Distributional solution

A distributional solution or solution in the distributional sense is a distribution that has the same property for partial differential equations as the regular distribution generated by a classical solution . It generalizes classic solutions in the sense that for every classic PDGL solution, the associated distribution is a distributional solution. ${\displaystyle u}$ ${\displaystyle u}$ ${\displaystyle T_{u}}$

definition

Be a distribution, and ${\displaystyle T}$ ${\displaystyle f\in C(\mathbb {R} ^{n},\mathbb {R} )}$

${\displaystyle L=\sum _{|\alpha |\leq k}a_{\alpha }(x){\frac {\partial ^{\alpha }}{\partial x^{\alpha }}}}$

a linear partial differential operator . is called a distributional solution of the partial differential equation if and only if: ${\displaystyle T}$ ${\displaystyle Lu=f}$

${\displaystyle LT=T_{f}}$

In this equation, a linear partial differential operator on distributions is understood. ${\displaystyle L}$

If there is a distributional solution of , and there is a locally integrable function , so that (ie if there is a regular distribution that is generated by), then the function is sometimes also called (instead of ) a distributional solution. ${\displaystyle T}$ ${\displaystyle LT=T_{f}}$ ${\displaystyle u\colon \mathbb {R} ^{n}\to \mathbb {R} }$ ${\displaystyle T=T_{u}}$ ${\displaystyle T}$ ${\displaystyle u}$ ${\displaystyle u}$ ${\displaystyle T_{u}}$

properties

Distributional solution from a classic solution

Is a classic solution from ${\displaystyle u}$

${\displaystyle Lu=f}$,

so the regular distribution generated by it is always a distributional solution. ${\displaystyle u}$ ${\displaystyle T_{u}}$

Classic solution from a distribution solution

Be , ie is a Green function for . If - times is continuously differentiable and - times continuously differentiable, and if then , where is the level of the highest derivative that occurs in, then is ${\displaystyle LT_{{\tilde {G}}(x-\xi )}=\delta _{\xi }}$ ${\displaystyle {\tilde {G}}(x-\xi )}$ ${\displaystyle Lu=f}$ ${\displaystyle f}$ ${\displaystyle m}$ ${\displaystyle {\tilde {G}}}$ ${\displaystyle n}$ ${\displaystyle n+m\geq k}$ ${\displaystyle k}$ ${\displaystyle L}$

${\displaystyle u(x)={\tilde {G}}*f}$

a classic solution from

${\displaystyle Lu=f}$.

literature

• Lars Hörmander : The Analysis of Linear Partial Differential Operators. Volume 1: Distribution Theory and Fourier Analysis. Second edition. Springer-Verlag, Berlin et al. 1990, ISBN 3-540-52345-6 ( basic teaching of mathematical sciences 256).
• Werner: functional analysis, chapter on locally convex spaces