An existence result for elliptic equations with VMO-coefficients

Abstract

In this paper we prove an existence and uniqueness theorem for the Dirichlet problem for linear elliptic equations (in unbounded domains) whose leading coefficients are locally VMO and satisfy a suitable condition at infinity

Some uniqueness results for singular elliptic problems

“…As far as we know, if Ω is a bounded set and p > n, Pucci proved a uniqueness result for the problem (1.2) when the leading coefficients belong to the class L ∞ (Ω) and the lower order coefficients a i , a satisfy certain local summability conditions with a > 0 (see [23]). This result was later on generalized to unbounded domains (see the introduction of [8]). …”

“…Next the uniqueness results for (1.2) of [17,31] have been generalized to the unbounded domains in [8].…”

The Dirichlet problem for elliptic equations in weighted Sobolev spaces on unbounded domains of the plane

“…This latter condition has been dropped in [15], [16], where the coefficients a i and a are required to satisfy some suitable summability hypotheses. Then the above results have also been extended to the case of unbounded open sets (see [3], [4]). …”

“…More recently, if n = 2, the problem (1.2) has been studied in [5] with analogous assumptions to those required in [15], [16], and in [6] with hypotheses similar to those considered in [3], [4].…”

The Dirichlet Problem for elliptic equations in unbounded domains of the plane

“…Recently, in two profound papers [9] and [10], it has been pointed out that the solvability of (1.1) holds, for every p ∈ ]1, +∞ [, under the assumption that the a ij 's are of class V MO , a i = a = 0 (note that V MO contains W 1,n (Ω)) and, later, the condition a i = a = 0 has been removed in [19] and [20]. If the set Ω is unbounded, problem (1.1) has for instance been studied in [16], where the leading coefficients satisfy restrictions similar to those of [13] and p = 2 , in [3], where the investigation has been extended to the case p ∈ ]1, +∞[, and in [4] and [5], where the a ij 's verify assumptions similar to those of [9] and [10].…”