nLab
distributional solution of a PDE

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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 }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Functional analysis

Contents

Idea

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.

References

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

Revised on November 23, 2017 09:27:03 by Urs Schreiber (94.220.49.59)