Orlicz–Minkowski flows

Abstract

We study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if they exist, solve the regular Orlicz–Minkowski problems. As an application, we obtain old and new existence results for the regular even Orlicz–Minkowski problems; the corresponding $$L_p$$ L p version is the even $$L_p$$ L p -Minkowski problem for $$p>-n-1$$ p > - n - 1 . Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz–Minkowski problems; the $$L_p$$ L p versions are the even $$L_p$$ L p -Minkowski problem for $$p>0$$ p > 0 and the $$L_p$$ L p -Minkowski problem for $$p>1$$ p > 1 . In the final section, we use a curvature flow with no global term to solve a class of $$L_p$$ L p -Christoffel–Minkowski type problems.

Deforming a hypersurface by a class of generalized fully nonlinear curvature flows

“…Another main issue is the study of the problem on the existence of the prescribed polynomial of the principal curvature radii of the hypersurface. Urbas [29], Gerhart [15], Chow-Tsai [12], Bryan-Ivaki-Scheuer [4] shed light on the convergence for the flow with the speed of symmetric polynomial of the principal curvature radii of the hypersurface. In this paper, we consider a class of generalized fully nonlinear curvature flow of convex hypersurfaces M t parameterized by smooth map X(•, t) :…”

Inverse Gauss Curvature Flows and Orlicz Minkowski Problem

“…The Gauss curvature flow was first introduced and studied by Firey [11] to model the shape change of worn stones. Since then, many scholars have found that using curvature flow to study the hypersurfaces is a very effective tool, such as solving the Minkowski-type problems and geometric inequalities in convex geometric analysis etc., see e.g., [2,3,5,7,8,17,18,27,29] and the references therein.…”

“…Compared with the geometric flows in [3,5,7,8,29], the flow we construct in the present paper is more complex due to its inclusion of functions φ, |∇Ψ| and γ(t), which is reflected in the fact that priori estimates are more difficult to obtain.…”

On the Musielak-Orlicz-Gauss image problem

“…The Orlicz-Minkowski problem [23] is related to the case when G = t n /n and ϕ is a non-homogeneous function. Solutions to the Orlicz-Minkowski problem can be found in, e.g., [2,9,26,35,40,45,57,58,59,63,64]. When G = t q /n and ϕ(t) = t p for 0 = p ∈ R, the general dual Orlicz-Minkowski problem reduces to the L p dual Minkowski problem [50]; contributions to this problem can be seen in, e.g., [3,10,11,14,33,38,41,55].…”