Lipschitz Estimates Research Articles

The Dirichlet problem for prescribed principal curvature equations

In this paper the Dirichlet problem for the equation which prescribes the ith principal curvature of the graph of a function u is considered. A Comparison principle is obtained within the class of semiconvex subsolutions by a local perturbation procedure combined with a fine Lipschitz estimate on the elliptic operator. Existence of solutions is stated for the Dirichlet problem with boundary conditions in the viscosity sense; further assumptions guarantee that no loss of boundary data occurs. Some conditions relating the geometry of the domain and the prescribing data which are sufficient for existence and uniqueness of solutions are presented.

Viscosity solutions of general viscous Hamilton–Jacobi equations

We present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton–Jacobi equations.

Norm Comparison Estimates for the Composite Operator

This paper obtains the Lipschitz and BMO norm estimates for the composite operator𝕄s∘Papplied to differential forms. Here,𝕄sis the Hardy-Littlewood maximal operator, andPis the potential operator. As applications, we obtain the norm estimates for the Jacobian subdeterminant and the generalized solution of the quasilinear elliptic equation.

Some inequalities for Hausdorff operators

In this paper, we give the sufficient and necessary conditions for the boundedness of Hausdorff operators on various function spaces. Moreover, we consider Lipschitz estimates for the commutator of Hausdorff operators. We extend some known results. Mathematics subject classification (2010): 26D10, 26D15, 46E30.

Weighted Lipschitz estimates for commutators of one-sided operators on one-sided Triebel-Lizorkin spaces

Weighted Lipschitz estimates for commutators of one-sided operators on one-sided Triebel-Lizorkin spaces

On [formula omitted] estimates in homogenization of elliptic equations of Maxwellʼs type

For a family of second-order elliptic systems of Maxwellʼs type with rapidly oscillating periodic coefficients in a C1,α domain Ω, we establish uniform estimates of solutions uε and ∇×uε in Lp(Ω) for 1<p⩽∞. The proof relies on the uniform W1,p and Lipschitz estimates for solutions of scalar elliptic equations with periodic coefficients.

Some results on general quadratic reflected BSDEs driven by a continuous martingale

We study the well-posedness of general reflected BSDEs driven by a continuous martingale, when the coefficient f of the driver has at most quadratic growth in the control variable Z, with a bounded terminal condition and a lower obstacle which is bounded above. We obtain the basic results in this setting: comparison and uniqueness, existence, stability. For the comparison theorem and the special comparison theorem for reflected BSDEs (which allows one to compare the increasing processes of two solutions), we give intrinsic proofs which do not rely on the comparison theorem for standard BSDEs. This allows to obtain the special comparison theorem under minimal assumptions. We obtain existence by using the fixed point theorem and then a series of perturbations, first in the case where f is Lipschitz in the primary variable Y, and then in the case where f can have slightly-superlinear growth and the case where f is monotonous in Y with arbitrary growth. We also obtain a local Lipschitz estimate in BMO for the martingale part of the solution.

Boundary behavior of solutions of elliptic equations in nondivergence form

A new proof for boundary Lipschitz estimate is shown for solutions of elliptic equations in nondivergence form. Under additional condition, the boundary differentiability is obtained.

Estimates for weighted Bergman projections on pseudo-convex domains of finite type in ℂn

In this paper, we investigate the regularity properties of weighted Bergman projections for smoothly bounded pseudo-convex domains of finite type in . The main result is obtained for weights equal to a non-negative rational power of the absolute value of a special defining function of the domain: we prove (weighted) Sobolev- and Lipschitz estimates for domains in (or, more generally, for domains having a Levi form of rank and for “decoupled” domains) and for convex domains. In particular, for these defining functions, we generalize results obtained by Bonami and Grellier and Chang and Li. We also obtain a general (weighted) Sobolev- estimate.

Large time behavior of periodic viscosity solutions for uniformly parabolic integro-differential equations

In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we construct solutions of the form $\lambda t + v(x)$. We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (i) the fact that we handle the case of "mixed operators" for which the required ellipticity comes from a combination of the properties of the local and nonlocal terms and (ii) the treatment of the superlinear case (in the gradient variable). Lipschitz estimates previously proved by the authors (2012) and Strong Maximum principles proved by the third author (2012) play a crucial role in the analysis.

Homogenization of elliptic systems with Neumann boundary conditions

For a family of second-order elliptic systems with rapidly oscillating periodic coefficients in a $C^{1,\alpha }$ domain, we establish uniform $W^{1,p}$ estimates, Lipschitz estimates, and nontangential maximal function estimates on solutions with Neumann boundary conditions.

Increasing stability in an inverse problem for the acoustic equation

In this work, we study the inverse boundary value problem of determining the refractive index in the acoustic equation. It is known that this inverse problem is ill-posed. Nonetheless, we show that the ill-posedness decreases when we increase the frequency and the stability estimate changes from logarithmic type for low frequencies to a Lipschitz estimate for large frequencies.

Weighted Estimates for Commutators of -Dimensional Rough Hardy Operators

We establish the weighted estimates for the commutators and which are generated by the -dimensional rough Hardy operators and central BMO functions on the weighted Lebesgue spaces, the weighted Herz spaces and the weighted Morrey-Herz spaces. Furthermore, the weighted Lipschitz estimates are also obtained.

Lipschitz estimates for convex functions with respect to vector fields

We present Lipschitz continuity estimates for a class of convex functionswith respect to Hormander vector fields. These results have been recently obtained in collaboration with M. Scienza, [22].

Transonic shock reflection problems for the self-similar two-dimensional nonlinear wave system

We discuss the regular transonic shock reflections for a model problem of multidimensional conservation laws, the nonlinear wave system. We consider two incident shocks that create reflected shocks, and the state behind the reflected shocks becomes subsonic. We present the existence of the global solution to this configuration, and provide an analysis to handle a degeneracy occurred in the problem and Lipschitz estimates near the sonic boundary. We further implement Lax–Liu positive schemes and Strang splitting, and obtain linear correlations of the incident shock strength and the reflected shock strength. The result obtained in this paper develops a mathematical theory of transonic shock reflection problems. Furthermore the numerical result provides better understanding of the solution structure. This paper provides an application of an important physical problem.

Formula omitted] regularity of solutions of some degenerate fully non-linear elliptic equations

In the present paper, a class of fully non-linear elliptic equations are considered, which are degenerate as the gradient becomes small. Hölder estimates obtained by the first author (2011) are combined with new Lipschitz estimates obtained through the Ishii–Lions method in order to get C1,α estimates for solutions of these equations.

Stability estimates for the solution to the problem of determining the kernel of a viscoelastic equation

For an integrodifferential equation corresponding to a two-dimensional viscoelastic problem, we study the problem of defining the spatial part of the kernel involved in the integral term of the equation. The support of the sought function is assumed to belong to a compact domain Ω. As information for solving this inverse problem, the traces of the solution to the direct Cauchy problem and its normal derivative are given for some finite time interval on the boundary of Ω. An important feature in the statement of the problem is the fact that the solution of the direct problem corresponds to the zero initial data and a force impulse in time localized on a fixed straight line disjoint with Ω. The main result of the article consists in obtaining a Lipschitz estimate for the conditional stability of the solution to the inverse problem under consideration.

Lipschitz regularity of solutions for mixed integro-differential equations

We establish new Hölder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii–Lionsʼs method. We thus extend the Hölder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local–nonlocal interaction, e.g. the fractional diffusion may give the ellipticity in one direction and the classical diffusion in the complementary one.

Boundedness for commutators of [formula omitted]-dimensional rough Hardy operators on Morrey–Herz spaces

In this paper, we study central BMO estimates for commutators of rough Hardy operators HΩ,βb on Morrey–Herz spaces. Furthermore, we also establish the Lipschitz estimates for HΩ,βb on some function spaces, such as the Lebesgue spaces, the Herz spaces and the Morrey–Herz spaces.

A priori estimate for non-uniform elliptic equations

A priori estimate for non-uniform elliptic equations with periodic boundary conditions is concerned. The domain considered consists of two sub-regions, a connected high permeability region and a disconnected matrix block region with low permeability. Let ϵ denote the size ratio of one matrix block to the whole domain. It is shown that in the connected high permeability sub-region, the Hölder and the Lipschitz estimates of the non-uniform elliptic solutions are bounded uniformly in ϵ. But Hölder gradient estimate and L p estimate of the second order derivatives of the solutions in general are not bounded uniformly in ϵ.