The augmented operator of a surjective partial differential operator with constant coefficients need not be surjective

Abstract

For $d\ge 3$ we give an example of a constant coefficient surjective differential operator $P(D):{\mathcal{D}}^{\prime }(X)\to {\mathcal{D}}^{\prime }(X)$ over some open subset $X\subset {\mathbb{R}}^d$ such that $P^{+}(D):{\mathcal{D}}^{\prime }(X\times \mathbb{R})\to {\mathcal{D}}^{\prime }(X\times \mathbb{R})$ is not surjective, where $P^{+}(x_1,...,x_{d+1}):=P(x_1,...,x_d)$. This answers in the negative a problem posed by Bonet and Domański in \cite[Problem 9.1]{Bonet}.

References in corpus (1)

• Every P-convex subset of ${\mathbb{R}}^2$ is already strongly P-convex

Cited by in corpus (4)

• Surjectivity of differential operators and linear topological invariants for spaces of zero solutions
• Linear topological invariants for kernels of convolution and differential operators
• An approximation theorem of Runge type for kernels of certain non-elliptic partial differential operators
• Quantitative Runge type approximation theorems for zero solutions of certain partial differential operators