scholarly journals Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients

2018 ◽  
Vol 38 (12) ◽  
pp. 6073-6090
Author(s):  
Serena Dipierro ◽  
◽  
Aram Karakhanyan ◽  
Enrico Valdinoci ◽  
◽  
...  
Keyword(s):  
Free Boundary ◽  
Discontinuous Coefficients ◽  
Boundary Points ◽  
Divergence Type

scholarly journals The local structure of the free boundary in the fractional obstacle problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Matteo Focardi ◽  
Emanuele Spadaro
Keyword(s):  
Hausdorff Dimension ◽  
Free Boundary ◽  
Hausdorff Measure ◽  
Local Structure ◽  
Obstacle Problem ◽  
List Type ◽  
Minkowski Content ◽  
Local Finiteness ◽  
Lower Dimensional

AbstractBuilding upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results:(i)local finiteness of the {(n-1)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\mathcal{H}}^{n-1}}-rectifiability of the free boundary,(iii)classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most {(n-2)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {k\geq 2}, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than {k+1}.

scholarly journals On classification of classical and half-orientable horseshoes in terms of boundary points

2010 ◽  
pp. 549-566
Author(s):  
S. V. Gonchenko ◽  
◽  
A. S. Gonchenko ◽  
M. I. Malkin ◽  
◽  
...  
Keyword(s):  
Boundary Points

Spreading under shifting climate by a free boundary model: Invasion of deteriorated environment

2020 ◽  
pp. 2050077
Author(s):  
Yuanyang Hu ◽  
Xinan Hao ◽  
Yihong Du
Keyword(s):  
Free Boundary ◽  
Space Dimension ◽  
Complete Classification ◽  
Time Dynamics ◽  
Favorable Environment ◽  
Asymptotic Speed ◽  
Long Time ◽  
Parameterized Family ◽  
Boundary Model

In this paper, we consider a free boundary model in one space dimension which describes the spreading of a species subject to climate change, where favorable environment is shifting away with a constant speed [Formula: see text] and replaced by a deteriorated yet still favorable environment. We obtain two threshold speeds [Formula: see text] and a complete classification of the long-time dynamics of the model, which reveals significant differences between the cases [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. For example, when [Formula: see text], for a suitably parameterized family of initial functions [Formula: see text] increasing continuously in [Formula: see text], we show that there exists [Formula: see text] such that the species vanishes eventually when [Formula: see text], it spreads with asymptotic speed [Formula: see text] when [Formula: see text], it spreads with forced speed [Formula: see text] when [Formula: see text], and it spreads with speed [Formula: see text] when [Formula: see text]. Moreover, in the last case, while the spreading front propagates with asymptotic speed [Formula: see text], the profile of the population density function [Formula: see text] approaches a propagating pair consisting of a traveling wave with speed [Formula: see text] and a semi-wave with speed [Formula: see text].

A free boundary problem arising in the modelling of internal oxidation of binary alloys

1995 ◽  
Vol 6 (3) ◽  
pp. 225-245
Author(s):  
Bei Hu ◽  
Jianhua Zhang
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Internal Oxidation ◽  
Free Boundary Problem ◽  
Binary Alloys ◽  
Local Existence ◽  
Discontinuous Coefficients ◽  
Parabolic Partial Differential Equation ◽  
One Dimensional ◽  
A Free Boundary Problem

A one-dimensional free boundary problem arising in the modelling of internal oxidation of binary alloys is studied in this paper. The free boundary of this problem is determined by the equation u = 0, where u is the solution of a parabolic partial differential equation with discontinuous coefficients across the free boundary. Local existence, uniqueness and the regularity of the free boundary are established. Global existence is also studied.

scholarly journals Properties of the free boundary near the fixed boundary of the double obstacle problems

2021 ◽  
Author(s):  
Jinwan Park
Keyword(s):  
Free Boundary ◽  
Obstacle Problem ◽  
Nonlinear Operator ◽  
Main Idea ◽  
Global Solutions ◽  
Obstacle Problems ◽  
Local Regularity ◽  
Fixed Boundary ◽  
New Type

In this paper, we study the tangential touch and [Formula: see text] regularity of the free boundary near the fixed boundary of the double obstacle problem for Laplacian and fully nonlinear operator. The main idea to have the properties is regarding the upper obstacle as a solution of the single obstacle problem. Then, in the classification of global solutions of the double problem, it is enough to consider only two cases for the upper obstacle, [Formula: see text] The second one is a new type of upper obstacle, which does not exist in the study of local regularity of the free boundary of the double problem. Thus, in this paper, a new type of difficulties that come from the second type upper obstacle is mainly studied.

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