Harnack inequality for the elliptic \(p(x)\)-Laplacian with a three-phase exponent \(p(x)\) (Q2206365)

The authors consider an elliptic \(p(x)\)-Laplacian with a piecewise constant three-phase exponent in the plane with three phases joining at a point, a Harnack inequality is proved and the Hölder continuity of the solution is established. Note that the regularity of solutions to elliptic and parabolic equations has been studied thus far in the case of a continuous exponent (with a logarithmic or close-to-logarithmic modulus of continuity) or in the two-phase case with a sufficiently smooth phase boundary. More complicated geometries were not investigated, although they are obviously of interest for applications. The importance of the present work relies that one of the first steps taken in this direction.