1,720,979 research outputs found
Stability theorems for Gagliardo-Nirenberg-Sobolev inequalities
No full textIn this paper we investigate the quantitative stability for Gagliardo-Nirenberg-
Sobolev inequalities. The main result is a reduction theorem, which states that, to solve the
problem of the stability for Gagliardo-Nirenberg-Sobolev inequalities, one can consider only the
class of radial decreasing functions
Equilibrium shapes of charged liquid droplets and related problems: (mostly) a review.
No full textWe review some recent results on the equilibrium shapes of charged liquid drops. We show that the natural variational model is ill-posed and how this can be overcome by either restricting the class of competitors or by adding penalizations in the functional. The original contribution of this note is twofold. First, we prove existence of an optimal distribution of charge for a conducting drop subject to an external 10 electric field. Second, we prove that there exists no optimal conducting drop in this setting
Brunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian
No full textWe prove that that the 1-Riesz capacity satisfies a Brunn-Minkowski inequality, and that the capacitary function of the 1/2-Laplacian is level set convex
A spectral shape optimization problem with a nonlocal competing term
Full text linkWe study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is small enough. Eventually we show that for certain regimes of the Riesz repulsion, regular minimizers do not exist
Compact Sobolev embeddings and torsion functions
Full text linkFor a general open set, we characterize the compactness of the embedding for the homogeneous Sobolev space D1,p → Lq in 0
terms of the summability of its torsion function. In particular, for 1 ≤ q < p we obtain that the embedding is continuous if and only if it is compact. The proofs crucially exploit a torsional Hardy inequality that we investigate in detail
On a class of weighted Gauss-type isoperimetric inequalities and applications to symmetrization
No full textWe solve a class of weighted isoperimetric problems with Gaussian type weight. As a consequence, we prove a comparison result for the solutions of degenerate elliptic equations
A spectral shape optimization problem with a nonlocal competing term
No full textWe study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is small enough. Eventually we show that for certain regimes of the Riesz repulsion, regular minimizers do not exist
On equilibrium shapes of charged flat drops
Full text linkThe equilibrium shapes of two-dimensional charged, perfectly conducting liquid drops are governed by a geometric variational problem that involves a perimeter term modeling line tension and a capacitary term modeling Coulombic repulsion. Here we give a complete explicit solution to this variational problem. Namely, we show that at fixed total charge a ball of a particular radius is the unique global minimizer among all sufficiently regular sets in the plane. For sets whose area is also fixed, we show that balls are the only minimizers if the charge is less than or equal to a critical charge, while for larger charge minimizers do not exist. Analogous results hold for drops whose potential, rather than charge, is fixed. © 2017 Wiley Periodicals, Inc
REIFENBERG FLATNESS FOR ALMOST MINIMIZERS OF THE PERIMETER UNDER MINIMAL ASSUMPTIONS
Full text linkThe aim of this note is to prove that almost minimizers of the perimeter are Reifenberg flat, for a very weak notion of minimality. The main observation is that smallness of the excess at some scale implies smallness of the excess at all smaller scales
Existence and stability for a non-local isoperimetric model of charged liquid drops
Full text linkWe consider a variational problem related to the shape of charged liquid drops at equilibrium. We show that this problem never admits local minimizers with respect to L 1 perturbations preserving the volume. However, we prove that the ball is stable under small C1,1 perturbations when the charge is small enough
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