07-18
Stability of the positivity of biharmonic Green\'s functions under perturbations of the domain
Preprint series: 07-18, Preprints
- MSC:
- 35J40 Boundary value problems for higher-order, elliptic equations
- 35B50 Maximum principles
Abstract: In general, higher order elliptic equations and boundary value problems like the biharmonic equation or the linear clamped plate boundary value problem do not enjoy neither a maximum principle nor a comparison principle or -- equivalently -- a positivity preserving property. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem {from} being written as a system of second order boundary value problems. On the other hand, the biharmonic Green\'s function in balls $B\subset\mathbb{R}^n$ under Dirichlet (i.e. clamped) boundary conditions is known explicitly and is positive. Previously it was shown that this property also remains under small regular perturbations of the domain, if $n=2$. In the present paper, such a stability result is proved for $n\ge 3$.
Keywords: biharmonic Green\'s function, positivity, domain perturbations, dimension > 2
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