Parabolic and Elliptic Equations with VMO Coefficients

By Nicolai V. Krylov. 2005

297 citations recorded by the source corpus.

An Lp-theory of divergence and non-divergence form elliptic and parabolic equations is presented. The main coefficients are supposed to belong to the class VMOx, which, in particular, contains all functions independent of x. Weak uniqueness of the martingale problem associated with such equations is obtained.

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Citations in the corpus

On the Lp-Solvability of Higher Order Parabolic and Elliptic Systems with BMO Coefficients (2010, methodology): “For systems in the whole space, which corresponds to interior estimates, the mean oscillation estimates follow rather easily by adapting the techniques in [24, 26] to higher order systems.”
On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights (2016, unknown): “For this approach, see, for instance, [40, 42, 20].”
Parabolic systems with polynomial growth and regularity (2011, background): “As recently noticed by Krylov [55], in the case of linear parabolic equations it is not necessary to assume coefficients e(x, t) which are VMO with respect to both variables, but rather, functions e(x, t) which are only measurable with respect to the time variable can be still considered when provin”
Boundary Estimates in Elliptic Homogenization (2015, background): “We remark that theW 1,p estimates (local or global) for operators with nonsmooth coefficients in nonsmooth domains are of interest in their own rights and have been studied extensively in recent years (see [11, 45, 8, 39, 9, 34, 10, 41, 13, 29, 16, 18] and their references).”
Elliptic and parabolic equations for measures (2009, background): “It is worth noting that some of the conditions on A employed above can be relaxed by using the results in [107], [108], [109].”
Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains (2011, methodology): “Differed from the arguments in [9, 12, 10, 35], which are based on singular integrals and commutator estimates, the crucial step in [27] is to establish certain interior mean oscillation estimates of solutions to equations with ‘simple’ coefficients, which are measurable in some directions and indep”
Lp solvability of divergence type parabolic and elliptic systems with partially BMO coefficients (2011, methodology): “In a recent interesting paper [15], Krylov treated in a unified way the Lp solvability of both divergence and non-divergence form parabolic equations with leading coefficients measurable in the time variable and VMO in the spatial variables, which is denoted as VMOx.”
L-estimates for time fractional parabolic equations with coefficients measurable in time (2018, methodology): “2), it is possible to consider more general classes of coefficients than those in [7, 8, 4].”
Lorentz estimates for degenerate and singular evolutionary systems (2013, background): “In the elliptic linear case div [ A(x)Du ] = divF, (2.5) if the matrix A(x) ≡ A is constant, Calderón-Zygmund estimates follow from the classic work of Stampacchia [35, 36].”
Uniform Regularity Estimates in Parabolic Homogenization (2013, methodology): “It was in fact proved in [9, 25] that the constant C in (2.”
Elliptic equations with measurable coefficients in Reifenberg domains (2010, result): “In what concerns divergence and non-divergence form elliptic and parabolic operators whose coefficients are measurable in one variable and VMO with respect to the others, we would like to mention a series of recent works [11,12,17–19] where D.”