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

该项目涉及椭圆和抛物型方程的二阶解的性质，如估计绿色的功能和调和或热量措施，唯一的爆破解决方案，半线性椭圆方程，正则性的解决方案，拟线性和非线性方程弱结构假设。第1部分的项目是致力于解决方案的边界附近的行为。特别注意系数的条件，这保证了调和和热量措施的绝对连续性关于表面上的边界措施。这样的迪尼型条件是已知的方程在发散形式。研究非发散形式的方程需要相当不同的技术。第1部分还包括半线性方程解的唯一性问题，它在边界上爆炸。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率。我们期望爆破解在边界的一部分附近必须有相同的增长率。该项目的第2部分涉及关于解的梯度具有最小光滑度的拟线性方程的解的正则性，以及关于二阶导数不具有非线性条件的完全非线性方程的解的正则性。 该项目的一个重要部分涉及非光滑系数偏微分方程解的边界性质。这种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。解的边界性质特别重要，因为它们对给定物体与其环境的相互作用（热发射、辐射等）“负责”。在许多情况下，为了在一个给定的区域内获得理想的效果，人们只能在其边界上采取行动。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加可预测和有效。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。在这个项目中，我们还讨论了非线性方程解的正则性。这类正结果不仅从理论的角度来看是有趣的，它们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.