Partial Regularity of the Minimizers of Quadratic Functionals With Vmo Coefficients

Abstract

The paper investigates the partial regularity of the minimizers for quadratic functionals whose integrands have V M O coefficients in principal part and nonlinear terms that are Carathéodory functions. We use some majorizations for the functional, rather than the well known Euler equation associated to it.

Gradient estimates for elliptic systems in non-smooth domains

“…There have been many research activities in elliptic systems with discontinuous coefficients (see [2,10,12,16,22,[24][25][26]). However, most results are concerned with interior estimates on BMO spaces, VMO space, Hölder spaces, Morrey spaces, or boundary estimates for p close 2.…”

“…We would like to point out that the class VMO is strictly contained in the class BMO with small semi-norms which we are treating in this paper. Thus our present result and approach can be used to refine the results in [10,12,23,25] for non-linear equations/systems.…”

Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms

“…Speaking about the case q = p it is worth saying that there is a quite extensive literature about equations and systems with VMO coefficients. The interested reader can consult [3][4][5]7,8,[17][18][19][20][21][22][23][24], and the references therein.…”

C 0,γ-Regularity for Vector-Valued Minimizers of Quasilinear Functionals

“…In some special cases the Hausdorff dimension of the so-called singular set Ω \ Ω u can be estimated. Finally we recall the recent papers [7] and [18] (this paper extends [4] and [7]) where the authors obtained partial regularity results for minimizers of vectorial quasilinear (in the second paper with added nonlinear part which has some sublinear growth) functionals such as in (1) where the coefficients A αβ ij = A αβ ij (x, u) satisfy well known that V MO dependence on the variable x and the continuous one on u.…”

“…These examples imply that, in general, we cannot expect (without some additional structural assumptions) everywhere regularity of vector-valued minimizers even if the coefficients are regular. For such reason partial regularity properties for minimizers of vectorial functionals have been studied by many authors (see [7,8,11,14,18] and references therein). Here "partial regularity" means that the minimizer u of functional A is regular in some specified sense, in an open subset Ω u ⊂ Ω such that |Ω \ Ω u | = 0.…”