2020 **Abstract:** The Poisson problem consists in finding an immersed surface Σ ⊂ R m minimising Germain's elastic energy (known as Willmore energy in geometry) with prescribed boundary, boundary Gauss map and area. This problem represents a non-linear model for the equilibrium state of thin, clamped elastic plates originating from the work of S. Germain and S.D. Poisson in the early XIX century. We present a solution to this problem in the case of boundary data of class C 1,1 and when the boundary curve is simple and closed. T…

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“…Prescribing additional conditions and showing existence has also been very successful, i.e. the isoperimetric ratio in [23] and [36], the area in [25], boundary conditions in [6,32,35] and [9]. Finally, the Willmore conjecture was shown to be true in [28].…”

confidence: 99%

“…Prescribing additional conditions and showing existence has also been very successful, i.e. the isoperimetric ratio in [23] and [36], the area in [25], boundary conditions in [6,32,35] and [9]. Finally, the Willmore conjecture was shown to be true in [28].…”

confidence: 99%

“…For the Helfrich functional an existence result for branched immersions in R 3 was found in [19]. This is somehow related to [11], where an area constraint was imposed in order to minimise the Willmore functional.…”

confidence: 90%

“…There are several applications of the Helfrich energy in e. g. biology in modelling red blood cells or lipid bilayers (see e. g. [8,27,34,36]) and in modelling thin elastic plates (see e. g. [24]). As noted in [11,34], the Willmore functional was already considered in the early 19th century (see [24,37]) and has been periodically reintroduced according to the availability of more powerful mathematical tools e.g. in the early 1920s (see [45]) and most successfully by Willmore (see [46]) in the 1960s.…”

confidence: 99%

“…In the case where the curve and the tangent plane are prescribed along the boundary, there are substantial existence and regularity results for minimizers by Schätzle [15] and Da Lio, Palmurella and Rivière [4]. It is clearly of interest to develop an analogous theory for the free boundary problems in the case of curved supporting surfaces or curves.…”

confidence: 99%