几何分析综述2022(英文版)

A Free Boundary Problems on RCD Spaces Chungkwong Chan Department of Mathematics， Sun Yat-sen University， Guangzhou， China Huichun Zhang Department of Mathematics， Sun Yat-sen University， Guangzhou， China Xiping Zhu Department of Mathematics， Sun Yat-sen University， Guangzhou， China Abstract We survey some recent work on free boundary problems on metric measure spaces with generalized Ricci curvature bounded from below. 1 Introduction Let (X， d， μ) be a complete metric space (X， d) equipped with a Radon measure μ with supp(μ) = X. In the last twenty years， a generalized notion of lower Ricci bounds， called the Riemannian curvature-dimension condition RCD(K，N) for some K ∈ R and N ∈ [1，+∞]， has been established on non-smooth metric measure space (X， d， μ) (see [Stu06a， Stu06b， LV09， LV07， Gig13， AGS14a， Gig15， AGS14b， EKS15， AMS16， CM16]). The parameters K ∈ R and N ∈ [1，+∞] play the role of “Ricci curvature . K and dimension in Riemannian geometry. The main examples in the class of RCD(K，N) spaces include the Ricci limit spaces in the Cheeger-Colding theory [CC96， CC97， CC00] and finite dimensional Alexandrov spaces with curvature bounded from below (see [Pet11] and [ZZ10， Appendix A]). There have been growing interest and a plenty of results on the linear aspect of geometric analysis on RCD(K，N) spaces， see [Amb18] for a recent survey on this topic. In very recent several years， some developments for nonlinear aspect of geometric analysis on RCD(K，N) spaces have appeared， such as [ZZ18， GJZ22， MS22， Gig22] for harmonic maps from RCD(K，N) spaces， [MS21] for minimal surfaces and [CZZ21] for free boundary problems on RCD(K，N) spaces. In fact， these three nonlinear elliptic problems share very strong analogies. We also mention that Ding [Ding21] studied the theory minimal surfaces in Ricci limit spaces. In this survey， we describe the existence and regularity results for the Bernoulli free boundary problems on an RCD(K，N) spaces (X， d， μ). Let Ω X be a bounded domain. According to [Che99， Sha00， HK00， AGS13， AGS14a]， the Sobolev space W1，2(Ω) is well-defined. The Bernoulli free boundary problem can be represented as a minimization of functional for u = (u1， u2， ， um) ∈ W1，2(Ω，Rm). It is not hard to get the existence of the minimizer of J under the Dirichlet condition. We will introduce the main regularity results in [CZZ21] in the followings: the Lipschitz continuity of the solution u; the local finiteness of (N . 1)-dimensional Hausdorff measure of the free boundary. and the corresponding Euler-Lagrange equation; the partial regularity of the free boundary it is the union of a Cα manifold and a set of singular points， whose Hausdorff dimension is no more than N . 3. Remark 1.1 We should mention S. Lin’s recent work [Lin22b] on obstacle problems on RCD(K，N) spaces. 2 Preliminaries Let (X， d) be a complete and separable metric space， equipped with a (nonnegative) Radon measure μ such that， where BR(x0) is the open ball of center x0 ∈ X and radius R. We shall denote by for any p ∈ [1，∞] and any μ-measurable set 2.1 Calculus on RCD(K，N) metric measure spaces Given a function f ∈ C(X)， the pointwise Lipschitz constant (see [Che99]) of f at x is defined by where we put lipf(x) = 0 if x is isolated. Clearly， lipf is a μ-measurable function on X. The Cheeger energy， denoted by， is defined ([Che99， AGS14a]) by (2.1) where the infimum is taken over all sequences of Lipschitz functions converging to f in L2(X). In general， Ch is a convex and lower semi-continuous functional on L2(X). Given any open set Ω， we denote by Liploc(Ω) the set of locally Lipschitz continuous functions on Ω. For any and f ∈ Liploc(Ω)， its W1，pnorm is defined by The Sobolev space W1，p(Ω) is defined by the closure of the set under the W1，p-norm. Spaces is defined by the closure of under the W1，p-norm. We say a function for every open subset. Notice that， the domain of the Cheeger energy. For any， the minimal weak upper gradient of f is denoted by See [Che99， AGS13， AGS14a] for the definition of minimal weak upper gradient. In [Che99]， it is shown that for any， it holds . Throughout this paper， we always assume that (X， d， μ) is an RCD(K，N) space with N ∈ [1，+∞) and K ∈ R. For explicit definitions， one can refer to [Gig15， AMS16， EKS15] and the survey [Amb18]. Here we only recall some basic properties [LV09， AGS14b， AGMR15， EKS15] as follows: (X， d) is a locally compact length space. In particular， for any p， q ∈ X， there is a shortest curve connecting them; If N > 1， then the generalized Bishop-Gromov inequality holds. In particular， it implies a local measure doubling property holds; The W1，2(Ω) is a Hilbert space， and the inner product for can be given by (see [Gig15]): It was proved [Gig15] that the limit exists in L1(Ω). The local Poincaré inequality (see [KS