Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains

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.     

Parabolic equations with measurable coefficients II

We prove the existence and uniqueness of solutions in Sobolev spaces to second-order parabolic equations in non-divergence form. The coefficients (except one of them) of second-order terms of the equations are measurable in both time and one spatial variables, and VMO (vanishing mean oscillation) in other spatial variables.     

Solvability of parabolic equations in divergence form with partially BMO coefficients

We prove the solvability of second order parabolic equations in divergence form with leading coefficients aij measurable in (t,x¹) and having small BMO (bounded mean oscillation) semi-norms in the other variables. Additionally we assume a¹¹ is measurable in x¹ and has small BMO semi-norms in the other variables. The corresponding results for the Cauchy problem are also established. Parabolic equations in Sobolev spaces with mixed norms are also considered under the same conditions of the coefficients.     

Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms

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.     

Parabolic and Elliptic Equations with VMO Coefficients

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.     

Regularity for quasi-linear degenerate elliptic equations with VMO coefficients

In this paper we establish an interior regularity of weak solution for quasi-linear degenerate elliptic equations under the subcritical growth if its coefficient matrix A（x, u） satisfies a VMO condition in the variable x uniformly with respect to all u, and the lower order item B（x, u, △↓u） satisfies the subcritical growth （1.2）. In particular, when F（x） ∈ L^q（Ω） and f（x） ∈ L^γ（Ω） with q,γ 〉 for any 1 〈 p 〈 ＋∞, we obtain interior HSlder continuity of any weak solution of （1.1） u with an index κ = min{1 - n/q, 1 - n/γ}.     

Parabolic equations with VMO coefficients in generalized Morrey spaces

In this paper, the authors establish the regularity in generalized Morrey spaces of solutions to parabolic equations with VMO coefficients by means of the theory of singular integrals and linear commutators.     

Nonlocal fractional elliptic equations and applications

The L_p-coercive properties of a nonlocal fractional elliptic equation is studied. Particularly, it is proved that the fractional elliptic operator generated by this equation is sectorial in L_p space and also is a generator of an analytic semigroup. Moreover, by using the L_p-separability properties of the given elliptic operator the maximal regularity of the corresponding nonlocal fractional parabolic equation is established.     

Regularity of solutions of divergence form elliptic equations

The aim of this paper is to study local regularity in the Morrey spaces of the first derivatives of the solutions of an elliptic second order equation in divergence form

where is assumed to be in some spaces and the coefficients belong to the space

     

Regular solutions for nonlinear elliptic equations,with convective terms,in Orlicz spaces

We establish some existence and regularity results to the Dirichlet problem, for a class of quasilinear elliptic equations involving a partial differential operator, depending on the gradient of the solution. Our results are formulated in the Orlicz–Sobolev spaces and under general growth conditions on the convection term. The sub- and supersolutions method is a key tool in the proof of the existence results.     

A higher integrability result for nondivergence elliptic equations

The aim of this paper is to establish a higher integrability result of the second derivatives of solutions to nondivergence elliptic equations of the type . We assume that the coefficients a _ij are bounded and have small BMO-norm.      

Infinitely many solutions for a class of degenerate elliptic equations

Let X = (X1, ···, Xm) be an infinitely degenerate system of vector fields. The aim of this paper is to study the existence of infinitely many solutions for the sum of operators X =sum ( ) form j=1 to m Xj Xj.     

The existence and regularity of multiple solutions for a class of infinitely degenerate elliptic equations

Let X = (X₁, …, X_m) be an infinitely degenerate system of vector fields. We study the existence and regularity of multiple solutions of the Dirichlet problem for a class of semi‐linear infinitely degenerate elliptic operators associated with the sum of square operator δ_X = ∑_{j = 1}^m X_j^* X_j (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Graphs of maps between manifolds in trace spaces and with vanishing mean oscillation

We give a positive answer to a question raised by Alberti in connection with a recent result by Brezis and Nguyen. We show the existence of currents associated with graphs of maps in trace spaces that have vanishing mean oscillation. The degree of such maps may be written in terms of these currents, of which we give some structure properties. We also deal with relevant examples.     

Hardy–Sobolev critical elliptic equations with boundary singularities

Unlike the non-singular case s=0, or the case when 0 belongs to the interior of a domain Ω in (n3), we show that the value and the attainability of the best Hardy–Sobolev constant on a smooth domain Ω,

when 0<s<2, , and when 0 is on the boundary ∂Ω are closely related to the properties of the curvature of ∂Ω at 0. These conditions on the curvature are also relevant to the study of elliptic partial differential equations with singular potentials of the form:

where f is a lower order perturbative term at infinity and f(x,0)=0. We show that the positivity of the sectional curvature at 0 is relevant when dealing with Dirichlet boundary conditions, while the Neumann problems seem to require the positivity of the mean curvature at 0.     

The Dirichlet problem for non‐divergence parabolic equations with discontinuous in time coefficients

We consider the Dirichlet problem for non‐divergence parabolic equation with discontinuous in t coefficients in a half space. The main result is weighted coercive estimates of solutions in anisotropic Sobolev spaces. We give an application of this result to linear and quasi‐linear parabolic equations in a bounded domain. In particular, if the boundary is of class C^1,δ , δ ∈ [0, 1], then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)