A Talenti-type comparison theorem for the \(p\)-Laplacian on \(\operatorname{RCD}(K,N)\) spaces and some applications (Q6611122)

In this article, the author obtains Talenti comparison theorems, together with the corresponding rigidity and almost-rigidity results, in the non-smooth setting of \(\mathsf{RCD}(K,N)\) spaces (i.e., finite-dimensional metric measure spaces satisfying lower Riemannian Ricci curvature bounds). \medskip\N\NLet \((\mathrm{X},\mathrm{d},\mathfrak m)\) be a given \(\mathsf{RCD}(K,N)\) space with \(K>0\), \(1<N<\infty\) and \(\mathfrak m(\mathrm{X})=1\). Let \(\Omega\subseteq\mathrm{X}\) be an open set with \(0<\mathfrak m(\Omega)<1\). Let \(p,q\in(1,\infty)\) be conjugate exponents and \(f\in L^q(\Omega)\). Assume that \(u\in W^{1,p}_0(\Omega)\) is a weak solution to the Dirichlet problem\N\[\N\left\{\begin{array}{ll} -\mathscr L_p u=f\\\Nu=0 \end{array}\quad\begin{array}{ll} \text{ in }\Omega,\\\N\text{ on }\partial\Omega, \end{array}\right.\N\]\Nwhere \(\mathscr L_p\) denotes the \(p\)-Laplace operator. Moreover, assume that \(w\in W^{1,p}([0,r),\mathfrak m_{K,N})\) is a weak solution to\N\[\N\left\{\begin{array}{ll} -\mathscr L_p w=f^\star\quad\text{ in }[0,r),\\\Nw(r)=0, \end{array}\right.\N\]\Nwhere:\N\begin{itemize}\N\item \((J_{K,N},\mathrm{d}_{eu},\mathfrak m_{K,N})\) is the 1D model space for the \(\mathsf{CD}\) condition of parameters \(K\) and \(N\);\N\item \(r\in J_{K,N}\) is chosen so that \(\mathfrak m_{K,N}([0,r))=\mathfrak m(\Omega)\);\N\item \(f^\star\) is the Schwarz symmetrization of \(f\).\N\end{itemize}\NIn the spirit of [\textit{G. Talenti}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 3, 697--718 (1976; Zbl 0341.35031)], it is proved in Theorem 1.1 that the Schwarz symmetrization \(u^\star\) of \(u\) satisfies the sharp inequality\N\[\Nu^\star(x)\leq w(x)\quad\text{ for every }x\in[0,r]\N\]\Nand that \(\int_\Omega|\nabla u|^p\,\mathrm{d}\mathfrak m\leq\int_0^r|\nabla w|^p\,\mathrm{d}\mathfrak m_{K,N}\). Furthermore, the author proves the associated rigidity result in Theorem 3.5 (i.e.\ if \(u^\star=w\) then the space is a spherical suspension) and almost rigidity result in Theorem 3.18 (i.e., the equality case is almost achieved in the \(L^p\)-sense exactly when the space is close to a spherical suspension in the measured Gromov-Hausdorff sense). In the case \(p=2\), these results were previously obtained in [\textit{A. Mondino} and \textit{M. Vedovato}, Calc. Var. Partial Differ. Equ. 60, No. 4, Paper No. 157, 43 p. (2021; Zbl 1480.53060)]. \medskip\N\NFinally, the author applies the theorems discussed above to obtain (in Theorems 1.2 and 4.7) a reverse Hölder's inequality for the first Dirichlet eigenfunctions of the \(p\)-Laplacian, together with the corresponding rigidity and almost-rigidity statements, thus extending previous results by [\textit{M. A. Gunes} and \textit{A. Mondino}, Proc. Am. Math. Soc. 151, No. 1, 295--311 (2023; Zbl 1507.53043)].