Elliptic equations with VMO a, b$\,\in L_{d}$, and c$\,\in L_{d/2}$
N. V. Krylov
备注：
19 p, two references added, few glitches are corrected and numerous comments by Hongjie Dong are taken into account, this version is accepted by the Trans. AMS
We consider elliptic equations with operators $L=a^{ij}D_{ij}+b^{i}D_{i}-c$ with $a$ being almost in VMO, $b\in L_{d}$ and $c\in L_{q}$, $c\geq0$, $d>q\geq d/2$. We prove the solvability of $Lu=f\in L_{p}$ in bounded $C^{1,1}$-domains, $1<p\leq q$, and of $λu-Lu=f$ in the whole space for any $λ>0$. Weak uniqueness of the martingale problem associated with such operators is also obtained.
An $L_{p}$-theory of divergence and non-divergence form elliptic and parabolic equations is presented. The main coefficients are supposed to belong to the class $VMO_{x}$, which, in particular, contains all functions independent of $x$. Weak uniqueness of the martingale problem associated with such equations is obtained.
The solvability in $W^{2}_{p}(\bR^{d})$ spaces is proved for second-order elliptic equations with coefficients which are measurable in one direction and VMO in the orthogonal directions in each small ball with the direction depending on the ball. This generalizes to a very large extent the case of equations with continuous or VMO coefficients.
We establish partial regularity for vector-valued solutions to inhomogeneous elliptic systems in divergence form where the coefficients are possibly discontinuous with respect to $x$. More precisely, we assume a VMO-condition with respect to the $x$ and continuity with respect to $u$ and prove Hölder continuity of the solutions outside of singular sets.
We prove the solvability in Sobolev spaces $W^{1,2}_p$, $p>d+1$, of the terminal-boundary value problem for a class of fully nonlinear parabolic equations, including parabolic Bellman's equations, in bounded cylindrical domains with VMO ``coefficients''. The solvability in $W^{2}_p$, $p>d$, of the corresponding elliptic boundary-value problem is also obtained.
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 $p=2$.
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."
In this paper, we prove $L^p$ estimates for the fractional derivatives of solutions to elliptic fractional partial differential equations whose coefficients are $VMO$. In particular, our work extends the optimal regularity known in the second order elliptic setting to a spectrum of fractional order elliptic equations.
We prove mixed $L_{p}(L_{q})$-estimates, with $p,q\in(1,\infty)$, for higher-order elliptic and parabolic equations on the half space $\R^{d+1}_{+}$ with general boundary conditions which satisfy the Lopatinskii--Shapiro condition. We assume that the elliptic operators $A$ have leading coefficients which are in the class of vanishing mean oscillations in both the time and the space variable. In the proof, we produce mean oscillation estimates for equations on the half space with general boundary conditions.
We prove that for any homogeneous, second order, constant complex coefficient elliptic system $L$, the Dirichlet problem in $\mathbb{R}^{n}_{+}$ with boundary data in BMO is well-posed in the class of functions $u$ with $dμ_u(x',t):=|\nabla u(x',t)|^2\,t\,dx'dt$ being a Carleson measure. We establish a Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for smooth null-solutions $u$ of such systems satisfying the said Carleson measure condition. These imply that BMO can be characterized as the collection of nontangential pointwise traces of smooth null-s…
We prove higher regularity for nonlinear nonlocal equations with possibly discontinuous coefficients of VMO-type in fractional Sobolev spaces. While for corresponding local elliptic equations with VMO coefficients it is only possible to obtain higher integrability, in our nonlocal setting we are able to also prove a substantial amount of higher differentiability, so that our result is in some sense of purely nonlocal type. By embedding, we also obtain higher Hölder regularity for such nonlocal equations.
We study the obstacle problem with an elliptic operator in divergence form. 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 in the case where the coefficients are in VMO.
We prove weighted mixed $L_{p}(L_{q})$-estimates, with $p,q\in(1,\infty)$, for higher-order elliptic and parabolic equations on the half space $\mathbb{R}^{d+1}_{+}$ and on domains with general boundary conditions which satisfy the Lopatinskii--Shapiro condition. We assume that the elliptic operators $A$ have leading coefficients which are in the class of vanishing mean oscillations both in the time and the space variables, and that the boundary conditions have variable leading coefficients. The proofs are based on and generalize the estimates recently obtained by the authors in [DG17]. [DG17]…
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 $p>2$.
We present a result about solvability in $W^{2}_{p}$, $p>d$, in the whole space $\bR^{d}$ of Bellman's equations with VMO ``coefficients''. Parabolic equations are touched upon as well.
Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except $a^{11}$) are measurable in both time and one spatial variable, and VMO in the other spatial variables. The coefficient $a^{11}$ is measurable in time and VMO in the spatial variables. The unique solvability of equations in the whole space is applied to solving Dirichlet and oblique derivative problems for parabolic equations defined in a half-space.
An $L_{q}(L_{p})$-theory of divergence and non-divergence form parabolic equations is presented. The main coefficients are supposed to belong to the class $VMO_{x}$, which, in particular, contains all measurable functions depending only on $t$. The method of proving simplifies the methods previously used in the case $p=q$.
We show that if $Ω$ is a vanishing chord-arc domain and $L$ is a divergence-form elliptic operator with Hölder-continuous coefficient matrix, then $\log k_L \in VMO$, where $k_L$ is the elliptic kernel for $L$ in the domain $Ω$. This extends the previous work of Kenig and Toro in the case of the Laplacian.
We prove that boundary value problems for fully nonlinear second-order parabolic equations admit $L_{p}$-viscosity solutions, which are in $C^{1+α}$ for an $α\in(0,1)$. The equations have a special structure that the "main" part containing only second-order derivatives is given by a positive homogeneous function of second-order derivatives and as a function of independent variables it is measurable in the time variable and, so to speak, VMO in spatial variables.
In this paper, we establish interior $C^{1,α}$ estimates for solutions of the linearized Monge-Amp$\grave{e}$re equation $$\mathcal{L}_ϕu:=\mathrm{tr}[ΦD^2 u]=f,$$ where the density of the Monge-Amp$\grave{e}$re measure $g:=\mathrm{det}D^2ϕ$ satisfies a $\mathrm{VMO}$-type condition and $Φ:=(\mathrm{det}D^2ϕ)(D^2ϕ)^{-1}$ is the cofactor matrix of $D^2ϕ$.
In this paper, we establish global $W^{2,p}$ estimates for solutions of the linearized Monge-Amp$\grave{e}$re equation $$\mathcal{L}_ϕu:=\mathrm{tr}[ΦD^2 u]=f,$$ where the density of the Monge-Amp$\grave{e}$re measure $g:=\mathrm{det}D^2ϕ$ satisfies a $\mathrm{VMO}$-type condition, and $Φ:=(\mathrm{det}D^2ϕ)(D^2ϕ)^{-1}$ is the cofactor matrix of $D^2ϕ$.
We study a class of nonlocal double phase problems with discontinuous coefficients. A local self-improving property and a higher Hölder continuity result for weak solutions to such problems are obtained under the assumptions that the associated coefficient functions are of type VMO (vanishing mean oscillation) and that the principal coefficient depends not only on the variables but also on the solution itself.
This paper is a comprehensive study of $L_p$ estimates for time fractional wave equations of order $α\in (1,2)$ in the whole space, a half space, or a cylindrical domain. We obtain weighted mixed-norm estimates and solvability of the equations in both non-divergence form and divergence form when the leading coefficients have small mean oscillation in small cylinders.
We prove the unique solvability of second order elliptic equations in non-divergence form in Sobolev spaces. The coefficients of the second order terms are measurable in one variable and VMO in other variables. From this result, we obtain the weak uniqueness of the martingale problem associated with the elliptic equations.
We prove weighted and mixed-norm Sobolev estimates for fully nonlinear elliptic and parabolic equations in the whole space under a relaxed convexity condition with almost VMO dependence on space-time variables. The corresponding interior and boundary estimates are also obtained.
This work is concerned with both higher integrability and differentiability for linear nonlocal equations with possibly very irregular coefficients of VMO-type or even coefficients that are merely small in BMO. In particular, such coefficients might be discontinuous. While for corresponding local elliptic equations with VMO coefficients such a gain of Sobolev regularity along the differentiability scale is unattainable, it was already observed in previous works that gaining differentiability in our nonlocal setting is possible under less restrictive assumptions than in the local setting. In th…
We consider solutions $u\in W^{1,p}\big(Ω;\mathbb{R}^{N}\big)$ of the $p$-Laplacian PDE \begin{equation} \nabla\cdot\big(a(x)|Du|^{p-2}Du\big)=0,\notag \end{equation} for $x\inΩ\subseteq\mathbb{R}^{n}$, where $Ω$ is open and bounded. More generally, we consider solutions of the elliptic system \begin{equation} \nabla\cdot\left(a(x)g'\big(a(x)|Du|\big)\frac{Du}{|Du|}\right)=0\text{, }x\inΩ\notag \end{equation} as well as minimizers of the functional \begin{equation} \int_Ωg\big(a(x)|Du|\big)\ dx.\notag \end{equation} In each case, the coefficient map $a\ : \ Ω\rightarrow\mathbb{R}$ is only assu…
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 consider a divergence form hypoelliptic operator consisting of a system of real smooth vector fields $X_{1},..., X_{q}$ satisfying Hörmander condition in some domain $Ω\subseteq\erren$. Interior $L^{p}$ estimates, $2\leq p<\infty$, can be obtained for weak solutions of the equation $X_j^T(a^{ij}X_iu)=X_j^T F^j,$ by assuming that the coefficients $a^{ij}$ belong locally to the space $VMO_X$ with respect to the Carnot--Caratheodory metric induced by the vector fields.
We consider uniformly elliptic and parabolic second-order equations with bounded zeroth-order and bounded VMO leading coefficients and possibly growing first-order coefficients. We look for solutions which are summable to the $p$-th power with respect to the usual Lebesgue measure along with their first and second-order derivatives with respect to the spatial variable.
The unique solvability of parabolic equations in Sobolev spaces with mixed norms is presented. The second order coefficients (except $a^{11}$) are assumed to be only measurable in time and one spatial variable, and VMO in the other spatial variables. The coefficient $a^{11}$ is measurable in one spatial variable and VMO in the other variables.
In this paper, we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients. We begin with an approach given by N. V. Krylov to parabolic equations in the whole space with VMO$_x$ coefficients. We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions, weighted $L_p$ estimates with Muckenhoupt ($A_p$) weights, non-local elliptic and parabolic equations, as well as fully nonlinear elliptic and parabolic equations.
We study the Dirichlet problem in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order, with bounded, complex-valued coefficients. Our main result gives a sharp condition on the local mean oscillations of the coefficients of the differential operator and the unit normal to the boundary (which is automatically satisfied if these functions belong to space VMO) guaranteeing that the solution operator associated with this problem is an isomorphism.
We consider elliptic equations with operators $L=a^{ij}D_{ij}+b^{i}D_{i}-c$ with $a$ being almost in VMO, $b$ in a Morrey class containing $ L_{d}$, and $c\geq0$ in a Morrey class containing $L_{d/2}$. We prove the solvability in Sobolev spaces of $Lu=f\in L_{p}$ in bounded $C^{1,1}$-domains, and of $λu-Lu=f$ in the whole space for any $λ>0$. Weak uniqueness of the martingale problem associated with such operators is also discussed.
Solutions of the 3D inhomogeneous incompressible Navier-Stokes system with initial velocity in $VMO^{-1}$
University of Bath（巴斯大学）；School of Mathematical Science, Peking University（北京大学数学科学学院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
We present some results concerning the solvability of linear elliptic equations in bounded domains with the main coefficients almost in VMO, the drift and the free terms in Morrey classes containing $L_{d}$, and bounded zeroth order coefficient. We prove that the second-order derivatives of solutions are in a local Morrey class containing $W^{2}_{p,loc}$. Actually, the exposition is given for fully nonlinear equations and encompasses the above mentioned results, which are new even if the main part of the equation is just the Laplacian.
We establish the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like $H(v,Dv,D^{2}v,x)=0$ in smooth domains without requiring $H$ to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of $H$ at points at which $|D^{2}v|\leq K$, where $K$ is any given constant. For large $|D^{2}v|$ some kind of relaxed convexity assumption with respect to $D^{2}v$ mixed with a VMO condition with respect to $x$ are still imposed. The solutions are sought in Sobolev classes.
Let Omega be a non-empty open proper and connected subset of R^n. Consider p elliptic Schrödinger type operator L_{E}u=A_{E}u+V in Omega, and the linear parabolic operator L_{P}u=A_{P}u+Vu in Omega x (0,T), where the coefficients of A_{E} and A_{P} are in VMO and the potential V satisfies a reverse-Hölder condition. The aim of this paper is to obtain a priori estimates for the operators L_{E} and L_{P} in weighted Sobolev spaces involving the distance to the boundary and weights in a local-A class.
In this paper we study boundary value problems for higher order elliptic differential operators in divergence form. We establish well posedness for problems with boundary data in Besov spaces $\dot B^{p,p}_s$, $p\leq 1$, given well posedness for appropriate values of $s$ and $p>1$. We work with smoothness parameter $s$ between $0$ and $1$; this allows us to consider inhomogeneous differential equations. Combined with results of Maz'ya, I. Mitrea, M. Mitrea, and Shaposhnikova, this allows us to establish new well posedness results for higher order operators whose coefficients are in or close to…
We consider the Robin boundary value problem $\mathrm{div} (A \nabla u) = \mathrm{div} \mathbf{f}+F$ in $Ω$, $\mathcal{C}^1$ domain, with $(A \nabla u - \mathbf{f})\cdot \mathbf{n} + αu = g$ on $Γ$, where the matrix $A$ belongs to $VMO (\mathbb{R}^3) $, and discover the uniform estimates on $\|u\|_{W^{1,p}(Ω)}$, with $1 < p < \infty$, independent on $α$. At the difference with the case $p = 2,$ which is simpler, we call here the weak reverse Hölder inequality. This estimates show that the solution of Robin problem converges strongly to the solution of Dirichlet (resp. Neumann) problem in corre…
We establish the existence of solutions of fully nonlinear elliptic second-order equations like $H(v,Dv,D^{2}v,x)=0$ in smooth domains without requiring $H$ to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of $H$ at points at which $|D^{2}v|\leq K$, where $K$ is any given constant. For large $|D^{2}v|$ some kind of relaxed convexity assumption with respect to $D^{2}v$ mixed with a VMO condition with respect to $x$ are still imposed. The solutions are sought in Sobolev classes. We also establish the solvability without almost any c…
We study qualitative properties of solutions to double divergence form elliptic equations (or stationary Kolmogorov equations) on~$\mathbb{R}^d$. It is shown that the Harnack inequality holds for nonnegative solutions if the diffusion matrix $A$ is nondegenerate and satisfies the Dini mean oscillation condition and the drift coefficient $b$ is locally integrable to a power $p>d$. We establish new estimates for the $L^p$-norms of solutions and obtain a generalization of the known theorem of Hasminskii on the existence of a probability solution to the stationary Kolmogorov equation to the case w…
We study nonlinear elliptic equations in divergence form $${\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G.$$ When ${\mathcal A}$ has linear growth in $Du$, and assuming that $x\mapsto{\mathcal A}(x,ξ)$ enjoys $B^α_{\frac{n}α, q}$ smoothness, local well-posedness is found in $B^α_{p,q}$ for certain values of $p\in[2,\frac{n}α)$ and $q\in[1,\infty]$. In the particular case ${\mathcal A}(x,ξ)=A(x)ξ$, $G=0$ and $A\in B^α_{\frac{n}α,q}$, $1\leq q\leq\infty$, we obtain $Du\in B^α_{p,q}$ for each $p<\frac{n}α$. Our main tool in the proof is a more general result, that holds also if ${\m…
In this paper, we study parabolic equations in divergence form with coefficients that are singular degenerate as some Muckenhoupt weight functions in one spatial variable. Under certain conditions, weighted reverse Hölder's inequalities are established. Lipschitz estimates for weak solutions are proved for homogeneous equations with singular degenerate coefficients depending only on one spatial variable. These estimates are then used to establish interior, boundary, and global weighted estimates of Calderón-Zygmund type for weak solutions, assuming that the coefficients are partially VMO (vani…
Let $Ω$ be a Lipschitz domain in $\mathbb R^d$, and let $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$ be a strongly elliptic operator on $Ω$. We suppose that $\varepsilon$ is small and the function $A$ is Lipschitz in the first variable and periodic in the second, so the coefficients of $\mathcal A^\varepsilon$ are locally periodic and rapidly oscillate. Given $μ$ in the resolvent set, we are interested in finding the rates of approximations, as $\varepsilon\to0$, for $(\mathcal A^\varepsilon-μ)^{-1}$ and $\nabla(\mathcal A^\varepsilon-μ)^{-1}$ in the operator topology o…
We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $Ω$ in $\mathbb{R}^n$, $n \ge 2$: $$ -\sum_{i,j=1}^n a^{ij}D_{ij} u + b \cdot D u + cu = f \;\;\text{ in $Ω$} \quad \text{and} \quad u=0 \;\;\text{ on $\partial Ω$} $$ and $$ - {\rm div} \left( A D u \right) + {\rm div}(ub) + cu = {\rm div} F \;\;\text{ in $Ω$} \quad \text{and} \quad u=0 \;\;\text{ on $\partial Ω$} , $$ where $A=[a^{ij}]$ is symmetric, uniformly elliptic, and of vanishing mean oscillation (VMO). The main purposes of this paper is to study uni…
We prove an existence result for a quasilinear elliptic equation satisfying natural growth conditions. As a consequence, we deduce an existence result for a quasilinear elliptic equation containing a singular drift. A key tool, in the proof, is the study of an auxiliary variational inequality playing the role of "natural constraint"
