Lipschitz Continuous Coefficients Research Articles

Strong Unique Continuation Properties of Generalized Baouendi–Grushin Operators

This paper gives a quantitative control of the order of zero of a weak solution to perturbations of the Baouendi–Grushin operator, which generalizes the result due to Aronszaijn, Krzywicki, and Szarski valid for elliptic operators in divergence form with Lipschitz continuous coefficients.

Subsolutions of Elliptic Operators in Divergence Form and Application to Two-Phase Free Boundary Problems

Let be a divergence form operator with Lipschitz continuous coefficients in a domain, and let be a continuous weak solution of in. In this paper, we show that if satisfies a suitable differential inequality, then is a subsolution of away from its zero set. We apply this result to prove regularity of Lipschitz free boundaries in two-phase problems.

A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient

The existence of a mean-square continuous strong solution is established for vector-valued Itô stochastic differential equations with a discontinuous drift coefficient, which is an increasing function, and with a Lipschitz continuous diffusion coefficient. A scalar stochastic differential equation with the Heaviside function as its drift coefficient is considered as an example. Upper and lower solutions are used in the proof.

Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem

We deal with the Cauchy problem for a strictly hyperbolic second-order operator with non-regular coefficients in the time variable. It is well-known that the problem is well-posed in L 2 in case of Lipschitz continuous coefficients and that the log-Lipschitz continuity is the natural threshold for the well-posedness in Sobolev spaces which, in this case, holds with a loss of derivatives. Here, we prove that any intermediate modulus of continuity between the Lipschitz and the log-Lipschitz one leads to an energy estimate with arbitrary small loss of derivatives. We also provide counterexamples to show that the following classification: modulus of continuity → loss of derivatives is sharp Lipschitz → no loss , intermediate → arbitrary small loss , log - Lipschitz → finite loss .

The Poincaré problem in [formula omitted]-Sobolev spaces—I: codimension one degeneracy

We study a degenerate oblique derivative problem in Sobolev spaces W 2 , p ( Ω ) , ∀ p > 1 , for uniformly elliptic operators with Lipschitz continuous coefficients. The vector field prescribing the boundary condition becomes tangential to ∂ Ω at the points of a non-empty set and is of emergent type.

Time-inhomogeneous stochastic processes on the {$p$}-adic number field

The theory of stochastic differential equation on the field of $p$-adics is initiated by Kochubei. In this article, we will focus on a class of random walks with a certain integrability to develop the theory of stochastic analysis in a way similar to the existing theory of stochastic analysis based on the Brownian motion. In fact, for any random walk in the class, we can introduce the notion of stochastic integral with respect to the random walk and justify the existence of the solution of the stochastic differential equations based on the random walk, where the stochastic differential equations admit not only Lipschitz continuous path dependent coefficients but also continuous coefficients growing at most linearly with respect to $p$-adic norm. Finally, we will see an example of stochastic process which can be covered by Dirichlet space theory and obtained also by solving stochastic differential equation.

Chebyshev rational matrix approximation with a priori error bounds for linear and Riccati matrix equations

This paper deals with initial value problems for Lipschitz continuous coefficient matrix Riccati equations. Using Chebyshev polynomial matrix approximations the coefficients of the Riccati equation are approximated by matrix polynomials in a constructive way. Then using the Fröbenius method developed in [1], given an admissible error ϵ > 0 and the previously guaranteed existence domain, a rational matrix polynomial approximation is constructed so that the error is less than ϵ in all the existence domain. The approach is also considered for the construction of matrix polynomial approximations of nonhomogeneous linear differential systems avoiding the integration of the transition matrix of the associated homogeneous problem.

On estimating the diffusion coefficient: parametric versus nonparametric

We consider the following problem: estimate the Lipschitz continuous diffusion coefficient σ 2 from the path of a 1-dimensional diffusion process sampled at times i/n, i=0,…,n , when we believe that σ 2 actually belongs to a smaller regular parametric set Σ 0. By introducing random normalizing factors in the risk function, we obtain confidence sets which can be essentially better than the minimax rate n −1/3 of estimation for Lipschitz functions in diffusion models. With a prescribed confidence level α n , we show that the best possible attainable (random) rate is ( logα n −1 /n) 2/5 . We construct an optimal estimator and an optimal random normalizing factor in the sense of Lepski (1999). This has some consequences for classical estimation: our procedure is adaptive w.r.t. Σ 0 and enables us to test the hypothesis that σ 2 is parametric against a family of local alternatives with prescribed 1st and 2nd-type error probabilities.

On non-degenerate quasi-linear stochastic partial differential equations

We prove a limit theorem for non-degenerate quasi-linear parabolic SPDEs driven by space-time white noise in one space-dimension, when the diffusion coefficient is Lipschitz continuous and the nonlinear drift term is only measurable. Hence we obtain an existence and uniqueness and a comparison theorem, which generalize those in [2], [4], [5] to the case of non-degenerate SPDEs with measurable drift and Lipschitz continuous diffusion coefficients.

Weakly hyperbolic equations with Lipschitz or Hölder continuous coefficients with respect to time

We consider second order weakly hyerbolic equations, with Lipschitz or Holder continuous coefficients with respect to time, concerning the well posedness of the Cauchy problem and the propagation of the singularities in the framework of Gevrey classes.

Asymptotic estimates of green's function and the “difference step” in the case of Lipschitz-continuous coefficients

A study is made of the elementary hyperbolic equation u t ϱ( ζ) u ζ =0 ϱ( ζ) is assumed to be Lipschitz continuous. Asymptotic estimates are obtained for Green's function and the “difference step” for difference schemes of maximum odd order of accuracy 2 k−1, k= O(lnτ−1), τ is the time step. The natural constraint g9( ζ) τ/ h⩽1 is imposed on the ratio of mesh steps. The basic asymptotic estimates are obtained by the saddle-point method. The main difficulty lies in the fact that, with n⪢ k ω , the integrand contains, instead of the power of a single function, as in the case of constant coefficients, the product of essentially different factors. Using the Lipschitz continuity, we obtain estimates in L 1, close to optimal, and identical with the estimates for the case of constant coefficients.