Non-smooth Coefficients Research Articles

H-distributions on Hörmander spaces

We define an important microlocal tool – H-distributions, a generalization of H-measures, on Hörmander Bp,k spaces. We also consider applications to linear partial differential equations with non-smooth coefficients and to some semilinear equations.

Optimal polynomial stability of a string with locally distributed Kelvin–Voigt damping and nonsmooth coefficient at the interface

The purpose of this paper is to investigate the stabilization of one‐dimensional wave equation with localized internal viscoelastic damping of Kelvin–Voigt type in a bounded domain. In this work, we consider only one damping Kelvin–Voigt mechanism acting in the internal of the body. Using frequency domain arguments combined with the multiplier method, we prove that the energy of the system has the optimal decay of type t−4.

Gradient bounds for solutions to irregular parabolic equations with (p, q)-growth

We provide quantitative gradient bounds for solutions to certain parabolic equations with unbalanced polynomial growth and non-smooth coefficients.

Identification of the timewise thermal conductivity in a 2D heat equation from local heat flux conditions

ABSTRACT The aim of this paper is to identify numerically the timewise thermal conductivity coefficients in the two-dimensional heat equation in a rectangular domain using initial and Dirichlet boundary conditions and the local heat flux as over-specification conditions. The measurement data represented by the local heat flux is shown to ensure the unique solvability of the inverse problem solution. The two-dimensional inverse problem is discretized using an alternating direction explicit method. The resulting constrained optimization problem is minimized iteratively by employing the MATLAB subroutine. Exact and noisy input data are inverted numerically. The root mean square error values for various noise levels p for both smooth and non-smooth continuous timewise thermal conductivity coefficients Examples are compared. Numerical results are presented and discussed in order to illustrate the performance of the inversion for thermal conductivity components identification. This study will be significant to researchers working on computational and mathematical methods for solving inverse coefficient identification problems with applications in heat transfer and porous media.

On solvability in the small of higher order elliptic equations in grand-Sobolev spaces

ABSTRACT This work deals with the mth order elliptic equation with non-smooth coefficients in grand-Sobolev space generated by the norm of the grand-Lebesgue space . These spaces are non-separable, and therefore, to use classical methods for treating solvability problems in these spaces, you need to modify these methods. To this aim, we consider some subspace, where the infinitely differentiable functions are dense. Then we prove that this subspace is invariant with respect to the singular integral operator and with respect to the multiplication operator by a function from . Finally, using classical method of parametrics, we prove the existence in the small of the solution to the considered equation in .

Asymptotic Approximations for Solutions of Sturm–Liouville Differential Equations with a Large Complex Parameter and Nonsmooth Coefficients

This article deals with fundamental solutions of the Sturm–Liouville differential equations $$\left( P\left( x\right) y^{\prime }\right) ^{\prime }+\left( R\left( x\right) -\xi ^{2}Q\left( x\right) \right) y=0$$ with a complex parameter $$\xi $$ , $$\left\vert \xi \right\vert \gg 1$$ , whose coefficients $$P\left( x\right) $$ and $$Q\left( x\right) $$ are positive piecewise continuous functions while $$\left( P\left( x\right) Q\left( x\right) \right) ^{\prime }$$ and $$R\left( x\right) $$ belong to $$L^{p}$$ , $$p>1$$ . Two methods are suggested to reduce the problem to special Volterra or Volterra–Hammerstein integral equations with “small” integral operators, which can be treated by iterations that constitute asymptotic scales or by direct numerical methods. Asymptotic analysis of the iterations permits constructing uniform asymptotic approximations for fundamental solutions and their derivatives. A uniform with respect to integration limits analog of Watson’s lemma for finite range Laplace integrals is proved in this connection. When the coefficients belong to $$C^{N}$$ on intervals of continuity, a new simple direct method is suggested in order to derive explicit uniform asymptotic approximations of fundamental solutions as $$\operatorname{Re} \left( \xi \right) \gg 1$$ by a certain recurrent procedure. This method is expected to be more efficient than the WKB method. Theorems estimating errors of such approximations are given. Efficacy of the suggested uniform approximations is demonstrated by computing approximations for a combination of Bessel functions of large indices and arguments.

A transmission problem of a system of weakly coupled wave equations with Kelvin–Voigt dampings and non-smooth coefficient at the interface

The purpose of this work is to investigate the stabilization of a system of weakly coupled wave equations with one or two locally internal Kelvin–Voigt damping and non-smooth coefficient at the interface. The main novelty in this paper is that the considered system is a coupled system and that the geometrical situations covered (see Remarks 5.6, 5.12) are richer than all previous results, even for simple wave equation with Kelvin–Voigt damping. Firstly, using a unique continuation result, we prove that the system is strongly stable. Secondly, we show that the system is not always exponentially stable, instead, we establish some polynomial energy decay estimates. Further, we prove that a polynomial energy decay rate of order $$t^{-1/2}$$ is optimal in some sense.

Recovery of non-smooth radiative coefficient from nonlocal observation by diffusion system

Abstract The heat conduction process in composite medium can be modeled by a parabolic equation with discontinuous radiative coefficient. To detect the composite medium characterized by such a non-smooth coefficient from measurable information about the heat distribution, we consider a nonlinear inverse problem for parabolic equation, with the average measurement of temperature field in some time interval as the inversion input. We firstly establish the uniqueness for this nonlinear inverse problem, based on the property of the direct problem and the known uniqueness result for linear inverse source problem. To solve the inverse problem from a nonlinear operator equation, the differentiability and the tangential condition of this nonlinear map is analyzed. An iterative process called two-point gradient method is proposed by minimizing data-fit term and the penalty term alternatively, with rigorous convergence analysis in terms of the tangential condition. Numerical simulations are presented to illustrate the effectiveness of the proposed method.

On the approximate controllability of parabolic problems with non-smooth coefficients

In this paper we prove the existence of approximate controls for certain classes of parabolic problems with non-smooth coefficients and discuss as examples the problem of approximate controllability for the heat flow in heterogeneous media such as, periodic composites, perforated domains or periodic microstructures separated by rough interfaces.

АСИМПТОТИЧЕСКОЕ РЕШЕНИЕ ВОЗМУЩЕННОЙ ПЕРВОЙ КРАЕВОЙ ЗАДАЧИ С НЕГЛАДКИМ КОЭФФИЦИЕНТОМ

АСИМПТОТИЧЕСКОЕ РЕШЕНИЕ ВОЗМУЩЕННОЙ ПЕРВОЙ КРАЕВОЙ ЗАДАЧИ С НЕГЛАДКИМ КОЭФФИЦИЕНТОМ

Well-Posed Solvability of the Neumann Problem for a Generalized Mangeron Equation with Nonsmooth Coefficients

For a fourth-order generalized Mangeron equation with nonsmooth coefficients defined on a rectangular domain, we consider the Neumann problem with nonclassical conditions that do not require matching conditions. We justify the equivalence of these conditions to classical boundary conditions for the case in which the solution to the problem is sought in an isotropic Sobolev space. The problem is solved by reduction to a system of integral equations whose well-posed solvability is established based on the method of integral representations. The well-posed solvability of the Neumann problem for the generalized Mangeron equation is proved by the method of operator equations.

Analysis of Markov Chain Approximation for Diffusion Models with Non-Smooth Coefficients

Diffusion models with non-smooth coefficients often appear in financial applications, with examples including but not limited to threshold models for financial variables, the pricing of occupation time derivatives and shadow rate models for interest rate dynamics. To calculate the expected value of a discounted payoff under general state-dependent discounting and monitoring of barrier crossing, continuous time Markov chain (CTMC) approximation can be applied. In a recent work, Zhang and Li (2018, Operations Research, forthcoming) established sharp convergence rates of CTMC approximation for diffusion models with smooth coefficients but non-smooth payoff functions, and proposed grid design principles to ensure nice convergence behaviors. However, their theoretical analysis fails to obtain sharp convergence rates when model coefficients lack smoothness. Moreover, it is unclear how to design the grid of CTMC to remedy the inferior convergence behaviors resulting from non-smooth model coefficients. In this paper, we introduce new ways for the theoretical analysis of CTMC approximation for general diffusion models with non-smooth coefficients. We prove that convergence of option price is only first order in general. However, strikingly, if all the discontinuous points of the model coefficients and the payoff function are in the midway between two grid points, second order convergence in the maximum norm is restored and in this case, delta and gamma have second order convergence at almost all grid points except those next to the discontinuous points. Numerical experiments are conducted that confirm the validity of our theoretical results. We also compare the CTMC approximation approach with properly designed grids to a classical numerical PDE scheme for diffusion models with non-smooth coefficients, where the finite difference method is applied separately in each region with smooth coefficients and continuous pasting of the value function is enforced at the discontinuities. We show that our approach is superior to the latter in terms of both the convergence rate and the simplicity of implementation.

Stabilization of the wave equations with localized Kelvin–Voigt type damping under optimal geometric conditions

The purpose of this note is to investigate the stabilization of the wave equation with Kelvin–Voigt damping in a bounded domain. Damping is localized via a non-smooth coefficient in a suitable subdomain. We prove a polynomial stability result in any space dimension, provided that the damping region satisfies some geometric conditions. The main novelty of this note is that the geometric situations covered here are richer than that considered in [25], [22], [16] and include in particular an example where the damping region is not localized in a neighborhood of the whole or a part of the boundary.

Recovery of Nonsmooth Coefficients Appearing in Anisotropic Wave Equations

We study the problem of unique recovery of a nonsmooth one-form $\mathcal{A}$ and a scalar function $q$ from the Dirichlet to Neumann map, $\Lambda_{\mathcal{A},q}$, of a hyperbolic equation on a R...

Nonsmooth pseudodifferential boundary value problems on manifolds

We study pseudodifferential boundary value problems in the context of the Boutet de Monvel calculus or Green operators, with nonsmooth coefficients on smooth compact manifolds with boundary. In order to have a definition that is independent of the choice of (smooth) coordinates, we prove that nonsmooth Green operators are invariant under smooth coordinate transformations.

Weighted gradient estimates for higher order elliptic systems with non-smooth coefficients

In this paper we prove the regularity estimates on the weighted Lorentz spaces for the higher-order elliptic system of equations with non-smooth coefficients on a Lipschitz domain. As a by product, we obtain the regularity estimate on the Lorentz–Morrey spaces. Our results extend previous results to the larger class of Muckenhoupt weights.

Variational approach for the Stokes and Navier–Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds

The purpose of this paper is to show well-posedness results for Dirichlet problems for the Stokes and Navier–Stokes systems with $$L^{\infty }$$ -variable coefficients in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. First, we refer to the Dirichlet problem for the nonsmooth coefficient Stokes system on Lipschitz domains in compact Riemannian manifolds and show its well-posedness by employing a variational approach that reduces the boundary value problem of Dirichlet type to a variational problem defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. We show also the equivalence between some transmission problems for the nonsmooth coefficient Stokes system in complementary Lipschitz domains on compact Riemannian manifolds and their mixed variational counterparts, and then their well-posedness in $$L^2$$ -based Sobolev spaces by using the remarkable Necas–Babuska–Brezzi technique (see Babuska in Numer Math 20:179–192, 1973; Brezzi in RAIRO Anal Numer R2:129–151, 1974; Necas in Rev Roum Math Pures Appl 9:47–69, 1964). As a consequence of these well-posedness results we define the layer potential operators for the nonsmooth coefficient Stokes system on Lipschitz surfaces in compact Riemannian manifolds, and provide their main mapping properties. These properties are used to construct explicitly the solution of the Dirichlet problem for the Stokes system. Further, we combine the well-posedness of the Dirichlet problem for the nonsmooth coefficient Stokes system with a fixed point theorem to show the existence of a weak solution to the Dirichlet problem for the nonsmooth variable coefficient Navier–Stokes system in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. The well developed potential theory for the smooth coefficient Stokes system on compact Riemannian manifolds (cf. Dindos and Mitrea in Arch Ration Mech Anal 174:1–47, 2004; Mitrea and Taylor in Math Ann 321:955–987, 2001) is also discussed in the context of the potential theory developed in this paper.

Layer Potentials and Poisson Problems for the Nonsmooth Coefficient Brinkman System in Sobolev and Besov Spaces

In this paper we obtain well-posedness results for Poisson problems with Dirichlet, Neumann, or mixed boundary conditions for the Brinkman system with measurable coefficients and data in $$L^p$$ -based Sobolev and Besov spaces in Lipschitz domains on a compact Riemannian manifold M of dimension $$m\ge 2$$ . For the mixed problem we refer to partially vanishing traces on Ahlfors regular sets. We exploit the continuity property of an operator related to the variational formulation of such a boundary value problem on complex interpolation scales of $$L^p$$ -based Sobolev spaces defined on M or on a Lipschitz domain of M, $$p\in (1,\infty )$$ , and the property that this operator is an isomorphism for $$p=2$$ . Then the stability of the quality of being isomorphism on complex interpolation scales leads to the extension of the well-posedness results of analyzed boundary value problems from $$p\!=\!2$$ to p in a neighborhood of 2. First, we focus on a variational approach that reduces boundary problems of transmission, Dirichlet and mixed type for the Brinkman system to equivalent mixed variational formulations with data in $$L^p$$ -based Sobolev and Besov spaces. For $$p=2$$ , such a mixed variational formulation is well-posed. The mixed variational formulation is further expressed in terms of a linear continuous operator on $$H^{1,q}\times L^q$$ -Sobolev spaces for any $$q\in (1,\infty )$$ , which is also invertible on the solution space corresponding to $$q=2$$ . Working on complex interpolation scales allows us to extend the invertibility of the operator for $$q=2$$ to a neighborhood of 2, and then to extend the well-posedness result to $$L^p$$ -based Sobolev spaces with p in a neighborhood of 2. Well-posedness results for the analyzed transmission problems allow us to define the layer potentials for the nonsmooth coefficient Brinkman system and to obtain their properties in $$L^p$$ -based Sobolev and Besov spaces. Then the solution of the Poisson problem of Dirichlet type is constructed explicitly in terms of such layer potentials. Finally, the Poisson problem of Neumann type is also analyzed and the corresponding well-posedness result in $$L^p$$ -based Sobolev and Besov spaces is also obtained. In addition, we determine the unique solution of the Neumann problem in the case $$p=2$$ , by using a layer potential approach. We extend the well-posedness results obtained in Kohr and Wendland (Boundary value problems for the Brinkman system with measurable coefficients in Lipschitz domains on compact Riemannian manifolds: a variational approach, 2018) for boundary problems for the nonsmooth coefficient Brinkman system in $$L^2$$ -based Sobolev spaces on Lipschitz domains in compact Riemannian manifolds to a more general setting of $$L^p$$ -based Sobolev and Besov spaces.

Regularity problem for 2m-order quasilinear parabolic systems with non smooth in time principal matrix. (A(t),m)-caloric approximation method

Partial regularity of solutions to a class of $2m$-order quasilinear parabolic systems and full interior regularity for $2m$-order linear parabolic systems with non smooth in time principal matrices is proved in the paper. The coefficients are assumed to be bounded and measurable in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the $(A(t),m)$-caloric approximation method, $m\geq 1$. It is both an extension of the $A(t)$-caloric approximation applied by the authors earlier to study regularity problem for systems of the second order with non-smooth coefficients and an extension of the $A$-polycaloric lemma proved by V. Bogelein in \cite{Bo} to systems of $2m$-order.

Multigrid algorithm based on hybrid smoothers for variational and selective segmentation models

ABSTRACTAutomatic segmentation of an image to identify all meaningful parts is one of the most challenging as well as useful tasks in a number of application areas. This is widely studied. Selective segmentation, less studied, aims to use limited user-specified information to extract one or more interesting objects (instead of all objects). Constructing a fast solver remains a challenge for both classes of model. However, our primary concern is on selective segmentation. In this work, we develop an effective multigrid algorithm, based on a new non-standard smoother to deal with non-smooth coefficients, to solve the underlying partial differential equations of a class of variational segmentation models in the level-set formulation. For such models, non-smoothness (or jumps) is typical as segmentation is only possible if edges (jumps) are present. In comparison with previous multigrid methods which were shown to produce an acceptable mean smoothing rate for related models, the new algorithm can ensure a small and global smoothing rate that is a sufficient condition for convergence. Our rate analysis is by local Fourier analysis and, with it, we design the corresponding iterative solver, improving on an ineffective line smoother. Numerical tests show that the new algorithm outperforms multigrid methods based on competing smoothers.