New applications of Hadamard-in-the-mean inequalities to incompressible variational problems
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arXiv
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| Format: | Preprint |
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2024
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| author | Bevan, Jonathan Kružík, Martin Valdman, Jan |
| author_facet | Bevan, Jonathan Kružík, Martin Valdman, Jan |
| contents | Let $\mathbb{D}(u)$ be the Dirichlet energy of a map $u$ belonging to the Sobolev space $H^1_{u_0}(Ω;\mathbb{R}^2)$ and let $A$ be a subclass of $H^1_{u_0}(Ω;\mathbb{R}^2)$ whose members are subject to the constraint $\det \nabla u = g$ a.e. for a given $g$, together with some boundary data $u_0$. We develop a technique that, when applicable, enables us to characterize the global minimizer of $\mathbb{D}(u)$ in $A$ as the unique global minimizer of the associated functional $F(u):=\mathbb{D}(u)+ \int_Ω f(x) \, \det \nabla u(x) \, dx$ in the free class $H^1_{u_0}(Ω;\mathbb{R}^2)$. A key ingredient is the mean coercivity of $F(φ)$ on $H^1_0(Ω;\mathbb{R}^2)$, which condition holds provided the `pressure' $f \in L^{\infty}(Ω)$ is `tuned' according to the procedure set out in \cite{BKV23}. The explicit examples to which our technique applies can be interpreted as solving the sort of constrained minimization problem that typically arises in incompressible nonlinear elasticity theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18467 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New applications of Hadamard-in-the-mean inequalities to incompressible variational problems Bevan, Jonathan Kružík, Martin Valdman, Jan Analysis of PDEs 49J40, 65K10 Let $\mathbb{D}(u)$ be the Dirichlet energy of a map $u$ belonging to the Sobolev space $H^1_{u_0}(Ω;\mathbb{R}^2)$ and let $A$ be a subclass of $H^1_{u_0}(Ω;\mathbb{R}^2)$ whose members are subject to the constraint $\det \nabla u = g$ a.e. for a given $g$, together with some boundary data $u_0$. We develop a technique that, when applicable, enables us to characterize the global minimizer of $\mathbb{D}(u)$ in $A$ as the unique global minimizer of the associated functional $F(u):=\mathbb{D}(u)+ \int_Ω f(x) \, \det \nabla u(x) \, dx$ in the free class $H^1_{u_0}(Ω;\mathbb{R}^2)$. A key ingredient is the mean coercivity of $F(φ)$ on $H^1_0(Ω;\mathbb{R}^2)$, which condition holds provided the `pressure' $f \in L^{\infty}(Ω)$ is `tuned' according to the procedure set out in \cite{BKV23}. The explicit examples to which our technique applies can be interpreted as solving the sort of constrained minimization problem that typically arises in incompressible nonlinear elasticity theory. |
| title | New applications of Hadamard-in-the-mean inequalities to incompressible variational problems |
| topic | Analysis of PDEs 49J40, 65K10 |
| url | https://arxiv.org/abs/2412.18467 |