In this paper we study the classical external Bernoulli problem set in an annular domain $\Omega$ of the plain.\\ We focus on the curvature of the free boundary $\Gamma$ (outer component of the boundary of our domain) and establish a one-to-one correspondence between positive/negative curvature arcs of $\Gamma$ and of the curve $\gamma$ representing the data, extending a method pushed forward by A. Acker. Moreover we show that the positive curvature arcs on the free boundary bend less than the corresponding arcs on the inner curve, \ie{} the maximum attained by the curvature is greater on the $\gamma$ than on $\Gamma$. Thus we can draw the following conclusions: the geometry of $\Gamma$ is simpler than that of $\gamma$ (an already known result); the shape of $\Gamma$ is alleviated with respect to that of $\gamma$.
On the curvature of free boundaries with a Bernoulli-type condition / R. MAGNANINI; S. CECCHINI. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - STAMPA. - 68:(2008), pp. 940-950.
On the curvature of free boundaries with a Bernoulli-type condition
MAGNANINI, ROLANDO;
2008
Abstract
In this paper we study the classical external Bernoulli problem set in an annular domain $\Omega$ of the plain.\\ We focus on the curvature of the free boundary $\Gamma$ (outer component of the boundary of our domain) and establish a one-to-one correspondence between positive/negative curvature arcs of $\Gamma$ and of the curve $\gamma$ representing the data, extending a method pushed forward by A. Acker. Moreover we show that the positive curvature arcs on the free boundary bend less than the corresponding arcs on the inner curve, \ie{} the maximum attained by the curvature is greater on the $\gamma$ than on $\Gamma$. Thus we can draw the following conclusions: the geometry of $\Gamma$ is simpler than that of $\gamma$ (an already known result); the shape of $\Gamma$ is alleviated with respect to that of $\gamma$.File | Dimensione | Formato | |
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