Partial differential equations Definition

Local solvability

Builds on Linear PDE Bump function Smooth function

Definition

Let $\Omega \subseteq \R^n$ be open and let $P$ be a linear differential operator on $\Omega$ with smooth complex coefficients. A distribution on an open set $U$ is a continuous linear functional on the space of test functions on $U$ . The operator $P$ is locally solvable at $x_0 \in \Omega$ if there is an open neighbourhood $U \subseteq \Omega$ of $x_0$ with the following property. For every smooth function $f$ on $\Omega$ there is a distribution $u$ on $U$ with $P u = f$ in $U$ , in the sense of distributions. It is locally solvable in $\Omega$ if it is locally solvable at every point of $\Omega$ .

Remarks

Example: $P = \partial_1$ on $\R^n$ is locally solvable at every point: $u(x) = \int_0^{x_1} f(s, x_2, \dots, x_n) \, ds$ is smooth and satisfies $\partial_1 u = f$ . More generally, every nonzero operator with constant coefficients is locally solvable, by the Malgrange–Ehrenpreis theorem. Non-example: Lewy's operator $L = \partial_x + i \partial_y - 2 i (x + i y) \partial_t$ on $\R^3$ with variables $(x, y, t)$ . There is a smooth function $f$ on $\R^3$ such that $L u = f$ has no distribution solution in any open subset of $\R^3$ , so $L$ is locally solvable at no point. Some texts state the definition with slightly different classes of right-hand sides or of solutions.

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Bump function concept
Linear PDE concept
Smooth function concept
Local solvability definition

Tag	`DQ0TSVK7ZT`
URI	`m2://DQ0TSVK7ZT`
Kind	definition
Writing	W0 stub
MSC	35A01, 35A27
Defines	Local solvability
Uses	Linear PDE, Bump function, Smooth function
Changed	2 November 2026

Local solvability

Definition

Remarks

Reading path

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