Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions

Let \(\mathbb {H}^{n}=\mathbb {C}^{n}\times \mathbb {R}\) be the n-dimensional Heisenberg group, \(Q=2n+2\) be the homogeneous dimension of \(\mathbb {H}^{n}\). We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of Lions (Rev Mat Iberoam 1:145–201, 1985) to the setting of the Heisenberg group \(\mathbb {H}^{n}\). Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space \({ HW}^{1,Q}(\mathbb {H}^{n}) \) on the entire Heisenberg group \(\mathbb {H}^{n}\). Our results improve the sharp Trudinger–Moser inequality on domains of finite measure in \(\mathbb {H}^{n}\) by Cohn and Lu (Indiana Univ Math J 50(4):1567–1591, 2001) and the corresponding one on the whole space \(\mathbb {H}^n\) by Lam and Lu (Adv Math 231:3259–3287, 2012). All the proofs of the concentration-compactness principles for the Trudinger–Moser inequalities in the literature even in the Euclidean spaces use the rearrangement argument and the Polyá–Szegö inequality. Due to the absence of the Polyá–Szegö inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of Q- Laplacian subelliptic equations on \(\mathbb {H}^{n}\):

$$\begin{aligned} -\mathrm {div}\left( \left| \nabla _{\mathbb {H}}u\right| ^{Q-2} \nabla _{\mathbb {H}}u\right) +V(\xi ) \left| u\right| ^{Q-2}u=\frac{f(u) }{\rho (\xi )^{\beta }} \end{aligned}$$

with nonlinear terms f of maximal exponential growth \(\exp (\alpha t^{\frac{Q}{Q-1}})\) as \(t\rightarrow +\infty \). All the results proved in this paper hold on stratified groups with the same proofs. Our method in this paper also provide a new proof of the classical concentration-compactness principle for Trudinger-Moser inequalities in the Euclidean spaces without using the symmetrization argument.

     

L ∞-Error estimate for an approximation of a parabolic variational inequality

Summary Almost optimalL -convergence of an approximation of a variational inequality of parabolic type is proved under regularity assumptions which are met by the solution of a one phase Stefan problem. The discretization employs piecewise linear finite elements in space and the backward Euler scheme in time. By means of a maximum principle the problem is reduced to an error estimate for an auxiliary parabolic equation. The latter bound is obtained by using the smoothing property of the Galerkin method.     

Moment inequality and Hölder inequality for BSDEs

Under the Lipschitz and square integrable assumptions on the generator g of BSDEs, this paper proves that if g is positively homogeneous in （y, z） and is decreasing in y, then the Moment inequality for BSDEs with generator g holds in general, and if g is positively homogeneous and sub-additive in （y, z）, then the HSlder inequality and Minkowski inequality for BSDEs with generator g hold in general.     

Harnack’s principle for quasiminimizers

Abstract We study Harnack type properties of quasiminimizers of the -Dirichlet integral on metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. We show that an increasing sequence of quasiminimizers converges locally uniformly to a quasiminimizer, provided the limit function is finite at some point, even if the quasiminimizing constant and the boundary values are allowed to vary in a bounded way. If the quasiminimizing constants converge to one, then the limit function is the unique minimizer of the -Dirichlet integral. In the Euclidean case with the Lebesgue measure we obtain convergence also in the Sobolev norm. Keywords: Metric space, doubling measure, Poincaré inequality, Newtonian space, Harnack inequality, Harnack convergence theorem Mathematics Subject Classification (2000): 49J52, 35J60, 49J27     

Local-global principle for étale cohomology

Letk be a field of characteristic different from 2 and ø be a galoisian cohomological class ofk, with values in /2. J. K. Arason proved that ø is killed by a cup-product power of (–1) if and only if the restriction of ø is zero in all the real closedk-extensions. In this paper, we extend such a local-global principle to semilocal rings with 2 a unit, étale cohomology replacing Galois cohomology.     

Levitin–Polyak well-posedness by perturbations for the split inverse variational inequality problem

In this paper, we extend the notion of Levitin–Polyak wellposedness by perturbations to the split inverse variational inequality problem. We derive metric characterizations of Levitin–Polyak wellposedness by perturbations. Under mild conditions, we prove that the Levitin–Polyak well-posedness by perturbations of the split inverse variational inequality problem is equivalent to the existence and uniqueness of its solution.     

Relative isoperimetric inequality and¶linear isoperimetric inequality for minimal submanifolds

We prove an optimal relative isoperimetric inequality

Dynamic programming principle for stochastic control problems driven by general Lévy noise

We extend the proof of the dynamic programming principle (DPP) for standard stochastic optimal control problems driven by general Lévy noise. Under appropriate assumptions, it is shown that the DPP still holds when the state process fails to have any moments at all.     

Characterizations of Levitin–Polyak well-posedness by perturbations for the split variational inequality problem

The purpose of this paper is to investigate Levitin–polyak well-posedness by perturbations of the split variational inequality problem in reflexive Banach spaces. Furi-Vignoli-type characterizations are established for the well-posedness. We prove that the weak generalized Levitin–Polyak well-posedness by perturbations is equivalent to the nonemptiness and boundedness of the solution set of the problem. Finally, we discuss the relations between the Levitin–Polyak well-posedness by perturbations of the split variational inequality problem and the Levitin–Polyak well-posedness by perturbations of the split minimization problem when the split variational inequality problem arises from the split minimization problem.     

Concentration-compactness principles for Moser–Trudinger inequalities: new results and proofs

We are concerned with the best exponent in Concentration-Compactness principles for the borderline case of the Sobolev inequality. We present a new approach, which both yields a rigorous proof of the relevant principle in the standard case when functions vanishing on the boundary are considered, and enables us to deal with functions with unrestricted boundary values.     

The invariance principle for ϕ-mixing sequences

Summary In this paper we investigate the invariance principle for -mixing sequences, satisfying restrictions on the variances which are a weak form of stationarity. No mixing rate is assumed. For -mixing strictly stationary sequences we give a necessary and sufficient condition for the invariance principle.     

Inclusion–exclusion principle for belief functions

The inclusion–exclusion principle is a well-known property in probability theory, and is instrumental in some computational problems such as the evaluation of system reliability or the calculation of the probability of a Boolean formula in diagnosis. However, in the setting of uncertainty theories more general than probability theory, this principle no longer holds in general. It is therefore useful to know for which families of events it continues to hold. This paper investigates this question in the setting of belief functions. After exhibiting original sufficient and necessary conditions for the principle to hold, we illustrate its use on the uncertainty analysis of Boolean and non-Boolean systems in reliability.     

A comparison theorem and an inequality of redhefferf†

For equations of the form w″+B(e^z)w = 0, where B(ζ) is a rational function which is analytic on 0<|ζ|∞, we determine the regions where the bulk of the zeros of a solution must be located. In the special case of the general Mathieu equation, these results complement earlier results of E. Hille (1924) who considered the special case of real Mathieu equations     

Stolarsky’s invariance principle for projective spaces

We show that Stolarsky’s invariance principle, known for point distributions on the Euclidean spheres, can be extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane.     

On the Picard principle for Δ?+?μ

Given a (local) Kato measure?μ on ${{\mathbb{R}^d} \setminus \{0\},\,d \ge 2}$ , let ${{\mathcal H}_0^{\Delta+\mu}(U)}$ be the convex cone of all continuous real solutions u?≥ 0 to the equation Δu?+?u μ?=?0 on the punctured unit ball U satisfying ${\lim_{|x|\to 1} u(x)=0}$ . It is shown that ${{\mathcal H}_0^{\Delta+\mu}(U)\ne \{0\}}$ if and only if the operator ${f\mapsto \int_U G(\cdot,y)f(y)\,d\mu(y)}$ , where G denotes the Green function on U, is bounded on ${\mathcal L^2(U,\mu)}$ and has a norm which is at most one. Moreover, extremal rays in ${{\mathcal H}_0^{\Delta+\mu}(U)}$ are characterized and it is proven that Δ?+?μ satisfies the Picard principle on U, that is, that ${{\mathcal H}_0^{\Delta+\mu}(U)}$ consists of one ray, provided there exists a suitable sequence of shells in U such that, on these shells,?μ is either small or not too far from being radial. Further, it is shown that the verification of the Picard principle can be localized. Several results on L ²-(sub)eigenfunctions and 3G-inequalities which are used in the paper, but may be of independent interest, are proved at the end of the paper.     

Some Lower Bounds in the B. and M. Shapiro Conjecture for Flag Varieties

The B. and M. Shapiro conjecture stated that all solutions of the Schubert Calculus problems associated with real points on the rational normal curve should be real. For Grassmannians, it was proved by Mukhin, Tarasov, and Varchenko. For flag varieties, Sottile found a counterexample and suggested that all solutions should be real under certain monotonicity conditions. In this paper, we compute lower bounds on the number of real solutions for some special cases of the B. and M. Shapiro conjecture for flag varieties, when Sottile’s monotonicity conditions are not satisfied.     

Extremal Lyapunov exponents: an invariance principle and applications

We propose a new approach to analyzing dynamical systems that combine hyperbolic and non-hyperbolic (“center”) behavior, e.g. partially hyperbolic diffeomorphisms. A number of applications illustrate its power.