Hongjie Dong, Seick Kim
We construct the Green function for second-order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition. We show that the Green's function is BMO in the domain and establish logarithmic pointwise bounds. We also obtain pointwise bounds for first and second derivatives of the Green's function.
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Weighted Hessian estimates in Orlicz spaces for nondivergence elliptic operators with certain potentials
本文针对含满足反向Hölder条件非负势的非散度型椭圆方程，证明了Orlicz空间中的内部加权Hessian估计。
We prove the natural weighted Calderón and Zygmund estimates for solutions to elliptic and parabolic obstacle problems in nondivergence form with discontinuous coefficients and irregular obstacles. We also obtain Morrey regularity results for the Hessian of the solutions and Hölder continuity of the gradient of the solutions.
Given two elliptic operators L and M in nondivergence form, with coefficients A_L(x), A_M(x) and drift terms b_L(x), b_M(x), respectively, satisfying a Carleson measure disagreement condition in a Lipschitz domain Omega in R^{n+1}, then their harmonic measures are mutually absolutely continuous. As an application of this, a new approximation argument and known results we obtain necessary and sufficient conditions for a single operator L (in divergence or nondivergence form) to have regular harmonic measure with respect to Lebesgue measure. The results are sharp in all cases.
We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in Caffarelli's 1977 paper "The regularity of free boundaries in higher dimensions."
On nondivergence form linear parabolic and elliptic equations with degenerate coefficients
本文研究了上半空间中退化抛物型和椭圆型方程在加权混合范数Sobolev空间中的唯一可解性，探讨了函数空间的最优性。
Characterizations of diffusion matrices in homogenization of elliptic equations in nondivergence-form
本文刻画了非散度形式线性椭圆方程周期均匀化中产生 $\mathcal{O}(\varepsilon^2)$ 最优 $L^\infty$ 收敛率的扩散矩阵，证实了相关猜想，并给出了二维对角矩阵的完整刻画及高维研究。
We study optimal convergence rates in the periodic homogenization of linear elliptic equations of the form $-A(x/\varepsilon):D^2 u^{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We show that the optimal rate for the convergence of $u^{\varepsilon}$ to the solution of the corresponding homogenized problem in the $W^{1,p}$-norm is $\mathcal{O}(\varepsilon)$. We further obtain optimal gradient and Hessian bounds with correction terms taken into account in the $L^p$-norm. We then provide an explicit $c$-bad diffusion matrix and use it to perform various numerical experi…
Green function and invariant measure estimates for nondivergence form elliptic homogenization
本文研究了非散度形式椭圆方程随机正则化中抛物型格林函数和不变测度的定量估计，得出了扩散过程的局部 CLT 和环境过程的遍历性估计。
We consider the existence and asymptotics for the fundamental solution of an elliptic operator in nondivergence form, ${\mathcal L}(x,\del_x)=a_{ij}(x)\del_i\del_i$, for $n\geq 3$. We assume that the coefficients have modulus of continuity satisfying the square Dini condition. For fixed $y$, we construct a solution of ${\mathcal L}Z_y(x)=0$ for $0<|x-y|<\e$ with explicit leading order term which is $O(|x-y|^{2-n}e^{I(x,y)})$ as $x\to y$, where $I(x,y)$ is given by an integral and plays an important role for the fundamental solution: if $I(x,y)$ approaches a finite limit as $x\to y$, then we ca…
Quantitative homogenization and large-scale regularity for nondivergence-form equations under a critical ellipticity moment
本文针对非散度型椭圆方程，在临界椭圆矩条件下建立定量均匀化估计，构造二次校正子，证明大尺度正则性与Liouville定理，并识别有效矩阵。
We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold $M$, with the sectional curvature bounded from below by $-κ$ for $κ\geq 0$. In the elliptic case, Wang and Zhang \cite{WZ} recently extended the results of \cite{Ca} to nonlinear elliptic equations in nondivergence form on such $M$, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative {\it viscosity solutions} to nonlinear uniformly {\it parabolic equations} in nondivergence form on $M$. The Harnack inequali…
We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional to a power of the microscopic length scale, assuming a finite range of dependence. The results are new even for linear equations. The arguments rely on a new geometric quantity which is controlled in part by adapting elements of the regularity theory for the Monge-Ampère equation.
Interior pointwise $C^{1}$ and $C^{1,1}$ regularity of solutions for general semilinear elliptic equation in nondivergence form
该研究在更弱假设下推导非散度型一般半线性椭圆方程$L^n$-粘性解的内点逐点$C^1$与$C^{1,1}$正则性，推广了非齐次项相关结果，给出无界系数下的正则性判定条件。
We prove quantitative estimates for the stochastic homogenization of linear uniformly elliptic equations in nondivergence form. Under strong independence assumptions on the coefficients, we obtain optimal estimates on the subquadratic growth of the correctors with stretched exponential-type bounds in probability. Like the theory of Gloria and Otto \cite{GO1,GO2} for divergence form equations, the arguments rely on nonlinear concentration inequalities combined with certain estimates on the Green's functions and derivative bounds on the correctors. We obtain these analytic estimates by developin…
We propose a least-squares method involving the recovery of the gradient and possibly the Hessian for elliptic equation in nondivergence form. As our approach is based on the Lax--Milgram theorem with the curl-free constraint built into the target (or cost) functional, the discrete spaces require no inf-sup stabilization. We show that standard conforming finite elements can be used yielding apriori and aposteriori convergnece results. We illustrate our findings with numerical experiments with uniform or adaptive mesh refinement.
Boundary differentiability is shown for solutions of nondivergence elliptic equations with unbounded drift
Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three
该研究构造特定系数矩阵证明三维非散度型一致椭圆方程不存在仅依赖椭圆性的内部 $W^{1,1}$ 及更高次 $W^{1,p}$ 估计，解决了相关开放问题并得到非局部有界变差的极限函数。
We define the fractional powers $L^s=(-a^{ij}(x)\partial_{ij})^s$, $0 < s < 1$, of nondivergence form elliptic operators $L=-a^{ij}(x)\partial_{ij}$ in bounded domains $Ω\subset\mathbb{R}^n$, under minimal regularity assumptions on the coefficients $a^{ij}(x)$ and on the boundary $\partialΩ$. We show that these fractional operators appear in several applications such as fractional Monge--Ampère equations, elasticity, and finance. The solution $u$ to the nonlocal Poisson problem $$\begin{cases} (-a^{ij}(x) \partial_{ij})^su = f&\hbox{in}~Ω\\ u=0&\hbox{on}~\partialΩ\end{cases}$$ is characterized…
The solvability in Sobolev spaces $W^{1,2}_p$ is proved for nondivergence form second order parabolic equations for $p>2$ close to 2. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, and almost VMO (vanishing mean oscillation) with respect to the other coordinates. This implies the $W^{2}_p$-solvability for the same $p$ of nondivergence form elliptic equations with leading coefficients measurable in two coordinates and VMO in the others. Under slightly different assumptions, we also obtain the solvability results when $p=2$.
It is known that solutions to second order uniformly elliptic and parabolic equations, either in divergence or nondivergence (general) form, are Hölder continuous and satisfy the interior Harnack inequality. We show that even in the one-dimensional case ($x\in R^1$), these properties are not preserved for equations of mixed divergence-nondivergence structure: for elliptic equations. \begin{equation*} D_i(a^1_{ij}D_ju)+a^2_{ij}D_{ij}u=0, \end{equation*} and parabolic equations \begin{equation*} p\partial_t u=D_i(a_{ij}D_ju), \end{equation*} where $p=p(t,x)$ is a bounded strictly positive functi…
Singular semilinear elliptic equations in nondivergence form
本文研究非散度型二阶椭圆算子驱动的奇异半线性方程 -Pu = f/u^γ，在区域和系数正则性假设下证明了解的存在性与 L^1 唯一性，方法上结合了非线性 Gagliardo--Nirenberg 不等式、Green 函数估计和 Kato 型不等式的新变体。
Interior Schauder estimates for fractional elliptic equations in nondivergence form
本文研究非散度型分数阶椭圆算子的非局部 Poisson 问题，通过退化/奇异延拓、新型粘性解及 Caffarelli 扰动方法建立精确内部 Schauder 估计，并证明 Hopf 引理、Harnack 不等式与 Hölder 正则性。
We consider the Dirichlet problem Lu = 0 in D u = g on E = boundary of D for two second order elliptic operators L_k(u) = \sum_{i,j=1}^n a_k^{ij}(x) \partial_{ij} u(x), k=0,1, in a bounded Lipschitz domain D in R^n. The coefficients a_k^{ij} belong to the space of bounded mean oscillation BMO with a suitable small BMO modulus. We assume that L_0 is regular in L^p(E,ds) for some p, 1<p<\infty, that is, |Nu|_{L^p}< C |g|_{L^p} for all continuous boundary data g. Here ds is the surface measure on E and Nu is the nontangential maximal operator. The aim of this paper is to establish sufficient cond…
We prove that any homogeneous order one solution to 3-d nondivergence elliptic equations must be linear.
We give a simplified presentation of the obstacle problem approach to stochastic homogenization for elliptic equations in nondivergence form. Our argument also applies to equations which depend on the gradient of the unknown function. In the latter case, we overcome difficulties caused by a lack of estimates for the first derivatives of approximate correctors by modifying the perturbed test function argument to take advantage of the spreading of the contact set.
We extend the Caffarelli-Świech-Winter $C^{1,α}$ regularity estimates to $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form with superlinear growth in the gradient and unbounded coefficients. As an application, in addition to the usual $W^{2,p}$ results, we prove the existence of positive eigenvalues for proper operators with nonnegative unbounded weight, in particular for Pucci's operators with unbounded coefficients.
For a second-order elliptic equation of nondivergence form in the plane, we investigate conditions on the coefficients which imply that all strong solutions have first-order derivatives that are Lipschitz continuous or differentiable at a given point. We assume the coefficients have modulus of continuity satisfying the square-Dini condition, and obtain additional conditions associated with a dynamical system that is derived from the coefficients of the elliptic equation. Our results extend those of previous authors who assume the modulus of continuity satisfies the Dini condition.
In this paper we present examples of nondivergence form second order elliptic operators with continuous coefficients such that $L$ has an irregular boundary point that is regular for the Laplacian. Also for any eigenvalue spread <1 of the matrix of the coefficients we provide an example of operator with discontinuous coefficients that has regular boundary points nonequivalent to Laplacian's (we give examples for each direction of nonequivalence). All examples are constructed for each dimension starting with 3.
Homogenization of nondivergence-form elliptic equations with discontinuous coefficients and finite element approximation of the homogenized problem
本文研究满足 Cordes 条件的不连续周期系数非散度型椭圆方程均匀化，证明不变测度、校正子与最优收敛率，并提出基于有限元逼近不变测度的有效系数及均匀化解的数值格式。
We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of $\mathbb{R}^n$. The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the $p$-Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as w…
We develop a new, unified approach to the following two classical questions on elliptic PDE: the strong maximum principle for equations with non-Lipschitz nonlinearities, and the at most exponential decay of solutions in the whole space or exterior domains. Our results apply to divergence and nondivergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in…
We prove the doubling property of L-caloric measure corresponding to the second order parabolic equation in the whole space and in Lipschitz domains. For parabolic equations in the divergence form, a weaker form of the doubling property follows easily from a recent result, the backward Harnack inequality, and known estimates of Green's function. Our method works for both the divergence and nondivergence cases. Moreover, the backward Harnack inequality and estimates of Green's function are not needed in the course of proof.
For $n\geq 3$, we study the existence and asymptotic properties of the fundamental solution for elliptic operators in nondivergence form, ${\mathcal L}(x,\partial_x)=a_{ij}(x)\partial_i\partial_j+b_k(x)\partial_k$, where the $a_{ij}$ have modulus of continuity $ω(r)$ satisfying the square-Dini condition and the $b_k$ are allowed mild singularities of order $r^{-1}ω(r)$. A singular integral is introduced that controls the existence of the fundamental solution. We give examples that show the singular drift $b_k\partial_k$ may act as a perturbation that does not dramatically change the fundamenta…
The solvability in Sobolev spaces with special mixed norms is proved for nondivergence form second order parabolic equations. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, and be almost in VMO (vanishing mean oscillation) with respect to the other coordinates. This solvability result implies the weak uniqueness of solutions of the corresponding stochastic Itô equations in the class of "good" solutions (which is nonempty). This also implies uniqueness of a Green's function in the class of "good" ones (which is always nonempty)…
We prove the $W^{1,2}_{p}$-solvability of second order parabolic equations in nondivergence form in the whole space for $p\in (1,\infty)$. The leading coefficients are assumed to be measurable in one spatial direction and have vanishing mean oscillation (VMO) in the orthogonal directions and the time variable in each small parabolic cylinder with the direction depending on the cylinder. This extends a recent result by Krylov [17] for elliptic equations and removes the restriction that $p>2$.
In this short note we review some facts about elliptic differential operators on Riemannian manifolds.
We study Green's matrices for divergence form, second order strongly elliptic systems with bounded measurable coefficients in two dimensional domains. We establish existence, uniqueness, and pointwise estimates of the Green's matrices.
We construct the Green function for second order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition and the domain has $C^{1,1}$ boundary. We also obtain pointwise bounds for the Green functions and its derivatives.
We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green's function for constant coefficients equations.
On the approximation of the principal eigenvalue for a class of nonlinear elliptic operators
本文提出有限差分法计算非线性椭圆算子主特征值及对应的特征函数，通过非线性min-max优化问题证明方法收敛性，并通过实例验证有效性。
On the smoothness of solutions of fully nonlinear second order equations in the plane
本文研究了平面内全非线性二阶椭圆方程解的正则性，通过发散型和非发散型方程理论，证明了解在内部的高阶光滑性。
Wiener's criterion for the regularity of a boundary point with respect to the Dirichlet problem for the Laplace equation has been extended to various classes of elliptic and parabolic partial differential equations. They include linear divergence and nondivergence equations with discontinuous coefficients, equations with degenerate quadratic form, quasilinear and fully nonlinear equations, as well as equations on Riemannian manifolds, graphs, groups, and metric spaces. A common feature of these equations is that all of them are of second order, and Wiener type characterizations for higher orde…
By using some recent results for divergence form equations, we study the $L_p$-solvability of second-order elliptic and parabolic equations in nondivergence form for any $p\in (1,\infty)$. The leading coefficients are assumed to be in locally BMO spaces with suitably small BMO seminorms. We not only extend several previous results by Krylov and Kim [14]-[18] to the full range of $p$, but also deal with equations with more general coefficients.
The aim of this paper is to investigate Green's function for parabolic and elliptic systems satisfying a possibly nonlocal Robin-type boundary condition. We construct Green's function for parabolic systems with time-dependent coefficients satisfying a possibly nonlocal Robin-type boundary condition assuming that weak solutions of the system are locally Hölder continuous in the interior of the domain, and as a corollary we construct Green's function for elliptic system with a Robin-type condition. Also, we obtain Gaussian bound for Robin Green's function under an additional assumption that weak…
In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients in $\mathbb{R}^n$ and for the corresponding Green functions in arbitrary open sets. We impose certain non-homogeneous versions of de Giorgi-Nash-Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions.
We study the existence of the Green function for an elliptic system in divergence form $-\nabla\cdot a\nabla$ in $\mathbb{R}^d$, with $d>2$. The tensor field $a=a(x)$ is only assumed to be bounded and $λ$-coercive. For almost every point $y \in \mathbb{R}^d$, the existence of a Green's function $G(a; \cdot, y)$ centered in $y$ has been proven in [J. Conlon, A. Giunti and F.Otto, "Green's function for elliptic systems: Delmotte-Deuschel bounds", 2017]. In this paper, we show that the set of points $y \in \mathbb{R}^d$ for which $G(a; \cdot, y)$ does not exist has zero $p$-capacity, for an expon…
Nonlocal problems for higher-order elliptic operators in dihedral and plane angles are considered. The Green formula is obtained, which leads to adjoint problems that take the form of nonlocal transmission problems in dihedral and plane angles. This allows us to establish necessary conditions of Fredholm solvability and sufficient conditions of one-valued solvability for nonlocal problems in dihedral angles.
Note on Green's functions of non-divergence elliptic operators with continuous coefficients
本文改进了Kim和Lee的结果，证明当非散度型椭圆算子系数具有Dini平均振荡时，其格林函数在极点附近与冻结常系数方程的格林函数具有相同的渐近行为。
We establish existence and pointwise estimates of fundamental solutions and Green's matrices for divergence form, second order strongly elliptic systems in a domain $Ω\subseteq \mathbb{R}^n$, $n \geq 3$, under the assumption that solutions of the system satisfy De Giorgi-Nash type local Hölder continuity estimates. In particular, our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation.
On regularity of averaged Green's functions in homogenization of elliptic PDE
本文利用Bourgain的洞见，在更一般情形下证明了随机系数椭圆偏微分方程均质化中平均格林函数的二阶导数估计。
Essentials of Real Analysis and Morrey-Sobolev spaces for second-order elliptic and parabolic PDEs with singular first-order coefficients
本文简要介绍了Hardy-Littlewood极大函数定理等实分析结果在具奇异一阶系数的二阶椭圆与抛物型PDE的Sobolev及Morrey-Sobolev空间理论中的应用，并在最后一节给出了新结果。
We construct the fundamental solution or Green function for a divergence form elliptic system in two dimensions with bounded and measurable coefficients. We consider the elliptic system in a Lipschitz domain with mixed boundary conditions. Thus we specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We require a corkscrew or non-tangential accessibility condition on the set where we specify Dirichlet boundary conditions. Our proof proceeds by defining a variant of the space $BMO(\partial Ω)$ that is adapted to the boundary conditions and showing tha…
This article is about the $\mathbb{Z}^d$-periodic Green function $G_n(x,y)$ of the multiscale elliptic operator $Lu=-{\rm div}\left( A(n\cdot) \cdot \nabla u \right)$, where $A(x)$ is a $\mathbb{Z}^d$-periodic, coercive, and Hölder continuous matrix, and $n$ is a large integer. We prove here pointwise estimates on $G_n(x,y)$, $\nabla_x G_n(x,y)$, $\nabla_y G_n(x,y)$ and $\nabla_x \nabla_y G_n(x,y)$ in dimensions $d \geq 2$. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.
We establish global Gaussian estimates for the Green's matrix of divergence form, second order parabolic systems in a cylindrical domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local Hölder estimate. From these estimates, we also derive global estimates for the Green's matrix for elliptic systems with bounded measurable coefficients in two dimensional domains. We present a unified approach valid for both the scalar and vectorial cases and discuss several applications of our result.
Boundary estimates and Green function's expansion for elliptic systems with random coefficients
本文研究了具有随机系数的椭圆系统边界估计及格林函数展开，推导了退火意义下的衰减和校正估计，建立了与其他领域的联系。
In three-dimensional case, we consider two classical operators: Schrodinger operator and an operator in the divergence form. For slowly-decaying oscillating potentials, we establish spatial asymptotics of the Green's function. The main term in this asymptotics involves vector-valued analytic function whose behavior is studied away from the spectrum. The absolute continuity of the spectrum is established as a corollary. For the operator in the divergence form, we consider the wave equation and establish existence of wave operators.
We prove that for an open domain $D \subset \mathbb{R}^d $ with $d \geq 2 $ , for every (measurable) uniformly elliptic tensor field $a$ and for almost every point $y \in D$ , there exists a unique Green's function centred in $ y $ associated to the vectorial operator $ -\nabla \cdot a\nabla $ in D. In particular, when $d > 2$ this result also implies the existence of the fundamental solution for elliptic systems, i.e. the Green function for $ -\nabla \cdot a\nabla $ in $ \mathbb{R}^d $. Moreover, introducing an ensemble $\langle\cdot \rangle$ over the set of uniformly elliptic tensor fields, …
For a given second-order linear elliptic operator $L$ which admits a positive minimal Green function, and a given positive weight function $W$, we introduce a family of weighted Lebesgue spaces $L^p(ϕ_p)$ with their dual spaces, where $1\leq p\leq \infty$. We study some fundamental properties of the corresponding (weighted) Green operators on these spaces. In particular, we prove that these Green operators are bounded on $L^p(ϕ_p)$ for any $1\leq p\leq \infty$ with a uniform bound. We study the existence of a principal eigenfunction for these operators in these spaces, and the simplicity of th…
We construct Green's functions for elliptic operators of the form $\mathcal{L}u=-\text{div}(A\nabla u+bu)+c\nabla u+du$ in domains $Ω\subseteq\mathbb R^n$, under the assumption $d\geq\text{div}b$, or $d\geq\text{div}c$. We show that, in the setting of Lorentz spaces, the assumption $b-c\in L^{n,1}(Ω)$ is both necessary and optimal to obtain pointwise bounds for Green's functions. We also show weak type bounds for Green's functions and their gradients. Our estimates are scale invariant and hold for general domains $Ω\subseteq\mathbb R^n$. Moreover, there is no smallness assumption on the norms …
We construct Green's function for second order elliptic operators of the form $Lu=-\nabla \cdot (\mathbf{A} \nabla u + \boldsymbol{b} u)+ \boldsymbol c \cdot \nabla u+ du$ in a domain and obtain pointwise bounds, as well as Lorentz space bounds. We assume that the matrix of principal coefficients $\mathbf A$ is uniformly elliptic and bounded and the lower order coefficients $\boldsymbol{b}$, $\boldsymbol{c}$, and $d$ belong to certain Lebesgue classes and satisfy the condition $d - \nabla \cdot \boldsymbol{b} \ge 0$. In particular, we allow the lower order coefficients to be singular. We also …
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