Caccioppoli inequality

A Caccioppoli inequality is a local energy estimate obtained by testing an elliptic equation with a cutoff times the solution. It controls the gradient on a smaller ball by the function and forcing on a larger ball.

For $\mathrm{div}(ADu)+b\cdot Du+cu=f$ with bounded measurable coefficients and uniformly elliptic $A$, testing with ${\eta }^{2}u$ and using Young inequality gives $\int {\eta }^2|Du{|}^2\le C\int (1+|D\eta {|}^2)u^2+C\int {\eta }^2{f}^2$. No sign condition on $c$ is needed for this interior estimate. A cutoff equal to one on $U$ and supported in $V$ yields the displayed bound; the constant depends only on dimension, ellipticity, coefficient bounds and the separation of $U$ from the boundary of $V$.

Solutions of one fixed bounded-coefficient uniformly elliptic equation $Lu=f$ that are bounded in global $L^2$ have a subsequence converging strongly in $H^1$ on every relatively compact subdomain. Local energy bounds and the Rellich-Kondrachov compactness theorem first give local strong $L^2$ and weak $H^1$ convergence on a smooth exhaustion. Pass to the weak equation to identify the limit. Differences solve $L(u_k-u)=0$, so the interior Caccioppoli inequality upgrades the local strong $L^2$ convergence to strong gradient convergence. Global weak $L^2$ convergence preserves the original global bound.

For a weak solution of a uniformly elliptic homogeneous divergence-form elliptic operator, testing with ${\eta }^2(u-c)$ and choosing the gradient of the cutoff function in $B_{2R}\setminus B_R$ gives ${\int }_{B_R}|\nabla u{|}^2\le C{R}^{-2}{\int }_{B_{2R}\setminus B_R}|u-c{|}^2$. The arbitrary constant $c$ can be chosen as an annular average, which makes the estimate suitable for a hole-filling argument.

For a nonnegative bounded weak subsolution and $\beta >1$, test with ${\eta }^2(u+\epsilon {)}^{\beta -1}$. Uniform ellipticity and Cauchy-Schwarz inequality give $\int |Du{|}^2{u}^{\beta -2}{\eta }^2\le 4{\Lambda }_{\ast }^2/[{\lambda }^2(\beta -1{)}^2]\int u^{\beta }|D\eta {|}^2$, with ${\Lambda }_{\ast }$ an operator norm bound for the coefficient matrix. The Sobolev chain rule and $\epsilon \downarrow 0$ justify fractional powers and zero values. An entrywise bound $\Lambda$ yields ${\Lambda }_{\ast }\le n\Lambda$.

For a nonnegative $H^1$ weak solution of a homogeneous divergence-form elliptic equation on $B_1\subset {\mathbb{R}}^{\ell }$, $\ell \ge 2$, with symmetric measurable $\lambda I\le A\le \Lambda I$, set $R=\Lambda /\lambda$ and $\alpha =1+1/\ell$. There is $D_{\ell }\ge 1$ independent of $q\ge 2$ such thatTest the equation with a cutoff times $u^{q-1}$, first regularized or truncated, to get $\int {\zeta }^2|\nabla u^{q/2}{|}^2\le [q/(q-1){]}^2{R}^2\int u^q|\nabla \zeta {|}^2$. Apply a fixed-domain Sobolev embedding theorem to $\zeta u^{q/2}$ and take the $2/q$ power. Keeping the constant inside that power is crucial for Moser iteration.

For a positive weak supersolution $u$ of a uniformly elliptic divergence-form equation, testing with ${\eta }^2/u$ givesThe constant depends only on the ellipticity bounds.