Abstract
We consider the class of planar maps with Jacobian prescribed to be a fixed radially symmetric function f and which, moreover, fixes the boundary of a ball; we then study maps which minimise the 2p-Dirichlet energy in this class. We find a quantity lambda [f] which controls the symmetry, uniqueness and regularity of minimisers: if lambda [f]le 1 then minimisers are symmetric and unique; if lambda [f] is large but finite then there may be uncountably many minimisers, none of which is symmetric, although all of them have optimal regularity; if lambda [f] is infinite then generically minimisers have lower regularity. In particular, this result gives a negative answer to a question of Hélein (Ann. Inst. H. Poincaré Anal. Non Linéaire 11(3):275–296, 1994). Some of our results also extend to the setting where the ball is replaced by {mathbb {R}}^2 and boundary conditions are not prescribed.
Highlights
Given a domain Ω ⊂ Rn, an orientation-preserving diffeomorphism u0 : Ω → u0(Ω) and a continuous stored-energy function W : Ω × GL+(n) → R, a typical problem in nonlinear elastostatics is tominimise W [u] ≡ W (x, Du) dx, Ω among all u ∈ u0 + W01,∞(Ω, Rn); (1.1)see for instance [2,3]
One is sometimes led to consider non-coercive energy functions and this is the case, for instance, when W depends on Du only through Ju ≡ det Du, see e.g. [18,32,34] for examples in the study of elastic crystals and
We see that the existence and regularity of solutions to (1.1) can be approached by studying the existence and regularity of solutions to the prescribed Jacobian equation
Summary
Given a domain Ω ⊂ Rn, an orientation-preserving diffeomorphism u0 : Ω → u0(Ω) and a continuous stored-energy function W : Ω × GL+(n) → R, a typical problem in nonlinear elastostatics is tominimise W [u] ≡ W (x, Du) dx, Ω among all u ∈ u0 + W01,∞(Ω, Rn); (1.1). If f is Hölder-continuous or smoother one can find transport maps with optimal regularity and, in this setting, there is a rich well-posedness theory [21]. Apart from the nonlinear character of the Jacobian, the main obstacle in studying existence and regularity of solutions to (1.2) is the underdetermined nature of the equation, as transport maps are far from unique. As we are interested in Sobolev regularity of solutions to (1.2), in this paper we will investigate whether minimisation of the np-Dirichlet energy is an appropriate selection criterion. From the Direct Method and the weak continuity of the Jacobian, it follows that if (1.2) admits a solution in W 1,np, there is at least one np-energy minimiser.
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