论文标题
通过广义量度理论拉普拉斯定义的热方程解决方案的近似值
An Approximation of Solutions to Heat Equations defined by Generalized Measure Theoretic Laplacians
论文作者
论文摘要
我们考虑由$ [0,1] $的广义量度理论laplacian定义的热方程。该方程式描述了在棒中的热扩散，因此棒的质量分布是由非原子的borel概率测量$μ$给出的，我们不假定存在严格的正质量密度。我们表明，弱度量收敛意味着在强烈的回避意义上，相应的广义拉普拉斯人的收敛性。我们证明，相对于统一规范的强烈半群收敛，这意味着溶液均匀地收敛到相应的热方程。例如，这提供了对栏上热量扩散的数学模型的解释，该模型具有间隙，即对相应的热方程的解决方案的表现就像在这些间隙上有足够小的质量的棒上的热流相近。
We consider the heat equation defined by a generalized measure theoretic Laplacian on $[0,1]$. This equation describes heat diffusion in a bar such that the mass distribution of the bar is given by a non-atomic Borel probabiliy measure $μ$, where we do not assume the existence of a strictly positive mass density. We show that weak measure convergence implies convergence of the corresponding generalized Laplacians in the strong resolvent sense. We prove that strong semigroup convergence with respect to the uniform norm follows, which implies uniform convergence of solutions to the corresponding heat equations. This provides, for example, an interpretation for the mathematical model of heat diffusion on a bar with gaps in that the solution to the corresponding heat equation behaves approximately like the heat flow on a bar with sufficiently small mass on these gaps.
