Hypoelliptic Research Articles

Functional calculus on non-homogeneous operators on nilpotent groups

We study the functional calculus associated with a hypoelliptic left-invariant differential operator mathcal {L} on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator mathcal {L}_0 on the ‘local’ contraction G_0 of G, as well as of the corresponding Rockland operator mathcal {L}_infty on the ‘global’ contraction G_infty of G. We provide asymptotic estimates of the Riesz potentials associated with mathcal {L} at 0 and at infty , as well as of the kernels associated with functions of mathcal {L} satisfying Mihlin conditions of every order. We also prove some Mihlin–Hörmander multiplier theorems for mathcal {L} which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the ‘Plancherel measure’ associated with mathcal {L} from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.

Hardy–Littlewood–Sobolev inequalities for a class of non-symmetric and non-doubling hypoelliptic semigroups

In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order evolution equation which displays some challenging features. In the opening of his 1967 hypoellipticity paper Hörmander discussed a general class of degenerate Ornstein–Uhlenbeck operators that includes Kolmogorov’s as a special case. In this note we combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of Hardy–Littlewood–Sobolev type in the situation when the drift matrix has nonnegative trace. Our work has been influenced by ideas of E. Stein and Varopoulos in the framework of symmetric semigroups. One of our objectives is to show that such ideas can be pushed to successfully handle the present degenerate non-symmetric setting.

Multiple solutions for elliptic equations with quasilinear perturbation

We study elliptic equations with quasilinear perturbation{Δu+12κuΔu2+ε∑i,j=1N(aij(u)Diju+12Dzaij(u)DiuDju)+f(u)+εh(u)=0,inΩ,u=0,on∂Ω, where Ω⊂RN,N≥3 is a smooth bounded domain, κ∈{0,1}, ε is a small parameter. Multiplicity result is proved by elliptic regularization method.

Lp-spectrum of degenerate hypoelliptic Ornstein-Uhlenbeck operators

We describe the spectrum of degenerate hypoelliptic Ornstein-Uhlenbeck operators A=∑i,j=1nqijDij+∑i,j=1nbijxjDi in Lp(Rn), 1≤p<+∞, and in C0(Rn). We show that the spectrum of A is the sum of (−∞,0] and the spectrum of the drift term. Our result gives a complete picture of the spectral properties of Ornstein-Uhlenbeck operators in Lp spaces.

Smoothing properties of fractional Ornstein-Uhlenbeck semigroups and null-controllability

We study fractional hypoelliptic Ornstein-Uhlenbeck operators acting on L2(Rn) satisfying the Kalman rank condition. We prove that the semigroups generated by these operators enjoy Gevrey regularizing effects. Two byproducts are derived from this smoothing property. On the one hand, we prove the null-controllability in any positive time from thick control subsets of the associated parabolic equations posed on the whole space. On the other hand, by using interpolation theory, we get global L2 subelliptic estimates for the these operators.

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.

Geometric microlocal analysis in Denjoy–Carleman classes

A systematic geometric theory for the ultradifferentiable (non-quasianalytic and quasianalytic) wavefront set similar to the well-known theory in the classic smooth and analytic setting is developed. In particular an analogue of Bony's Theorem and the invariance of the ultradifferentiable wavefront set under diffeomorphisms of the same regularity is proven using a Theorem of Dyn'kin about the almost-analytic extension of ultradifferentiable functions. Furthermore we prove a microlocal elliptic regularity theorem for operators defined on ultradifferentiable vector bundles. As an application we show that Holmgren's theorem and several generalizations hold for operators with quasianalytic coefficients.

How to smooth a crinkled map of space-time: Uhlenbeck compactness for L ∞ connections and optimal regularity for general relativistic shock waves by the Reintjes-Temple equations.

We present the authors' new theory of the RT-equations ('regularity transformation' or 'Reintjes-Temple' equations), nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections Γ to optimal regularity, one derivative smoother than the Riemann curvature tensor Riem(Γ). As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at general relativistic shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations by application of elliptic regularity theory in L p spaces. The theory and results announced in this paper apply to arbitrary L ∞ connections on the tangent bundle of arbitrary manifolds , including Lorentzian manifolds of general relativity.

Analyzing the role of the Inf-Sup condition for parameter identification in saddle point problems with application in elasticity imaging

We study the inverse problem of parameter identification in general saddle point problems. For saddle point problems, the use of elliptic regularization is an essential component. Saddle point problems, after discretization, lead to a non-invertible system, whereas the regularized saddle point problems result in an invertible system. Regularization methods, in the context of saddle point problems, have also been used to mitigate the role of the Inf-Sup condition, synonymously, also called the Babuska-Brezzi condition. This work aims to analyze the impact of regularizing the saddle point problem on the inverse problem. We investigate the inverse problem by using the output least-squares objective. To exploit the use of regularization fully, we work under the assumption that the solution map is nonempty. We regularize the saddle point problem and consider a family of optimization problems using the output least-squares objective for the regularized saddle point problem where some noise contaminates the whole data set. We give a complete convergence analysis showing that the optimization problems, given for the regularized output least-squares, approximate the original problem suitably. We also provide the first-order and the second-order adjoint method for the computation of the first-order and the second-order derivatives of the output least-squares objective. We present some heuristic numerical results. In the context of the elasticity imaging inverse problem, we conduct detailed numerical experiments on synthetic data (to study the role of the regularization parameter) as well as on phantom data.

Regularization methods for elliptic quasi-variational inequalities in Banach spaces

ABSTRACT In this paper, we deal with a class of ill-posed quasi-variational inequalities with contaminated data by employing the elliptic regularization in Banach spaces. Firstly, we get an existence result for quasi-variational inequalities and show that the sequence of bounded regularized solutions converges strongly to a solution of the original quasi-variational inequalities under mild coercivity conditions. Next, we give the boundedness of regularized solutions for quasi-variational inequalities. Finally, an example is presented to illustrate our main results.

Quantitative unique continuation for hyperbolic and hypoelliptic equations

We review recent results of the authors concerning quantitative unique continuation estimates for operators with coefficients that are analytic in some (or all the) variables. We describe several applications for wave-like equations, but also equations based on hypoelliptic operators. These proceedings are a survey of the general results in [LL19] together with applications to wave equations [LL16] and to hypoelliptic equations [LL17, LL20b].

Analytic regularity for solutions to sums of squares: an assessment

We present a brief survey on the state of the theory of the real analytic regularity (real analytic hypoellipticity) for the solutions to sums of squares of vector fields satisfying the Hormander condition.

Spectral Stability of the $${\overline{\partial }}$$-Neumann Laplacian: The Kohn–Nirenberg Elliptic Regularization

In this paper we study spectral stability of the $$\bar{\partial }$$ -Neumann Laplacian under the Kohn–Nirenberg elliptic regularization. We obtain quantitative estimates for stability of the spectrum of the $$\bar{\partial }$$ -Neumann Laplacian when either the operator or the underlying domain is perturbed.

The heat equation with rough boundary conditions and holomorphic functional calculus

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded H∞-calculus on weighted Lp-spaces for power weights which fall outside the classical class of Ap-weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data.

On a Non–Local Problem for a Multi–Term Fractional Diffusion-Wave Equation

Abstract This paper deals with the multi-term generalisation of the time-fractional diffusion-wave equation for general operators with discrete spectrum, as well as for positive hypoelliptic operators, with homogeneous multi-point time-nonlocal conditions. Several examples of the settings where our nonlocal problems are applicable are given. The results for the discrete spectrum are also applied to treat the case of general homogeneous hypoelliptic left-invariant differential operators on general graded Lie groups, by using the representation theory of the group. For all these problems, we show the existence, uniqueness, and the explicit representation formulae for the solutions.

Implicit shock tracking using an optimization-based high-order discontinuous Galerkin method

A novel framework for resolving discontinuous solutions of conservation laws, e.g., contact lines, shock waves, and interfaces, using implicit tracking and a high-order discontinuous Galerkin (DG) discretization was introduced in [39]. Central to the framework is an optimization problem whose solution is a discontinuity-aligned mesh and the corresponding high-order approximation to the flow that does not require explicit meshing of the unknown discontinuity surface. The method was shown to deliver highly accurate solutions on coarse, high-order discretizations without nonlinear stabilization and recover optimal convergence rates O(hp+1) even for problems with discontinuous solutions. This work extends the implicit tracking framework such that robustness is improved and convergence accelerated. In particular, we introduce an improved formulation of the central optimization problem and an associated sequential quadratic programming (SQP) solver. The new error-based objective function penalizes violation of the DG residual in an enriched test space and is shown to have excellent tracking properties. The SQP solver simultaneously converges the nodal coordinates of the mesh and DG solution to their optimal values and is equipped with a number of features to ensure robust, fast convergence: Levenberg-Marquardt approximation of the Hessian with weighted elliptic regularization, backtracking line search based on the ℓ1 merit function, and rigorous convergence criteria. We use the proposed method to solve a range of inviscid conservation laws of varying difficulty. We show the method is able to deliver accurate solutions on coarse, high-order meshes and the SQP solver is robust and usually able to drive the first-order optimality system to tight tolerances.

Nonexistence Results for the Hyperbolic-Type Equations on Graded Lie Groups

In this paper, we deal with systems of wave and pseudo-hyperbolic equations. Some semilinear equations for hypoelliptic operators on $${\mathbb {R}}^{n}$$ , Heisenberg groups, stratified Lie groups and graded Lie groups are studied. In particular, we obtain nonexistence results for nonlinear hyperbolic and pseudo-hyperbolic equations and systems on graded Lie groups. Also, we show Kato-type exponents for systems of pseudo-hyperbolic equations for Rockland operators on graded Lie groups.

QUADRATIC DIFFERENTIAL EQUATIONS: PARTIAL GELFAND–SHILOV SMOOTHING EFFECT AND NULL-CONTROLLABILITY

AbstractWe study the partial Gelfand–Shilov regularizing effect and the exponential decay for the solutions to evolution equations associated with a class of accretive non-selfadjoint quadratic operators, which fail to be globally hypoelliptic on the whole phase space. By taking advantage of the associated Gevrey regularizing effects, we study the null-controllability of parabolic equations posed on the whole Euclidean space associated with this class of possibly non-globally hypoelliptic quadratic operators. We prove that these parabolic equations are null-controllable in any positive time from thick control subsets. This thickness property is known to be a necessary and sufficient condition for the null-controllability of the heat equation posed on the whole Euclidean space. Our result shows that this geometric condition turns out to be a sufficient one for the null-controllability of a large class of quadratic differential operators.

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

We define globally hypoelliptic smooth $ \mathbb R^k $ actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When $ k = 1 $, strong global rigidity is conjectured by Greenfield-Wallach and Katok: every globally hypoelliptic flow is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces [9]. In this paper we conjecture that among homogeneous $ \mathbb R^k $ actions ($ k\ge 2 $) on homogeneous spaces globally hypoelliptic actions exist only on nilmanifolds. We obtain a partial result towards this conjecture: we show non-existence of globally hypoelliptic $ \mathbb R^2 $ actions on homogeneous spaces $ G/\Gamma $, with at least one quasi-unipotent generator, where $ G = SL(n, \mathbb R) $. We also show that the same type of actions on solvmanifolds are smoothly conjugate to homogeneous actions on nilmanifolds.

On the Uniform Convergence of the Weak Galerkin Finite Element Method for a Singularly-Perturbed Biharmonic Equation

For the biharmonic equation or this singularly-perturbed biharmonic equation, lower order nonconforming finite elements are usually used. It is difficult to construct high order $$C^1$$ conforming, or nonconforming elements, especially in 3D. A family of any quadratic or higher order weak Galerkin finite elements is constructed on 2D polygonal grids and 3D polyhedral grids for solving the singularly-perturbed biharmonic equation. The optimal order of convergence, up to any order the smooth solution can have, is proved for this method, in a discrete $$H^2$$ norm. Under a full elliptic regularity $$H^4$$ assumption, the $$L^2$$ convergence achieves the optimal order as well, in 2D and 3D. Numerical tests are presented verifying the theory.