Fully discrete hyperbolic initial boundary value problems with nonzero initial data

The stability theory for hyperbolic initial boundary value problems relies most of the time on the Laplace transform with respect to the time variable. For technical reasons, this usually restricts the validity of stability estimates to the case of zero initial data. In this article, we consider the class of non-glancing finite difference approximations to the hyperbolic operator. We show that the maximal stability estimates that are known for zero initial data and nonzero boundary source term extend to the case of nonzero initial data in \^a 2 . The main novelty of our approach is to cover finite difference schemes with an arbitrary number of time levels. As an easy corollary of our main trace estimate, we recover former stability results in the semigroup sense by Kreiss [Kre68] and Osher [Osh69b]

Monocentric analysis of restricting the budget share of housing alone or with transportation

Considering the prolonged rise of energy price and the still elevated housing prices, the policy to limit the share of housing expenses in the households' budget, so as to secure their solvability, has been criticized. Supposedly, it induces people to get farther from the city center in search for cheaper housing prices, but with subsequent increased transportation costs that are often disregarded during the house search process. Therefore, to improve the well-being of households, it has been advocated to set a constraint on the share of both housing and transportation expenditure. The paper is purported to analyze and compare the effects of the two policies in terms of: 1. Well-being of the households; 2. Land-use: city size and density curve; 3. Solvability of the households; 4. Transportation costs. The analysis is carried out within the classical monocentric model of urban economics. After setting a general analysis, an applied model is specified to capture the effects of each policy in straightforward formulae. It is shown that constraining housing expenses may increase the well-being of households. Besides, both policies prove effective in reducing urban sprawl and hereby energy consumption. Thus the choice of the optimal policy will depend on the local authority's objectives.monocentric model, urban economics, housing expenses, transportation expenses, housing policy

Amplitude equations for weakly nonlinear surface waves in variational problems

Among hyperbolic Initial Boundary Value Problems (IBVP), those coming from a variational principle 'generically' admit linear surface waves, as was shown by Serre [J. Funct. Anal. 2006]. At the weakly nonlinear level, the behavior of surface waves is expected to be governed by an amplitude equation that can be derived by means of a formal asymptotic expansion. Amplitude equations for weakly nonlinear surface waves were introduced by Lardner [Int. J. Engng Sci. 1983], Parker and co-workers [J. Elasticity 1985] in the framework of elasticity, and by Hunter [Contemp. Math. 1989] for abstract hyperbolic problems. They consist of nonlocal evolution equations involving a complicated, bilinear Fourier multiplier in the direction of propagation along the boundary. It was shown by the authors in an earlier work [Arch. Ration. Mech. Anal. 2012] that this multiplier, or kernel, inherits some algebraic properties from the original IBVP. These properties are crucial for the (local) well-posedness of the amplitude equation, as shown together with Tzvetkov [Adv. Math., 2011]. Properties of amplitude equations are revisited here in a somehow simpler way, for surface waves in a variational setting. Applications include various physical models, from elasticity of course to the director-field system for liquid crystals introduced by Saxton [Contemp. Math. 1989] and studied by Austria and Hunter [Commun. Inf. Syst. 2013]. Similar properties are eventually shown for the amplitude equation associated with surface waves at reversible phase boundaries in compressible fluids, thus completing a work initiated by Benzoni-Gavage and Rosini [Comput. Math. Appl. 2009]

Monocentric analysis of restricting the budget share of housing alone or with transportation

Considering the prolonged rise of energy price and the still elevated housing prices, the policy to limit the share of housing expenses in the households' budget, so as to secure their solvability, has been criticized. Supposedly, it induces people to get farther from the city center in search for cheaper housing prices, but with subsequent increased transportation costs that are often disregarded during the house search process. Therefore, to improve the well-being of households, it has been advocated to set a constraint on the share of both housing and transportation expenditure. The paper is purported to analyze and compare the effects of the two policies in terms of: 1. Well-being of the households; 2. Land-use: city size and density curve; 3. Solvability of the households; 4. Transportation costs. The analysis is carried out within the classical monocentric model of urban economics. After setting a general analysis, an applied model is specified to capture the effects of each policy in straightforward formulae. It is shown that constraining housing expenses may increase the well-being of households. Besides, both policies prove effective in reducing urban sprawl and hereby energy consumption. Thus the choice of the optimal policy will depend on the local authority's objectives