论文标题
在任意和分形域上线性和非线性波方程的dirichlet边界有价值问题
Dirichlet boundary valued problems for linear and nonlinear wave equations on arbitrary and fractal domains
论文作者
论文摘要
强烈阻尼的线性波方程和具有同质dirichlet边界条件的非线性WESTERVELT方程的较弱的拟态性结果在任意的三维域或任何两个维度域上证明了可以通过相同的几何常数降级的NTA域限制的限制获得的任何二维域。由于与均匀的dirichlet边界条件的弱配方相对应的功能的MOSCO收敛，因此获得了二维结果。 在Sobolev扩展域组成的一类可接受的域名中也可以处理非均匀的Dirichlet条件在$ \ mathbb {r}^3 $中或在$ \ mathbb {r}^2 $中的nta域的收敛序列上，其边界规律性没有其他条件的域。
The weak well-posedness results of the strongly damped linear wave equation and of the non linear Westervelt equation with homogeneous Dirichlet boundary conditions are proved on arbitrary three dimensional domains or any two dimensional domains which can be obtained by a limit of NTA domains caractarized by the same geometrical constants. The two dimensional result is obtained thanks to the Mosco convergence of the functionals corresponding to the weak formulations for the Westervelt equation with the homogeneous Dirichlet boundary condition. The non homogeneous Dirichlet condition is also treated in the class of admissible domains composed on Sobolev extension domains of $\mathbb{R}^n$ with a $d$-set boundary $n-1\leq d<n$ preserving Markov's local inequality.The obtained Mosco convergence also alows to approximate the solution of the Westervelt equation on an arbitrary domain by solutions on a converging sequence of domains without additional conditions on their boundary regularity in $\mathbb{R}^3$, or on a converging sequence of NTA domains in $\mathbb{R}^2$.
