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In this paper we prove uniqueness theorems for mappings $F\in W_{\text{loc}}^{1,n}(\mathbb{B}^{n};\mathbb{R}^{n})$ of finite distortion $1\leq K(x)=\Vert \mathit{DF}(x)\Vert ^{n}/J_{F}(x)$ satisfying some integrability conditions. These types of theorems fundamentally state that if a mapping defined in $\mathbb{B}^{n}$ has the same boundary limit $a$ on a ‘relatively large’ set $E\subset \unicode[STIX]{x2202}\mathbb{B}^{n}$, then the mapping is constant. Here the size of the set $E$ is measured in terms of its $p$-capacity or equivalently its Hausdorff dimension.

The aim of this paper is to provide a rigorous variational formulation forthe detection of points in 2-d biological images. To this purposewe introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals forwhich we prove the Γ-convergence to the initial one.

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UNIQUENESS THEOREMS FOR MAPPINGS OF FINITE DISTORTION

Variational approximation for detectingpoint-like target problems *

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2 results

UNIQUENESS THEOREMS FOR MAPPINGS OF FINITE DISTORTION

Variational approximation for detectingpoint-like target problems*

Variational approximation for detectingpoint-like target problems *