Editing Partial differential equation (section)

== Weak solutions ==
{{Main|Weak solution}}
Weak solutions are functions that satisfy the PDE, yet in other meanings than regular sense. The meaning for this term may differ with context, and one of the most commonly used definitions is based on the notion of [[Schwartz distribution|distributions]].

An example{{sfn|Evans|1998|loc=chpt. 6. Second-Order Elliptic Equations}} for the definition of a weak solution is as follows: 

Consider the boundary-value problem given by:
<math display="block">\begin{align}
Lu&=f \quad\text{in }U,\\ 
u&=0 \quad \text{on }\partial U,
\end{align}
</math> 
where <math display="block">Lu=-\sum_{i,j}\partial_j (a^{ij}\partial_i u)+\sum_{i}b^{i}\partial_i u + cu </math> denotes a second-order partial differential operator in '''divergence form'''. 

We say a [[Sobolev space|<math>u\in H_{0}^{1}(U)</math>]] is a weak solution if
<math display="block">\int_{U} [\sum_{i,j}a^{ij}(\partial_{i}u)(\partial_{j}v)+\sum_{i}b^i (\partial_{i}u) v +cuv]dx=\int_{U} fvdx</math> for every <math>v\in H_{0}^{1}(U)</math>, which can be derived by a formal integral by parts.

An example for a weak solution is as follows:
<math display="block">\phi(x)=\frac{1}{4\pi} \frac{1}{|x|} </math> is a weak solution satisfying 
<math display="block"> \nabla^2 \phi=\delta 
 \text{ in }R^3</math> in distributional sense, as formally, <math display="block"> \int_{R^3}\nabla^2 \phi(x)\psi(x)dx=\int_{R^3} \phi(x)\nabla^2 \psi(x)dx=\psi(0)\text{ for }\psi\in C_{c}^{\infty}(R^3).</math>