Oblique derivative problem for uniformly elliptic operators with VMO coefficients and applications

Elliptic equations with BMO coefficients in Lipschitz domains

“…We would like to point out that our assumption that A is (δ, R)-vanishing refines the assumption in other papers (see e.g. [2,5,6,13,14,16,17,21,23,26,29,30,31]) that A is in VMO (cf. [27]).…”

“…Their main tool is an appropriate representation for the Green function on the upper half space. In [10], Caffarelli and Peral used a method of application (see section 1 of [10]) to give an alternative proof of W 1,p regularity in the case of a linear elliptic equation, which uses the general theory of singular integrals (see [1,2,5,6,11,12,13,14,16,17,21,22,23,24,25,28,29,30,31]). More precisely, they deduced the following:…”

“…There has been much research activity (see, e.g., [1,2,5,6,10,12,13,14,16,17,23,24,25,26,28,29,30,31,33]) concerning the classical Calderón-Zygmmund estimates established in [11]. In this paper, we will employ the method used in [33] to investigate the minimal requirements on the coefficients and the smoothness requirement on the domains for the following Dirichlet problem:…”

Regularity for a class of degenerate elliptic equations with discontinuous coefficients under natural growth

“…The VMO functions, introduced by Sarason [24], are appropriate subclasses with a number of good properties, which are not shared by general bounded measurable functions and BMO functions. Recent research related to this problem is the developed L p -Schauder theory for linear and nonlinear elliptic and degenerate elliptic equations with a VMO ∩ L ∞ coefficient [1,4,14,17,19,22], and work on Morrey spaces regularity for local minimizers of functionals [6]. So it seems that a natural generalization is to assume that the coefficient matrix A ij (x, u) is a vanishing mean oscillation in x ∈ Ω uniformly with respect to u.…”

-Solvability for the Parabolic Poincaré Problem

“…l preserves its sign near to the set of tangency); emergent -the ℓ-curves first enter into Ω and after the contact go out of the domain (ℓ changes its sign from − to + near to the set of tangency); submergent -al contrary of the previous case ℓ changes its sign from + to − (under a sign of ℓ we always mean the sign of the scalar product (ℓ · ν) and ν is the unit outer normal to ∂Ω). A various results about elliptic and parabolic Poincaré problems in Hölder and H s spaces are presented in [10], [12], [11] [13], [16], [17], [18], [9].…”