Existence et effet régularisant pour des équations de Schrödinger non linéaires généralisées : le cas de l'intéraction quadratique

We improve the result obtained by C. E. Kenig, G. Ponce and L. Vega, 1998, Inventionnes Math., on generalized nonlinear Schrödinger equations in the case where the interactions are quadratic. We establish the well-posedness of the Cauchy problem for such equations in weighted Sobolev spaces H s,σ (R n) for s > n/2 + 2 and σ > 1.Nous améliorons le resultat obtenu par C. E. Kenig, G. Ponce and L. Vega, 1998, Inventionnes Math., sur les équations de Schrödinger non linéaires généralisées dans le cas où les interactions sont quadratiques. Nous démontrons le caractère bien posé de telles équations dans des espaces de Sobolev à poids H s,σ (R n) avec s > n/2 + 2 et σ > 1

On the Fefferman-Phong inequality

We show that the number of derivatives of the symbol needed to establish the Fefferman-Phong inequality is bounded by n+4+ improving thus the bound 2n+4+ obtained by N. Lerner and Y. Morimoto in [1]. We also give an abstract result. We even show that in the case of classical symbols, that is S21,0, the number of needed derivatives is bounded by n2 + 4 + .

On a shape derivative formula for star-shaped domains using Minkowski deformation

We consider the shape derivative formula for a volume cost functional studied in previous papers where we used the Minkowski deformation and support functions in the convex setting. In this work, we extend it to some non-convex domains, namely the star-shaped ones. The formula happens to be also an extension of a well-known one in the geometric Brunn-Minkowski theory of convex bodies. At the end, we illustrate the formula by applying it to some model shape optimization problem

Existence et effet régularisant pour des équations de Schrödinger non linéaires généralisées : le cas de l'intéraction quadratique

We improve the result obtained by C. E. Kenig, G. Ponce and L. Vega, 1998, Inventionnes Math., on generalized nonlinear Schrödinger equations in the case where the interactions are quadratic. We establish the well-posedness of the Cauchy problem for such equations in weighted Sobolev spaces H s,σ (R n) for s > n/2 + 2 and σ > 1.Nous améliorons le resultat obtenu par C. E. Kenig, G. Ponce and L. Vega, 1998, Inventionnes Math., sur les équations de Schrödinger non linéaires généralisées dans le cas où les interactions sont quadratiques. Nous démontrons le caractère bien posé de telles équations dans des espaces de Sobolev à poids H s,σ (R n) avec s > n/2 + 2 et σ > 1

Community Management for Improved Sustainability: Case Studies of Three Rural Community Water Supply and Sanitation Projects in Honduras

A Professional Project Report submitted in partial fulfillment of the requirements for the Degree of Master of Water Resources, Water Resources Program, University of New Mexico.The sustainability of a rural water supply and sanitation project is essential to the health of a community. Three rural communities in Honduras organized themselves and requested water supply and sanitation projects. With the help of the local and national governments and technical volunteers, they constructed the systems, were trained in operation and maintenance, and educated in hygiene and sanitation. The goal was self-sustaining projects after the technicians left. Upon visiting the projects of Nueva Vida, Miramar, and Las Flores after 3, 4, and 13 years of operation, respectively, I observed that the communities did not appear to be heading in the direction of sustainability. This study includes an assessment of the three projects, which found that the lack of a strong managing body for the system, the reliance on emergency maintenance, and the lack of chlorine all constrained the sustainability of the systems. External agencies imposed the current management structure, including the rules and regulations for the system, onto these communities. The communities did not play a role in designing the management structure of the project and are therefore reluctant to comply entirely. In order to prevent the failure of these systems, the communities need an external support agency (ESA) -- a government or private agency or non-governmental organization (NGO) -- for management and technical assistance. Given the technical and social design of the projects, community management, or community control and responsibility of the system with external support, will improve the sustainability of the three water supply projects

A shape optimization approach for a class of free boundary problems of Bernoulli type

summary:We are interested in an optimal shape design formulation for a class of free boundary problems of Bernoulli type. We show the existence of the optimal solution of this problem by proving continuity of the solution of the state problem with respect to the domain. The main tools in establishing such a continuity are a result concerning uniform continuity of the trace operator with respect to the domain and a recent result on the uniform Poincaré inequality for variable domains