# Existence of the solution to PDE

Consider a system of PDEs given by: $$\begin{eqnarray} z_x=a(x,y) \text{ and } z_y=b(x,y), \end{eqnarray}$$ where $$a,b \in \mathcal{D}'(\mathbb{R} \times \mathbb{R}).$$

The condition $$a_y=b_x$$ in the sense of distribution, is necessary for the existence of the solution of the above PDE(in the sense of distribution). Is this condition sufficient or do we need something extra?

Also what is the corresponding result in higher dimensions?

• If $a, b$ are smooth then they define a 1-form $a dx + b dy$ which is closed iff exists a smooth $z$ satisfying your equations (this is also known as Poincarè lemma). The condition for being closed is $a_y -b_x = 0$. I don't know in the non smooth case. Commented Sep 2, 2022 at 11:50

Let $$\{a_i\}_{i=1}^k\subset \mathcal{D}'(\mathbb{R}^n)$$, $$1\leq k \leq n$$. Then there exists a distribution $$z\in \mathcal{D}'$$ for which $$z_{x_i}=a_i, \quad (1\leq i \leq k)$$ if and only if distributions $$a_i$$ satisfy relations $${a_i}_{x_j} = {a_j}_{x_i}$$ for $$1\leq i,j\leq k, \ i\neq j$$.