distributional solution of a PDE


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from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Functional analysis



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

Pu=0, P u = 0 \,,

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

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

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

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

Pu f=0 u f(P *b)=0AAAfor allb𝒟 fP *bdvol=0AAAfor allb𝒟 (Pf)bdvol=0AAAfor allb𝒟 Pf=0 \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

Pu=δ P u = \delta

the generalized solutions uu are the fundamental solutions or Green's functions of PP.


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

Last revised on November 23, 2017 at 09:27:03. See the history of this page for a list of all contributions to it.