# Confusion about weak solution and distributions

Let $$f \in L^2(\Omega)$$ for some smooth domain $$\Omega \subset \mathbb{R}^n$$, and let $$u \in H^1_0(\Omega)$$ such that $$\sum_{i=1}^{n} \int_\Omega \partial_{x_i}u \partial_{x_i}v = \int_\Omega fv$$ for all $$v \in H^1_0(\Omega)$$.

Then I have questions about the following statements.

1) $$-\Delta u = f$$ holds in distributional sense.

Does this mean that if $$T_u$$ denotes the distribution "generated" by $$u$$ via

$$\langle T_u, \phi \rangle = \int_\Omega u \phi$$ for smooth test function $$\phi$$, then the (negative) distributional Laplacian of $$u$$, $$-\Delta T_u$$ is the same distribution as the distribution $$T_f$$ generated by $$f$$ as

$$\langle T_f,\phi\rangle = \int_{\Omega}f\phi?$$ $$\qquad$$

2) Since $$f\in L^2(\Omega)$$, the equation is $$-\Delta u = f$$ is true in $$L^2(\Omega)$$, hence almost everywhere in $$\Omega$$.

What does this statement mean?

All I know from 1) is that $$-\Delta T_u$$ and $$T_f$$ are the same mappings on the set of test functions $$\phi$$, but this statement seems to say that the pointwise negative Laplacian of $$u$$ is defined almost everywhere and square integrable, and equal to $$f$$. How is $$-\Delta u$$ defined for almost every $$x \in \Omega$$?

If $$u$$ has ordinary second order derivatives and $$\phi$$ is a test function, then $$\langle T_{-\Delta u}, \phi \rangle = \int_\Omega (-\Delta u) \phi = \int_\Omega Du \cdot D\phi.$$ The last integral makes sense even if $$u$$ belongs only to $$H^1(\Omega)$$, and the distribution $$T$$ defined by $$\langle T,\phi \rangle = \int_\Omega Du \cdot D\phi$$ is called the distributional Laplacian of $$u$$. If it happens that $$\int_\Omega Du \cdot D\phi = \int_\Omega f\phi$$ for all test functions $$\phi$$ then you have $$T = T_f$$. This is what is meant by $$f = -\Delta u$$ in the distributional sense.
Your second question requires some regularity theory to ensure that $$u$$ has second order weak derivatives.
• Am I right that your saying $u$ having ordinary second order derivatives can be replaced by $u \in H^2(\Omega)$?