Global Gradient Estimates for a General Type of Nonlinear Parabolic Equations

Abstract

We provide global gradient estimates for solutions to a general type of nonlinear parabolic equations, possibly in a Riemannian geometry setting. Our result is new in comparison with the existing ones in the literature, in light of the validity of the estimates in the global domain, and it detects several additional regularity effects due to special parabolic data. Moreover, our result comprises a large number of nonlinear sources treated by a unified approach, and it recovers many classical results as special cases.

A Proof of (3.7)

We recall that a metric $g_{ij}$ is said to be conformal (or, more precisely, conformal to the Euclidean metric) if

$$\begin{aligned} g_{ij}=\varphi \delta _{ij}, \end{aligned}$$

(A.19)

for some scalar factor $\varphi$. For instance, the metric of the Poincaré disk in the plane given in (3.6) is conformal, with factor $\varphi := \frac{4\lambda ^2}{ (1-|x|^{2})^{2}}$, being $|\cdot |$ the standard Euclidean norm.

The Laplacian operator (or, more precisely, the Laplace-Beltrami operator) possesses an explicit representation with respect to conformal metrics: roughly speaking, since conformal metrics preserve angles, an infinitesimal orthonormal frame is transformed into an infinitesimal orthogonal frame (the length of the vectors possibly being affected by the conformal factor $\varphi$), thus the new Laplacian (being computed as sum of second derivatives with respect to an orthonormal frame) remains the same possibly up to a “curvature” term which accounts for the variation of $\varphi$ (this additional term pops up because the Laplacian is a second order operator). The case of dimension 2 is somewhat special, since this additional term vanishes.

Here are the explicit computations underpinning this heuristic idea. From (A.19), we have that $g^{ij}=\varphi ^{-1}\delta ^{ij}$ and $\det g=\varphi ^{n}$. Hence, in local coordinates, the Laplacian with respect to the conformal metrics in (A.19) is

$$\begin{aligned}&\frac{1}{\sqrt{\det g}} \sum _{i,j=1}^n \partial _i\Big ( \sqrt{\det g} \;g^{ij} \partial _j\Big )= \frac{1}{\varphi ^{\frac{n}{2}}} \sum _{i,j=1}^n \partial _i\Big ( \varphi ^{\frac{n-2}{2}} \,\delta ^{ij} \partial _j\Big )= \frac{1}{\varphi ^{\frac{n}{2}}} \sum _{i=1}^n \partial _i\Big ( \varphi ^{\frac{n-2}{2}} \partial _i\Big )\\&\qquad \qquad = \frac{1}{\varphi ^{\frac{n}{2}}} \sum _{i=1}^n \left( \frac{n-2}{2}\,\varphi ^{\frac{n-4}{2}} \partial _i\varphi \, \partial _i +\varphi ^{\frac{n-2}{2}} \partial _{ii}\right) =\sum _{i=1}^n \left( \frac{n-2}{2\,\varphi ^2} \partial _i\varphi \,\partial _i +\frac{1}{\varphi } \partial _{ii}\right) . \end{aligned}$$

In dimension 2, this boils down to

$$\begin{aligned} \frac{1}{\varphi } \,\sum _{i=1}^n \partial _{ii}, \end{aligned}$$

which is a scalar multiple of the Euclidean Laplacian, and therefore (3.7) plainly follows.