Existence and regularity results for a class of equations with logarithmic growth. (September 2015) - Passarelli di Napoli, Antonia
Abstract: We study the existence and the regularity of the solutions of the following Dirichlet problem − div ( A ( x, ∇ u ) ) = f in Ω u = 0 on ∂ Ω where A ( x, ξ ) is slowly increasing at ∞ as | ξ | → + ∞ . More precisely, we consider the model case A ( x, ξ ) = M ( x ) ξ | ξ | log α ( 1 + | ξ | ), where M ( x ) is a bounded elliptic matrix and α > 0 is a fixed exponent. We will show that if f ∈ L n ( Ω ) then there exists a unique solution u ∈ W 0 1, 1 ( Ω ) ∩ L ∞ ( Ω ), such that ∇ u ∈ L log α L ( Ω ) . Moreover, we prove that there exists an exponent ε > 0 such that | ∇ u | log α + ε ( 1 + | ∇ u | ) ∈ L 1 ( Ω ) . In case the matrix M ( x ) is assumed to be Hölder continuous, we establish a higher differentiability result for the solution in the scale of fractional order Sobolev spaces which yields a higher integrability result in the scale of Lebesgue spaces.,