Qualitative Properties of Solutions to Second Order Elliptic and Parabolic Differential Equations

本课题研究二阶椭圆型和抛物型方程解的性质，如格林函数和调和量或热量的估计，半线性椭圆型方程爆破解的唯一性，弱结构假设下拟线性和非线性方程解的正则性。项目的第一部分致力于研究边界附近解的行为。特别注意系数的条件，这些条件保证了谐波测度和热测度相对于边界上的表面测度的绝对连续性。这种迷你型条件在散度形式方程中是已知的。研究非散度形式的方程需要完全不同的技术。第1部分还包括在边界上爆炸的半线性方程解的唯一性问题。我们最近的结果表明，在边界上消失的正解具有相同的衰减速率。我们期望边界附近的爆破解具有相同的增长率。本课题的第二部分讨论了具有最小光滑性的拟线性方程的解对解梯度的正则性，以及不具有凹性条件的完全非线性方程的解对二阶导数的正则性。本课题的一个重要部分是研究非光滑系数偏微分方程解的边界性质。这类方程出现在复合材料、多孔介质、化学、生物等不同过程的研究中。解的边界属性特别重要，因为它们“负责”给定对象与其环境（热辐射，辐射等）的相互作用。在许多情况下，为了在给定区域获得理想的效果，人们只能在其边界上行动。对边界附近解的行为的研究使我们能够使这种边界控制更具可预测性和有效性。半线性方程的爆破解与生物学、医学、核工程中的某些过程有关，在这些过程中，它们的控制不能被高估。本课题还讨论了非线性方程解的正则性。这类积极的结果不仅从理论的角度来看是有趣的，而且也为方程的数值解提供了背景。

The project deals with properties of solutions to elliptic and parabolic equations of second order, such as estimates for Green's functions and harmonic or caloric measures, uniqueness of blowup solutions to semilinear elliptic equations, regularity of solutions to quasilinear and nonlinear equations under weak structural assumptions. Part 1 of the project is devoted to the behavior of solutions near the boundary. Special attention is paid to the conditions on the coefficients, which guarantee the absolute continuity of harmonic and caloric measures with respect to the surface measure on the boundary. Such Dini-type conditions are known for equations in the divergence form. The study of equations in the non-divergence form needs quite different technique. Part 1 also includes the problem of uniqueness of solutions to semilinear equations, which blow up on the boundary. Our recent results show that positive solutions, which vanish on the boundary, have same rate of decay. We expect blowup solution near a portion of the boundary must have same rate of growth. Part 2 of the project deals with the regularity of solutions of quasilinear equations with minimal smoothness with respect to the gradient of solution, and of fully nonlinear equations without the concavity conditions with respect to the second derivatives. An essential part of the project deals with boundary properties of solutions of partial differential equations with non-smooth coefficients. Such equations appear in the investigation of different processes in composite materials, porous media, chemistry, biology, etc. The boundary properties of solutions are especially important, because they are "responsible" for interaction of a given object with its environment (heat emission, radiation, etc). In many situations, in order to get a desirable effect in a given region, one can only act on its boundary. The study of behavior of solutions near the boundary allows us to make such boundary control more predictable and effective. The blowup solutions of semilinear equations are associated with certain processes in biology, medicine, nuclear engineering, where their control cannot be overestimated. In this project, we also discuss the regularity of solutions of nonlinear equations. Positive results of such sort are interesting not only from the theoretical point of view, they also give the background for numerical solution of equations.