AI 中文总结
研究非散度形式变系数热方程柯西 - 狄利克雷问题，结合多种思想推导解的短时行为渐近公式，证明椭圆对应形式及相关渐近公式，并用公式描述热方程时间不变曲面，给出不同边界值情况的结果。
AI 中文摘要
我们考虑一个非散度形式的变系数热方程的柯西 - 狄利克雷问题。初始值假设为齐次的，而狄利克雷边界值是非恒定的。在此设定下，我们推导了解的短时行为的渐近公式，它扩展了著名的瓦拉丹公式。强调边界值允许消失或变号。该公式通过结合瓦拉丹的原始思想与埃文斯和石井关于粘性解理论的思想以及一些进一步的说明得到。顺便证明了该公式的椭圆对应形式，涉及预解式方程解的慢扩散行为。对于经典拉普拉斯算子的情况，我们证明了在与区域边界相切的球面上预解式方程解的热含量和平均值的渐近公式，这些公式扩展了第二作者和坂口史郎先前得到的某些公式到非恒定边界值的情况，公式涉及切点处的主曲率和狄利克雷边界值。然后我们用这些公式描述存在非恒定狄利克雷边界值时热方程的时间不变曲面。对于非负狄利克雷边界值给出了两个对称结果，当边界值变号时给出了一个不存在结果。
英文摘要
We consider a Cauchy-Dirichlet problem for a heat equation with variable coefficients in non-divergence form. The initial values are assumed to be homogeneous, while the Dirichlet boundary values are non-constant. In this setting, we derive an asymptotic formula for the short-time behavior of the solution, which extends the celebrated Varadhan formula. We stress the fact that the boundary values are allowed to vanish or change sign. The formula is obtained by combining the original ideas of Varadhan with those of Evans and Ishii, pertaining to the theory of viscosity solutions, and some further remarks. In passing, we prove the elliptic counterpart of the formula. This concerns the slow-diffusion behavior of the solutions of the resolvent equation. Furthermore, for the case of the classical Laplace operator, we prove asymptotic formulas for the heat content and the mean value of the solution of the resolvent equation on spheres touching the boundary of the domain. These extend to the case of non-constant boundary values, certain formulas previously obtained by the second author and S. Sakaguchi. The formulas involve the principal curvatures at the touching point and the Dirichlet boundary values. We then use our formulas to describe time-invariant surfaces for the heat equation, in presence of non-constant Dirichlet boundary values. We give two symmetry results in case of non-negative Dirichlet boundary values and a non-existence result, when those values change sign.
Comments28 pages
