Operators In Divergence Form Research Articles

Parabolic equations with singular divergence‐free drift vector fields

In this paper, we study an elliptic operator in divergence-form but not necessary symmetric. In particular, our results can be applied to elliptic operator $L=\nu\Delta+u(x,t)\cdot\nabla$, where $u(\cdot,t)$ is a time-dependent vector field in $\mathbb{R}^{n}$, which is divergence-free in distribution sense, i.e. $\nabla\cdot u=0$. Suppose $u\in L_{t}^{\infty}(\textrm{BMO}_{x}^{-1})$. We show the existence of the fundamental solution $\varGamma(x,t;\xi,\tau)$ of the parabolic operator $L-\partial_{t}$, and show that $\varGamma$ satisfies the Aronson estimate with a constant depending only on the dimension $n$, the elliptic constant $\nu$ and the norm $\left\Vert u\right\Vert _{L^{\infty}(\textrm{BMO}^{-1})}$. Therefore the existence and uniqueness of the parabolic equation $\left(L-\partial_{t}\right)v=0$ are established for initial data in $L^{2}$-space, and their regularity is obtained too. In fact, we establish these results for a general non-symmetric elliptic operator in divergence form.

On the homogenization of random stationary elliptic operators in divergence form

In this note we comment on the homogenization of a random elliptic operator in divergence form $-\nabla \cdot a\nabla$, where the coefficient field $a$ is distributed according to a stationary, but not necessarily ergodic, probability measure $P$. We generalize the well-known case for $P$ stationary and ergodic by showing that the operator $-\nabla \cdot a(\frac{\cdot}{\varepsilon})\nabla$ almost surely homogenizes to a constant-coefficient, random operator $-\nabla \cdot A_h\nabla$. Furthermore, we use a disintegration formula for $P$ with respect to a family of ergodic and stationary probability measures to show that the law of $A_h$ may be obtained by using the standard homogenization results on each probability measure of the previous family. We finally provide a more explicit formula for $A_h$ in the case of coefficient fields which are a function of a stationary Gaussian field.

Berry\u2013Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

We study the random conductance model on the lattice {mathbb {Z}}^d, i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral gap estimate. As a main result we obtain in dimension dge 3 quantitative central limit theorems for the random walk in form of a Berry–Esseen estimate with speed t^{-frac{1}{5}+varepsilon } for dge 4 and t^{-frac{1}{10}+varepsilon } for d=3. Additionally, in the uniformly elliptic case in low dimensions d=2,3 we improve the rate in a quantitative Berry–Esseen theorem recently obtained by Mourrat. As a central analytic ingredient, for dge 3 we establish near-optimal decay estimates on the semigroup associated with the environment process. These estimates also play a central role in quantitative stochastic homogenization and extend some recent results by Gloria, Otto and the second author to the degenerate elliptic case.

On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators

We approximate an elliptic problem with oscillatory coefficients using a problem of the same type, but with constant coefficients. We deliberately take an engineering perspective, where the information on the oscillatory coefficients in the equation can be incomplete. A theoretical foundation of the approach in the limit of infinitely small oscillations of the coefficients is provided, using the classical theory of homogenization. We present a comprehensive study of the implementation aspects of our method, and a set of numerical tests and comparisons that show the potential practical interest of the approach. The approach detailed in this article improves on an earlier version briefly presented in [C. Le Bris, F. Legoll and K. Li, C.R. Acad. Sci. Paris, Série I 351 (2013) 265–270].

On certain degenerate and singular elliptic PDEs II: Divergence-form operators, Harnack inequalities, and applications

By means of the Monge–Ampère real-analysis and PDE techniques associated to certain convex functions, an approach towards Harnack inequalities is developed that simultaneously extends the one for uniformly elliptic operators from the De Giorgi–Nash–Moser theory and the one for the linearized Monge–Ampère operator from the Caffarelli–Gutiérrez theory. Applications include regularity properties for solutions to divergence-form elliptic equations with power-like singularities and C2-estimates for solutions to the Monge–Ampère equation.

Long-time homogenization and asymptotic ballistic transport of classical waves

Consider an elliptic operator in divergence form with symmetric coefficients. If the diffusion coefficients are periodic, the Bloch theorem allows one to diagonalize the elliptic operator, which is key to the spectral properties of the elliptic operator and the usual starting point for the study of its long-time homogenization. When the coefficients are not periodic (say, quasi-periodic, almost periodic, or random with decaying correlations at infinity), the Bloch theorem does not hold and both the spectral properties and the long-time behavior of the associated operator are unclear. At low frequencies, we may however consider a formal Taylor expansion of Bloch waves (whether they exist or not) based on correctors in elliptic homogenization. The associated Taylor-Bloch waves diagonalize the elliptic operator up to an error term (an ``eigendefect''), which we express with the help of a new family of extended correctors. We use the Taylor-Bloch waves with eigendefects to quantify the transport properties and homogenization error over large times for the wave equation in terms of the spatial growth of these extended correctors. On the one hand, this quantifies the validity of homogenization over large times (both for the standard homogenized equation and higher-order versions). On the other hand, this allows us to prove asymptotic ballistic transport of classical waves at low energies for almost periodic and random operators.

Representation formulae of solutions of second order elliptic inequalities

We prove various representation formulae of solutions of second order differential inequalities on RN. As an outcome we obtain positivity results of the solutions of −Lu≥0, where L is a general second order elliptic operator in divergence form. Results of this type are very useful for studying related Liouville theorems of second order semilinear inequalities.

Failure of $$L^2$$ L 2 boundedness of gradients of single layer potentials for measures with zero low density

Consider a totally irregular measure \(\mu \) in \({{\mathbb {R}}}^{n+1}\), that is, the upper density \(\limsup _{r\rightarrow 0}\frac{\mu (B(x,r))}{(2r)^n}\) is positive \(\mu \)-a.e. in \({{\mathbb {R}}}^{n+1}\), and the lower density \(\liminf _{r\rightarrow 0}\frac{\mu (B(x,r))}{(2r)^n}\) vanishes \(\mu \)-a.e. in \({{\mathbb {R}}}^{n+1}\). We show that if \(T_\mu f(x)=\int K(x,y)\,f(y)\,d\mu (y)\) is an operator whose kernel \(K(\cdot ,\cdot )\) is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with Holder continuous coefficients, then \(T_\mu \) is not bounded in \(L^2(\mu )\). This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the n-dimensional Riesz transform.

The Dirichlet-to-Neumann operator for divergence form problems

We present a way of defining the Dirichlet-to-Neumann operator on general Hilbert spaces using a pair of operators for which each one's adjoint is formally the negative of the other. In particular, we define an abstract analogue of trace spaces and are able to give meaning to the Dirichlet-to-Neumann operator of divergence form operators perturbed by a bounded potential in cases where the boundary of the underlying domain does not allow for a well-defined trace. Moreover, a representation of the Dirichlet-to-Neumann operator as a first-order system of partial differential operators is provided. Using this representation, we address convergence of the Dirichlet-to-Neumann operators in the case that the appropriate reciprocals of the leading coefficients converge in the weak operator topology. We also provide some extensions to the case where the bounded potential is not coercive and consider resolvent convergence.

Eigenvalue estimates for a class of elliptic differential operators in divergence form

We compute estimates for eigenvalues of a class of linear second-order elliptic differential operators in divergence form (with Dirichlet boundary condition) on a bounded domain in a complete Riemannian manifold. Our estimates are based upon the Weyl’s asymptotic formula. As an application, we find a lower bound for the mean of the first k eigenvalues of the drifting Laplacian. In particular, we have extended for this operator a partial solution given by Cheng and Yang for the generalized conjecture of Pólya. We also derive a second-Yang type inequality due to Chen and Cheng, and other two inequalities which generalize results by Cheng and Yang obtained for a domain in the unit sphere and for a domain in the projective space.

Double phase anisotropic variational problems and combined effects of reaction and absorption terms

This paper deals with the existence of multiple solutions for the quasilinear equation −divA(x,∇u)+V(x)|u|α(x)−2u=f(x,u) in RN, which involves a general variable exponent elliptic operator in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like |ξ|q(x)−2ξ for small |ξ| and like |ξ|p(x)−2ξ for large |ξ|, where 1<α(⋅)≤p(⋅)<q(⋅)<N. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works A. Azzollini et al. (2014) [4] and N. Chorfi and V. Rădulescu (2016) [11] from cases where the exponents p and q are constant, to the case where p(⋅) and q(⋅) are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the Cerami compactness condition.

Efficient Methods for the Estimation of Homogenized Coefficients

The main goal of this paper is to define and study new methods for the computation of effective coefficients in the homogenization of divergence-form operators with random coefficients. The methods introduced here are proved to have optimal computational complexity and are shown numerically to display small constant prefactors. In the spirit of multiscale methods, the main idea is to rely on a progressive coarsening of the problem, which we implement via a generalization of the Green–Kubo formula. The technique can be applied more generally to compute the effective diffusivity of any additive functional of a Markov process. In this broader context, we also discuss the alternative possibility of using Monte Carlo sampling and show how a simple one-step extrapolation can considerably improve the performance of this alternative method.

Nonautonomous maximal Lp-regularity under fractional Sobolev regularity in time

We prove non-autonomous maximal $L^p$-regularity results on UMD spaces replacing the common H\"older assumption by a weaker fractional Sobolev regularity in time. This generalizes recent Hilbert space results by Dier and Zacher. In particular, on $L^q(\Omega)$ we obtain maximal $L^p$-regularity for $p \ge 2$ and elliptic operators in divergence form with uniform $VMO$-modulus in space and $W^{\alpha,p}$-regularity for $\alpha > \frac{1}{2}$ in time.

One-dimensional stochastic heat equation with discontinuous conductance

ABSTRACTWe introduce a new stochastic partial differential equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by a space–time white noise. Such equation could be used in mathematical modeling of diffusion phenomena in medium consisting of two kinds of materials and undergoing stochastic perturbations. We prove the existence of the solution and we present explicit expressions of its covariance and variance functions. Some regularity properties of the solution sample paths are also analyzed.

Gradient estimates for double phase problems with irregular obstacles

An irregular obstacle problem with non-uniformly elliptic operator in divergence form of (p,q)-growth is studied. We find an optimal regularity for such a double phase obstacle problem by essentially proving that the gradient of a solution is as integrable as both the gradient of the assigned obstacle function and the associated nonhomogeneous term in the divergence. Calderón–Zygmund type estimates are also obtained under minimal regularity requirements of the prescribed data.

Quantitative homogenization of degenerate random environments

We study discrete linear divergence-form operators with random coecients, also known as the random conductance model. We assume that the conductances are bounded, independent and stationary; the law of a conductance may depend on the orientation of the associated edge. We give a simple necessary and sucient condition for the relaxation of the environment seen by the particle to be diusive, in the sense of every polynomial moment. As a consequence, we derive polynomial moment estimates on the corrector. MSC 2010: 35B27, 35K65, 60K37.

Abstract approach to non homogeneous Harnack inequality in doubling quasi metric spaces

We develop an abstract theory to obtain Harnack inequality for non homogeneous PDEs in the setting of quasi metric spaces. The main idea is to adapt the notion of double ball and critical density property given by Di Fazio, Gutiérrez, Lanconelli, taking into account the right hand side of the equation. Then we apply the abstract procedure to the case of subelliptic equations in non divergence form involving Grushin vector fields and to the case of X-elliptic operators in divergence form.

A local Faber–Krahn inequality and applications to Schrödinger equations

ABSTRACTWe prove a local Faber–Krahn inequality for solutions u to the Dirichlet problem for Δ+V on an arbitrary domain Ω in ℝn. Suppose a solution u assumes a global maximum at some point x0∈Ω and u(x0)>0. Let T(x0) be the smallest time at which a Brownian motion, started at x0, has exited the domain Ω with probability ≥1∕2. For nice (e.g., convex) domains, but we make no assumption on the geometry of the domain. Our main result is that there exists a ball B of radius such thatprovided that n≥3. In the case n = 2, the above estimate fails and we obtain a substitute result. The Laplacian may be replaced by a uniformly elliptic operator in divergence form. This result both unifies and strenghtens a series of earlier results.

Singular parabolic equations with interior degeneracy and non smooth coefficients: The Neumann case

We establish Hardy - Poincare and Carleman estimates for non-smooth degenerate/singular parabolic operators in divergence form with Neumann boundary conditions. The degeneracy and the singularity occur both in the interior of the spatial domain. We apply these inequalities to deduce well-posedness and null controllability for the associated evolution problem.

The Dirichlet problem for second order parabolic operators in divergence form

We study parabolic operators H = ∂t − div λ,x A(x, t)∇ λ,x in the parabolic upper half space R n+2 + = {(λ, x, t) : λ > 0}. We assume that the coefficients are real, bounded, measurable, uniformly elliptic, but not necessarily symmetric. We prove that the associated parabolic measure is absolutely continuous with respect to the surface measure on R n+1 in the sense defined by A∞(dx dt). Our argument also gives a simplified proof of the corresponding result for elliptic measure.