From Modular Spaces to Boundary Value Problems: A Survey of Recent Advances

This survey explores the recent discovery of the connection between the geometry of modular spaces and partial differential equations with non-standard growth. Specifically, we demonstrate the solvability of the non-homogeneous Dirichlet problem for the variable exponent p(·)-Laplacian. We place special emphasis on the case when the variable exponent p(·) is unbounded and the boundary data is in the variable exponent Sobolev spaceW1,p(·)(Ω). Tracing the historical development from Riesz’s introduction of the Lp(·)-spaces through the contributions by Orlicz and Nakano to further advances at the end of the twentieth century, we examine the properties and geometriccharacteristics of modular spaces. We discuss the uniform convexity of modulars on Lp(·), ℓp(·), andW1,p(·) and discuss its essential role in the analysis of the Dirichlet problem for the p(·)-Laplacian when the variable exponent p is unbounded. It will be evident from our analysis that the Banach space structure is inadequate in treating this case.