Doyoon Kim, N. V. Krylov
We investigate the unique solvability of second order parabolic equations in non-divergence form in $W_p^{1,2}((0,T) \times \bR^d)$, $p \ge 2$. The leading coefficients are only measurable in either one spatial variable or time and one spatial variable. In addition, they are VMO (vanishing mean oscillation) with respect to the remaining variables.
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We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, or on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.
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.
We present a refined Green's function estimate of the time measurable parabolic operators on conic domains that involves mixed weights consisting of appropriate powers of the distance to the vertex and of the distance to the boundary.
In this paper we present a weighted $L_p$-theory of parabolic systems on a half space. The leading coefficients are assumed to be only measurable in $t$ and have small bounded mean oscillations (BMO) with respect to $x$, and the lower order coefficients are allowed to blow up near the boundary.
We establish the $L_p$-solvability for time fractional parabolic equations when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend a recent result in [6] to a large extent.
In this paper, we establish $L_p$ estimates and solvability for time fractional divergence form parabolic equations in the whole space when leading coefficients are merely measurable in one spatial variable and locally have small mean oscillations with respect to the other variables. The corresponding results for equations on a half space are also derived.
We consider linear elliptic and parabolic equations with measurable coefficients and prove two types of $L_{p}$-estimates for their solutions, which were recently used in the theory of fully nonlinear elliptic and parabolic second order equations in \cite{DKL}. The first type is an estimate of the $γ$th norm of the second-order derivatives, where $γ\in(0,1)$, and the second type deals with estimates of the resolvent operators in $L_{p}$ when the first-order coefficients are summable to an appropriate power.
We are dealing with possibly degenerate second-order parabolic operators whose coefficients are infinitely differentiable with respect to space variables and only measurable with respect to the time variable. We impose the Hörmander condition on the diffusion coefficients and prove that the solutions of the corresponding equations with right-hand sides which are infinitely differentiable in the space variables in a space-time domain have also this property.
In this article we present several results concerning uniqueness of $C$-viscosity and $L_{p}$-viscosity solutions for fully nonlinear parabolic equations. In case of the Isaacs equations we allow lower order terms to have just measurable bounded coefficients. Higher-order coefficients are assumed to be Hölder continuous in $x$ with exponent slightly less than $1/2$. This case is treated by using stability of maximal and minimal $L_{p}$-viscosity solutions.
We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which is assumed to be only measurable either in the time or the spatial variable. As functions of the other variables the coefficients have small bounded mean oscillation (BMO) semi-norms. The lower-order coefficients are allowed to blow up near the boundary with a certain optimal growth condition. As a corollary, we also obtain the corresponding results for ell…
$L_p$-estimates of the conormal derivative problem for parabolic equations with time measurable coefficients and $A_p$-weights
本文针对Reifenberg平坦域上带共法向导数边界条件的散度型抛物方程，在时间可测、空间小振荡系数下建立加权混合范数估计，并利用半时间导数估计解决正则性难题。
Green's function estimates for time measurable parabolic operators on polyhedrons and polyhedral cones
本文在三维多面体与多面锥区域上建立抛物型算子的格林函数估计，引入依赖顶点、边和边界距离的混合权重，并由区域几何明确给出权重参数的允许范围。
We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.
We establish existence, uniqueness, and arbitrary order Sobolev regularity results for the second order parabolic equations with measurable coefficients defined on the conic domains $D$ of the type $$ D(M):=\left\{x\in R^d :\,\frac{x}{|x|}\in M\right\}, \quad \quad M \subset S^{d-1}. $$ We obtain the regularity results by using a system of mixed weights consisting of appropriate powers of the distance to the vertex and of the distance to the boundary. We also provide the sharp ranges of admissible powers of the distance to the vertex and to the boundary.
We prove Gaussian upper and lower bounds for the fundamental solutions of a class of degenerate parabolic equations satisfying a weak Hormander condition. The bound is independent of the smoothness of the coefficients and generalizes classical results for uniformly parabolic equations
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.
We establish a new regularity property for weak solutions of parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally H{ö}lder continuous Lp valued functions for some p > 2.
Auscher, McIntosh and Tchamitchian studied the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on $\mathbb{R}^n$. In particular, in the case when $n=2$ they obtained Gaussian upper bound estimates for the heat kernel without imposing further assumption on the coefficients. We study the fundamental solutions of the systems of second order parabolic equations in the divergence form with bounded, measurable, time-independent coefficients, and extend their results to the systems of parabolic equations.
We present the unique solvability in Sobolev spaces of time fractional parabolic equations in divergence and non-divergence forms. The leading coefficients are merely measurable in $(t,x_1)$ for $a^{ij}$, $1 \leq i,j \leq d$, $(i,j) \neq (1,1)$. The coefficient $a^{11}$ is merely measurable locally either in $t$ or $x_1$. As functions of the remaining variables, the coefficients have small mean oscillations. We consider mixed norm Sobolev spaces with Muckenhoupt weights. Our results generalize previous work on parabolic equations with time fractional derivatives to a much larger class of coeff…
The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder. The directions in which the coefficients are only measurable vary depending on each cylinder. The corresponding elliptic problem is also considered.
Regularity theory for sub-critical $p$-parabolic systems with measurable coefficients
本文研究了具有可测系数的亚临界 $p$-抛物系统弱解的正则性理论，得出了局部有界性和梯度更高积分性的主要结果。
We consider fractional operators of the form $$\mathcal{H}^s=(\partial_t -\mathrm{div}_{x} ( A(x,t)\nabla_{x}))^s,\ (x,t)\in\mathbb R^n\times\mathbb R,$$ where $s\in (0,1)$ and $A=A(x,t)=\{A_{i,j}(x,t)\}_{i,j=1}^{n}$ is an accretive, bounded, complex, measurable, $n\times n$-dimensional matrix valued function. We study the fractional operators ${\mathcal{H}}^s$ and their relation to the initial value problem $$(λ^{1-2s}\mathrm{u}')'(λ) =λ^{1-2s}\mathcal{H} \mathrm{u}(λ), \quad λ\in (0, \infty),$$ $$\mathrm{u}(0) = u,$$ in $\mathbb R_+\times \mathbb R^n\times\mathbb R$. Exploring this type of r…
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 $p=2$.
For a family of second-order parabolic systems with bounded measurable, rapidly oscillating and time-dependent periodic coefficients, we investigate the sharp convergence rates of weak solutions in $L^2$. Both initial-Dirichlet and initial-Neumann problems are studied.
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。
We establish pointwise estimates for the Green function to the Dirichlet problem for parabolic equation with coefficients measurable in time variable. Using these estimate we obtain coercive estimates for this problem in anisotropic weighted Lebesgue spaces and prove the solvability theorems.
For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coeffcients and lower order terms from non-linear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.
We study both divergence and non-divergence form parabolic and elliptic equations in the half space $\{x_d>0\}$ whose coefficients are the product of $x_d^α$ and uniformly nondegenerate bounded measurable matrix-valued functions, where $α\in (-1, \infty)$. As such, the coefficients are singular or degenerate near the boundary of the half space. For equations with the conormal or Neumann boundary condition, we prove the existence, uniqueness, and regularity of solutions in weighted Sobolev spaces and mixed-norm weighted Sobolev spaces when the coefficients are only measurable in the $x_d$ direc…
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 $p=q$.
Sobolev estimates for parabolic and elliptic equations in divergence form with degenerate coefficients
本文研究上半空间中主系数为 $x_d^2a_{ij}$ 的退化散度型抛物与椭圆方程，在系数具有小有界平均振荡条件下，建立了加权混合范数 Sobolev 空间中解的适定性与正则性。
On one-space dimensional parabolic equations with measurable coefficients: Sobolev estimates and the Alexandrov maximum principle
本文研究一维可测系数抛物方程，构造反例证明Sobolev估计在临界指数处失效，并证明在2附近最优阶为κ的区间内估计成立且唯一可解，进而得到最优的Alexandrov极大值原理。
We prove Schauder estimates for solutions to both divergence and non-divergence type higher-order parabolic systems in the whole space and the half space. We also provide an existence result for divergence type systems in a cylindrical domain. All coefficients are assumed to be only measurable in the time variable and Hölder continuous in the spatial variables.
In this paper we prove a generalization of Montel's theorem for a class of first order elliptic equations with measurable coefficients involving Hodge-Dirac operators. We then apply this result to sequences of solutions of second order uniformly elliptic equations with measurable coefficients on divergence form and show that this results in a precompactness result for such sequences.
An observability for parabolic equations from a measurable set in time
本文针对凸域上的抛物型方程，通过时间单点定量唯一延拓，建立了在空间开子集与时间正测集乘积区域上的新可观测性估计，并应用于范数和时间最优控制问题的bang-bang性质。
We prove the interior null-controllability of one-dimensional parabolic equations with time independent measurable coefficients.
Several negative results are presented concerning the solvability in Sobolev classes of the Cauchy problem for the inhomogeneous second-order uniformly parabolic equations without lower order terms in one space dimension. The main coefficient is assumed to be a bounded measurable function of $(t,x)$ bounded away from zero. We also discuss upper and lower estimates of certain kind on the fundamental solutions of such equations.
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 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.
Gaussian estimates for fundamental solutions of higher-order parabolic equations with time-independent coefficients
研究高阶抛物方程的De Giorgi-Moser-Nash估计，获得基本解的高斯上界和Hölder正则性估计。
Under various conditions, we establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving Hölder semi-norms not with respect to all, but only with respect to some of the independent variables. A novelty of our results is that the coefficients are allowed to be merely measurable with respect to the other independent variables.
We give a unified approach to weighted mixed-norm estimates and solvability for both the usual and time fractional parabolic equations in nondivergence form when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend the previous result in [6] for unmixed $L_p$-estimates without weights.
We prove continuous dependence on Cauchy data for a backward parabolic operator whose coefficients are Log-Lipschitz continuous in time.
Stochastic and deterministic parabolic equations with bounded measurable coefficients in space and time: well-posedness and maximal regularity
本文针对散度形式线性随机抛物方程，在系数仅需有界可测的广泛条件下，通过引入路径弱解和新的调和分析工具，建立了适定性与极大正则性，并将Lions定理从T²推广至T^p。
On the modulus of continuity of solutions to nonlocal parabolic equations
本文在极小及稍强尾部条件下，分别量化并推导了非局部抛物型方程局部有界弱解的一般连续性模与Hölder连续性模，并在具有可测核的非局部$p$-Laplacian框架下证明了这些正则性估计。
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 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 are concerned in this paper with the degenerate fractional diffusion advection equations posed in bounded domains. Due to a suitable formulation, we show the existence of weak entropy solutions for measurable and bounded initial and Dirichlet boundary data. Moreover, we prove a $L^1-$type contraction property for weak entropy solutions obtained via parabolic perturbation. This is a weak selection principle which means that the weak entropy solutions are stable in this class.
We establish the existence and the pointwise bound of the fundamental solution for the stationary Stokes system with measurable coefficients in the whole space $\mathbb{R}^d$, $d \ge 3$, under the assumption that weak solutions of the system are locally Hölder continuous. We also discuss the existence and the pointwise bound of the Green function for the Stokes system with measurable coefficients on $Ω$, where $Ω$ is an unbounded domain such that the divergence equation is solvable. Such a domain includes, for example, half space and an exterior domain.
We prove uniqueness for backward parabolic equations whose coefficients are Osgood continuous in time for $t>0$ but not at $t=0$.
This work is devoted to the strong unique continuation problem for second order parabolic equations with nonsmooth coefficients. Introduction and bibliography have been revised.
We find minimal regularity conditions on the coefficients of a parabolic operator, ensuring that no nontrivial solution tends to zero faster than any exponential.
We construct fundamental solutions of second-order parabolic systems of divergence form with bounded and measurable leading coefficients and divergence free first-order coefficients in the class of $BMO^{-1}_x$, under the assumption that weak solutions of the system satisfy a certain local boundedness estimate. We also establish Gaussian upper bound for such fundamental solutions under the same conditions.
We establish two-sided Gaussian bounds for fundamental solutions of general non-divergence form parabolic operators with Hölder continuous coefficients. The result we obtain is essentially based on parametrix method.
We consider an oblique derivative problem in a wedge for nondivergence parabolic equations with discontinuous in $t$ coefficients. We obtain weighted coercive estimates of solutions in anisotropic Sobolev spaces.
We find new quantitative estimates on the space-time analyticity of solutions to linear parabolic equations with time-independent coefficients and apply them to obtain observability inequalities for its solutions over measurable sets.
We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori $\dot W^1_q$-estimates for any $q\in [2,\infty)$ when the coefficients are merely measurable functions in one fixed direction. For the system on bounded Lipschitz domains with a small Lipschitz constant, we obtain a $W^1_q$-estimate and prove the solvability for any $q\in (1,\infty)$ when the coefficients are merely measurable functions in one direction and have locally small mean oscillations in the orthogonal dir…
We obtain Calderón-Zygmund type estimates for parabolic equations with Orlicz growth, where nonlinearities involved in the equations may be discontinuous for the space and time variables. In addition, we consider parabolic systems with the Uhlenbeck structure.
We study the Neumann Green's function for second order parabolic systems in divergence form with time-dependent measurable coefficients in a cylindrical domain $\mathcal{Q}=Ω\times (-\infty,\infty)$, where $Ω\subset \mathbb{R}^n$ is an open connected set such that a multiplicative Sobolev embedding inequality holds there. Such a domain includes, for example, a bounded Sobolev extension domain, a special Lipschitz domain, and an unbounded domain with compact Lipschitz boundary. We construct the Neumann Green's function in $\mathcal{Q}$ under the assumption that weak solutions of the systems sat…
We obtain local boundedness of weak solutions of double phase quasilinear parabolic equations of the form \[u_t-\text{div} \left(|\nabla u|^{p-2}\nabla u+a(x,t)|\nabla u|^{q-2}\nabla u\right)=0,\] where, we have imposed the restrictions $\frac{2N}{N+2}<p<\infty$, $0\leq a(x,t)\leq M$ is measurable and $q < p\frac{N+1}{N-1}$.
We consider time-fractional parabolic equations with a Caputo time derivative of order $α\in(0,1)$. For such equations, we give an elementary proof of the weak maximum principle under no assumptions on the sign of the reaction coefficient. This proof is also extended for weak solutions, as well as for various types of boundary conditions and variable-coefficient variable-order multiterm time-fractional parabolic equations.
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