论文标题
在$ c^1- $ $ sentary点的组装上
On the assembly of $C^1-$stationary points of a polyconvex functional and finite BOP-theory
论文作者
论文摘要
在这项工作中，以下能量被认为是$ i（u）= \ int \ limits_b {\ frac {1} {2} {2} | \ nabla u |^2+ρ（\ det \ nabla u; dx}，$ w^{1,2}（b，\ mathbb {r}^2），$和$ρ：\ mathbb {r} \ rightarrow \ rightarrow \ mathbb {r} _0^+$ smooth and convex and convex and convex and $ρ（s）= 0 $ s）= 0 $ s $ s \ s \ s \ s \ s \ les $ $ p ys $ 0. $（M \ in \ Mathbb {n} \ setMinus \ {0 \}） - $以合适的方式覆盖地图作为边界条件。在这种情况下，我们构建了径向对称的$ m- $涵盖固定的能量点，至少$ c^1 $（在某些情况下，甚至$ c^\ infty），$并验证了更多精制的属性，这些固定点需要满足。我们这样做是通过遵循P. Bauman，N。C. Owen和D. Phillips（BOP）开发的，确认该方法在任意$ M = 2- $ MADE上仍然有效。有限性，不施加这种严格的条件，允许固定点的更丰富的可能行为类别，从而使完全确定它们更加困难。
In this work the following energy is considered $I(u)=\int\limits_B{\frac{1}{2}|\nabla u|^2+ρ(\det\nabla u)\;dx},$ where $B\subset\mathbb{R}^2$ denotes the unit ball, $u\in W^{1,2}(B,\mathbb{R}^2),$ and $ρ:\mathbb{R}\rightarrow\mathbb{R}_0^+$ smooth and convex with $ρ(s)=0$ for all $s\le0$ and $ρ$ becomes affine when $s$ exceeds some value $s_0>0.$ Additionally, we may impose $(M\in \mathbb{N}\setminus\{0\})-$covering maps as boundary conditions in a suitable fashion. For such situations we then construct radially symmetric $M-$covering stationary points of the energy, which are at least $C^1$ (in some circumstances even $C^\infty),$ and verify more refined properties, which these stationary points need to satisfy. We do so by following the strategy first and foremost developed by P. Bauman, N. C. Owen, and D. Phillips (BOP) confirming and generalising that the method remains valid beyond the $M=2-$case for an arbitrary $M.$ Furthermore, as far as we know, this is the first treatise of BOP-theory in finite elasticity. The finiteness, not imposing such strict conditions, allows for a richer class of possible behaviours of the stationary points, making it more difficult to completely determine them.
