Hausdorff Measure Estimates and Lipschitz Regularity in Inhomogeneous Nonlinear Free Boundary Problems

Moreira, Diego R.; Wang, Lihe

doi:10.1007/s00205-014-0739-8

Cited by 7 publications

(5 citation statements)

References 21 publications

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“…For other references related to the free boundary problem under consideration in this paper we would like to refer the reader to [3], [4], [5], [9], [10], [11], [27], [28], [30], [31], [32], [34], [35] and the references therein. This list is by no means exhaustive.…”

Section: Introductionmentioning

confidence: 99%

Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the $p(x)$-Laplacian

Lederman¹,

Wolanski²

2017

Interfaces Free Bound.

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Abstract. In this paper we study a one phase free boundary problem for the p(x)-Laplacian with non-zero right hand side. We prove that the free boundary of a weak solution is a C 1,α surface in a neighborhood of every "flat" free boundary point. We also obtain further regularity results on the free boundary, under further regularity assumptions on the data. We apply these results to limit functions of an inhomogeneous singular perturbation problem for the p(x)-Laplacian that we studied in [25]. IntroductionIn this paper we study the following inhomogeneous free boundary problem for the p(x)-Laplacian: u ≥ 0 andThe p(x)-Laplacian serves as a model for a stationary non-newtonian fluid with properties depending on the point in the region where it moves. For example, such a situation corresponds to an electrorheological fluid. These are fluids such that their properties depend on the magnitude of the electric field applied to it. In some cases, fluid and Maxwell's equations become uncoupled and a single equation for the p(x)-Laplacian appears (see [33]).The free boundary problem P (f, p, λ * ) appears, for instance, in the limit of a singular perturbation problem that may model high activation energy deflagration flames in a fluid with electromagnetic sensitivity (see [25]). When p(x) ≡ 2 (in which case the p(x)-Laplacian coincides with the Laplacian) this singular perturbation problem was introduced by Zeldovich and Frank-Kamenetski in order to model these kind of flames in [37]. In this latter case, the right hand side f may come from nonlocal effects as well as from external sources (see [23]).The free boundary problem considered in this paper also appears in an inhomogeneous minimization problem that we study in [26] where we prove that minimizers are weak solutions to P (f, p, λ * ).In the present article we prove that the free boundary ∂{u > 0} -with u a weak solution of P (f, p, λ * )-is a smooth hypersurface in a neighborhood of every "flat" free boundary point.Key words and phrases. Free boundary problem, variable exponent spaces, regularity of the free boundary, singular perturbation, inhomogeneous problem.

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Section: Introductionmentioning

confidence: 99%

Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the $p(x)$-Laplacian

Lederman¹,

Wolanski²

2017

Interfaces Free Bound.

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“…This is not an obstacle problem explicitly but the solutions are automatically constrained above zero, thus it is equivalent to the problem with the constraint u ≥ ψ ≡ 0 (flat obstacle problem). This problem is one of important interface models and has been generalized variously (for example, see [25] generalizing the first term, [3] the second term, or their references). Especially, Yamaura [33] considered a non-linearized case, namely the case that the first term is replaced to the area functional.…”

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confidence: 99%

Singular perturbation by bending for an adhesive obstacle problem

Miura

2016

Calc. Var.

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A free boundary problem arising from materials science is studied in the one-dimensional case. The problem studied here is an obstacle problem for the non-convex energy consisting of a bending energy, tension and an adhesion energy. If the bending energy, which is a higher order term, is deleted then "edge" singularities of the solutions (surfaces) may occur at the free boundary as Alt-Caffarelli type variational problems. The main result of this paper is to give a singular limit of the energy utilizing the notion of Γ-convergence, when the bending energy can be regarded as a perturbation. This singular limit energy only depends on the state of surfaces at the free boundary as seen in singular perturbations for phase transition models.2010 Mathematics Subject Classification. 35B25, and 35R35.

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“…As an alternative, some scholars use the Hausdorff distance model, the generalized Hausdorff distance model, or the median Hausdorff distance model to measure the position similarity [25,34]. This method not only measures the position difference of spatial objects, but also reflects differences between objects in terms of shape and overall distribution.…”

Section: Selection Of Hierarchical Matching Operatorsmentioning

confidence: 99%

Incrementally Detecting Change Types of Spatial Area Object: A Hierarchical Matching Method Considering Change Process

Wang

Zhang

Guan

2018

IJGI

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Detecting and extracting the change types of spatial area objects can track area objects' spatiotemporal change pattern and provide the change backtracking mechanism for incrementally updating spatial datasets. To respond to the problems of high complexity of detection methods, high redundancy rate of detection factors, and the low automation degree during incrementally update process, we take into account the change process of area objects in an integrated way and propose a hierarchical matching method to detect the nine types of changes of area objects, while minimizing the complexity of the algorithm and the redundancy rate of detection factors. We illustrate in details the identification, extraction, and database entry of change types, and how we achieve a close connection and organic coupling of incremental information extraction and object type-of-change detection so as to characterize the whole change process. The experimental results show that this method can successfully detect incremental information about area objects in practical applications, with the overall accuracy reaching above 90%, which is much higher than the existing weighted matching method, making it quite feasible and applicable. It helps establish the corresponding relation between new-version and old-version objects, and facilitate the linked update processing and quality control of spatial data.

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Hausdorff Measure Estimates and Lipschitz Regularity in Inhomogeneous Nonlinear Free Boundary Problems

Cited by 7 publications

References 21 publications

Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the $p(x)$-Laplacian

Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the $p(x)$-Laplacian

Singular perturbation by bending for an adhesive obstacle problem

Incrementally Detecting Change Types of Spatial Area Object: A Hierarchical Matching Method Considering Change Process

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