Non-smooth Coefficients Research Articles

Stochastic Partial Differential Equations and Invariant Manifolds in Embedded Hilbert Spaces

AbstractWe provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds for solutions of stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth coefficients. Furthermore, we establish a link between invariance of submanifolds for such SPDEs in Hermite Sobolev spaces and invariance of submanifolds for finite dimensional SDEs. This provides a new method for analyzing stochastic invariance of submanifolds for finite dimensional Itô diffusions, which we will use in order to derive new invariance results for finite dimensional SDEs.

Optimal semiclassical spectral asymptotics for differential operators with non-smooth coefficients

We consider differential operators defined as Friedrichs extensions of quadratic forms with non-smooth coefficients. We prove a two-term optimal asymptotic for the Riesz means of these operators and thereby also reprove an optimal Weyl law under certain regularity conditions. The methods used are then extended to consider more general admissible operators perturbed by a rough differential operator and to obtain optimal spectral asymptotics again under certain regularity conditions. For the Weyl law, we assume that the coefficients are differentiable with Hölder continuous derivatives, while for the Riesz means we assume that the coefficients are twice differentiable with Hölder continuous derivatives.

Green functions and smooth distances.

In the present paper, we show that for an optimal class of elliptic operators with non-smooth coefficients on a 1-sided Chord-Arc domain, the boundary of the domain is uniformly rectifiable if and only if the Green function G behaves like a distance function to the boundary, in the sense that is the density of a Carleson measure, where D is a regularized distance adapted to the boundary of the domain. The main ingredient in our proof is a corona decomposition that is compatible with Tolsa's -number of uniformly rectifiable sets. We believe that the method can be applied to many other problems at the intersection of PDE and geometric measure theory, and in particular, we are able to derive a generalization of the classical F. and M. Riesz theorem to the same class of elliptic operators as above.

Necessity of a logarithmic estimate for hypoellipticity of some degenerately elliptic operators

This paper extends a class of degenerate elliptic operators for which hypoellipticity requires more than a logarithmic gain of derivatives of a solution in every direction. Work of Hoshiro and Morimoto in late 80s characterized a necessity of a super-logarithmic gain of derivatives for hypoellipticity of a sum of a degenerate operator and some non-degenerate operators like Laplacian. The operators we consider are similar, but more general. We examine operators of the form L1(x)+g(x)L2(y), where L1(x) is one-dimensional and g(x) may itself vanish. The argument of the paper is based on spectral projections, analysis of a spectral differential equation, and interpolation between standard and operator-adapted derivatives. Unlike prior results in the literature, our methods do not require explicit analytic construction in the non-degenerate direction. In fact, our result allows non-analytic and even non-smooth coefficients for the non-degenerate part.

Stable reconstruction of discontinuous solutions to the Cauchy problem in steady-state anisotropic heat conduction with non-smooth coefficients

We study the recovery of the missing discontinuous/non-smooth thermal boundary conditions on an inaccessible portion of the boundary of the domain occupied by a solid from Cauchy data prescribed on the remaining boundary assumed to be accessible, in case of stationary anisotropic heat conduction with non-smooth/discontinuous conductivity coefficients. This inverse boundary value problem is ill-posed and, therefore, should be regularized. Consequently, a stabilising method is developed based on a priori knowledge on the solution to this inverse problem and the smoothing feature of the direct problems involved. The original problem is transformed into a control one which reduces to solving an appropriate minimisation problem in a suitable function space. The latter problem is tackled by employing an appropriate variational method which yields a gradient-type iterative algorithm that consists of two direct problems and their corresponding adjoint ones. This approach yields an algorithm designed to approximate specifically merely L2–boundary data in the context of a non-smooth/discontinuous anisotropic conductivity tensor, hence both the notion of solution to the direct problems involved and the convergence analysis of the approximate solutions generated by the algorithm proposed require special attention. The numerical implementation is realised for two-dimensional homogeneous anisotropic solids using the finite element method, whilst regularization is achieved by terminating the iteration according to two stopping criteria.

Stabilization of a coupled wave equation with one localized nonregular fractional Kelvin–Voigt damping with nonsmooth coefficients

In this paper, we study the stabilization of a coupled wave system formed by one localized nonregular fractional viscoelastic damping of Kelvin–Voigt type and localized nonsmooth coefficients. Our main aim is to prove that the ‐semigroup associated with this model is strong stability and decays polynomially at a rate of . By introducing a new system to deal with fractional Kelvin–Voigt damping, we obtain a new equivalent augmented system, so as to show the well‐posedness of the system based on Lumer–Phillips theorem. We achieve the strong stability for the ‐semigroup associated with this new model by using a general criteria of Arendt–Batty and then turn out a polynomial energy decay rate of order with the help of a frequency domain approach.

Analysis of Markov Chain Approximation for Diffusion Models with Nonsmooth Coefficients

In this study, we analyze the convergence of continuous-time Markov chain approximation for one-dimensional diffusions with nonsmooth coefficients. We obtain a sharp estimate of the convergence rate for the value function and its first and second derivatives, which is generally first order. To improve it to second order, we propose two methods: applying the midpoint rule that places all nonsmooth points midway between two neighboring grid points or applying harmonic averaging to smooth the model coefficients. We conduct numerical experiments for various financial applications to confirm the theoretical estimates. We also show that the midpoint rule can be applied to achieve second-order convergence for some jump-diffusion and two-factor short rate models.

Polynomial stabilization of non-smooth direct/indirect elastic/viscoelastic damping problem involving Bresse system

We consider an elastic/viscoelastic problem for the Bresse system with fully Dirichlet or Dirichlet–Neumann–Neumann boundary conditions. The physical model consists of three wave equations coupled in certain pattern. The system is damped directly or indirectly by global or local Kelvin–Voigt damping. Actually, the number of the dampings, their nature of distribution (locally or globally) and the smoothness of the damping coefficient at the interface play a crucial role in the type of the stabilization of the corresponding semigroup. Indeed, using frequency domain approach combined with multiplier techniques and the construction of a new multiplier function, we establish different types of energy decay rate (see the table of stability results at the end). Our results generalize and improve many earlier ones in the literature (see El Arwadi and Youssef, in: On the stabilization of the Bresse beam with Kelvin–Voigt damping, Applied Mathematics and Optimization, pp. 1–27, 2019) and in particular some studies done on the Timoshenko system with Kelvin–Voigt damping (see for instance Ghader and Wehbe in A transmission problem for the Timoshenko system with one local Kelvin–Voigt damping and non-smooth coefficient at the interface, 2020. arXiv:2005.12756; Tian and Zhang in Z Angew Math Phys 68:20, 2017; Zhao et al. Acta Math Sin Engl Ser 21:655–666, 2005).

On solvability in the small and Schauder-type estimates for higher order elliptic equations in grand Sobolev spaces (nonseparable case)

In this work, it is considered an elliptic operator L of mth order with nonsmooth coefficients in a non-standard grand Sobolev space on a bounded domain generated by the norm of the grand Lebesgue space . Under weaker restrictions on the coefficients of the operator, we prove the solvability (in the strong sense) in the small in and also establish interior Schauder-type estimates for these spaces. These estimates play the main role in establishing the Fredholmness of the Dirichlet problem for the equation Lu = f. The considered spaces are not separable, infinitely differentiable functions are not dense in them, and therefore many classical methods concerning Sobolev spaces are not applicable in this case. Nevertheless, it is possible to obtain the corresponding results under the assumption that the coefficients of the principal terms of the operator L are continuous, and the rest are essentially bounded in Ω.

Moment and polynomial bounds for ruin-related quantities in risk theory

We establish a novel numerical quantification method based upon mathematical programming for the ruin-related quantities on collective risk models with fairly general features, such as reflecting, non-homogeneous and multidimensional dynamics with non-smooth coefficients. Rather than seeking approximations with a great deal of computing effort as in the existing numerical approximation methods, the proposed method aims to instantly provide deterministic upper and lower bounds for such quantities as linear combinations of moments or in the explicit polynomial form, without random number generation or sophisticated numerical algorithms. Given that closed-form solutions are rarely available especially in finite-time problems, the upper and lower bounds have great potential to be useful information that can complement approximate solutions obtained from existing numerical methods. We present numerical results throughout to justify the theoretical developments and convergence results so as to illustrate the effectiveness of the proposed method with low computational complexity. We also demonstrate further improvements of the bounds by scaling the problem domain, employing piecewise polynomial test functions and the exponential tempering of the polynomial bases.

Inverse boundary value problems for polyharmonic operators with non-smooth coefficients

<p style='text-indent:20px;'>We consider inverse boundary value problems for polyharmonic operators and in particular, the problem of recovering the coefficients of terms up to order one. The main interest of our result is that it further relaxes the regularity required to establish uniqueness. The proof relies on an averaging technique introduced by Haberman and Tataru for the study of an inverse boundary value problem for a second order operator.</p>

Solvability in the small of $m$-th order elliptic equations in weighted grand Sobolev spaces

In this work we consider the Sobolev spaces generated by the norm of the power weighted grand Lebesgue spaces. It is considered $m$-th order elliptic equation with nonsmooth coefficients on bounded domain in $R^{n} $. This space is nonseparable and by using shift operator we define the separable subspace of it, in which infinitely differentiable functions are dense. The investigation needs to establish boundedness property of convolution regarding weighted grand Lebesgue spaces. Then on scheme of nonweighted case we establish solvability (strong sense) in the small of $m$-th order elliptic equations in power weighted grand Sobolev spaces. Note that in weighted spaces this question is considered for the first time in connection with certain mathematical difficulties.

A transmission problem for the Timoshenko system with one local Kelvin–Voigt damping and non-smooth coefficient at the interface

In this paper, we study the indirect stability of Timoshenko system with local or global Kelvin–Voigt damping, under fully Dirichlet or mixed boundary conditions. Unlike Zhao et al. (Acta Mathematica Sinica Engl Ser 21(3):655–666, 2004), Tian and Zhang (Mathematik und Physik 68(1), 2017), and Liu and Zhang (SIAM J Control Optim 56(6):3919–3947, 2018), in this paper, we consider the Timoshenko system with only one locally or globally distributed Kelvin–Voigt damping D [see System (1.1)]. Indeed, we prove that the energy of the system decays polynomially of type \(t^{-1}\) and that this decay rate is in some sense optimal. The method is based on the frequency domain approach combining with multiplier method.

Stability of a star-shaped network with local Kelvin-Voigt damping and non-smooth coefficient at interface

In this paper, we study the stability problem of a star-shaped network of elastic strings with a local Kelvin-Voigt damping. Under the assumption that the damping coefficients have some singularities near the transmission point, we prove that the semigroup corresponding to the system is polynomially stable and the decay rates depend on the speed of the degeneracy. This result improves the decay rate of the semigroup associated to the system on an earlier result of Z. Liu and Q. Zhang in [21] involving the wave equation with local Kelvin-Voigt damping and non-smooth coefficient at interface.

A weighted discontinuous Galerkin method for diffusion equations with non-smooth coefficients on distorted polygonal meshes

The elliptic equations with heterogeneous and anisotropic diffusion coefficients play an important role in many physical applications such as radiation hydrodynamics and reservoir simulations. Usually the distorted meshes are considered since the 解 of the diffusion part involves the hydro part. The discontinuous Galerkin (DG) method is an important class of methods in computational mathematics. Thefeatures of DG methods can be implemented flexibly on large deformation quadrilateral meshes anddiscontinuous coefficients. In this study, we present a weighted discontinuous Galerkin (WDG) method for numerically solving the elliptic 问题 with non-smooth coefficients. The WDG method is based on the symmetric interior penalty technique. We give the proof of the coercivity and continuity of bilinear forms in the WDG scheme. The convergence analysis for the energy norm is presented based on the coercivity and continuity. Next, we prove an error estimate in the $L^2$ norm based on the duality argument. Numerical examples demonstrate the validity of the WDG method for elliptic problems with discontinuous and anisotropic diffusion coefficients. Some types of distorted meshes such as random, sinusoidal, Shestakov, and Z meshes are used.

The Dirichlet problem for elliptic operators having a BMO anti-symmetric part

The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In particular, the coefficients are not necessarily bounded. We prove that the Dirichlet problem for elliptic equation ${\rm div}(A\nabla u)=0$ in the upper half-space $(x,t)\in\mathbb{R}^{n+1}_+$ is uniquely solvable when $n\ge2$ and the boundary data is in $L^p(\mathbb{R}^n,dx)$ for some $p\in (1,\infty)$. This result is equivalent to saying that the elliptic measure associated to $L$ belongs to the $A_\infty$ class with respect to the Lebesgue measure $dx$, a quantitative version of absolute continuity.

Averaging Principle and Normal Deviations for Multiscale Stochastic Systems

We study the asymptotic behavior for an inhomogeneous multiscale stochastic dynamical system with non-smooth coefficients. Depending on the averaging regime and the homogenization regime, two strong convergences in the averaging principle of functional law of large numbers type are established. Then we consider the small fluctuations of the system around its average. Nine cases of functional central limit type theorems are obtained. In particular, even though the averaged equation for the original system is the same, the corresponding homogenization limit for the normal deviation can be quite different due to the difference in the interactions between the fast scales and the deviation scales. We provide quite intuitive explanations for each case. Furthermore, sharp rates both for the strong convergences and the functional central limit theorems are obtained, and these convergences are shown to rely only on the regularity of the coefficients of the system with respect to the slow variable, and do not depend on their regularity with respect to the fast variable, which coincide with the intuition since in the limit equations the fast component has been totally averaged or homogenized out.

Stochastic impulse control of nonsmooth dynamics with partial observation and execution delay: Application to an environmental restoration problem

AbstractNonsmooth dynamics driven by stochastic disturbance arise in a wide variety of engineering problems. Impulsive interventions are often employed to control stochastic systems; however, the modeling and analysis subject to execution delay have been less explored. In addition, continuously receiving information of the dynamics is not always possible. In this article, with an application to an environmental restoration problem, a continuous‐time stochastic impulse control problem subject to execution delay under discrete and random observations is newly formulated and analyzed. The dynamics have a nonsmooth coefficient modulated by a Markov chain, and eventually attain an undesirable state like a depletion due to the nonsmoothness. The goal of the control problem is to find the most cost‐efficient policy that can prevent the dynamics from attaining the undesirable state. We demonstrate that finding the optimal policy reduces to solving a nonstandard system of degenerate elliptic equations, the Hamilton–Jacobi–Bellman equation (HJBE), which is rigorously and analytically verified in a simplified case. The associated Fokker–Planck equation (FPE) for the controlled dynamics is derived and solved explicitly as well. The model is finally applied to numerical computation of a recent river environmental restoration problem. The HJBE and FPE are successfully computed, and the optimal policy and the probability density functions are numerically obtained. The impacts of execution delay are discussed to deeper analyze the model.

Stability results of coupled wave models with locally memory in a past history framework via nonsmooth coefficients on the interface

In this paper, we investigate the stabilization of a locally coupled wave equations with local viscoelastic damping of past history type acting only in one equation via non smooth coefficients. First, using a general criteria of Arendt–Batty, we prove the strong stability of our system. Second, using a frequency domain approach combined with the multiplier method, we establish the exponential stability of the solution if and only if the two waves have the same speed of propagation. In case of different speed propagation, we prove that the energy of our system decays polynomially with rate t−1. Finally, we show the lack of exponential stability if the speeds of wave propagation are different.

Моделювання скінченних неоднорідностей дискретними особливостями

This work focuses on development of a mathematical apparatus that allows to perform an approximate description of inhomogeneities of finite sizes in a continuous bodies by arranging the sources given on sets of smaller dimensions. The structure and properties of source densities determine the adequacy of the model. The theory of differential forms and generalized functions underlies this study. The boundary value problems with nonsmooth coefficients are formulated. The solutions of such problems is sought in the form of weakly convergent series and as an alternative - an equivalent recurrent set of boundary value problems with jumps. A feature of this approach is the ability to consistently improve the adequacy of the description of inhomogeneity. This is important because it allows to qualitatively assess the impact of real characteristic properties on the accuracy of the model description. Reducing the dimensions of inhomogeneities allows the use of efficient methods such as the Green's function and boundary integral equations to obtain a semi-analytic solution for direct and inverse problems. The work is based on a number of partial problems that demonstrate the proposed approach in modeling of inhomogeneities. The problems of modeling of the set of finite defects in an oscillating elastic beam, the set of inhomogeneities of an arbitrary shape in an oscillating plate, fragile cracks in a two-dimensional elastic body under static loading are considered.