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Singular dimension of solution set of a class of p-Laplace equations

We consider the boundary value problem -Delta(p)u = F(x), u is an element of W(0)(1, p) (Omega) in a bounded open domain Omega subset of R(N), where F is an element of L(p)' (Omega), 1 < p < infinity, p ' = p/(p - 1). Let X(Omega, p) be the set of weak solutions u generated by all right-hand sides F. Define the singular dimension of the solution set as the supremum of Hausdorff dimension of singular sets of solutions in X(Omega, p), and denote it by s-dim X(Omega, p). We show that for p > 2 we have s-dim X(Omega, p) (N - pp ')(+), where r(+) = max{;0, r};. In the proof we exploit among others a regularity result for p-Laplace equations due to J. Simon [Sur des Equations aux Derivees Partielles Non Lineaires, These, Paris, 1977], involving Besov spaces. For 1 < p < 2, an estimate for the singular dimension of the solution set is obtained.

singular set ; fractal set ; singular dimension ; p-laplace equation

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