Sobolev Constant Research Articles

Minimal potential results for Schrödinger equations with Neumann boundary conditions

Abstract We consider the boundary value problem - Δ p ⁢ u = V ⁢ | u | p - 2 ⁢ u - C {-\Delta_{p}u=V|u|^{p-2}u-C} , where u ∈ W 1 , p ⁢ ( D ) {u\in W^{1,p}(D)} is assumed to satisfy Neumann boundary conditions, and D is a bounded domain in ℝ n {{\mathbb{R}^{n}}} . We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for the product of a sharp Sobolev constant and an L p {L^{p}} norm of V. When p = n {p=n} , Orlicz norms are used. In many cases, these inequalities are best possible. Applications to linear and non-linear eigenvalue problems are also discussed.

Calabi flow on toric varieties with bounded Sobolev constant, I

Abstract Let (X, P) be a toric variety. In this note, we show that the C0-norm of the Calabi flow φ(t) on X is uniformly bounded in [0, T) if the Sobolev constant of φ(t) is uniformly bounded in [0, T). We also show that if (X, P) is uniform K-stable, then the modified Calabi flow converges exponentially fast to an extremal Kähler metric if the Ricci curvature and the Sobolev constant are uniformly bounded. At last, we discuss an extension of our results to a quasi-proper Kähler manifold.

Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian

This paper is devoted to the asymptotic analysis of the optimal Sobolev constants in the semiclassical limit and in any dimension. We combine semiclassical arguments and concentration-compactness estimates to tackle the case when an electromagnetic field is added as well as a smooth boundary carrying a Robin condition. As a byproduct of the semiclassical strategy, we also get exponentially weighted localization estimates of the minimizers.

Asymptotics for the best Sobolev constants and their extremal functions

Let $\Omega$ be a bounded domain of $\mathbf{R}^{N},$ $N\geq2.$ Let, for $p>N,$ \[ \Lambda_{p}(\Omega):=\inf\left\{ \left\Vert \nabla u\right\Vert _{p}^{p}:u\in W_{0}^{1,p}(\Omega)\quad and\quad\left\Vert u\right\Vert _{\infty}=1\right\} . \] We first prove that \[ \lim_{p\rightarrow\infty}\Lambda_{p}(\Omega)^{\frac{1}{p}}=\frac{1}{\left\Vert \rho\right\Vert _{\infty}}, \] where $\rho$ denotes the distance function to the boundary. Then, we show that, up to subsequences, the extremal functions of $\Lambda_{p}(\Omega)$ converge (as $p\rightarrow\infty$) to the viscosity solutions of a specific Dirichlet problem involving the infinity Laplacian in the punctured $\Omega.$

TheW-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials

In this paper, we develop a new approach to prove the $W$-entropy formula for the Witten Laplacian via warped product on Riemannian manifolds and give a natural geometric interpretation of a quantity appeared in the $W$-entropy formula. Then we prove the $W$-entropy formula for the Witten Laplacian on compact Riemannian manifolds with time dependent metrics and potentials, and derive the $W$-entropy formula for the backward heat equation associated with the Witten Laplacian on compact Riemannian manifolds equipped with Lott's modified Ricci flow. We also extend our results to complete Riemannian manifolds with negative $m$-dimensional Bakry-Emery Ricci curvature, and to compact Riemannian manifolds with $K$-super $m$-dimensional Bakry-Emery Ricci flow. As application, we prove that the optimal logarithmic Sobolev constant on compact manifolds equipped with the $K$-super $m$-dimensional Bakry-Emery Ricci flow is decreasing in time.

Positive minimizers of the best constants and solutions to coupled critical quasilinear systems

In this paper, systems of quasilinear elliptic equations are investigated, which involve critical homogeneous nonlinearities and deferent Hardy-type terms. By variational methods and careful analysis, positive minimizers of the related best Sobolev constants are found and the existence of positive solutions to the systems is verified. The results are new even in the case p=2.

Ground States of Time-Harmonic Semilinear Maxwell Equations in $${\mathbb{R}^3}$$ R 3 with Vanishing Permittivity

We investigate the existence of solutions \({E:\mathbb{R}^3 \to \mathbb{R}^3}\) of the time-harmonic semilinear Maxwell equation $$\nabla \times (\nabla \times E) + V(x) E = \partial_E F(x, E) \quad {\rm in} \mathbb{R}^3$$where \({V:\mathbb{R}^3 \to \mathbb{R}}\), \({V(x) \leqq 0}\) almost everywhere on \({\mathbb{R}^3}\), \({\nabla \times}\) denotes the curl operator in \({\mathbb{R}^3}\) and \({F:\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}}\) is a nonlinear function in E. In particular we find a ground state solution provided that suitable growth conditions on F are imposed and the \({L^{3/2}}\) -norm of V is less than the best Sobolev constant. In applications, F is responsible for the nonlinear polarization and \({V(x) = -\mu\omega^2 \varepsilon(x)}\) where μ > 0 is the magnetic permeability, ω is the frequency of the time-harmonic electric field \({\mathfrak{R}\{E(x){\rm e}^{i\omega t}\}}\) and \({\varepsilon}\) is the linear part of the permittivity in an inhomogeneous medium.

On the Sobolev and Hardy constants for the fractional Navier Laplacian

We prove the coincidence of the Sobolev and Hardy constants relative to the “Dirichlet” and “Navier” fractional Laplacians of any real order m∈(0,n2) over bounded domains in Rn.

Hypercontractivity of quasi-free quantum semigroups

Hypercontractivity of a quantum dynamical semigroup has strong implications for its convergence behavior and entropy decay rate. A logarithmic Sobolev inequality and the corresponding logarithmic Sobolev constant can be inferred from the semigroupʼs hypercontractive norm bound. We consider completely-positive quantum mechanical semigroups described by a Lindblad master equation. To prove the norm bound, we follow an approach which has its roots in the study of classical rate equations. We use interpolation theorems for non-commutative spaces to obtain a general hypercontractive inequality from a particular -norm bound. Then, we derive a bound on the -norm from an analysis of the block diagonal structure of the semigroupʼs spectrum. We show that the dynamics of an N-qubit graph state Hamiltonian weakly coupled to a thermal environment is hypercontractive. As a consequence this allows for the efficient preparation of graph states in time by coupling at sufficiently low temperature. Furthermore, we extend our results to gapped Liouvillians arising from a weak linear coupling of a free-fermion systems.

On Fractional Laplacians

We compare two natural types of fractional Laplacians (− Δ) s , namely, the “Navier” and the “Dirichlet” ones. We show that for 0 < s < 1 their difference is positive definite and positivity preserving. Then we prove the coincidence of the Sobolev constants for these two fractional Laplacians.

The Yamabe problem on stratified spaces

We introduce new invariants of a Riemannian singular space, the local Yamabe and Sobolev constants, and then go on to prove a general version of the Yamabe theorem under that the global Yamabe invariant of the space is strictly less than one or the other of these local invariants. This rests on a small number of structural assumptions about the space and of the behavior of the scalar curvature function on its smooth locus. The second half of this paper shows how this result applies in the category of smoothly stratified pseudomanifolds, and we also prove sharp regularity for the solutions on these spaces. This sharpens and generalizes the results of Akutagawa and Botvinnik (GAFA 13:259–333, 2003) on the Yamabe problem on spaces with isolated conic singularities.

On the question of diameter bounds in Ricci flow

A question about Ricci flow is when the diameters of the manifold under the evolving metrics stay finite and bounded away from $0$. Topping (Comm. Anal. Geom. 13 (2005) 1039–1055) addresses the question with an upper bound that depends on the $L^{(n-1)/2}$ bound of the scalar curvature, volume and a local version of Perelman’s $\nu$ invariant. Here $n$ is the dimension. His result is sharp when Perelman’s F entropy is positive. In this note, we give a direct proof that for all compact manifolds, the diameter bound depends just on the $L^{(n-1)/2}$ bound of the scalar curvature, volume and the Sobolev constants (or positive Yamabe constant). This bound seems directly computable in large time for some Ricci flows. In addition, since the result in its most general form is independent of Ricci flow, further applications may be possible. A generally sharp lower bound for the diameters is also given, which depends only on the initial metric, time and $L^{\infty}$ bound of the scalar curvature. These results imply that, in finite time, the Ricci flow can neither turn the diameter to infinity nor zero, unless the scalar curvature blows up.

Quantum logarithmic Sobolev inequalities and rapid mixing

A family of logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced. The framework of non-commutative \documentclass[12pt]{minimal}\begin{document}$\mathbb {L}_p$\end{document}Lp-spaces is reviewed and the relationship between quantum logarithmic Sobolev inequalities and the hypercontractivity of quantum semigroups is discussed. This relationship is central for the derivation of lower bounds for the logarithmic Sobolev (LS) constants. Essential results for the family of inequalities are proved, and we show an upper bound to the generalized LS constant in terms of the spectral gap of the generator of the semigroup. These inequalities provide a framework for the derivation of improved bounds on the convergence time of quantum dynamical semigroups, when the LS constant and the spectral gap are of the same order. Convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory. We provide a number of examples, where improved bounds on the mixing time of several semigroups are obtained, including the depolarizing semigroup and quantum expanders.

Existence problems for the p-Laplacian

Abstract We consider a number of boundary value problems involving the p-Laplacian. The model case is - Δ p u = V | u | p - 2 u ${-\Delta _p u = V|u|^{p-2}u}$ for u ∈ W 0 1 , p ( D ) ${u\in W_0^{1,p}(D)}$ with D a bounded domain in ℝ n . We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of V, the measure of D, and a sharp Sobolev constant. In most cases, these inequalities are best possible. Applications to non-linear eigenvalue problems are also discussed.

Absolute continuity of the best Sobolev constant

Let λq≔inf{‖∇u‖Lp(Ω)p/‖u‖Lq(Ω)p:u∈W01,p(Ω)∖{0}} be the best Sobolev constant of the immersion W01,p(Ω)↪Lq(Ω), where Ω is a bounded and smooth domain of RN,1<p<N and 1≤q≤p⋆≔NpN−p. We prove that the function q∈[1,p⋆]⟼λq is absolutely continuous.

Multiple solutions for a nonhomogeneous Schrödinger–Maxwell system in [formula omitted

The paper considers the following nonhomogeneous Schrödinger–Maxwell system: (SM){−Δu+u+λϕ(x)u=|u|p−1u+g(x),x∈R3,−Δϕ=u2,x∈R3, where λ>0, p∈(1,5), and 0≤g(x)=g(|x|)∈L2(R3). There seem to be no results on the existence of multiple solutions to problem (SM) for p∈(1,3). In this paper, we find that there is a constant Cp>0 such that problem (SM) has at least two solutions for all p∈(1,5) provided that ‖g‖L2≤Cp, however, for p∈(1,2] we need λ>0 is small. Moreover, Cp=(p−1)2p[(p+1)Sp+12p]1/(p−1), where S is the Sobolev constant.

A Global Curvature Pinching Result of the First Eigenvalue of the Laplacian on Riemannian Manifolds

The paper starts with a discussion involving the Sobolev constant on geodesic balls and then follows with a derivation of a lower bound for the first eigenvalue of the Laplacian on manifolds with small negative curvature. The derivation involves Moser iteration.

The logarithmic Sobolev constant of the lamplighter

We give estimates on the logarithmic Sobolev constant of some finite lamplighter graphs in terms of the spectral gap of the underlying base. Also, we give examples of application.

Sharp higher-order Sobolev inequalities in the hyperbolic space $${\mathbb{H}^n}$$

In this paper, we obtain the sharp k-th order Sobolev inequalities in the hyperbolic space \({\mathbb{H}^n}\) for all k = 1, 2, 3, . . . . This gives an answer to an open question raised by Aubin in [Aubin, Princeton University Press, Princeton (1982), pp. 176–177] for \({W^{k,2}(\mathbb{H}^n)}\) with k > 1. In addition, we prove that the associated Sobolev constants are optimal.

Two isoperimetric inequalities for the Sobolev constant

In this note we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first such inequality is a variation on the classical Schwarz Lemma from complex analysis, similar to recent inequalities of Burckel, Marshall, Minda, Poggi-Corradini, and Ransford, while the second generalises an isoperimetric inequality for the first eigenfunction of the Laplacian due to Payne and Rayner.