1,720,994 research outputs found
Convex duality for principal frequencies
No full textWe consider the sharp Sobolev-Poincaré constant for the embedding of W01,2(Ω) into Lq(Ω). We show that such a constant exhibits an unexpected dual variational formulation, in the range 1 < q < 2. Namely, this can be written as a convex minimization problem, under a divergence-type constraint. This is particularly useful in order to prove lower bounds. The result generalizes what happens for the torsional rigidity (corresponding to q = 1) and extends up to the case of the first eigenvalue of the Dirichlet-Laplacian (i.e., to q = 2)
The fractional Makai–Hayman inequality
No full textWe prove that the first eigenvalue of the fractional Dirichlet–Laplacian of order s on a simply connected set of the plane can be bounded from below in terms of its inradius only. This is valid for 1 / 2 < s< 1 and we show that this condition is sharp, i.e., for 0 < s≤ 1 / 2 such a lower bound is not possible. The constant appearing in the estimate has the correct asymptotic behavior with respect to s, as it permits to recover a classical result by Makai and Hayman in the limit s↗ 1. The paper is as self-contained as possible
Long-time behavior for the porous medium equation with small initial energy
No full textWe study the long-time behavior for the solution of the Porous Medium Equation in an open bounded connected set, with smooth boundary. Homogeneous Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution converges to a nontrivial constant sign solution of a sublinear Lane-Emden equation, once suitably rescaled. We point out that the initial datum is allowed to be sign-changing. We also give a sufficient energetic criterion on the initial datum, which permits to decide whether convergence takes place towards the positive solution or to the negative one
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
Lipschitz regularity for orthotropic functionals with nonstandard growth conditions
No full textWe consider a model convex functional with orthotropic structure and super-quadratic nonstandard growth conditions. We prove that bounded local minimizers are locally Lipschitz, with no restrictions on the ratio between the highest and the lowest growth rates
On principal frequencies, volume and inradius in convex sets
Full text linkWe provide a sharp double-sided estimate for Poincaré–Sobolev constants on a convex set, in terms of its inradius and N- dimensional measure. Our results extend and unify previous works by Hersch and Protter (for the first eigenvalue) and of Makai, Pólya and Szegő (for the torsional rigidity), by means of a single proof
On Morrey's inequality in Sobolev-Slobodeckiĭ spaces
No full textWe study the sharp constant in the Morrey inequality for fractional Sobolev-Slobodeckiĭ spaces on the whole RN. By generalizing a recent work by Hynd and Seuffert, we prove existence of extremals, together with some regularity estimates. We also analyze the sharp asymptotic behaviour of this constant as we reach the borderline case sp=N, where the inequality fails. This can be done by means of a new elementary proof of the Morrey inequality, which combines: a local fractional Poincaré inequality for punctured balls, the definition of capacity of a point and Hardy's inequality for the punctured space. Finally, we compute the limit of the sharp Morrey constant for s↗1, as well as its limit for p↗∞. We obtain convergence of extremals, as well
A note on homogeneous Sobolev spaces of fractional order
No full textWe consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev–Slobodeckiĭ norm. We compare it to the fractional Sobolev space obtained by the K-method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible.Fil: Brasco, Lorenzo. Universita Di Ferrara. Dipartimento Di Física; ItaliaFil: Salort, Ariel Martin. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin
- …
