# nLab distributional solution of a PDE

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

Given a linear differential operator $P$, then a generalized solution or weak solution of the corresponding homogenous differential equation is a distribution $u$ (hence a “generalized function” via this prop) satisfying

$P u = 0 \,,$

where the differential operator on the left now acts via derivatives of distributions. This means that for all $b \in \mathcal{D}$ we have

$u(P^\ast(b)) = 0 \,,$

where $P^\ast$ is the formally adjoint differential operator.

This is such that in the case that $u = u_f$ happens to be a non-singular distribution given by an ordinary smooth function $f$, then $u_f$ is a generalized solution precisely if $f$ is an ordinary solution:

\begin{aligned} P u_f = 0 \;\; & \Leftrightarrow\;\; u_f(P^\ast b) = 0 \phantom{AAA}\text{for all}\, b \in \mathcal{D} \\ & \Leftrightarrow\;\; \int f P^\ast b \, dvol = 0 \phantom{AAA}\text{for all}\, b \in \mathcal{D} \\ & \Leftrightarrow\;\; \int (P f) b \, dvol = 0 \phantom{AAA}\text{for all}\, b \in \mathcal{D} \\ & \Leftrightarrow\;\; P f = 0 \end{aligned}

Similarly there are generalized solutions for the inhomogeneous equation, and in fact now the inhomogeneity may itself be a distribution. In particular for delta distribution-inhomogeneity

$P u = \delta$

the generalized solutions $u$ are the fundamental solutions or Green's functions of $P$.

## References

• Ram Kanwal, section 10.2 of Generalized Functions: Theory and Applications, Springer 2004