Interpolation Inequalities Research Articles

Inequalities involving Aharonov–Bohm magnetic potentials in dimensions 2 and 3

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions 2 and 3.

Higher-order interpolation inequalities with weights for radial functions

We consider higher-order interpolation inequalities of the Gagliardo–Nirenberg type with power weights for radial functions. We show that those inequalities hold for a better range of admissible power weights if we restrict ourselves to the space of radially symmetric functions. The key of the proof is to reduce the problem to a radial improvement for the weighted Hardy–Littlewood–Sobolev inequalities.

Best constants in Sobolev and Gagliardo\u2013Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations

In this paper the dependence of the best constants in Sobolev and Gagliardo–Nirenberg inequalities on the precise form of the Sobolev space norm is investigated. The analysis is carried out on general graded Lie groups, thus including the cases of mathbb {R}^n, Heisenberg, and general stratified Lie groups, in all these cases the results being new. The Sobolev norms may be defined in terms of Rockland operators, i.e. the hypoelliptic homogeneous left-invariant differential operators on the group. The best constants are expressed in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The orders of these equations can be high depending on the Sobolev space order in the Sobolev or Gagliardo–Nirenberg inequalities, or may be fractional. Applications are obtained also to equations with lower order terms given by different hypoelliptic operators. Already in the case of {mathbb {R}}^n, the obtained results extend the classical relations by Weinstein (Commun Math Phys 87(4):567–576 (1982/1983)) to a wide range of nonlinear elliptic equations of high orders with elliptic low order terms and a wide range of interpolation inequalities of Gagliardo–Nirenberg type. However, the proofs are different from those in Weinstein (Commun Math Phys 87(4):567–576 (1982/1983)) because of the impossibility of using the rearrangement inequalities already in the setting of the Heisenberg group. The considered class of graded groups is the most general class of nilpotent Lie groups where one can still consider hypoelliptic homogeneous invariant differential operators and the corresponding subelliptic differential equations.

On convergence of Chorin’s projection method to a Leray–Hopf weak solution

The projection method to solve the incompressible Navier–Stokes equations was first studied by Chorin (Math Comput, 1969) in the framework of a finite difference method and Temam (Arch Ration Mech Anal, 1969) in the framework of a finite element method. Chorin showed convergence of approximation and its error estimates in problems with periodic boundary conditions assuming existence of a $$C^5$$ -solution, while Temam demonstrated an abstract argument to obtain a Leray–Hopf weak solution in problems on a bounded domain with the no-slip boundary condition. In the present paper, the authors extend Chorin’s result with full details to obtain convergent finite difference approximation of a Leray–Hopf weak solution to the incompressible Navier–Stokes equations on an arbitrary bounded Lipschitz domain of $${{\mathbb {R}}}^3$$ with the no-slip boundary condition and an external force. We prove unconditional solvability of our implicit scheme and strong $$L^2$$ -convergence (up to a subsequence) under the scaling condition $$ h^{3-\alpha }\le \tau $$ (no upper bound is necessary), where $$h,\tau $$ are space, time discretization parameters, respectively, and $$\alpha \in (0,2]$$ is any fixed constant. The results contain a compactness method based on a new interpolation inequality for step functions.

Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds

In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several measure rigidity results related to optimal transport on Riemannian manifolds, which completely characterize CD(K,∞) condition and non-collapsed CD(K,N) condition on Riemannian manifolds with boundary. In particular, we reveal the measure rigidity of Riemannian interpolation inequality proved by Cordero-Erausquin, McCann and Schmuckenschläger. We prove that log-(semi)concave measures are the only reference measures so that displacement convexity holds on Riemannian manifolds. This is the first measure rigidity result concerning the synthetic dimension-free CD condition, which is new even on Rn. Using L1-optimal transportation theory, we also prove that CD condition yields geodesical convexity (with respect to the usual Riemannian distance).

The Asymptotically Sharp Geometric Rigidity Interpolation Estimate in Thin Bi-Lipschitz Domains

This work is part of a program of development of asymptotically sharp geometric rigidity estimates for thin domains. A thin domain in three dimensional Euclidean space is roughly a small neighborhood of regular enough two dimensional compact surface. We prove an asymptotically sharp geometric rigidity interpolation inequality for thin domains with little regularity. In contrast to that celebrated Friesecke et al. rigidity estimate [Commun. Pure Appl. Math. 55(11):1461–1506, 2002] for plates, our estimate holds for any proper rotations $\boldsymbol {R}\in SO(3)$ . Namely, the estimate bounds the $L^{p}$ distance of the gradient of any $\boldsymbol {y}\in W^{1,p}(\Omega ,\mathbb{R}^{3})$ field from any constant proper rotation $\boldsymbol {R}\in SO(3)$ , in terms of the average $L^{p}$ distance (nonlinear strain) of the gradient $\nabla \boldsymbol {y}$ from the rotation group $SO(3)$ , and the average $L^{p}$ distance of the field itself from the set of rigid motions corresponding to the rotation $\boldsymbol {R}$ . There are several remarkable facts about the estimate: 1. The constants in the estimate are sharp in terms of the domain thickness scaling for any thin domains with the required regularity. 2. In the special cases when the domain has positive or negative Gaussian curvature, the inequality reduces the problem of estimating the gradient $\nabla \boldsymbol {y}$ in terms of the prototypical nonlinear strain $\int _{\Omega }\mathrm{dist}^{p}(\nabla \boldsymbol {y}(x),SO(3))dx$ to the easier problem of estimating only the vector field $\boldsymbol {y}$ in terms of the nonlinear strain without any loss in the constant scalings as the Ansatze suggest. The later will be a geometric rigidity Korn-Poincare type estimate. This passage is major progress in the thin domain rigidity problem. 3. For the borderline energy scaling (bending-to-stretching), the estimate implies improved strong compactness on the vector fields for free. Finally, this being said, our new interpolation inequality reduces the problem of proving “any” geometric one well rigidity problem in thin domains to estimating the vector field itself instead of the gradient, thus reducing the complexity of the problem.

Absolute continuity and numerical approximation of stochastic Cahn–Hilliard equation with unbounded noise diffusion

In this article, we consider the absolute continuity and numerical approximation of the solution of the stochastic Cahn–Hilliard equation with unbounded noise diffusion. We first obtain the Hölder continuity and Malliavin differentiability of the solution of the stochastic Cahn–Hilliard equation by using the strong convergence of the spectral Gakerkin approximation. Then we prove the existence and strict positivity of the density function of the law of the exact solution for the stochastic Cahn–Hilliard equation with sublinear growth diffusion coefficient, which fills a gap for the existed result when the diffusion coefficient satisfies a growth condition of order 1/3<α<1. To approximate the density function of the exact solution, we propose a full discretization based on the spatial spectral Galerkin approximation and the temporal drift implicit Euler scheme. Furthermore, a general framework for deriving the strong convergence rate of the full discretization is developed based on the variation approach and the factorization method. Consequently, we obtain the sharp mean square convergence rates in both time and space via Sobolev interpolation inequalities and semigroup theories. To the best of our knowledge, this is the first result on the convergence rate of full discretizations for the considered equation.

A Remark on the Interpolation Inequality between Sobolev Spaces and Morrey Spaces

Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems related to integral estimates and regularity of solutions to elliptic and/or parabolic equations. The main purpose of our work is to provide an important observation concerning the $$L^p$$ -boundedness property in the context of interpolation inequalities between Sobolev and Morrey spaces, which may be useful for those working in this domain. We also construct a nontrivial counterexample, which shows that the range of admissible values of $$p$$ is optimal in a certain sense. Our proofs rely on integral representations and on the theory of maximal and sharp maximal functions.

Optimal exponentials of thickness in Korn’s inequalities for parabolic and elliptic shells

We establish Korn's interpolation inequalities and the rigidity results of the strain tensor of the middle surface for the parabolic and elliptic shells and show that the best constant in Korn's inequalities scales like $h^{3/2}$ for the parabolic shell and $h$ for the elliptic shell, removing the main assumption that the middle surface of the shell is given by one single principal coordinate in the literature and, in particular, including the closed elliptic shell.

An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow

<p style='text-indent:20px;'>In recent work of Nagasawa and the author, new interpolation inequalities between the deviation of curvature and the isoperimetric ratio were proved. In this paper, we apply such estimates to investigate the large-time behavior of the length-preserving flow of closed plane curves without a convexity assumption.

Global regularity for 2D fractional magneto-micropolar equations

The magneto-micropolar equations are important models in fluid mechanics and material sciences. This paper focuses on the global regularity problem on the 2D magneto-micropolar equations with fractional dissipation. We establish the global regularity for three important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ various technics including the regularization of generalized heat operators on the Fourier frequency localized functions, logarithmic Sobolev interpolation inequalities and the maximal regularity property of the heat operator.

Global in Time Results for a Parabolic Equation Solution in Non-rectangular Domains

This article deals with the parabolic equation ∂tw − c(t)∂2x w = f in D, D = { (t, x) ∈ R2 : t &gt; 0, φ1 (t) &lt; x &lt; φ2(t) } with φi : [0,+∞[→ R, i = 1, 2 and c : [0,+∞[→ R satisfying some conditions and the problem is supplemented with boundary conditions of Dirichlet-Robin type. We study the global regularity problem in a suitable parabolic Sobolev space. We prove in particular that for f ∈ L2(D) there exists a unique solution w such that w, ∂tw, ∂jw ∈ L2(D), j = 1, 2. Notice that the case of bounded non-rectangular domains is studied in [9]. The proof is based on energy estimates after transforming the problem in a strip region combined with some interpolation inequality. This work complements the results obtained in [19] in the case of Cauchy-Dirichlet boundary conditions

Fractional Sobolev spaces from a complex analytic viewpoint

For any non-zero function u in the Schwartz space S(Rn), we prove that s↦[u]s,22 can be extended to C as a transcendental meromorphic function, which establishes a connection between the Bourgain-Brezis-Mironescu's formula, Maz'ya-Shaponshikova's formula and the residues of the transcendental meromorphic function of s↦[u]s,22 at s=0 and s=1 separately. Moreover, we study the function properties of [u]s,22 and obtain the convergence rate version of the Bourgain-Brezis-Mironescu's formula and Maz'ya-Shaponshikova's formula. And we also obtain the following sharp interpolation inequality. For any u∈W1,2(Rn) and s∈(0,1), we have[u]s,22≤πn2+122s−1Γ(n2+s)Γ(1+s)sin⁡πs‖u‖22(1−s)‖∇u‖22s.

A Sobolev Interpolation Inequality and a Gross-Sobolev Logarithmic Inequality

A sharp integral inequality is proved and used to obtain a Sobolev interpolation inequality. Further, a new proof of a Gross-Sobolev logarithmic inequality is constructed on the basis of the Sobolev interpolation inequality.

Stabilization of the weakly coupled Schrödinger system

ABSTRACT In this paper, we investigate the energy decay for solutions of the weakly coupled dissipative Schrödinger system. Among the m-coupled equations, only one equation is directly damped. Under some assumptions about the damping and the coupling terms, it is shown that sufficiently smooth solutions of the system decay logarithmically with mixed boundary conditions, including the coupling of the Schrödinger system subject to Dirichlet and Robin type boundary conditions, respectively. The proof is based on some frequency estimates with an exponential loss on the resolvent operators, which will be solved by establishing an interpolation inequality for a suitable weakly coupled elliptic system.

Bakry–Émery curvature and model spaces in sub-Riemannian geometry

We prove comparison theorems for the sub-Riemannian distortion coefficients appearing in interpolation inequalities. These results, which are equivalent to a sub-Laplacian comparison theorem for the sub-Riemannian distance, are obtained by introducing a suitable notion of sub-Riemannian Bakry-\'Emery curvature. The model spaces for comparison are variational problems coming from optimal control theory. As an application we establish the sharp measure contraction property for 3-Sasakian manifolds satisfying a suitable curvature bound.

Improved Interpolation Inequalities and Stability

Abstract For exponents in the subcritical range, we revisit some optimal interpolation inequalities on the sphere with carré du champ methods and use the remainder terms to produce improved inequalities. The method provides us with lower estimates of the optimal constants in the symmetry breaking range and stability estimates for the optimal functions. Some of these results can be reformulated in the Euclidean space using the stereographic projection.

The relativistic Vlasov–Maxwell–Fokker–Planck system in the whole space

We are concerned with the Cauchy problem for the relativistic Vlasov–Maxwell–Fokker–Planck system in the whole space. For perturbative initial data with suitable regularity and integrability, we obtain the global classical solutions and the large time decay rates. For the proof, a new structure of the linearized relativistic Fokker–Planck operator is observed so that the coercivity estimates are established, crucial for the optimal large time decay rates is the pivotal Sobolev inequalities and interpolation inequalities as well as the structure of the coupled system. Moreover, it is shown that the energy without any weights decays in time with the polynomial rates varying directly with the derivatives of the solution, and the lower order derivatives of the solution except the magnetic field are further proved to decay faster than the optimal ones which coincide with the linearized equations without electro and magnetic fields, such a new phenomenon occurs only in the case of the relativistic Fokker–Planck model.

Optimality of logarithmic interpolation inequalities and extension criteria to the Navier–Stokes and Euler equations in Vishik spaces

We show the logarithmic interpolation inequality by means of the Vishik space $${\dot{V}}^{s}_{q,\sigma ,\theta }$$ which is larger than the homogeneous Besov space $${\dot{B}}^{s}_{q,\sigma }$$ . We emphasize that $${\dot{V}}^{s}_{q,\sigma ,\theta }$$ may be the largest normed space that satisfies the logarithmic interpolation inequality. As an application of this inequality, we prove that the strong solution to the Navier–Stokes and Euler equations can be extended if the scaling invariant quantity of vorticity in the Vishik space is finite. Namely, the Beirao da Veiga- and Beale–Kato–Majda-type regularity criteria are improved in the terms of the Vishik space.

Blow-up phenomena and local well-posedness for a generalized Camassa–Holm equation in the critical Besov space

In this paper we mainly study the Cauchy problem for a generalized Camassa–Holm equation in a critical Besov space. First, by using the Littlewood–Paley decomposition, transport equations theory, logarithmic interpolation inequalities and Osgood’s lemma, we establish the local well-posedness for the Cauchy problem of the equation in the critical Besov space $$B^{\frac{1}{2}}_{2,1}$$. Next we derive a new blow-up criterion for strong solutions to the equation. Then we give a global existence result for strong solutions to the equation. Finally, we present two new blow-up results and the exact blow-up rate for strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion.