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Branching time estimates in quasi-static evolution for the average distance functional
Abstract
In this paper, we analyze the discrete quasi-static irreversible with small steps evolution of a connected network related to an average distance functional minimization problem. Our main goal is to determine whether new branches may apear during the evolution, thus changing the topology. We give conditions for this and an upper bound for the time at which it must happen for a particular class of configurations. We will use extensively tools belonging to minimizing movements and optimal transportation theory with free Dirichlet regions. Then we will give some explicit examples of quasi-static evolution whose branching times will be estimated by direct computation, using both pure energy and mixed geometric/energy estimates.
... Given a domain Ω, let S ∈ A be an arbitrary element, and suppose there exists Q ∈ S angular and let be δ > 0 such that B(Q, δ) ∩ S is homeomorphic to (0, 1). Then the Voronoi cell V (Q) contains a triangle T Q with sides ρ Q > 0 and angleQ > 0. For the proof we refer to [9]. So we can apply the estimate of Proposition 2.4 to this configuration too, thus we have the following result: Proposition 2.7. ...
... For more details about this see [9]. Given a sequence {ε n } ↓ 0, accumulation points for Σ εn : [0, T ] −→ A are guaranteed to exist (for the proof see [4] and [9]). ...
... For more details about this see [9]. Given a sequence {ε n } ↓ 0, accumulation points for Σ εn : [0, T ] −→ A are guaranteed to exist (for the proof see [4] and [9]). ...
In this paper we consider the dynamic irreversible evolution of a connected network related to the average distance functional minimization problem, with suitable dissipation term. Our goal here will be finding which schemes admit locally stable points, and try to characterize the latter. Notions and tools belonging to minimizing movement theory and optimal transportation with free Dirichlet regions will be employed extensively. Finally, applying these results to mini-mizing movement evolutions, we will show that locally stable points are also stable points for a particular class of them.
... We aim to extend some results concerning geometric properties of optimal sets (for instance see [6], [7] and [8]) to the more general measures, and investigate the topological behavior of the minimizing movement evolution process, thus extending results in [12] to this more general case. ...
... Given a domain Ω, let S ∈ A be an arbitrary element, and suppose there exists Q ∈ S angular and let be δ > 0 such that B(Q, δ) ∩ S is homeomorphic to (0, 1). Then the Voronoi cell V (Q) contains a triangle T Q with sides ρ Q > 0 and angleQ > 0. For the proof we refer to [12]. The next result is a condition on the branching behaviors for Euler schemes. ...
... For the proof we refer to [12]. Now we can present an upper bound estimate for the branching time. ...
In this paper we consider the quasi-static irreversible evolution of a connected network re-lated to an average distance functional minimization problem. Our main goal is to extend some geometric properties of optimal sets to the coercive L p measure case, and determine whether a branches is exhibited during the a minimizing movement evolution, thus changing the topology. We would give a sufficient condition for the latter. Tools belonging to minimizing movements and optimal transportation theory with free Dirichlet regions will be used extensively. Finally, we will apply our results to find an upper bound for the branching time for a particular class of configurations.
... In this section we recall some results on the estimates (both lower and upper bound) for the functional F . Most proofs of this section can be found elsewhere, as in [9], but we will report them due to the important role played in this paper. First, we recall the definition of Voronoi cell: Given a subset S ⊂ S, its Voronoi cell is simply V (S ) := {x ∈ Ω : dist(x, S ) = dist(x, S)}. ...
... A subset of S is smooth is all its non endpoints are smooth. Now we can present a lower bound estimate: the proof is very similar to that found in [9], but we report it as the argument used is useful in the proof of the next result. ...
... Geometrically, one may imagine that a point P is angular when the tangent vectors form an angle (or a cuspid) here. The next result states that an angular point satisfy condition ( * ) of Proposition 2.5 too: For the proof, similar to that used in Proposition 2.5 (first statement), we refer to [9]. So we can apply the estimate of Proposition 2.5 to this configuration too, thus reassuming we have proven the following result: Then there exists ε 0 > 0 such that for any ε < ε 0 ...
In this paper we consider discrete dynamic irreversible evolutions with small steps of a con-nected network related to the average distance functional minimization problem, with suitable dissipation term. Our goal here will be finding which schemes admit locally stable points, and try to characterize the latter. Notions and tools belonging to minimizing movement theory and optimal transportation with free Dirichlet regions will be employed extensively. Finally, apply-ing these results to minimizing movement evolutions, we will study the role of locally stable points in the evolution context.
... Geometrically, one may imagine that a point P is angular when the tangent vectors form an angle (or a cuspid) here. The following result holds: For the proof we refer to [12]. So we can apply the same argument found in the proof of Proposition 2.4 to these points, essentially obtaining the same conclusion, i.e. adding a small segment of length ε the gain for F has order O(ε). ...
... The proof is easy, and done in [12]: the key argument is that in order to pass to the other region, the set must cross the border, thus creating a branching and changing the topology. Now we present an estimate relating the energy and the " free space " : Given a domain Ω, an element S 1 ∈ A, and suppose that there exists Q ∈ Ω and R > 0 such that the ball B(Q, R) ∩ S = ∅. ...
In this paper we consider the dynamic irreversible evolution of a connected network related to an average distance functional minimization problem, with associated dissipation term. Our goal is to determine whether and when new branches may appear. Tools belonging to mini-mizing movements and optimal transportation theory with free Dirichlet regions will be used extensively. Then we will show an application of conditions found to a particular class of config-urations, and give an upper bound estimate for the branching time for them.
We present an abstract framework for irreversible rate independent evolution processes of quasi-static nature. The main tool relies on the minimizing movement theory. In particular situations, stability and energy inequality of Mielke’s type are satisfied. Several examples are given, among which the obstacle erosion.
In the paper the problem of constructing an optimal urban transportation network in a city with given densities of population and of workplaces is studied. The network is modeled by a closed connected set of assigned length, while the optimality condition consists in minimizing the Monge-Kantorovich functional representing the total transportation cost. The cost of trasporting a unit mass between two points is assumed to be proportional to the distance between them when the transportation is carried out outside of the network, and negligible when it is carried out along the network. The same problem can be also viewed as finding an optimal Dirichlet zone minimizing the Monge-Kantorovich cost of transporting the given two measures. The paper basically studies qualitative topological and geometrical properties of optimal networks. A mild regularity result for optimal networks is also provided. Mathematics Subject Classification (2000): 49Q10 (primary), 49Q15, 49N60, 90B10 (secondary).
We discuss quasistatic evolution processes for capacitary measures and shapes in order to model debonding membranes. Minimizing
movements as well as rate-independent processes are investigated and some models are described, together with a series of
open problems.
A Dirichlet region for an optimal mass transportation problem is, roughly speaking, a zone in which the transportation cost is vanishing. We study the optimal transportation problem with an unknown Dirichlet region Σ which varies in the class of closed connected subsets having prescribed 1-dimensional Hausdorff measure. We show the existence of an optimal Σ
opt
and study some of its geometrical properties. We also present numerical computations which show the shape of Σ
opt
in some model examples.
We study the properties of the transport density measure in the Monge-Kantorovich optimal mass transport problem in the presence of so-called Dirichlet constraint, i.e. when some closed set is given along which the cost of transportation is zero. The Hausdorff dimension estimates, as well as summability and higher regularity properties of the transport density are studied. The uniqueness of the transport density is proven in the case when the masses to be transported are represented by measures absolutely continuous with respect to the Lebesgue measure.
We consider the optimization problem min{F(g):g∈X(Ω)}, whereF(g) is a variational energy associated to the obstaclegand the classX(Ω) of admissible obstacles is given byX(Ω)={g: Ω→: g⩽ψonΩ, ∫Ω g dx=c} withψ∈W1, p0(Ω) andc∈ fixed. Generally, this problem does not have a solution and it may happen that the “optimal” obstacle is of relaxed form. Under a monotonicity assumption onF, we prove the existence of a non-relaxed optimal obstacle in the familyX(Ω) through a new method based on the notions ofγandwγ-convergences.
Toader: A model for the quasi-static growth of brittle fracture: existence and approx-imation results
- G Dal
- R Maso
G. Dal Maso, R. Toader: A model for the quasi-static growth of brittle fracture: existence and approx-imation results. Arch. Ration. Mech. Anal. 162(2002) 101-135
New problems on minimizing movements In boundary value problems for partial differential equations
- De Giorgi
E. De Giorgi: New problems on minimizing movements. In boundary value problems for partial differential equations, Res. Notes Appl. math. vol. 29. Masson, Paris, pp. 81-98, 1993