论文标题
紧凑抗性空间上部分微分方程的近似
Approximation of partial differential equations on compact resistance spaces
论文作者
论文摘要
我们考虑在二次形式意义上均匀椭圆形和抛物线的电阻空间上的线性部分微分方程，并涉及抽象梯度和分歧项。我们的主要兴趣是为其独特的解决方案提供图形和度量图近似值。对于在单个紧凑抗性空间上具有不同系数的方程式的家族，我们证明，如果系数保持界限，则解决方案对空间中均匀收敛具有积累点。如果在一系列方程式中适当地收敛，则溶液沿子序列均匀地收敛。对于有限分支的局部电阻形式的特殊情况，我们还考虑了近似于内部有限分支的电阻空间序列。在适当的假设下，对近似空间上方程解的系数（扩展）的线性化，沿着有限分支的集合上的目标方程溶液均匀地累积甚至均匀收敛。结果涵盖了离散和度量图的近似值，并讨论了两者。
We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and metric graph approximations for their unique solutions. For families of equations with different coefficients on a single compact resistance space we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformly along a subsequence. For the special case of local resistance forms on finitely ramified sets we also consider sequences of resistance spaces approximating the finitely ramified set from within. Under suitable assumptions on the coefficients (extensions of) linearizations of the solutions of equations on the approximating spaces accumulate or even converge uniformly along a subsequence to the solution of the target equation on the finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed.
