CRITICAL POINTS OF SOLUTIONS TO A KIND OF LINEAR ELLIPTIC EQUATIONS IN MULTIPLY CONNECTED DOMAINS

In this paper, we mainly study the critical points and critical zero points of solutions u to a kind of linear elliptic equations with nonhomogeneous Dirichlet boundary conditions in a multiply connected domain Omega in R-2. Based on the delicate analysis about the distributions of connected components of the super-level sets {x is an element of Omega : u(x) > t} and sub-level sets {x is an element of Omega: u(x) < t} for some t, we obtain the geometric structure of interior critical point sets of u. Precisely, let Omega be a multiply connected domain with the interior boundary gamma(I) and the external boundary gamma(E), where u vertical bar gamma(I) = psi(1) (x), u vertical bar gamma(E) = psi(2) (x). When psi(1) (x) and psi(2)(x) have N-1 and N-2 local maximal points on gamma(I) and gamma(E) respectively, we deduce that Sigma(k)(i=1) m(i) <= N-1 + N-2, where m(1),..., m(k) are the respective multiplicities of interior critical points x(1),..., x(k) of u. In addition, when min gamma(E) psi(2)(x) >= max gamma(I) psi(1)(x) and u has only N-1 and N-2 equal local maxima relative to (Omega) over bar on gamma(I) and gamma(E) respectively, we develop a new method to show that one of the following three results holds Sigma(k)(i=1) m(i) = N-1 + N-2 or Sigma(k)(i=1) m(i) + 1 = N-1 + N-2 or Sigma(k)(i=1) m(i) + 2 = N-1 + N-2. Moreover, we investigate the geometric structure of interior critical zero points of u. We obtain that the sum of multiplicities of the interior critical zero points of u is less than or equal to the half of the number of its isolated zero points on the boundaries.

Automated summary

In this paper, the critical points and critical zero points of solutions to a kind of linear elliptic equations with nonhomogeneous Dirichlet boundary conditions in a multiply connected domain are studied. The geometric structures of interior critical point sets and interior critical zero point sets are obtained based on the analysis of the distributions of connected components of certain level sets. Some theorems regarding the structural properties are presented as well.