Abstract
We consider the minimization of an average-distance functional defined on a two-dimensional domain $\\Omega$ with an Euler elastica penalization associated with $\\partial\\Omega$, the boundary of $\\Omega$. The average distance is given by $$ \\int\_{\\Omega}\\operatorname{dist}^p(x,\\partial\\Omega)\\operatorname{d}x, $$ where $p\\geq 1$ is a given parameter and $\\operatorname{dist}(x,\\partial\\Omega)$ is the Hausdorff distance between ${x}$ and $\\partial\\Omega$. The penalty term is a multiple of the Euler elastica (i.e., the Helfrich bending energy or the Willmore energy) of the boundary curve ${\\partial\\Omega}$, which is proportional to the integrated squared curvature defined on $\\partial\\Omega$, as given by $$ \\lambda\\int{\\partial\\Omega} \\kappa{\\partial\\Omega}^2 \\operatorname{d}\\mathcal{H}\_{\\llcorner\\partial\\Omega}^1, $$ where $\\kappa{\\partial\\Omega}$ denotes the (signed) curvature of $\\partial\\Omega$ and $\\lambda>0$ denotes a penalty constant. The domain $\\Omega$ is allowed to vary among compact, convex sets of $\\mathbb{R}^2$ with Hausdorff dimension equal to two. Under no a priori assumptions on the regularity of the boundary $\\partial\\Omega$, we prove the existence of minimizers of $E{p,\\lambda}$. Moreover, we establish the $C^{1,1}$-regularity of its minimizers. An original construction of a suitable family of competitors plays a decisive role in proving the regularity.
Highlights
In 2D, the Willmore energy simplifies to be a multiple of the integrated squared curvature, which is commonly referred as the Euler elastica
In this paper we investigated the minimization problem for the average distance functional defined for a two-dimensional domain with respect to its boundary, subject to a penalty proportional to the Euler elastica of the boundary
We proved the existence and C1,1-regularity of minimizers, mainly relying on the method of contradictions by constructing suitable competitors
Summary
The curvature of boundaries plays an important role in many physical and biological models. Σ is assumed to be a connected set with its Hausdorff dimension equal to 1 and its one dimensional Hausdorff measure is to be bounded from above by a specified constant. Problems of this type are used in many modeling applications, such as urban planning and optimal pricing. We consider the average distance energy functional as a functional of the domain Ω with Σ = ∂Ω and penalized by the Euler elastica of ∂Ω, as given by. We are reducing our minimization problem to quite regular sets, i.e. domains Ω whose boundaries admit an H2 regular arc-length parameterization.
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More From: Interfaces and Free Boundaries, Mathematical Analysis, Computation and Applications
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