Completeness theorems related to BVPs satisfying the Lopatinskii condition for higher order elliptic equations

Abstract

In this paper, we consider a linear elliptic operator $E$ with real constant coefficients of order $2m$ in two independent variables without lower order terms. For this equation, we consider linear BVPs in which the boundary operators $T_1,\ldots,T_m$ are of order $m$ and satisfy the Lopatinskii-Shapiro condition with respect to $E$. We prove boundary completeness properties for the system $\{(T_1\omega_k,\ldots, T_m\omega_k)\}$, where $\{\omega_k\}$ is a system of polynomial solutions of the equation $Eu=0$.

Keywords

• Completeness theorems
• Lopatinskii condition
• Elliptic equations of higher order
• Partial differential equations with constant coefficients

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