Quasiconvex functions and nonlinear PDEs (Q2847036)

The paper deals with the operator \(L_0:\mathbb R^n\times S(n) \rightarrow\mathbb R\) defined as NEWLINE\[NEWLINE L_0( p, M)= \min \left\{ vMv^T;\,\,v \in\mathbb R^n, \,\,|v|=1,\,\,vp=0\right\} NEWLINE\]NEWLINE and the viscosity solutions of the following nonlinear differential problem: NEWLINE\[NEWLINE L_0( Du,D^2 u)=g(x), \,\,x \in \Omega, \quad v=h\,\,\text{on}\, \partial \Omega. NEWLINE\]NEWLINE In two dimensions this is the mean curvature operator. The main result of the paper is a comparison principle for a \(g\geq C_g>0\) and, when \(g=0\), for quasiconvex function i.e. function \(u :\Omega \rightarrow [-\infty,+\infty]\) such that: NEWLINE\[NEWLINE u(\lambda x+(1-\lambda)y) \leq \max (u(x),u(y)),\,\text{for all } x,y \in \Omega,\,\,0<\lambda<1. NEWLINE\]NEWLINE This property is equivalent to require that the sublevels of \(u\) are convex. Moreover, some results are also given for the constraint problem NEWLINE\[NEWLINEL_{\alpha}(Du,D^2 u)= \min \left\{ v D^2uMv^T;\,\, |v|=1,\,\,|vDu| \leq \alpha \right\}.NEWLINE\]