On a class of nonlinear variational inequalities: High concentration of the graph of weak solution via its fractional dimension and Minkowski content

Abstract

Weak continuous bounded solutions of a class of nonlinear variational inequalities associated to one-dimensional p-Laplacian are studied. It is shown that a kind of boundary behaviour of nonlinearity in the main problem produces a kind of high boundary concentration of the graph of solutions. It is verified by calculating lower bounds for the upper Minkowski-Bouligand dimension and Minkowski content of the graph of each solution and its derivative. Finally, the order of growth for singular behaviour of the $L^{p}$ norm of derivative of solutions is given.

Keywords

• Double obstacles
• nonlinear p-Laplacian
• graph
• fractional dimension
• Minkowski content
• singularity of derivative.