Li Chen, YanNan Liu, Jian Lu, Ni Xiang
In this paper we study the dual Orlicz-Minkowski problem, which is a generalization of the dual Minkowski problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial vectors, we obtain a new existence result of smooth even solutions to this problem for smooth even measures.
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In this paper the Orlicz-Minkowski problem for torsional rigidity, a generalization of the classical Minkowski problem, is studied. Using the flow method, we obtain a new existence result of solutions to this problem for general measures.
In this paper the dual Orlicz-Minkowski problem, a generalization of the $L_p$ dual Minkowski problem, is studied. By studying a flow involving the Gauss curvature and support function, we obtain a new existence result of solutions to this problem for smooth measures.
In this paper a generalized Gauss curvature flow about a convex hypersurface in the Euclidean $n$-space is studied. This flow is closely related to the Orlicz-Minkowski problem, which involves Gauss curvature and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As a byproduct, two existence results of solutions to the even Orlicz-Minkowski problem are obtained, one of which improves the known result.
In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic inverse Gauss curvature flows. By the stationary solutions of anisotropic flows, we obtain some new existence results for the dual Orlicz Minkowski type problem and even dual Orlicz Minkowski type problem for smooth measures, which is the most reasonable extension of the $L^p$ dual Minkowski problem from the dual point of view. The results of corresponding $L^p$ versions are $L^p$ dual Minkowski problem for $p>q$; and even $L^p$ dual Minkowski problem for $p>-1$, or $q<1$, or some ranges of $p<0<…
We study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if exist, solve the regular Orlicz-Minkowski problems. As an application, we obtain old and new results for the regular even Orlicz-Minkowski problems; the corresponding $L_p$ version is the even $L_p$-Minkowski problem for $p>-n-1$. Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz-Minkowski problems; the $L_p$ versions are the even $L_p$-Minkowski problem for $p>0$ and the $L_p…
Flow by Gauss Curvature to the Orlicz Minkowski Problem for q-torsional rigidity
本文首次提出并研究 q-扭转刚度的 Orlicz Minkowski 问题，利用高斯曲率流方法证明了 q>1 时光滑非偶解的存在性，填补了该领域的研究空白。
Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem
本文研究了无全局项的各向异性曲率流的长时间存在性，并利用其平稳解解决对偶Orlicz Christoffel-Minkowski型问题，涵盖多种已知问题并部分回答了一般指定曲率问题。
An Orlicz version of the $L_p$-Minkowski problem on $S^{n-1}$ is discussed corresponding to the case $-n<p<0$.
In this paper, combining the $p$-capacity for $p\in (1, n)$ with the Orlicz addition of convex domains, we develop the $p$-capacitary Orlicz-Brunn-Minkowski theory. In particular, the Orlicz $L_ϕ$ mixed $p$-capacity of two convex domains is introduced and its geometric interpretation is obtained by the $p$-capacitary Orlicz-Hadamard variational formula. The $p$-capacitary Orlicz-Brunn-Minkowski and Orlicz-Minkowski inequalities are established, and the equivalence of these two inequalities are discussed as well. The $p$-capacitary Orlicz-Minkowski problem is proposed and solved under some mild…
Flow by Gauss Curvature to the orlicz Chord Minkowski Problem
本文研究 orlicz 弦 Minkowski 问题，通过用连续函数 $\varphi$ 替换 $p$ 推广了 $L_p$ 弦 Minkowski 问题，并利用高斯曲率流方法证明了其光滑解的存在性。
The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski prob…
In this paper we study a normalised anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space R^n+1. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Amp`ere type equation. Our argument provides a parabolic proof in the smooth category for the existence of solutions to the Dual Orlicz-Minkowski problem introduced by Zhu, Xing and Ye.
In this paper, the extended Musielak-Orlicz-Gauss image problem is studied. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure $\widetilde{C}_{G,Ψ,λ}(Ω,\cdot)$ of convex body $Ω$ in $\mathbb{R}^{n+1}$ containing the origin (but the origin is not necessary in its interior). In particular, we provide solutions to the extended Musielak-Orlicz-Gauss image problem based on the study of suitably designed parabolic flows, and by the use of approximation technique (for general measures). Our parabolic flows involve two Musielak-Orlicz functions and hence contain many well-stud…
The general dual volume $\dveV(K)$ and the general dual Orlicz curvature measure $\deV(K, \cdot)$ were recently introduced for functions $G: (0, \infty)\times \sphere\rightarrow (0, \infty)$ and convex bodies $K$ in $\R^n$ containing the origin in their interiors. We extend $\dveV(K)$ and $\deV(K, \cdot)$ to more general functions $G: [0, \infty)\times \sphere\rightarrow [0, \infty)$ and to compact convex sets $K$ containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous depe…
For $K\subseteq \mathbb{R}^n$ a convex body with the origin $o$ in its interior, and $ϕ:\mathbb{R}^n\setminus\{o\}\rightarrow(0, \infty)$ a continuous function, define the general dual ($L_ϕ)$ Orlicz quermassintegral of $K$ by $$\mathcal{V}_ϕ(K)=\int_{\mathbb{R}^n \setminus K} ϕ(x)\,dx.$$ Under certain conditions on $ϕ$, we prove a variational formula for the general dual ($L_ϕ)$ Orlicz quermassintegral, which motivates the definition of $\widetilde{C}_{ϕ,\mathcal{V}}(K, \cdot)$, the general dual ($L_ϕ)$ Orlicz curvature measure of $K$. We pose the following general dual Orlicz-Minkowski probl…
An inverse Gauss curvature flow and its application to p-capacitary Orlicz-Minkowski problem
本文首次通过一类逆高斯曲率流证明了p∈(1,n)时p-容Orlicz-Minkowski问题光滑解的存在性，并在特殊情形下证明了解的唯一性。
In the present paper we initiate the study of the Musielak-Orlicz-Brunn-Minkowski theory for convex bodies. In particular, we develop the Musielak-Orlicz-Gauss image problem aiming to characterize the Musielak-Orlicz-Gauss image measure of convex bodies. For a convex body $K$, its Musielak-Orlicz-Gauss image measure, denoted by $\widetilde{C}_Θ(K, \cdot)$, involves a triple $Θ=(G, Ψ, λ)$ where $G$ and $Ψ$ are two Musielak-Orlicz functions defined on $S^{n-1}\times (0, \infty)$ and $λ$ is a nonzero finite Lebesgue measure on the unit sphere $S^{n-1}$. Such a measure can be produced by a variati…
In this paper we first obtain the existence of smooth solutions to Orlicz-Aleksandrov problem via a Gauss-like curvature flow.
In this paper, we obtain the existence result of smooth solutions to the Orlicz-Aleksandrov problem from the perspective of geometric flow. Furthermore, a special uniqueness result of solutions to this problem shall be discussed.
Uniqueness of solutions to a class of non-homogeneous curvature problems
本文利用 Heintze-Karcher 不等式证明一类各向同性混合 Christoffel-Minkowski 及 Minkowski 型问题的光滑凸解只能是中心球面，并推出 Orlicz-Minkowski 与 $L_p$-Gaussian-Minkowski 问题的唯一性。
In this work we estend a recent result of Kristály, Marzantowicz and Varga concerning the existence of three critical points certain non-smooth functionals. Using this result, we guarantee the existence of three solutions to a inclusion quasilinear problem, with non-homogeneous boundary condition through Orlicz Sobolev spaces.
We study long-time existence and asymptotic behaviour for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even $L_p$-Minkowski problems in $\mathbb{R}^{n+1}$ for $p>-n-1.$
We develop arguments on convexity and minimization of energy functionals on Orlicz-Sobolev spaces to investigate existence of solution to the equation $\displaystyle -\mbox{div} (ϕ(|\nabla u|) \nabla u) = f(x,u) + h \mbox{in} Ω$ under Dirichlet boundary conditions, where $Ω\subset {\bf R}^{N}$ is a bounded smooth domain, $ϕ: (0,\infty)\longrightarrow (0,\infty)$ is a suitable continuous function and $f: Ω\times {\bf R} \to {\bf R}$ satisfies the Carathéodory conditions, while $h$ is a measure.
We consider a sequence of positive smooth critical points of the Adams-Moser-Trudinger embedding of $H^m_0$ into Orlicz spaces. We study its concentration-compactness behavior and show that if the sequence is not precompact, then the liminf of the $H^m_0$-norms of the functions is greater than or equal to a positive geometric constant.
We study a non-homogeneous boundary value problem in a smooth bounded domain in $\mathbb{R}^N$. We prove the existence of at least two nonnegative and non-trivial weak solutions. Our approach relies on Orlicz-Sobolev spaces theory combined with adequate variational methods and a variant of Mountain Pass Lemma.
We consider the optimization problem of minimizing $\int_ΩG(|\nabla u|) dx$ in the class of functions $W^{1,G}(Ω)$, with a constrain on the volume of $\{u>0\}$. The conditions on the function $G$ allow for a different behavior at 0 and at $\infty$. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution $u$ is locally Lipschitz continuous and that the free boundary, $\partial\{u>0\}\cap Ω$, is smooth.
In this paper, we study the $L_p$-Gaussian Minkowski problem, which arises in the $L_p$-Brunn-Minkowski theory in Gaussian probability space. We use Aleksandrov's variational method with Lagrange multipliers to prove the existence of the logarithmic Gauss Minkowski problem. We construct a suitable Gauss curvature flow of closed, convex hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$, and prove its long-time existence and converges smoothly to a smooth solution of the normalized $L_p$ Gaussian Minkowski problem in cases of $p>0$ and $-n-1<p\leq 0$ with even prescribed function respectiv…
We prove that the set of smooth, $π$-periodic, positive functions on the unit circle for which the $L_{-2}$ Minkowski problem is solvable is dense in the set of all smooth, $π$-periodic, positive functions on the unit circle with respect to the $L^{\infty}$ norm. Furthermore, we obtain a necessary condition on the solvability of the even $ L_{-2}$ Minkowski problem. At the end, we prove uniqueness of the solutions up to an affine linear transformation.
We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature radii of the hypersurface. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of an elliptic equation, when the constants alpha, beta belong to a suitable range, provid…
We study the nonlinear eigenvalue problem $-{\rm div}(a(|\nabla u|)\nabla u)=λ|u|^{q(x)-2}u$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a bounded open set in $\RR^N$ with smooth boundary, $q$ is a continuous function, and $a$ is a nonhomogeneous potential. We establish sufficient conditions on $a$ and $q$ such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments. The abstract results of this paper are illustrated by the cases $a(t)=t^{p-2}\log (1+t^r)$ and $a(t)= t^{p-2} [\log (1+t)]^{-1}$.
An eigenvalue problem for a generalized polyharmonic operator in Orlicz-Sobolev spaces without the $Δ_2$-condition
本文研究了在无Δ₂条件的Orlicz-Sobolev空间中，广义多谐算子的特征值问题，证明了特征值序列趋于无穷大并给出了特征函数的正则性结果。
We study the boundary value problem $-{\rm div}(\log(1+ |\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a bounded domain in $\RR^N$ with smooth boundary. We distinguish the cases where either $f(u)=-λ|u|^{p-2}u+|u|^{r-2}u$ or $f(u)=λ|u|^{p-2}u-|u|^{r-2}u$, with $p$, $q>1$, $p+q<\min\{N,r\}$, and $r<(Np-N+p)/(N-p)$. In the first case we show the existence of infinitely many weak solutions for any $λ>0$. In the second case we prove the existence of a nontrivial weak solution if $λ$ is sufficiently large. Our approach relies on adequate variational methods …
We prove existence and stability of smooth entire strictly convex spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space. The proof is based on barrier constructions and local a priori estimates.
This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic problem $$ \left\{ \begin{array}{l} {\rm{div}} \left( ϕ(|\nabla u|)\nabla u\right) = a(x)f(u)\quad \mbox{in } Ω,\\ u\geq0\ \ \mbox{in}\ Ω,\ \ u=\infty\ \mbox{on}\ \partialΩ, \end{array} \right. $$ where either $Ω\subset \mathbb{R}^N$ with $N \geq 1$ is a smooth bounded domain or $Ω= \mathbb{R}^N$. The function $ϕ$ includes special cases appearing in mathematical models in nonlinear elasticity, plasticity, genera…
In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with speed $f r^α K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. If $α\ge n+1$, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if $f \equiv 1$. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by…
This work is concerned with the existence and multiplicity of solutions for the following class of quasilinear problems $$ -Δ_Φu+ϕ(|u|)u=f(u)~\text{in} ~Ω_λ, u(x)>0 ~\text{in}~Ω_λ, u=0~ \mbox{on} ~\partialΩ_λ, $$ where $Φ(t)=\int_0^{|t|} ϕ(s) s \, ds $ is an $N-$function, $Δ_Φ$ is the $Φ-$Laplacian operator, \linebreak $Ω_λ=λΩ,$ $Ω$ is a smooth bounded domain in $\mathbb{R}^N,$ $N \geq 2$, $λ$ is a positive parameter and $f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. Here, we use variational methods to get multiplicity of solutions by using of Lusternik-Schnirelmann category o…
Flow by Gauss curvature to the Minkowski problem of p-harmonic measure
本文利用高斯曲率流方法证明了p-调和测度Minkowski问题光滑解的存在性，推广了Jerison及Akman-Mukherjee的相关结果。
Existence of symmetric (resp. asymmetric) solutions to the $L_p$ Gaussian Minkowski problem for $p\leq 0$ (resp. $p\geq 1$) will be provided. Moreover, existence and uniqueness of smooth solutions to the problem for $p>n$ will also be proved without the restriction that the Gaussian volumes of convex bodies are not less than one-second.
The Dirichlet problem for a class of curvature equations in Minkowski space
针对 Minkowski 空间中的一类给定曲率方程，本文通过建立先验 C² 估计，证明了一般边界数据下光滑类空超曲面解的存在性。
In this paper, a generalization of the $L_{p}$-Christoffel-Minkowski problem is studied. We consider an anisotropic curvature flow and derive the long-time existence of the flow. Then under some initial data, we obtain the existence of smooth solutions to this problem for $c=1$.
In this paper, we introduce the so-called $L_p$ $q$-torsional measure for $p\in\mathbb{R}$ and $q>1$ by establishing the $L_p$ variational formula for the $q$-torsional rigidity of convex bodies without smoothness conditions. Moreover, we achieve the existence of solutions to the $L_p$ Minkowski problem $w.r.t.$ the $q$-torsional rigidity for discrete measure and general measure when $0<p<1$ and $q>1$.
Capillary $L_p$ Minkowski Flows
本文研究各向异性毛细管高斯曲率流的长时间行为，并利用流方法证明了半空间中两类毛细管 $L_p$ 闵可夫斯基问题光滑解的存在性。
\noindent In this paper we study existence of solution for a class of problem of the type $$ \left\{ \begin{array}{ll} -Δ_Φ{u}=f(u), \quad \mbox{in} \quad Ωu=0, \quad \mbox{on} \quad \partial Ω, \end{array} \right. $$ where $Ω\subset \mathbb{R}^N$, $N \geq 2$, is a smooth bounded domain, $f:\mathbb{R} \to \mathbb{R}$ is a continuous function verifying some conditions, and $Φ:\mathbb{R} \to \mathbb{R}$ is a N-function which is not assumed to satisfy the well known $Δ_2$-condition, then the Orlicz-Sobolev space $W^{1,Φ}_0(Ω)$ can be non reflexive. As main model we have the function $Φ(t)=(e^{t^{…
In this paper, we consider a large class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $\mathbb{R}^{n+1}$ with speed $ψu^αρ^δf^{-β}$, where $ψ$ is a smooth positive function on unit sphere, $u$ is the support function of the hypersurface, $ρ$ is the radial function, $f$ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. When $ψ=1$, we prove that the flow exists for all time and converges to infinity if $α+δ+β\le1, β>0$ and $α\le0$, while in case $α+δ+β>1,α,δ\le0$, the flow blo…
Smooth solutions to the chord log-Minkowski problem
本文研究了弦对数Minkowski问题的光滑解，通过非局部Monge-Ampère方程和非局部高斯曲率流方程，获得边界正则性和解的存在性。
An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems
该研究引入k阶(p,q)-混合曲率测度，提出推广型的(p,q)-Christoffel-Minkowski问题，借助扩张曲率流证明其光滑解存在性，并将讨论解的唯一性。
We consider a fully nonlinear partial differential equation associated to the intermediate $L^p$ Christoffel-Minkowski problem in the case $1<p<k+1$. We establish the existence of convex body with prescribed $k$-th even $p$-area measure on $\mathbb S^n$, under an appropriate assumption on the prescribed function. We construct examples to indicate certain geometric condition on the prescribed function is needed for the existence of smooth strictly convex body. We also obtain $C^{1,1}$ regularity estimates for admissible solutions of the equation when $ p\ge \frac{k+1}2$.
Curvature bound for $L_p$ Minkowski problem
本研究通过各向异性高斯曲率流的曲率估计，证明了$p \le -n+2$时$\mathbb{R}^{n+1}$中$L_p$ Minkowski问题解的$C^{1,1}$正则性，并验证了该结果的最优性，涵盖三维对数Minkowski问题情形。
The horospherical $p$-Christoffel-Minkowski and prescribed $p$-shifted Weingarten curvature problems in hyperbolic space
本文研究双曲空间中凸超曲面的对应几何问题，通过建立新满秩定理证明极限球面p-Christoffel-Minkowski问题存在光滑原点对称严格凸解，并提出给定p-偏移Weingarten曲率问题且证明其存在性结果。
We prove that if $ϕ\colon \mathbb{R}^2 \to \mathbb{R}^{1+2}$ is a smooth proper timelike immersion with vanishing mean curvature, then necessarily $ϕ$ is an embedding, and every compact subset of $ϕ(\mathbb{R}^2)$ is a smooth graph. It follows that if one evolves any smooth self-intersecting spacelike curve (or any planar spacelike curve whose unit tangent vector spans a closed semi-circle) so as to trace a timelike surface of vanishing mean curvature in $\mathbb{R}^{1+2}$, then the evolving surface will either fail to remain timelike, or it will fail to remain smooth. We show that, even allow…
$L_p$ Minkowski problem and Brunn-Minkowski inequality for dual quermassintegrals
本文研究 $L_p$ 对偶 Brunn-Minkowski 理论中的核心问题，包括对偶拟质量积分的 $L_p$ Minkowski 问题和 $L_p$ Brunn-Minkowski 不等式，建立了无对称假设的 $C^0$ 估计，证明了光滑解的唯一性，并研究了原点对称凸体的 $L_p$ Brunn-Minkowski 不等式。
On the uniqueness of solutions to the isotropic $L_{p}$ dual Minkowski problem
本文证明了在特定参数范围内，单位球面是各向同性 $L_p$ 对偶 Minkowski 问题的唯一光滑严格凸解。
We study the motion of smooth, closed, strictly convex hypersurfaces in Rn+1 expanding in the direction of their normal vector field with speed depending on the k-th elementary symmetric polynomial of the principal radii of curvature. As an application, we gives a unifed fow approach to Lp-Christoffel-Minkowski problem for p > 1.
We study a generalized Stokes system with Orlicz growth which is nonstandard in a non-smooth domain. Our purpose is to derive a Calderon-Zygmund type estimate of the gradient of a solution and the pressure to such a system like (1.1) under a small BMO non-linearity and a sufficient flatness on the boundary of the domain. In the process, we overcome not only lack of Lipschitz regularity for the corresponding limiting problem, but also the fact that the associated structure depends only on the symmetric part of the gradient for the desired global estimate, which is new even in the literature of …
On The Minkowski Problem With Respect To A Mixed Euclidean-Gaussian Density
本文研究对应混合欧氏-高斯体积的闵可夫斯基问题，采用流方法证明对称情形下蒙日-安培型方程光滑归一化解的存在性，经逼近得到原点对称解的存在性。
Existence of solution for two classes of quasilinear systems defined on a non-reflexive Orlicz-Sobolev Spaces
本文研究定义在非自反 Orlicz-Sobolev 空间上的两类拟线性系统，通过无 Palais-Smale 条件的环绕定理、集中紧性引理及 Rabinowitz 鞍点定理证明非平凡解的存在性。
In this paper we introduce a new class of quasilinear elliptic equations driven by the so-called double phase operator with variable exponents. We prove certain properties of the corresponding Musielak-Orlicz Sobolev spaces (an equivalent norm, uniform convexity, Radon-Riesz property with respect to the modular) and the properties of the new double phase operator (continuity, strict monotonicity, (S$_+$)-property). In contrast to the known constant exponent case we are able to weaken the assumptions on the data. Finally we show the existence and uniqueness of corresponding elliptic equations w…
A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains
本文证明了凸域上2-Hessian特征函数的严格对数凹性与扭转问题解的1/2-凸性，由此建立Brunn--Minkowski不等式，并构造反例说明该性质对3-Hessian方程失效。
Existence of multiple radial solutions for nonlinear equation involving the mean curvature operator in Lorentz-Minkowski space
该研究利用Szulkin非光滑泛函临界点理论，证明了Lorentz-Minkowski空间中含平均曲率算子的凹凸型非线性方程Dirichlet问题多径向解的存在性，并拓展至非线性项依赖解梯度的情形。
The current work focuses on the Gaussian-Minkowski problem in dimension 2. In particular, we show that if the Gaussian surface area measure is proportional to the spherical Lebesgue measure, then the corresponding convex body has to be a centered disk. As an application, this ``uniqueness'' result is used to prove the existence of smooth small solutions to the Gaussian-Minkowski problem via a degree-theoretic approach.
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