Elliptic partial differential equations for real analytic functions (Q1968584)

The problem of existence of a continuous linear right inverse operator for a given partial differential operators is studied. Suppose \(K\subset\mathbb{R}^N\) is a compact convex set with nonempty interior, \(\partial K\) is the boundary of \(K,\) \(A(K)\) is the space of all real analytic functions on \(K\) and the partial differential operator \(P(D): A(K)\to A(K)\) has the principal part \(P_M(z)\) of the characteristic polynomial. The following three theorems are proved. I. If \(\partial K\in C^{1,1}\) then the nonzero elliptic operator \(P(D)\) admits a continuous linear right inverse one. II. If the elliptic polynomial satisfies the condition \(P_m(z)=0\), \(z\in\mathbb{C}^N\setminus\{0\}\Rightarrow \langle z,\nabla P_m(z)\rangle\neq 0\) then \(P(D)\) admits a continuous linear right inverse operator if and only if \(\partial K\in C^{1,1}\). III. For \(N=2\) a nonzero constant coefficients operator \(P(D)\) admits a continuous linear right inverse if and only if \(\partial K\in C^{1,1}\) or the zeros of \(P_m(z)\) are contained in \(C\mathbb{R}^2=\{\zeta x \mid \zeta\in\mathbb{C}\), \(x\in \mathbb{R}^2 \}\). The theory of extremal plurisubharmonic functions is used.