Lipschitz Estimates Research Articles

The twin blow-up method for Hamilton-Jacobi equations in higher dimension

In this paper, we show how to extend the twin blow-up method recently developed by the authors (Comptes Rendus. Math., 2024), in order to obtain a new comparison principle for an evolution coercive Hamilton-Jacobi equation posed in a domain of an Euclidian space of any dimension and supplemented with a boundary condition. The method allows dealing with the case where tangential variables and the variable corresponding to the normal gradient of the solution are strongly coupled at the boundary. We elaborate on a method introduced by P.-L. Lions and P. Souganidis (Atti Accad. Naz. Lincei, 2017). Their argument relies on a {single} blow-up procedure after rescaling the semi-solutions to be compared while two simultaneous blow-ups are performed in this work, one for each variable of the classical doubling variable technique. A one-sided Lipschitz estimate satisfied by a combination of the two blow-up limits plays a key role.

[formula omitted]-wave-like properties for entropy solutions to scalar parabolic–hyperbolic conservation laws

In this paper, we consider qualitative properties for entropy solutions to one-dimensional Cauchy problems (CP) for scalar parabolic–hyperbolic conservation laws. Since the equations have both properties of hyperbolic equations and those of parabolic equations, it is difficult to investigate the behavior of solutions to (CP). In our previous works, we focused on the traveling wave structure instead of the self-similar structure. In fact, we succeeded in constructing shock wave type traveling waves with multiple discontinuity. Moreover, we constructed rarefaction wave type sub-, super-solutions to (CP) and investigated their properties.In the present paper, we investigate “N-wave-like properties” for entropy solutions to (CP) while we are not able to construct an analogue of N-waves. In particular, we derive generalized one-sided Lipschitz estimates (Oleinik type entropy estimates) and decay estimates for entropy solutions to (CP). Based on the decay estimates, we discuss the asymptotic profiles of entropy solutions to (CP) under some specific setting.

Local existence and uniqueness of solutions to the time-dependent Kohn–Sham equations coupled with classical nuclear dynamics

We prove the short-time existence and uniqueness of solutions to the initial-value problem associated with a class of time-dependent Kohn–Sham equations coupled with Newtonian nuclear dynamics, combining Yajima's theory for time-dependent Hamiltonians with Duhamel's principle, based on suitable Lipschitz estimates. We consider a pure power exchange term within a generalisation of the so-called local-density approximation, identifying a range of exponents for the existence and uniqueness of H2 solutions to the Kohn–Sham equations.

Uniform boundary estimates for Neumann problems in parabolic homogenization

For a family of second-order parabolic systems with periodic, oscillating and time-dependent coefficients, we establish the uniform boundary Hölder and Lipschitz estimates for Neumann problems in [Formula: see text] and [Formula: see text] cylinders, respectively, by using the convergence rate method. Moreover, we establish the uniform [Formula: see text] estimates for initial-Neumann problems in [Formula: see text] cylinders for [Formula: see text] by using the real-variable method. As a by-product, we also obtain the Gaussian estimates for Neumann function and its derivatives.

Characterisations for the depletion of reactant in a one-dimensional dynamic combustion model

In this paper, a novel observation is made on a one-dimensional compressible Navier–Stokes model for the dynamic combustion of a reacting mixture of γ-law gases (γ>1) with a discontinuous Arrhenius reaction rate function, on both bounded and unbounded domains. We show that the mass fraction of the reactant (denoted as Z) satisfies a weighted gradient estimate Zy/Z∈Lt∞Ly2, provided that at time zero the density is Lipschitz continuous and bounded strictly away from zero and infinity. Consequently, the graph of Z cannot form cusps or corners near the points where the reactant in the combustion process is completely depleted at any instant, and the entropy of Z is bounded from above. The key ingredient of the proof is a new estimate based on the Fisher information, first exploited by Bies and Cieślak (2014) and Cieślak et al. (2023) with applications to PDEs in chemorepulsion and thermoelasticity. Along the way, we also establish a Lipschitz estimate for the density.

Reiterated homogenization of parabolic systems with several spatial and temporal scales

We consider quantitative estimates in the homogenization of second-order parabolic systems with periodic coefficients that oscillate on multiple spatial and temporal scales,∂t−div(A(x,t,x/ε1,…,x/εn,t/ε1′,…,t/εm′)∇), where εℓ=εαℓ,εk′=εβk,ℓ=1,...,n,k=1,...,m, with 0<α1<...<αn<∞ and 0<β1<...<βm<∞. The convergence rate in the homogenization is derived in the L2 space, and the large-scale interior and boundary Lipschitz estimates are also established. In the case n=m=1, such issues have been addressed by Geng and Shen (2020) [12] based on an interesting scale reduction technique developed therein. Our investigation relies on a quantitative reiterated homogenization theory.

Lipschitz estimate for elliptic equations with oscillatory coefficients

&lt;p&gt;Regularity for elliptic equations with oscillatory coefficients was concerned. Problem domains were periodic and consisted of a connected region with normal permeability and a disconnected matrix block subset with high permeability. Coefficients of the elliptic equations depending on the permeability of the domains were highly oscillatory. Let $ {\epsilon}\in(0, 1) $ be the periodic size of domain, $ {\epsilon}\mu\in(0, 1) $ the size ratio of a matrix block to the whole domain, and $ \omega^2\in(1, \infty) $ the permeability ratio of the disconnected matrix block subset to the connected sub-region of the domains. This work presented Lipschitz estimate uniformly in $ {\epsilon}, \mu, \omega $ for the Green's functions and the solutions of the elliptic equations.&lt;/p&gt;

Reiterated homogenization of elliptic systems with Neumann boundary conditions

This paper is devoted to the quantitative estimates in reiterated homogenization of second‐order elliptic systems of linear elasticity, where the coefficients oscillate on multiple separated scales. We establish the uniform boundary Lipschitz estimate and the uniform estimate for the problem. The convergence rate in the space is also obtained as a by‐product.

Uniform W1,p estimates and large-scale regularity for Dirichlet problems in perforated domains

In this paper we study the Dirichlet problem for Laplace's equation in a domain ωε,η perforated periodically with small holes in Rd, where ε represents the scale of the minimal distances between holes and η the ratio between the scale of sizes of holes and ε. We establish W1,p estimates for solutions with bounding constants depending explicitly on ε and η. The proof relies on a large-scale Lipschitz estimate for harmonic functions in perforated domains. The results are optimal for d≥2.

Borderline Gradient Continuity for the Normalized p-Parabolic Operator

In this paper, we prove gradient continuity estimates for viscosity solutions to Delta _{p}^N u - u_t= f in terms of the scaling critical L(n+2,1 ) norm of f, where Delta _{p}^N is the game theoretic normalized p-Laplacian operator defined in (1.2) below. Our main result, Theorem 2.5 constitutes borderline gradient continuity estimate for u in terms of the modified parabolic Riesz potential textbf{P}^{f}_{n+1} as defined in (2.9) below. Moreover, for f in L^{m} with m>n+2, we also obtain Hölder continuity of the spatial gradient of the solution u, see Theorem 2.6 below. This improves the gradient Hölder continuity result in Attouchi and Parviainen (Commun Contemp Math 20(4):1750035, 2018) which considers bounded f. Our main results Theorem 2.5 and Theorem 2.6 are parabolic analogues of those in Banerjee and Munive (Commun Contemp Math 22(8):1950069, 2020). Moreover differently from that in Attouchi and Parviainen (Commun Contemp Math 20(4):1750035, 2018), our approach is independent of the Ishii–Lions method which is crucially used in Attouchi and Parviainen (Commun Contemp Math 20(4):1750035, 2018) to obtain Lipschitz estimates for homogeneous perturbed equations as an intermediate step.

Perturbations of singular fractional SDEs

We obtain well-posedness results for a class of ODE with a singular drift and additive fractional noise, whose right-hand-side involves some bounded variation terms depending on the solution. Examples of such equations are reflected equations, where the solution is constrained to remain in a rectangular domain, as well as so-called perturbed equations, where the dynamics depend on the running extrema of the solution. Our proof is based on combining the Catellier–Gubinelli approach based on Young nonlinear integration, with some Lipschitz estimates in p-variation for maps of Skorokhod type, due to Falkowski and Słominski. An important step requires to prove that fractional Brownian motion, when perturbed by sufficiently regular paths (in the sense of p-variation), retains its regularization properties. This is done by applying a variant of the stochastic sewing lemma.

Uniform Lipschitz estimates up to the boundary for singular perturbation problem for some nonlinear elliptic PDEs with unbounded ingredients

We prove uniform up to the boundary gradient estimates for one phase nonlinear inhomogeneous singular perturbation problems with unbounded measurable ingredients governed by fully nonlinear elliptic equations. We present similar results for quasilinear PDEs with bounded RHS. Our proof is based on the Lipschitz regularity up to the boundary for free boundary problems (FBP) found in Braga and Moreira (2022).

On the solution stability of parabolic optimal control problems

The paper investigates stability properties of solutions of optimal control problems constrained by semilinear parabolic partial differential equations. Hölder or Lipschitz dependence of the optimal solution on perturbations are obtained for problems in which the equation and the objective functional are affine with respect to the control. The perturbations may appear in both the equation and in the objective functional and may nonlinearly depend on the state and control variables. The main results are based on an extension of recently introduced assumptions on the joint growth of the first and second variation of the objective functional. The stability of the optimal solution is obtained as a consequence of a more general result obtained in the paper–the metric subregularity of the mapping associated with the system of first order necessary optimality conditions. This property also enables error estimates for approximation methods. A Lipschitz estimate for the dependence of the optimal control on the Tikhonov regularization parameter is obtained as a by-product.

Homogenization theory of elliptic system with lower order terms for dimension two

In this paper, we consider the homogenization problem for generalized elliptic systems$ \mathcal{L}_{ \varepsilon} = -\operatorname{div}[A(x/ \varepsilon)\nabla+V(x/ \varepsilon)]+B(x/ \varepsilon)\nabla+c(x/ \varepsilon)+\lambda I $with dimension two. Precisely, we will establish the $ W^{1, p} $ estimates, Hölder estimates, Lipschitz estimates and $ L^p $ convergence results for $ \mathcal{L}_{ \varepsilon} $ with dimension two. The operator $ \mathcal{L}_{ \varepsilon} $ has been studied by Qiang Xu with dimension $ d\geq 3 $ in [22,23] and the case $ d = 2 $ is remained unsolved. As a byproduct, we will construct the Green functions for $ \mathcal{L}_{ \varepsilon} $ with $ d = 2 $ and their convergence rates.

Parabolic homogenization with an interface

This paper considers a family of second-order parabolic equations in divergence form with rapidly oscillating and time-dependent periodic coefficients and an interface between two periodic structures. Following a framework initiated by Blanc, Le Bris and Lions and a generalized two-scale expansion in divergence form of elliptic homogenization with an interface by Josien, we can determine the effective (or homogenized) equation with the coefficient matrix being piecewise constant and discontinuous across the interface. Moreover, we obtain the $O(\varepsilon)$ convergence rates in $L^{2(d+2)/{d}}_{x,t}$ with $\varepsilon$-smoothing method and the uniform interior Lipschitz estimates via compactness argument.

Weighted Lipschitz estimates for commutators of multilinear Calderón–Zygmund operators with Dini type kernels

In this paper, we discuss the properties for commutators and iterated commutators generated by the multilinear ω-CZO and weighted Lipschitz functions on Lebesgue space.

Mean field games master equations with nonseparable Hamiltonians and displacement monotonicity

In this manuscript we propose a structural condition on nonseparable Hamiltonians, which we term displacement monotonicity condition, to study second-order mean field games master equations. A rate of dissipation of a bilinear form is brought to bear a global (in time) well-posedness theory, based on a priori uniform Lipschitz estimates on the solution in the measure variable. Displacement monotonicity being sometimes in dichotomy with the widely used Lasry–Lions monotonicity condition, the novelties of this work persist even when restricted to separable Hamiltonians.

Some estimates for $$p$$-adic fractional integral operator and its commutators on $$p$$-adic Herz spaces with rough kernels

In this note we study the boundedness of \(p\)-adic fractional integral operator with rough kernels on \(p\)-adic Herz spaces. Moreover, we establish Lipschitz estimates for commutators of \(p\)-adic fractional integral operator with rough kernels on Herz spaces. In addition, we also obtain central bounded mean oscillations\((C{\dot{M}}O)\) estimate for commutators of \(p\)-adic fractional integral operator with rough kernels on \(p\)-adic Herz spaces. As an application, we characterize \(p\)-adic Herz space in terms of wavelets in continuously differentiable functions \(({\mathcal {C}}^{1}({\mathbb {Q}}_p^n))\) with compact support.

Estimates for p-adic fractional integral operator and its commutators on p-adic Morrey–Herz spaces

This research investigates the boundedness of a p-adic fractional integral operator on p-adic Morrey–Herz spaces. In particular, p-adic central bounded mean oscillations (Cdot{M}O) and Lipschitz estimate for commutators of the p-adic fractional integral operator are provided as well.

Compactness and large-scale regularity for Darcy's law

This paper is concerned with the quantitative homogenization of the steady Stokes equations with the Dirichlet condition in a periodically perforated domain. Using a compactness method, we establish the large-scale interior C1,α and Lipschitz estimates for the velocity as well as the corresponding estimates for the pressure. These estimates, when combined with the classical regularity estimates for the Stokes equations, yield the uniform Lipschitz estimates. As a consequence, we also obtain the uniform Wk,p estimates for 1<p<∞.