On a phase field problem driven by interface area and interface curvature ∗ Xiaofeng Ren †‡ Department of Mathematics and Statistics The George Washington University Washington, DC 20052, USA Juncheng Wei § Department of Mathematics Chinese University of Hong Kong Hong Kong, PRC Abstract A two component system driven by both interface area and interface curvature is studied with a new phase field model. We show that if the curvature impact in the system is strong enough, there exist bubble profiles. A bubble profile describes a pattern of an inner core of one component surround by an outer membrane of the other component. It is a radial solution to a fourth order nonlinear PDE. We show the existence of such profiles in all dimensions, although the profile is unstable if the dimension is greater than two. 1 Introduction The bending energy plays a central role in the study of vesicle membranes formed by certain am- phiphilic molecules [4, 5]. In the isotropic case it may be expressed as a surface integral [13, 22] E b =  Γ {a 1 + a 2 (κ − c 0 ) 2 + a 3 G} ds. (1.1) Here Γ is a closed surface in R 3 representing a vesicle membrane, κ is the mean curvature of the surface, and G is the Gauss curvature of the surface. The constant a 1 represents the surface tension caused by the interaction effects between the vesicle material and the ambient fluid; a 2 is the bending rigidity and a 3 is the stretching rigidity, both of which are determined by the interaction properties of the amphiphilic molecules. The last constant c 0 is the spontaneous curvature describing an asymmetric effect. In (1.1) the integral of the first term a 1 leads to the area of the surface Γ. The integral of the third term a 3 G gives a topological invariant due to the Gauss-Bonnet Theorem. We may therefore ignore the third quantity. The most interesting part in (1.1) is the second term. In the case c 0 = 0, it is equation to a 2 times  Γ κ 2 ds. (1.2) * Abbreviated title: Infterface area and interface curvature † Corresponding author. Phone: 1 435 797-0755; Fax: 1 435 797-1822; E-mail: ren@math.usu.edu ‡ Supported in part by NSF grant DMS-0509725, DMS-0754066. § Supported in part by an Earmarked Grant of RGC of Hong Kong. 1