Weak solutions to quasilinear elliptic obstacle problems

Document Type : Original Article

Authors

Hasnae El Hammar ¹

Allalou Mouad ¹

Mohamed El ouaarabi ²

Abderrahmane Raji ¹

¹ Laboratory LMACS‎, ‎Faculty of Science and Technics‎, ‎Sultan Moulay Slimane University‎, ‎BP 523‎, ‎23000‎, ‎Beni Mellal‎, ‎Morocco

² Fundamental and Applied Mathematics Laboratory‎, ‎Faculty of Sciences Aïn Chock‎, ‎Hassan II University‎, ‎BP 5366‎, ‎20100‎, ‎Casablanca‎, ‎Morocco

10.22034/kjm.2024.484764.3351

Abstract

We study a class of obstacle problems in Sobolev spaces of the form
\begin{gather*}
\begin{cases}
\displaystyle\int_{\Omega}\Big(a(\vert D \varpi\vert) D \varpi ):D(\mathcal{U}-\varpi)+ \left\langle \varpi\vert \varpi\vert^{r-2}, \mathcal{U}-\varpi\right\rangle\Big) \mathrm{dy} \geq 0, \\\\
\mathcal{U}\in \wp_{\ell,g}.
\end{cases}
\end{gather*}
We prove the existence of a weak solution via Young measure theory and a theorem of Kinderlehrer and Stampacchia. Since our operator does not satisfy the property of monotonicity which is necessary in the proof, we suppose another condition to overcome this situation.

Keywords

Obstacle problem

weak solution

Theorem of Kinderlehrer and Stampacchia

Young measures

Subjects

• 35 Partial differential equations

---