Implicit Partial Differential Equations pp 205-216 | Cite as
The Case of Potential Wells
Chapter
Abstract
The problem of potential wells can be described as follows. Consider the minimization problem ,
Where \(
\Omega \subset {\mathbb{R}^n}
\) is a bounded open set \(
\varphi \in {W^{1,p}}\left( {\Omega ;{\mathbb{R}^n}} \right)
\) is a given map, and \(
f:{\mathbb{R}^{n \times n}} \to {\mathbb{R}_ + }
\)
is such that
$$
(P)inf\left\{ {\int_\Omega {f(Du(x))dx:u \in \varphi + W_0^{1,p}(\Omega ;{\mathbb{R}^n})} } \right\},
$$
$$
f\left( \xi \right) = 0 \Leftrightarrow \xi \in E = \mathop {{\text{ }}U}\limits_{i = 1}^N SO\left( n \right){A_i}.
$$
Keywords
Convex Hull Strict Inequality Representation Formula Reverse Inclusion Relative Interior
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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© Springer Science+Business Media New York 1999