Linear and Semilinear Partial Differential Equations

摘要Tataru 9622942 拟研究的重点是三个相互关联的主题，即（i）唯一的延续的解决方案，非椭圆型偏微分方程，（ii）半线性（主要是双曲型）偏微分方程和（iii）双曲型混合初边值问题。唯一延拓的研究主要集中在半线性问题和无界位势问题上。 在半线性双曲型方程领域，主要的工作将是研究一类（主要是）双曲型偏微分算子的加权Sobolev空间。这些空间适应双曲算子似乎是自然的设置为研究半线性方程，它似乎是关键，以进一步取得进展的领域。 一个主要目标是了解这些空间的乘法结构，包括零形式估计。 双曲边值问题研究的第一个目的是完成PI对此类问题解的迹正则性的研究。这是第二部分的关键一步，其目标是了解双曲边值问题解的Eschenhartz型估计，并随后将其用于半线性双曲边值问题的研究。 PI希望，从长远来看，这项研究计划也可以导致准线性双曲方程研究的进展。 偏微分方程是描述依赖于一个以上变量的量的变化率之间的关系的方程。 这类方程被用来模拟物理学、经济学等领域的许多现象。本项目的目的是研究某些偏微分方程解的一些特殊性质。 部分工作的动机是在控制理论和逆问题中出现的问题;有一个试图重建信息的系统或控制系统的方程的状态，基于部分观测。该项目的另一个目标是更好地理解一些迄今为止被证明极难研究的基本方程;例如，描述相对论时空结构的爱因斯坦方程。

Abstract Tataru 9622942 The proposed research focuses on three interconnecting subjects, namely (i) unique continuation for solutions to nonelliptic partial differential equations, (ii) semilinear (mostly hyperbolic) partial differential equations and (iii) hyperbolic mixed initial-boundary value problems. The research in unique continuation is oriented towards semilinear problems and problems with unbounded potentials. In the area of semilinear hyperbolic equations, the main effort will be directed towards the study of a class of weighted Sobolev spaces attached to (mostly) hyperbolic partial differential operators. These spaces adapted to the hyperbolic operator seem to be the natural setting for the study of semilinear equations; it appears that they are the key to further progress in the field. A main goal is to understand the multiplicative structure of these spaces, including null forms estimates. The first aim of the research on hyperbolic boundary value problems is to complete the ongoing PI's study of the trace regularity properties for solutions to such problems. This is a key step for the second part, whose goal is to understand the Strichartz type estimates for solutions to hyperbolic boundary value problems, and subsequently, to use this in the study of semilinear hyperbolic boundary value problems. It is the PI's hope that in the long run this research program could also lead to progress in the study of quasilinear hyperbolic equations. A partial differential equation is an equation which describes the relation between rates of change of quantities which depend on more than one variable. Such equations are used to model many phenomenas in physics, economics, etc. The aim of this project is to investigate some specific properties of solutions to certain partial differential equations. Part of the work is motivated by problems arising in control theory and in inverse problems; there one tries to reconstruct information about either the state of a system or th e equations governing the system, based on partial observations. Another goal of the project is to reach a better understanding of some fundamental equations which insofar have proved extremely difficult to study; for instance, Einstein's equations, which describe the structure of the relativistic space-time.