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On a class of nonlinear variational inequalities: high concentration of the graph of weak solution via its fractional dimension and Minkowski content. (English) Zbl 1189.35138

Summary: Weak continuous bounded solutions of a class of nonlinear variational inequalities associated to the 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.

MSC:

35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
34B15 Nonlinear boundary value problems for ordinary differential equations
28A75 Length, area, volume, other geometric measure theory
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