Preprint

Inserted: 23 jul 2026
Last Updated: 23 jul 2026

Year: 2026

Abstract:

In this paper, we consider the following obstacle minimization problem with linear term: \[\begin{equation} \label{min Main} \min\bigg\{\int_{\Omega} w\,\frac{|\nabla u|^p}{p}\, +\,f\,u\,:\,u \in W^{1,p}(\Omega),\,\,\,u \geq 0,\,\,\,\,u=g\,\,\,\,\mbox{on}\,\,\,\partial\Omega \bigg\} \end{equation}\] where $1<p<\infty$. Assume $w \in C^{0,\sigma}(\Omega)$ and $f \in L^q(\Omega)$ with $q \geq \frac{p}{p-1}$. If $q>N$, we will prove via an approximation argument that the solution $u$ of this problem is locally $C^{1,\alpha}$, where the exponent $\alpha \in (0,1]$ will be given explicitly in terms of $N$, $p$, $q$ and $\sigma$.

Moreover, we will show that $u$ enjoys higher regularity on the free boundary $\partial\{u>0\}$; it is locally of class $C^{1,\min\bigg\{\frac{1-\frac{N}{q}}{p-1},1\bigg\}}$. If $p\geq 2$, $w \in C^1(\Omega)$, $f \in L^\infty(\Omega)$ and $f \geq f_\star>0$, then we establish a nondegeneracy property, which implies that the regularity of the solution $u$ on the free boundary is optimal. Finally, we prove that the free boundary has zero Lebesgue measure.

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