Geodesic
On Friedrichs inequalities for a disk
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 2, pp. 48-61 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the Friedrichs inequality for functions defined on a disk of unit radius $\Omega$ and equal to zero on almost all boundary except for an arc $\gamma_\varepsilon$ of length $\varepsilon$, where $\varepsilon$ is a small parameter. Using the method of matched asymptotic expansions, we construct a two-term asymptotics for the Friedrichs constant $C(\Omega,\partial\Omega\backslash\overline\gamma_\varepsilon)$ for such functions and present a strict proof of its validity. We show that $C(\Omega,\partial\Omega\backslash\overline\gamma_\varepsilon)=C(\Omega,\partial\Omega)+\varepsilon^2C(\Omega,\partial\Omega)(1+O(\varepsilon^{2/7}))$. The calculation of the asymptotics for the Friedrichs constant is reduced to constructing an asymptotics for the minimum value of the operator $-\Delta$ in the disk with Neumann boundary condition on $\gamma_\varepsilon$ and Dirichlet boundary condition on the remaining part of the boundary.
Keywords: Friedrichs inequality, small parameter, eigenvalue, asymptotics. 
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R. R. Gadyl'shin; E. A. Shishkina. On Friedrichs inequalities for a disk. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 2, pp. 48-61. http://geodesic.mathdoc.fr/item/TIMM_2012_18_2_a4/

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