral functionals of the form λk1 (Ω)⋯ λkp (Ω)), for increasing functions f satisfying some suitable bi-Lipschitz type condition. © 2013 Springer Science+Business Media New York
The closure of planar diffeomorphisms in Sobolev spaces
Full text linkWe characterize the (sequentially) weak and strong closure of planar diffeomorphisms in the Sobolev topology and we show that they always coincide. We also provide some sufficient condition for a planar map to be approximable by diffeomorphisms in terms of the connectedness of its counter-images, in the spirit of Young's characterisation of monotone functions. We finally show that the closure of diffeomorphisms in the Sobolev topology is strictly contained in the class INV introduced by Müller and Spector
On the structure of measures constrained by linear PDEs
No full textThe aim of this note is to present some recent results on the structure of the singular part of measures satisfying a PDE constraint and to describe some applications
Regularity of Minimizers for a Model of Charged Droplets
No full textWe investigate properties of minimizers of a variational model describing the shape of charged liquid droplets. The model, proposed by Muratov and Novaga, takes into account the regularizing effect due to the screening of free counterionions in the droplet. In particular we prove partial regularity of minimizers, a first step toward the understanding of further properties of minimizers
The sharp quantitative isocapacitary inequality
No full textWe prove a sharp quantitative form of the classical isocapacitary inequality. Namely, we show that the difference between the capacity of a set and that of a ball with the same volume bounds the square of the Fraenkel asymmetry of the set. This provides a positive answer to a conjecture of Hall, Hayman, and Weitsman (J. Analyse Math.'91)
The behavior of harmonic functions at singular points of RCD spaces
No full textIn this note we investigate the behavior of harmonic functions at singular points of RCD(K, N) spaces. In particular we show that their gradient vanishes at all points where the tangent cone is isometric to a cone over a metric measure space with non-maximal diameter. The same analysis is performed for functions with Laplacian in LN+epsilon. As a consequence we show that on smooth manifolds there is no a priori estimate on the modulus of continuity of the gradient of harmonic functions which depends only on lower bounds of the sectional curvature. In the same way we show that there is no a priori Calderon-Zygmund theory for the Laplacian with bounds depending only on lower bounds of the sectional curvature
Positive solutions to the sublinear Lane-Emden equation are isolated
No full textWe prove that on a smooth bounded set, the positive least energy solution of the Lane-Emden equation with sublinear power is isolated. As a corollary, we obtain that the first (Formula presented.) eigenvalue of the Dirichlet-Laplacian is not an accumulation point of the (Formula presented.) spectrum, on a smooth bounded set. Our results extend to a suitable class of Lipschitz domains, as well
REGULARITY OF CAPILLARITY DROPLETS WITH OBSTACLE
No full textIn this paper we study the regularity properties of Λ-minimizers of the capillarity energy in a half space with the wet part constrained to be confined inside a given planar region. Applications to a model for nanowire growth are also provided
Regularity of the free boundary for the two-phase Bernoulli problem
Full text linkWe prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian
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