engleski

Generating singularities of weak solutions of p-Laplace equations on fractal sets

We study $p$-Laplace equations $-\Delta_p u=F(x)$ possessing weak solutions in the Sobolev space $W_0^{; ; 1, p}; ; (\Omega)$, $\Omega\subset\mathbb{; ; R}; ; ^N$, that are singular on prescribed fractal sets having large Hausdorff dimension. With an appropriate choice of $F\in L^{; ; p'}; ; (\Omega)$ the Hausdorff dimension of singular set of the weak solution can be made arbitrarily close to $N-pp'$ if $pp'<N$. For $p=2$, that is, for the classical Laplace equation, the bound $N-4$ is optimal, provided $N\ge4$. Moreovoer, there exist maximally singular solutions, that is, such that the bound is achieved. The proof is obtained via an explicit lower a-priori bound of supersolutions corresponding to special choice of right hand-sides that are singular near a fractal set.

p-Laplace equation; weak solution; singular set; fractal set

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