engleski

Loss of regularity of weak solutions of p-Laplace equations for p \neq 2

If $1<p<\infty$ and $p\ne2$ then the exponent $\gamma_c=p/|p-2|$ is critical for the pointwise loss of regularity of the $p$-Laplace equation $-\Delta_p u=F(x)$, $u\in W_0^{; ; 1, p}; ; (\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{; ; R}; ; ^N$, and $F\in L^{; ; p'}; ; (\Omega)$. By this we mean the following: if $1<p<2$ and $N$ is large enough, and the right-hand side $F$ has a singularity of order $\gamma>\gamma_c$ at some point $a\in\Omega$, that is, $F(x)\simeq|x-a|^{; ; -\gamma}; ; $ in a neighbourhood of $a$, then at the same point the weak solution $u$ has singularity of order which is larger than $\gamma$. The value of $\gamma_c$ is optimal. For $p>2$ we have the loss of regularity in the sense that if $F(x)=C|x|^m$ with $m>0$, then $u(x)=u(0)+D|x|^{; ; \mu}; ; $ with $\mu<m$, provided $m>\gamma_c$. We show that the $p$-Laplace operator is not hypoelliptic for $p\in(1, \infty)\setminus\{; ; 1+1/n:n\in2\mathbb{; ; N}; ; -1\}; ; $.

p-Laplacian; regularity; singularity; hypoellipticity

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