Divergence-form elliptic operator ($Lu=D_i(a^{ij}{D}_{j}u)$)

A divergence-form elliptic operator is $Lu=D_i(a^{ij}{D}_{j}u)$, with bounded measurable coefficients whose symmetric part has a uniform positive lower bound. Its weak formulation involves only first weak derivatives, so differentiability of the coefficients is unnecessary.

On an interval, the weak equation $(a{u}^{\prime }{)}^{\prime }=0$ makes $a{u}^{\prime }$ constant almost everywhere. If $\lambda \le a\le \Lambda$, the derivative energy on concentric subintervals satisfies $E(r)\le (\Lambda /\lambda {)}^2(r/R)E(R)$. This supplies dyadic energy decay even though a one-dimensional annulus is disconnected.

If $Lu=\mathrm{div}F+g$ and $L$ contains $\mathrm{div}(bu)+du$, then $L(u-k)=\mathrm{div}(F-kb)+(g-kd)$. Thus shifting a weak solution to make it nonnegative changes the forcing unless the operator annihilates constants. This bookkeeping is essential in applying the Weak Harnack inequality to upper and lower oscillations.

For a bounded measurable coefficient matrix $A$, an $H^1$ weak solution of $\mathrm{div}(A\nabla u)=0$ satisfies $\int A\nabla u\cdot \nabla \phi =0$ for every compactly supported smooth test function. By density this identity extends to compactly supported $H^1$ tests. If $A$ is symmetric and $\lambda I\le A\le \Lambda I$, Caccioppoli inequalities follow without differentiating $A$.

For $Lu=D_i(a^{ij}{D}_{j}u)$, a weak supersolution satisfies $Lu\le 0$ as a distribution, equivalently $\int a^{ij}{D}_{j}u{D}_i\varphi \ge 0$ for every nonnegative compactly supported test function $\varphi$. The Weak Harnack inequality applies to nonnegative weak supersolutions with this sign convention.

For the sign convention $Lu=D_i(a^{ij}{D}_{j}u)$, a weak subsolution satisfies $Lu\ge 0$ as a distribution, equivalently $\int a^{ij}{D}_{j}u{D}_i\varphi \le 0$ for every nonnegative compactly supported test function $\varphi$. A weak supersolution has the reversed inequality.

For a uniformly elliptic divergence-form elliptic operator with lower-order coefficients in the supercritical Lebesgue spaces, truncation tests and Moser iteration bound the positive part of a weak subsolution on an interior ball by its Lp norm on a larger ball and the forcing norms. In particular, every weak subsolution of the homogeneous equation is locally bounded above. This is the preliminary estimate needed when a strong maximum principle is phrased using essential suprema.