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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£¬ ¦Ì) be a complete metric space (X£¬ d) equipped with a Radon measure ¦Ì with supp(¦Ì) = 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 ¡Ê R and N ¡Ê [1£¬+¡Þ]£¬ has been established on non-smooth metric measure space (X£¬ d£¬ ¦Ì) (see [Stu06a£¬ Stu06b£¬ LV09£¬ LV07£¬ Gig13£¬ AGS14a£¬ Gig15£¬ AGS14b£¬ EKS15£¬ AMS16£¬ CM16]). The parameters K ¡Ê R and N ¡Ê [1£¬+¡Þ] 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£¬ ¦Ì). Let ¦¸ X be a bounded domain. According to [Che99£¬ Sha00£¬ HK00£¬ AGS13£¬ AGS14a]£¬ the Sobolev space W1£¬2(¦¸) is well-defined. The Bernoulli free boundary problem can be represented as a minimization of functional for u = (u1£¬ u2£¬ £¬ um) ¡Ê W1£¬2(¦¸£¬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 C¦Á 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 ¦Ì such that£¬ where BR(x0) is the open ball of center x0 ¡Ê X and radius R. We shall denote by for any p ¡Ê [1£¬¡Þ] and any ¦Ì-measurable set 2.1 Calculus on RCD(K£¬N) metric measure spaces Given a function f ¡Ê 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 ¦Ì-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 ¦¸£¬ we denote by Liploc(¦¸) the set of locally Lipschitz continuous functions on ¦¸. For any and f ¡Ê Liploc(¦¸)£¬ its W1£¬pnorm is defined by The Sobolev space W1£¬p(¦¸) 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£¬ ¦Ì) is an RCD(K£¬N) space with N ¡Ê [1£¬+¡Þ) and K ¡Ê 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 ¡Ê 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(¦¸) 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(¦¸). The local Poincaré inequality (see [KS