Inequalities, which are essentially saying, that solutions of elliptic PDEs cannot oscillate too much. To be more precise, given a solution \(u\) of an elliptic equation on a domain \(\Omega\), and \(U\subset V\subset \Omega\) denote proper subsets, then

\[ \lVert u\rVert_V \lesssim \lVert u\rVert_U^\delta \lVert u\rVert_\Omega^{1-\delta}, \]

for a suitable \(\delta\in (0,1)\). However, the concrete norms and domains vary from case to case.

Links Link to heading

Inequalities for harmonic functions Link to heading

• \(L^\infty\)-three balls inequality by Korevaar and Meyers

Inequalities for elliptic PDEs with Lipschitz coefficients Link to heading

• \(L^2\)-version by Alessandrini et al.

Inequalities for elliptic PDEs with piecewise Lipschitz coefficients Link to heading

• \(L^2\)-version on parabolic domains with a discontinuity on a hyperplane by Francini et al.
• \(L^2\)-version on bounded domains with a discontinuity on a hypersurface by Cârstea and Wang