Spherical floating bodies

Abstract

Several affirmative answers are given in any dimension for Ulam's question about bodies floating stable in every direction if the body floats like a ball and its floating body is spherical.

Equilibrium of immersed hyperelastic solids

“…The specific choice of η is indeed tailored to let |D y | = η exactly correspond to the case of barely floating solids. In fact, one can extend the discussion leading to (14) to the specific case of energy E from (29) in order to derive that a critical point of E fulfills…”

“…The rigid body case has been investigated in [11], both at the equilibrium level and for harmonic motions. In more recent years, a question by S. Ulam [16,Problem 19] triggered investigations on the stability of convex bodies of given density [8,14,22]. Moreover, attention has been given to the capillary case, where the fluid surface is not assumed to be flat and contact conditions arise [4,17,18,19,21].…”

Equilibrium of immersed hyperelastic solids

“…Another relevant issue is the stability of local minimizers (imagine a loaded ship at see). This issue is classical in the case of a rigid convex body [9,15,24] and, to our knowledge, completely open in the case of a deformable body. Note once again that our analysis of the ship situation from Section 9 left open the question on how to check local minimality of the configuration y * , given information on Ω and W .…”

“…The rigid body case has been investigated in [11], both at the equilibrium level and for harmonic motions. In more recent years, a question by S. Ulam [17,Problem 19] triggered investigations on the stability of convex bodies of given density [9,15,24]. Moreover, attention has been given to the capillary case, where the fluid surface is not assumed to be flat and contact conditions arise [5,18,19,20,23].…”

“…All such solids are necessarily floating, see (15) in Section 5. Thus, in particular, the energy E is coercive on A ∩ {E ≤ E(ŷ)} and A is closed with respect to the weak W 1,p -topology.…”