A generalised Picone identity of $p(x)$-sub-Laplacian for general vector fields and applications
Abimbola Abolarinwa University of Lagos
Picone identity named after Mauro Picone (1885-1977) is classical in the theory of homogeneous linear second order differential equations yielding many results. In this talk, I will present a new generalised variable exponent Picone type identity for horizontal $p(x)$-Laplacian of general vector fields. The identity generalises several known results in literature. Then as an application, we will study the indefinite weighted Dirichlet eigenvalue problem for horizontal $p(x)$-sub-Laplacian on smooth manifolds and discuss some properties of the first eigenvalue and its corresponding eigenfunctions such as uniqueness, simplicity, monotonicity and isolation in the context of variable exponent Sobolev spaces. Further applications also yield Hardy type inequalities and Caccioppoli estimates with variable exponent.