Inequalities For Vector Fields Research Articles

Soap bubbles and convex cones: optimal quantitative rigidity

We consider a class of recent rigidity results in a convex cone Σ ⊆ R N \Sigma \subseteq \mathbb {R}^N . These include overdetermined Serrin-type problems for a mixed boundary value problem relative to Σ \Sigma , Alexandrov’s soap bubble-type results relative to Σ \Sigma , and Heintze-Karcher’s inequality relative to Σ \Sigma . Each rigidity result is obtained here by means of a single integral identity and holds true under weak integral overdeterminations in possibly non-smooth cones. Optimal quantitative stability estimates are obtained in terms of an L 2 L^2 -pseudodistance. In particular, the optimal stability estimate for Heintze-Karcher’s inequality is new even in the classical case Σ = R N \Sigma = \mathbb {R}^N . Stability bounds in terms of the Hausdorff distance are also provided. Several new results are established and exploited, including a new Poincaré-type inequality for vector fields whose normal components vanish on a portion of the boundary and an explicit (possibly weighted) trace theory – relative to the cone Σ \Sigma – for harmonic functions satisfying a homogeneous Neumann condition on the portion of the boundary contained in ∂ Σ \partial \Sigma . We also introduce new notions of uniform interior and exterior sphere conditions relative to the cone Σ ⊆ R N \Sigma \subseteq \mathbb {R}^N , which allow to obtain (via barrier arguments) uniform lower and upper bounds for the gradient in the mixed boundary value-setting. In the particular case Σ = R N \Sigma =\mathbb {R}^N , these conditions return the classical uniform interior and exterior sphere conditions (together with the associated classical gradient bounds of the Dirichlet setting).

Weighted $p(\cdot)$-Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications

Weighted $p(\cdot)$-Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications

A New Proof of Gaffney’s Inequality for Differential Forms on Manifolds-with-Boundary: The Variational Approach à La Kozono-Yanagisawa

Let (\({\cal M}\), g0) be a compact Riemannian manifold-with-boundary. We present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over \({\cal M}\), via a variational approach à la Kozono-Yanagisawa [Lr-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains, Indiana Univ. Math. J. 58 (2009), 1853–1920], combined with global computations based on the Bochner technique.

Fractional Korn and Hardy-type inequalities for vector fields in half space

We prove a fractional Hardy-type inequality for vector fields over the half space based on a modified fractional semi-norm. A priori, the modified semi-norm is not known to be equivalent to the standard fractional semi-norm and in fact gives a smaller norm, in general. As such, the inequality we prove improves the classical fractional Hardy inequality for vector fields. We will use the inequality to establish the equivalence of a space of functions (of interest) defined over the half space with the classical fractional Sobolev spaces, which amounts to prove a fractional version of the classical Korn’s inequality.

Whole-Body Control With (Self) Collision Avoidance Using Vector Field Inequalities

This work uses vector field inequalities (VFI) to prevent robot self-collisions and collisions with the workspace. Differently from previous approaches, the method is suitable for both velocity and torque-actuated robots. We propose a new distance function and its corresponding Jacobian in order to generate a VFI to limit the angle between two Pl\"ucker lines. This new VFI is used to prevent both undesired end-effector orientations and violation of joints limits. The proposed method is evaluated in a realistic simulation and on a real humanoid robot, showing that all constraints are respected while the robot performs a manipulation task.

Dynamic Active Constraints for Surgical Robots Using Vector-Field Inequalities

Robotic assistance allows surgeons to perform dexterous and tremor-free procedures, but robotic aid is still underrepresented in procedures with constrained workspaces, such as deep brain neurosurgery and endonasal surgery. In these procedures, surgeons have restricted vision to areas near the surgical tooltips, which increases the risk of unexpected collisions between the shafts of the instruments and their surroundings. In this work, our vector-field-inequalities method is extended to provide dynamic active-constraints to any number of robots and moving objects sharing the same workspace. The method is evaluated with experiments and simulations in which robot tools have to avoid collisions autonomously and in real-time, in a constrained endonasal surgical environment. Simulations show that with our method the combined trajectory error of two robotic systems is optimal. Experiments using a real robotic system show that the method can autonomously prevent collisions between the moving robots themselves and between the robots and the environment. Moreover, the framework is also successfully verified under teleoperation with tool-tissue interactions.

Fractional integral operator for L1 vector fields and its applications

This paper studies fractional integral operator for vector fields in weighted L1. Using the estimates on fractional integral operator and Stein-Weiss inequalities, we can give a new proof for a class of Caffarelli-Kohn-Nirenberg inequalities and establish new div-curl inequalities for vector fields.

Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition in variable exponent Sobolev spaces

In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincare inequalities for vector fields satisfying Hormander’s condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian. Our results are only stated and proved for vector fields satisfying Hormander’s condition, but they also hold for Grushin vector fields as well with obvious modifications.

New identity and Korn's inequalities on a surface

We establish an identity satisfied by smooth-enough vector fields defined on a surface S⊂R3 with values in R3. As consequences of this identity, we establish several new Korn inequalities for vector fields that vanish on the entire boundary of this surface.

Hardy-type inequalities for vector fields with vanishing tangential components

Hardy-type inequalities for vector fields with vanishing tangential components

Limiting Sobolev inequalities for vector fields and canceling linear differential operators

The estimate \|{D^{k-1}u}\|_{L^{n/(n-1)}} \le \|{A(D)u}\|_{L^1} is shown to hold if and only if A(D) is elliptic and canceling. Here A(D) is a homogeneous linear differential operator A(D) of order k on \mathbb R^n from a vector space V to a vector space E . The operator A(D) is defined to be canceling if \bigcap_{\xi \in \mathbb R^n \setminus \{0\}} A(\xi)[V]=\{0\}. This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential operator L(D) of order k on \mathbb R^n from a vector space E to a vector space F is introduced. It is proved that L(D) is cocanceling if and only if for every f \in L^1(\mathbb R^n; E) such that L(D)f=0 , one has f \in \dot{W}^{-1, n/(n-1)}(\mathbb R^n; E) . The results extend to fractional and Lorentz spaces and can be strengthened using some tools of J. Bourgain and H. Brezis.

Hardy-Sobolev inequalities for vector fields and canceling linear differential operators

Given a homogeneous k-th order differential operator $A (D)$ on $\mathbb{R}^n$ between two finite dimensional spaces, we establish the Hardy inequality $$\int_{\mathbb{R}^n} \frac{\lvert D^{k-1}u\rvert}{\lvert x \rvert} \,\mathrm{d} x \leq C \int_{\mathbb{R}^n} \lvert A(D)u\rvert $$ and the Sobolev inequality $$\lVert D^{k-n} u\rVert_{L^{\infty}(\mathbb{R}^n)}\leq C \int_{\mathbb{R}^n} \lvert A(D)u\rvert $$ when $A(D)$ is elliptic and satisfies a recently introduced cancellation property. We also study the necessity of these two conditions.

Formula omitted]-theory for vector potentials and Sobolevʼs inequalities for vector fields

In a three-dimensional bounded possibly multiply-connected domain, we prove the existence and uniqueness of vector potentials in L p -theory, associated with a divergence-free function and satisfying some boundary conditions. We also present some results concerning scalar potentials and weak vector potentials. Furthermore, various Sobolev-type inequalities are given.

An estimate for the distance of a complex valued Sobolev function defined on the unit disc to the class of holomorphic functions

Let ƒ : B → ℂ denote a Sobolev function of class defined on the unit disc. We show that the distance of ƒ to the class of all holomorphic functions measured in the norm of the space is bounded by the Lp-norm of theWirtinger derivative . As a consequence we obtain a Korn type inequality for vector fields .

Weighted multilinear Poincare inequalities for vector fields of Hoermander type

Weighted multilinear Poincare inequalities for vector fields of Hoermander type

$L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains

We show that every L r -vector field on Ω can be uniquely decomposed into two spaces with scalar and vector potentials, and the harmonic vector space via operators rot and div, where Ω is a bounded domain in R 3 with the smooth boundary ∂Ω. Our decomposition consists of two kinds of boundary conditions such as u . v |∂Ω = 0 and u x ν|∂Ω = 0, where ν denotes the unit outward normal to ∂Ω. Our results may be regarded as an extension of the well-known de Rham-Hodge-Kodaira decomposition of C ∞ -forms on compact Riemannian manifolds into L r -vector fields on Ω. As an application, the generalized Biot-Savart law for the incompressible fluids in Ω is obtained. Furthermore, various bounds of u in L r for higher derivatives are given by means of rot u and div u.

Morse-type inequalities for dynamical systems and the Witten Laplacian

We provide an analytic proof of Morse-type inequalities for vector fields determining a Morse decomposition with normally hyperbolic dynamics. In the demonstration we reduce the problem to the gradient case using a Morse–Bott Lyapunov function and then apply Schrödinger operator techniques. This yields an explicit construction of the cohomology complex of the manifold in terms of the invariant sets of the Morse decomposition associated with the vector field.

THE ŁOJASIEWICZ EXPONENT AT AN EQUILIBRIUM POINT OF A STANDARD CNN IS 1/2

In the sixties, Łojasiewicz proved a fundamental inequality for vector fields defined by the gradient of an analytic function, which gives a lower bound on the norm of the gradient in a neighborhood of a (possibly) non-isolated critical point. The inequality involves a number belonging to (0, 1), which depends on the critical point, and is known as the Łojasiewicz exponent. In this paper, a class of vector fields which are defined on a hypercube of ℝn, is considered. Each vector field is the gradient of a quadratic function in the interior of the hypercube, however it is discontinuous on the boundary of the hypercube. An extended Łojasiewicz inequality for this class of vector fields is proved, and it is also shown that the Łojasiewicz exponent at each point where a vector field vanishes is equal to 1/2. The considered fields include a class of vector fields which describe the dynamics of the output trajectories of a standard Cellular Neural Network (CNN) with a symmetric neuron interconnection matrix. By applying the extended Łojasiewicz inequality, it is shown that each output trajectory of a symmetric CNN has finite length, and as a consequence it converges to an equilibrium point. Furthermore, since the Łojasiewicz exponent at each equilibrium point of a symmetric CNN is equal to 1/2, it follows that each (state) trajectory, and each output trajectory, is exponentially convergent toward an equilibrium point, and this is true even in the most general case where the CNN possesses infinitely many nonisolated equilibrium points. In essence, the obtained results mean that standard symmetric CNNs enjoy the property of absolute stability of exponential convergence.

A Riemannian version of Korn's inequality

We prove a Korn type inequality for vector fields on a Riemann manifold. This inequality includes the special cases proved in the literature for domains in $\mathbb{R}^3$ . If the domain is convex, we can considerably weaken the needed assumption on the boundary values.

Novikov-type inequalities for vector fields with non-isolated zero points

Novikov-type inequalities for vector fields with non-isolated zero points