Eigenfunctions and harmonic functions in convex and concave domains. (English) Zbl 0851.35008

Jerison, David

Chatterji, S. D. (ed.), Proceedings of the international congress of mathematicians, ICM ’94, August 3-11, 1994, Zürich, Switzerland. Vol. II. Basel: Birkhäuser. 1108-1117 (1995).

Summary: Let \(\Omega\) be a bounded, convex, open subset of \(\mathbb{R}^N\). This paper concerns the behavior of positive harmonic functions that vanish on \(\partial\Omega\). We consider both the case in which the function is defined in \(\Omega\) and the case in which the function is defined in the complement of \(\Omega\). We also discuss eigenfunctions defined in \(\Omega\). The theme is to study how the shape of \(\Omega\) influences the size of solutions to these basic elliptic equations. The simplest measure of the shape of \(\Omega\) is its eccentricity, which we define as \[ \text{ecc } \Omega= \text{diameter }\Omega/\text{inradius } \Omega. \] Our estimates will fall into two categories, global and local estimates. The global estimates are those estimates on the size of solutions that are uniform as the eccentricity tends to infinity. The local estimates are those estimates that are valid when the eccentricity is bounded above. Local estimates are valuable only because they are valid uniformly up to the boundary. One main focus will be on the normal derivative of the solution at the boundary and its interaction with the Gauss curvature of the boundary.
For the entire collection see [Zbl 0829.00015].

MSC:

35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25	Boundary value problems for second-order elliptic equations
35P05	General topics in linear spectral theory for PDEs

Keywords:

positive harmonic functions; estimates on the size of solutions