A priori estimate

Remarks

Example: let $\Omega \subseteq \R^n$ be bounded and open, $f \in L^2(\Omega)$, and let $u \in H^1_0(\Omega)$ be a weak solution of $-\Delta u = f$. Testing with $u$ gives $\int_\Omega \abs{\nabla u}^2 = \int_\Omega f u \le \norm{f}_{L^2} \norm{u}_{L^2}$. By the Poincaré inequality, $\norm{u}_{L^2} \le C_P \norm{\nabla u}_{L^2}$, hence $\norm{\nabla u}_{L^2} \le C_P \norm{f}_{L^2}$. The constant depends only on $\Omega$.

Non-example: a bound $\norm{u}_X \le C(u)$ with a constant that depends on the particular solution is not an a priori estimate. It gives no uniform control over a family of solutions.

The Schauder estimates and Calderón–Zygmund estimates are a priori estimates in Hölder and $L^p$ norms. Combined with the Leray–Schauder fixed point method or the method of continuity, a priori estimates yield existence.