Rotationally symmetric solutions to the \(L_p\)-Minkowski problem
Let \(f\) be a smooth positive function on the unit sphere \(\mathbb{S}^n\) in \(\mathbb{R}^{n+1}\). The authors study the equation \[ \det (\nabla^2 H +HI) = \frac{f}{H^{n+2}} \quad\text{on}\quad \mathbb{S}^n \leqno(1) \] where \(H\) is the support function of a bounded convex body \(K\) in \(\mathbb{R}^{n+1}\), \(I\) is the identity matrix, and \(\nabla^2 H\) denotes the covariant second derivatives of \(H\) with respect to an orthonormal frame on \(\mathbb{S}^n\). This equation arises in several problems, including the \(L_p\)-Minkowski problem with \(p=n-1\) studied by \textit{E.~Lutwak} [J. Differ. Geom. 38, No. 1, 131--150 (1993; Zbl 0788.52007)], the centroaffine Minkowski problem studied by \textit{K.-S.~Chou} and \textit{X.-J.~Wang} [Adv. Math. 205, No. 1, 33--83 (2006; Zbl 1245.52001)], self-similar solutions of anisotropic curve shortening flow studied by \textit{B.~Andrews} [Calc. Var. Partial Differ. Equ. 7, No. 4, 315--371 (1998; Zbl 0931.53030)] and \textit{K.-S.~Chou} and \textit{X.-P.~Zhu} [The curve shortening problem. Boca Raton, FL: Chapman \& Hall/CRC (2001; Zbl 1061.53045)] in the case \(n=1\), and in connection with the critical case of the Blaschke-Santalo inequality studied by \textit{E.~Lutwak} and \textit{G.~Zhang} [J. Differ. Geom. 47, No. 1, 1--16 (1997; Zbl 0906.52003)]. The existence problem for this equation is rather delicate. First, when \(f\) is constant, the support function of any ellipsoid centred at the origin is a solution of \((1)\). So one cannot obtain a priori estimates for solutions without additional assumptions on \(f\). Second, there exists a Kazdan-Warner type obstruction to the existence of solutions, namely for any \((n+1)\times(n+1)\) matrix \(B\), the vector field \(\xi(x)=Bx-(x^TBx)x\), \(x\in\mathbb{S}^n\), must satisfy \[ \int_{\mathbb{S}^n} \frac{\nabla_\xi f}{H^{n+1}}=0 \leqno(2) \] whenever \(H\) is a \(C^3\) solution of \((1)\). The authors prove the following results. Theorem 1. There exist positive constants \(C_n\), \(\tilde C_n\), depending only on \(n\), such that any solution \(H\) of \((1)\) satisfies \[ C_n \sqrt{f_{min}} \leq |K| \leq \tilde C_n \sqrt{f_{max}} \] where \[ |K|= \frac{1}{n+1} \int_{\mathbb{S}^n} H \det (\nabla^2 H +HI) \] is the volume of the corresponding convex body. Theorem 2. Under some assumptions on \(f\), including in particular rotational symmetry of \(f\), there exist positive constants \(C\) and \(\tilde C\) depending only on \(n\) and \(f\), such that any rotationally symmetric solution of \((1)\) satisfies \[ C\leq H \leq \tilde C. \leqno(3) \] An existence theorem for rotationally symmetric solutions follows with the aid of this a priori estimate, under some assumptions on \(f\). It is interesting that even in the case \(n=1\), the authors' conditions on \(f\) are different to those appearing in several other papers. The proof of the a priori estimates \((3)\) uses a blow-up technique coupled with condition (2). In the final section the authors prove an existence result for rotationally symmetric solutions to the \(L_p\)-Minkowski problem, in the supercritical case of the Blaschke-Santalo inequality.
scientific article; zbMATH DE number 6126877