Small Lipschitz Constant Research Articles

Backward stochastic Volterra integral equations with time delayed generators

AbstractIn this paper, we study backward stochastic Volterra integral equations of type‐I with time delayed generators. We derive an existence and uniqueness result under a small time horizon or a small Lipschitz constant. We provide with two example that this condition is necessary. However, this equation has a unique solution for arbitrary time horizon and Lipschitz constant. Finally, with additive assumptions, we derive a path continuity property.

Time-delayed generalized BSDEs

We prove the existence and uniqueness of the solution of a BSDE with time-delayed generators in the small delay setting (or equivalently small Lipschitz constant), which employs the Stieltjes integral with respect to an increasing continuous stochastic process. Moreover, we obtain a result of continuity of the solution with regard to the increasing process, assuming only uniform convergence, but not in variation. We also prove the existence in the case of arbitrary delay by imposing monotonicity and linearity on generators. Lastly, we provide an application of the theoretical framework within an insurance based example.

Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant

Let Omega subset {mathbb {R}}^d be a C^1 domain or, more generally, a Lipschitz domain with small Lipschitz constant and A(x) be a d times d uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume u is harmonic in Omega , or with greater generality u solves {text {div}}(A(x)nabla u)=0 in Omega , and u vanishes on Sigma = partial Omega cap B for some ball B. We study the dimension of the singular set of u in Sigma , in particular we show that there is a countable family of open balls (B_i)_i such that u|_{B_i cap Omega } does not change sign and K backslash bigcup _i B_i has Minkowski dimension smaller than d-1-epsilon for any compact K subset Sigma . We also find upper bounds for the (d-1)-dimensional Hausdorff measure of the zero set of u in balls intersecting Sigma in terms of the frequency. As a consequence, we prove a new unique continuation principle at the boundary for this class of functions and show that the order of vanishing at all points of Sigma is bounded except for a set of Hausdorff dimension at most d-1-epsilon .

Ulam–Hyers stability and exponentially dichotomic equations in Banach spaces

For finite-dimensional linear differential systems with bounded coefficients, we prove that their exponential dichotomy on R is equivalent to their Ulam–Hyers stability on R with uniqueness. We also consider abstract non-autonomous evolution equations which are exponentially bounded and exponentially dichotomic and prove that Ulam–Hyers stability with uniqueness is maintained when perturbing them with a nonlinear term having a sufficiently small Lipschitz constant.

Global finite-dimensional observer-based stabilization of a semilinear heat equation with large input delay

We study global finite-dimensional observer-based stabilization of a semilinear 1D heat equation with globally Lipschitz semilinearity in the state variable. We consider Neumann actuation and point measurement. Using dynamic extension and modal decomposition we derive nonlinear ODEs for the modes of the state. We propose a controller that is based on a nonlinear finite-dimensional Luenberger observer. Our Lyapunov H 1 -stability analysis leads to LMIs, which are shown to be feasible for a large enough observer dimension and small enough Lipschitz constant. Next, we consider the case of a constant input delay r > 0 . To compensate the delay, we introduce a chain of M sub-predictors that leads to a nonlinear closed-loop ODE system, coupled with nonlinear infinite-dimensional tail ODEs. We provide LMIs for H 1 -stability and prove that for any r > 0 , the LMIs are feasible provided M and the observer dimension N are large enough and the Lipschitz constant is small enough. Numerical examples demonstrate the efficiency of the proposed approach.

Parabolic rectifiability, tangent planes and tangent measures

We define rectifiability in \(\mathbb{R}^{n}\times\mathbb{R}\) with a parabolic metric in terms of \(C^1\) graphs and Lipschitz graphs with small Lipschitz constants and we characterize it in terms of approximate tangent planes and tangent measures. We also discuss relations between the parabolic rectifiability and other notions of rectifiability.

Gradient estimates in anisotropic Lorentz spaces to general elliptic equations of p-growth

We prove a global mixed-norm gradient estimate in the framework of anisotropic Lorentz spaces to general elliptic equations of p-growth under weak regularity data. It is assumed that the principle coefficients are small BMO seminorm in x-variable, and the boundary of underlying domain is locally C0,1-smooth with a small Lipschitz constant.

On Concavity of Solutions of the Nonlinear Poisson Equation

We consider the nonlinear Poisson equation $-\Delta u = f(u)$ in domains $\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions on $\partial \Omega$. We show (for monotonically increasing concave $f$ with small Lipschitz constant) that if $D^2 u$ is negative semi-definite on the boundary, then $u$ is concave. A conjecture of Saint Venant from 1856 (proven by Polya in 1948) is that among all domains $\Omega$ of fixed measure, the solution of $-\Delta u =1$ assumes its largest maximum when $\Omega$ is a ball. We extend this to $-\Delta u =f(u)$ for monotonically increasing $f$ with small Lipschitz constant.

Unique Continuation at the Boundary for Harmonic Functions in C1 Domains and Lipschitz Domains with Small Constant

AbstractLet be a domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if u is a function harmonic in and continuous in , which vanishes in a relatively open subset ; moreover, the normal derivative vanishes in a subset of with positive surface measure; then u is identically zero. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Elliptic Theory for Sets with Higher Co-dimensional Boundaries

Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let Γ ⊂ R n \Gamma \subset \mathbb {R}^n be an Ahlfors regular set of dimension d > n − 1 d>n-1 (not necessarily integer) and Ω = R n ∖ Γ \Omega = \mathbb {R}^n \setminus \Gamma . Let L = − div ⁡ A ∇ L = - \operatorname {div} A\nabla be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix A A are bounded from above and below by a multiple of dist ⁡ ( ⋅ , Γ ) d + 1 − n \operatorname {dist}(\cdot , \Gamma )^{d+1-n} . We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the Hölder continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or L p L^p estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to L L , establish its doubling property, non-degeneracy, change-of-the-pole formulas, and, finally, the comparison principle for local solutions. In another article to appear, we will prove that when Γ \Gamma is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator L L for which the harmonic measure given here is absolutely continuous with respect to the d d -Hausdorff measure on Γ \Gamma and vice versa. It thus extends Dahlberg’s theorem to some sets of codimension higher than 1.

Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law

Abstract We study the numerical approximation of the invariant measure of a viscous scalar conservation law, one-dimensional and periodic in the space variable and stochastically forced with a white-in-time but spatially correlated noise. The flux function is assumed to be locally Lipschitz continuous and to have at most polynomial growth. The numerical scheme we employ discretizes the stochastic partial differential equation (SPDE) according to a finite-volume method in space and a split-step backward Euler method in time. As a first result, we prove the well posedness as well as the existence and uniqueness of an invariant measure for both the semidiscrete and the split-step scheme. Our main result is then the convergence of the invariant measures of the discrete approximations, as the space and time steps go to zero, towards the invariant measure of the SPDE, with respect to the second-order Wasserstein distance. We investigate rates of convergence theoretically, in the case where the flux function is globally Lipschitz continuous with a small Lipschitz constant, and numerically for the Burgers equation.

Ulam-Hyers stability and exponentially stable evolution equations in Banach spaces

We show that uniformly exponentially stable abstract linear evolution equations are Ulam-Hyers stable on [a,\infty). Moreover, we prove that this property is maintained when perturbing this type of equations with a nonlinear term having a small Lipschitz constant. These results complement the literature on Ulam-Hyers stability, a special relation having with some works of I. A. Rus.

A general approach to nonautonomous shadowing for nonlinear dynamics

Given a nonautonomous and nonlinear differential equation(0.1)x′=A(t)x+f(t,x)t≥0, on an arbitrary Banach space X, we formulate very general conditions for the associated linear equation x′=A(t)x and for the nonlinear term f:[0,+∞)×X→X under which the above system satisfies an appropriate version of the shadowing property. More precisely, we require that x′=A(t)x admits a very general type of dichotomy, which includes the classical hyperbolic behavior as a very particular case. In addition, we require that f is Lipschitz in the second variable with a sufficiently small Lipschitz constant. Our general framework enables us to treat various settings in which no shadowing result has been previously obtained. Moreover, we are able to recover and refine several known results. We also show how our main results can be applied to the study of the shadowing property for higher order differential equations. Finally, we conclude the paper by presenting discrete time versions of our results.

Shadowing for nonautonomous difference equations with infinite delay

We formulate sufficient conditions under which a large class of semilinear nonautonomous difference equations with infinite delay is Hyers–Ulam stable. These conditions require that the nonautonomous linear part admits an exponential dichotomy and the nonlinear perturbations are uniformly Lipschitz continuous with a sufficiently small Lipschitz constant. In the more general case when the linear part admits a shifted exponential dichotomy, we are able to provide sufficient conditions for the existence of a certain weighted form of the shadowing property.

Hyers–Ulam Stability for Nonautonomous Semilinear Dynamics on Bounded Intervals

We prove that any nonautonomous semilinear dynamics on a bounded interval with the property that the nonlinear part is Lipschitz with a sufficiently small Lipschitz constant exhibits Hyers–Ulam stability.

Impulsive consensus of one-sided Lipschitz nonlinear multi-agent systems with Semi-Markov switching topologies

Many real systems involve not only parameter changes but also sudden variations in environmental conditions, which often causes unpredictable topologies switching. This paper investigates the impulsive consensus problem of the one-sided Lipschitz nonlinear multi-agent systems (MASs) with Semi-Markov switching topologies. Different from the existing modeling methods of the Markov chain, the Semi-Markov chain is adopted to describe this kind of randomly occurring changes reasonably. To cope with the communication and control cost constraints in the multi-agent systems, the distributed impulsive control method is applied to address the leader–follower consensus problem. Beyond that, to obtain a wider nonlinear application range, the one-sided condition is delicately developed to the controller design, and the results are different from the ones obtained in the traditional method with the Lipschitz condition (note that the existing results are usually only applicable to the case with small Lipschitz constant). Based on the characteristics of cumulative distribution functions, the theory of Lyapunov-like function and impulsive differential equation, the asymptotically mean square consensus of multi-agent systems is maintained with the proposed impulsive control protocol. Finally, an explanatory simulation is presented to validate the correctness of the proposed approach conclusively.

Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients

In this paper, we give a short proof of the Lorentz estimates for gradients of very weak solutions to the linear parabolic equations with the Muckenhoupt class A q -weights u t − div ( A ( x , t ) ∇ u ) = div ( F ) , in a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 , where A has a small mean oscillation, and Ω is a Lipchistz domain with a small Lipschitz constant.

General p-curl systems and duality mappings on Sobolev spaces for Maxwell equations

We study a general p-curl system arising from a model of type-II superconductors. We show several trace theorems that hold on either a Lipschitz domain with small Lipschitz constant or on a C^{1,1} domain. Certain duality mappings on related Sobolev spaces are computed and used to establish surjectivity results for the p-curl system. We also solve a nonlinear boundary value problem for a general p-curl system on a C^{1,1} domain and provide a variational characterization of the first eigenvalue of the p-curl operator.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/116/abstr.html

On the W 2 p Estimate for Oblique Derivative Problem in Lipschitz Domains

Abstract The aim of this paper is to establish $W^2_p$ estimate for non-divergence form 2nd-order elliptic equations with the oblique derivative boundary condition in domains with small Lipschitz constants. Our result generalizes those in [ 16] and [ 17], which work for $C^{1,\alpha }$ domains with $\alpha> 1-1/p$. As an application, we also obtain a solvability result. An extension to fully nonlinear elliptic equations with the oblique derivative boundary condition is also discussed.

Elliptic quasi-variational inequalities under a smallness assumption: uniqueness, differential stability and optimal control

We consider a quasi-variational inequality governed by a moving set. We employ the assumption that the movement of the set has a small Lipschitz constant. Under this requirement, we show that the quasi-variational inequality has a unique solution which depends Lipschitz-continuously on the source term. If the data of the problem is (directionally) differentiable, the solution map is directionally differentiable as well. We also study the optimal control of the quasi-variational inequality and provide necessary optimality conditions of strongly stationary type.