THE VARIABLE EXPONENT DE GIORGI CLASS AND THE VARIABLE EXPONENT HÕLDER CONTINUITY

For a bounded domain f2 in R n (n >= 2), we prove that the variable exponent De Giorgi class m ( x ) (f2, M, gamma, gamma 1 , 8) is a subset of the variable exponent H & ocirc;lder space C 0 , alpha ( x ) loc (f2). It gives an improvement for the earlier result, which states that m ( x ) (f2, M, gamma, gamma 1 , 8) subset of C 0 , alpha loc (f2), where alpha is a constant. This result can be applied to get C 0 , alpha ( x ) loc (f2) regularity for certain solutions of variational problems and nonlinear elliptic equations with variable exponents.