Non-Classical Solutions to the p-Laplace Equation

Abstract

Abstract: In this talk we will consider the $p$-Laplace equation, $div(|Du|^{p-2}Du) = 0$. In particular, we will focus on very weak solutions, i.e. solutions $u$ to the $p$-Laplace equation with $u \in W^{1,q}$, $\max\{1,p-1\} < q < p$. In 1994, T. Iwaniec and C. Sbordone showed that if $q$ is sufficiently close to $p$, then very weak solutions belong to $W^{1,p}$, and thus are classical solutions. They con-jectured the same to happen for any $\max\{1,p-1\} < q$. In this talk, I will present a positive result which shows that Iwaniec $\&$ Sbordone's conjecture is true if the gradient of $u$ belongs to suitable cones, and next I will sketch the construction of a counterexample for this conjecture if this additional condition is not fulfilled. This is based on a joint work with Maria Colombo.

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