β Scribed by E.N. Dancer; D. Daners
No coin nor oath required. For personal study only.
We prove domain perturbation theorems for linear and nonlinear elliptic equations under Robin boundary conditions. The theory allows very singular perturbation of domains. In particular, it includes cutting holes, parts degenerating to a set of measure zero such as the dumbbell problem, or wildly oscillating boundaries. In the last case we show that the limiting problem is the Dirichlet problem. 1997 Academic Press behave as the domains 0 n approach an open bounded set 0. Here, 2 is the Laplace operator, & the outer unit normal to the boundary 0 n of 0 n and ; 0 >0 a constant. Unlike in the case of Dirichlet or Neumann boundary conditions the boundary conditions in the limit may be different from the original ones. This depends on how the domains 0 n approach 0. We shall prove three perturbation results, each featuring a different behaviour when passing to the limit.
First, we suppose that the boundary of the original problem is only modified in the neighbourhood of a very small set, that is, a set of capacity zero. Examples of that type include rounding off corners, cutting or drilling small holes of any shape. It also includes examples where part of a domain degenerates such as in the dumbbell problem, where the handle shrinks to a line as in Fig. 1. The limiting problem to consider is (1.1) on the domain article no.
π SIMILAR VOLUMES
## Abstract We obtain the __L__~__p__~β__L__~__q__~ maximal regularity of the Stokes equations with Robin boundary condition in a bounded domain in β^__n__^ (__n__β©Ύ2). The Robin condition consists of two conditions: __v__ β __u__=0 and Ξ±__u__+Ξ²(__T__(__u__, __p__)__v__ β γ__T__(__u__, __p__)__v__,