nLab
distributional solution of a PDE

Contents

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

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

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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 }

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          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

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