Search

In 2006, Arnold, Falk, and Winther developed finite element exterior calculus, using the language of differential forms to generalize the Lagrange, Raviart–Thomas, Brezzi–Douglas–Marini, and Nédélec finite element spaces for simplicial triangulations. In a recent paper, Licht asks whether, on a single simplex, one can construct bases for these spaces that are invariant with respect to permuting the vertices of the simplex. For scalar fields, standard bases all have this symmetry property, but for vector fields, this question is more complicated: Such invariant bases may or may not exist, depending on the polynomial degree of the element. In dimensions two and three, Licht constructs such invariant bases for certain values of the polynomial degree r, and he conjectures that his list is complete, that is, that no such basis exists for other values of r. In this paper, we show that Licht's conjecture is true in dimension two. However, in dimension three, we show that Licht's ideas can be extended to give invariant bases for many more values of r; we then show that this new larger list is complete. Along the way, we develop a more general framework for the geometric decomposition ideas of Arnold, Falk, and Winther. [ABSTRACT FROM AUTHOR]

We study differential complexes of Kolmogorov–Alexander–Spanier type on metric measure spaces associated with unbounded non-local operators, such as operators of fractional Laplacian type. We define Hilbert complexes, observe invariance properties and obtain self-adjoint non-local analogues of Hodge Laplacians. For d-regular measures and operators of fractional Laplacian type we provide results on removable sets in terms of Hausdorff measures. We prove a Mayer–Vietoris principle and a Poincaré lemma and verify that in the compact Riemannian manifold case the deRham cohomology can be recovered. [ABSTRACT FROM AUTHOR]

We derive several properties of the heat equation with the Hodge operator associated with the Rumin’s complex on Heisenberg groups and prove several properties of the fundamental solution. As an application, we use the heat kernel for Rumin’s differential forms to construct a Calderón reproducing formula on Rumin’s forms.

On an n-dimensional smooth manifold, we consider higher-order normals of two types, i.e., the spaces that complement the tangent space of orders 1 or r − 1 to the tangent space of order r. We prove that the derivatives of some basic vectors in the direction of the given first-order (second-order) basis vectors are equal to the values on these vectors of the first-order (second-order) differentials of the first vectors. Using the differentials of basic tangent vectors of the first and second orders, we construct mappings from the set of first-order tangent vectors to the set of second- and third-orders normal vectors. Also, we introduce mappings that generate horizontal second- and third-order vectors for the canonical first- and second- order affine connections, respectively. [ABSTRACT FROM AUTHOR]

Let (N, g) be a complete noncompact Riemannian manifold with Ricci curvature bounded from below. In this paper, we study the gradient estimates of positive solutions to a class of nonlinear elliptic equations Δ u (x) + a (x) u (x) log u (x) + b (x) u (x) = 0 on N where a(x) is C2-smooth while b(x) is C1 and its parabolic counterparts (Δ − ∂ ∂ t ) u (x , t) + a (x , t) u (x , t) log u (x , t) + b (x , t) u (x , t) = 0 on N × [0, ∞) where a(x, t) and b(x, t) are C2 with respect to x ∊ N while are C1 with respect to the time t. In contrast with lots of similar results, here we do not assume the coefficients of equations are constant, so our results can be viewed as extensions to several classical estimates. [ABSTRACT FROM AUTHOR]

For the first main result of this note, we give a new characterization of Nakano q-positivity for the Riemannian flat vector bundle in the sense of L 2 estimate for the d operator. We also give a new characterization of the positivity of the curvature operator associated to the Riemannian metric on manifold in a local version in the sense of L 2 estimate. Then we prove two results parallel to Liu–Yang–Zhou’s answer to Lempert’s question about Hermitian holomorphic vector bundle. We also study the boundary convexity of relative compact domain in certain Riemann manifold with convexity restriction. [ABSTRACT FROM AUTHOR]

In this article, we establish a Gaffney type inequality, in Wℓ,p{W}^{\ell ,p}-Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind noncompact manifolds). Here, p∈]1,∞[p\in ]1,\infty {[} and ℓ=1,2\ell =1,2 depending on the order of the differential form we are considering. The proof relies on the structure of the Rumin’s complex of differential forms in contact manifolds, on a Sobolev-Gaffney inequality proved by Baldi-Franchi in the setting of the Heisenberg groups and on some geometric properties that can be proved for sub-Riemannian contact manifolds with bounded geometry.

A property of weak stationarity of a matrix valued differential form at superdensity points of its vanishing set is proved. This result is then applied in the context of the Maurer–Cartan equation. [ABSTRACT FROM AUTHOR]

The present work has as a first goal to extend the previous results in Cazacu et al. (J Funct Anal 283(10):109659, 2022) to weighted uncertainty principles with nontrivial radially symmetric weights applied to curl-free vector fields. Part of these new inequalities generalize the family of Caffarelli-Kohn-Nirenberg (CKN) inequalities studied by Catrina and Costa in Catrina and Costa (J Differ Equ 246(1):164–182, 2009) from scalar fields to curl-free vector fields. We will apply a new representation of curl-free vector fields developed by Hamamoto in Hamamoto and Takahashi (J Funct Anal 280(1):108790, 2021). The newly obtained results are also sharp and minimizers are completely described. Secondly, we prove new sharp second order interpolation functional inequalities for scalar fields with radial weights generalizing the previous results in Cazacu et al. (J Funct Anal 283(10):109659, 2022). We apply new factorization methods being inspired by our recent work (Cazacu et al. in J Differ Equ 302, 533–549, 2021). The main novelty in this case is that we are able to find a new independent family of minimizers based on the solutions of Kummer's differential equations. We point out that the two types of weighted inequalities under consideration (first order inequalities for curl-free vector fields vs. second order inequalities for scalar fields) represent independent families of inequalities unless the weights are trivial. [ABSTRACT FROM AUTHOR]

We develop a theory of finite element systems, for the purpose of discretizing sections of vector bundles, in particular those arising in the theory of elasticity. In the presence of curvature, we prove a discrete Bianchi identity. In the flat case, we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress–displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility conditions. Cochains with coefficients in rigid motions are given a key role in relating continuous and discrete elasticity complexes. [ABSTRACT FROM AUTHOR]

In this paper, via a new Hardy type inequality, we establish some cohomology vanishing theorems for free boundary compact submanifolds M n with n ≥ 2 immersed in the Euclidean unit ball B n + k under one of the pinching conditions | Φ | 2 ≤ C , | A | 2 ≤ C ~ , or | Φ | ≤ R (p , | H |) , where A (Φ) is the (traceless) second fundamental form, H is the mean curvature, C , C ~ are positive constants and R(p, |H|) is a positive function. In particular, we remove the condition on the flatness of the normal bundle, solving the first question, and partially answer the second question on optimal pinching constants proposed by Cavalcante, Mendes and Vitório. [ABSTRACT FROM AUTHOR]

Let M be a smooth manifold of dimension 2n, and let OM be the dense open subbundle in ∧2T*M of 2-covectors of maximal rank. The algebra of Diff M -invariant smooth functions of first order on OM is proved to be isomorphic to the algebra of smooth Sp(Ωx)-invariant functions on , x being a fixed point in M , and Ωx a fixed element in (OM)x. The maximum number of functionally independent invariants is computed. [ABSTRACT FROM AUTHOR]

In this short note, polynomial growth ancient solutions to the heat equation of the Hodge Laplacian on a complete manifold are studied. Under the assumption that the manifold has polynomial volume growth, it is shown that such solutions can be explicitly expressed into a polynomial of time t with time independent forms as coefficients. As a result, it is concluded that the dimension of the space of all such solutions with a fixed growth order is necessarily bounded by that of the corresponding space of harmonic forms [ABSTRACT FROM AUTHOR]

This note provides an affirmative answer to a question of Viterbo concerning the existence of nondiffeomorphic contact forms that share the same Reeb vector field. Starting from an observation by Croke–Kleiner and Abbondandolo that such contact forms define the same total volume, we discuss various related issues for the wider class of geodesible vector fields. In particular, we define an Euler class of a geodesible vector field in the associated basic cohomology and give a topological characterisation of vector fields with vanishing Euler class. We prove the theorems of Gauß–Bonnet and Poincaré–Hopf for closed, oriented 2-dimensional orbifolds using global surfaces of section and the volume determined by a geodesible vector field. This volume is computed for Seifert fibred 3-manifolds and for some transversely holomorphic flows. [ABSTRACT FROM AUTHOR]

We prove that if P(D) is some constant coefficient partial differential operator and f is a scalar field such that P(D)f vanishes in a given open set, then the integrals of f over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields. [ABSTRACT FROM AUTHOR]

Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension n, arbitrary polynomial degree r, and arbitrary differential form degree k. The study of finite element exterior calculus began with the P r Λ k and P r - Λ k families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between P r Λ k and P ˚ r + k + 1 - Λ n - k and between P r - Λ k and P ˚ r + k Λ n - k . In this article, we show that this duality relationship is, in essence, Hodge duality of differential forms on the standard n-sphere, disguised by a change of coordinates. We remove the disguise, giving explicit correspondences between the P r Λ k , P r - Λ k , P ˚ r Λ k and P ˚ r - Λ k spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between P r Λ k and P ˚ r + k + 1 - Λ n - k and between P r - Λ k and P ˚ r + k Λ n - k . These isomorphisms can be implemented via a simple computation, which we illustrate with examples. [ABSTRACT FROM AUTHOR]

Isotropic hyper-elasticity, altogether with the equilibrium equations and the usual boundary conditions, are formulated directly on the body ℬ, a three-dimensional compact and orientable manifold with boundary equipped with a mass measure. Pearson–Sewell–Beatty pressure potential on the boundary is recovered, using the Poincaré formula. The existence of such a potential requires conditions, which are formulated as non-holonomic constraints on the configuration space. [ABSTRACT FROM AUTHOR]

We study the Littlewood–Paley–Stein functions associated with Hodge-de Rham and Schrödinger operators on Riemannian manifolds. Under conditions on the Ricci curvature, we prove their boundedness on L p for p in some interval (p 1 , 2 ] and make a link to the Riesz Transform. An important fact is that we do not make assumptions of doubling measure or estimates on the heat kernel in this case. For p > 2 , we give a criterion to obtain the boundedness of the vertical Littlewood–Paley–Stein function associated with Schrödinger operators on L p . [ABSTRACT FROM AUTHOR]

We consider a compact Riemann surface R of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate R into two subsets: a connected Riemann surface Σ , and the union O of a finite collection of simply connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on O to the Bergman space of holomorphic forms on Σ is an isomorphism onto the exact one-forms, when restricted to the orthogonal complement of the set of forms on all of R. We then apply this to prove versions of the Plemelj–Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on Σ by elements of Bergman space and Dirichlet space on fixed regions in R containing Σ . [ABSTRACT FROM AUTHOR]

We investigate the problem of Poincaré duality for L p differential forms on bounded subanalytic submanifolds of R n (not necessarily compact). We show that, when p is sufficiently close to 1 then the L p cohomology of such a submanifold is isomorphic to its singular homology. In the case where p is large, we show that L p cohomology is dual to intersection homology. As a consequence, we can deduce that the L p cohomology is Poincaré dual to L q cohomology, if p and q are Hölder conjugate to each other and p is sufficiently large. [ABSTRACT FROM AUTHOR]

In this paper, we consider the tangent bundle of a Riemannian manifold (M, g) with g-natural metrics and among all of these metrics, we specify those with respect to which the unit tangent sphere bundle with induced g-natural metric is totally geodesic. Also, we equip the unit tangent sphere bundle T1M with g-natural contact (paracontact) metric structures, and we show that such structures are totally geodesic K-contact (K-paracontact) submanifolds of TM, if and only if the base manifold (M, g) has positive (negative) constant sectional curvature. Moreover, we establish a condition for g-natural almost contact B-metric structures on T1Msuch that these structures be totally geodesic submanifolds of TM.

Let R be a compact Riemann surface and Γ be a Jordan curve separating R into connected components Σ 1 and Σ 2 . We consider Calderón–Zygmund type operators T (Σ 1 , Σ k) taking the space of L 2 anti-holomorphic one-forms on Σ 1 to the space of L 2 holomorphic one-forms on Σ k for k = 1 , 2 , which we call the Schiffer operators. We extend results of Max Schiffer and others, which were confined to analytic Jordan curves Γ , to general quasicircles, and prove new identities for adjoints of the Schiffer operators. Furthermore, let V be the space of anti-holomorphic one-forms which are orthogonal to L 2 anti-holomorphic one-forms on R with respect to the inner product on Σ 1 . We show that the restriction of the Schiffer operator T (Σ 1 , Σ 2) to V is an isomorphism onto the set of exact holomorphic one-forms on Σ 2 . Using the relation between this Schiffer operator and a Cauchy-type integral involving Green's function, we also derive a jump decomposition (on arbitrary Riemann surfaces) for quasicircles and initial data which are boundary values of Dirichlet-bounded harmonic functions and satisfy the classical algebraic constraints. In particular we show that the jump operator is an isomorphism on the subspace determined by these constraints. [ABSTRACT FROM AUTHOR]