Linear and Semilinear Partial Differential Equations

本文主要研究三个相互联系的问题，即(i)非椭圆型偏微分方程解的唯一延拓，（ii）半线性（主要是双曲型）偏微分方程解的唯一延拓，（iii）双曲型混合初边值问题。唯一延拓的研究方向是半线性问题和无界势问题。在半线性双曲方程的领域，主要的工作将是研究一类附加（大部分）双曲偏微分算子的加权Sobolev空间。这些适用于双曲算子的空间似乎是研究半线性方程的自然环境；它们似乎是该领域进一步发展的关键。主要目标是理解这些空间的乘法结构，包括null形式估计。研究双曲型边值问题的首要目的是完成正在进行的PI对这类问题解的迹正则性的研究。这是第二部分的关键步骤，第二部分的目标是了解双曲型边值问题解的Strichartz型估计，并随后将其用于半线性双曲型边值问题的研究。PI希望从长远来看，这个研究项目也能在拟线性双曲方程的研究中取得进展。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。这些方程被用来模拟物理学、经济学等领域的许多现象。本课题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反问题中出现的问题；在那里，人们试图重建关于系统状态或控制系统的方程的信息，基于部分观察。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.