reason is the lack of good asymptotic estimates for the solutions to the Navier-
Stokes equations in the exterior of a rotating body. Consequently, we approximate
our original problem with a sequence of problems, formulated in bounded domains,
EN , N N, obtained by intersecting the exterior of Ω with a ball of radius N. On
each problem in EN we apply the strategy mentioned above to show solvability for
each N, and then pass to the limit N to show the existence of a solution to
the original problem. Of course, for this method to work, it is essential to obtain
estimates that are independent of N, which requires a careful study of the involved
Even though our result can be embedded in the literature of solid-liquid in-
teraction, we also believe it provides an important contribution to the theory of
nonlinear elasticity, inasmuch as it presents a solution to the free traction prob-
lem with a physically meaningful boundary condition that is non-trivial. We recall
that the free traction problem of nonlinear elasticity was first treated in a series of
papers by Signorini [36, 37, 38], where various properties of the solutions were
shown by means of what is today known as Signorini’s perturbation method. Sig-
norini did not, however, show existence, which was first obtained by Stoppelli
[39, 40, 41, 42, 43], under suitable regularity assumptions. Later on, the existence
results of Stopelli were completed and improved by various authors in [30], [7],
[8], [47], [32] and [31]; see also [45] and [26]. Characteristic for all of these results
is that they deal only with so-called dead loads, i.e., with data (the applied surface
forces) that do not depend on the resulting deformation of the elastic solid, namely,
they are functions of the spatial variable of the reference configuration only. Such
data are, with few exceptions, physically absurd. In fact, in most applications,
the data do depend on the deformation of body. This type of data are called live
loads. Live loads are intrinsically difficult to treat from a mathematical point of
view, due to the compatibility conditions (for the data) associated with the free
traction problem. More specifically, denoting by σE the first Piola-Kirchhoff stress
tensor and by Ω R3 the (stress-free) reference configuration of the body, the free
traction problem of nonlinear elasticity reads
= f in Ω,
σE(Φ) · n = g on ∂Ω,
where Φ : Ω
is the (unknown) deformation, f the applied body force, and g
the applied surface force. In order for a solution to (1.2) to exist, the data (f, g)
must satisfy the compatibility condition
f Φ(x) dx =
g Φ(x) dS. (1.3)
The fact that this relation involves the unknown solution Φ, complicates the task
of proving an existence theorem. This becomes challenging even in the case of dead
loads, and naturally even more so when f and g also depend Φ, i.e., the case of
live loads. There are only few existence results for the free traction problem with
live loads. One is due to Valent, see the monograph [46] and the papers cited
therein, who, however, needs further conditions imposed on the data that are very
difficult to verify in non-trivial
In view of this, we believe our result
fact, such conditions are also required in the case of dead loads. More specifically, in this
case the data (f, g) must have no axis of equilibrium, see for example [7].
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