Viscosity supersolutions of the evolutionary $ p$-Laplace Equation

Juan J. Manfredi
University of Pittsburgh

Abstract: Consider the parabolic $ p$-Laplacian equation

$\displaystyle \frac{\partial v}{\partial t} = \nabla \cdot (\vert\nabla v\vert^{p-2}\nabla v)$    

We study the regularity of the viscosity supersolutions and their spatial gradients. We give a new proof of the existence of $ \nabla v$ in Sobolev's sense and of the validity of the equation

$\displaystyle \int\!\!\!\!\int_{\Omega} \biggl(-v \frac{\partial \varphi}{\part...
...ert\nabla v\vert^{p-2} \nabla v, \ \nabla \varphi\rangle \biggr) dx \ dt \geq 0$    

for all test functions $ \varphi \geq 0$. Here $ \Omega$ is the underlying domain in $ {\mathbb{R}}^{n+1}$ and $ v$ is a bounded viscosity supersolution in $ \Omega$. The first step of our proof is to establish the above inequality for the so-called infimal convolution $ v_{\epsilon}$. The function $ v_{\epsilon}$ has the advantage of being differentiable with respect to all its variables $ x_{1}, x_{2}, \cdots, x_{n}$, and $ t$, while the original $ v$ is merely lower semicontinuous to begin with. The second step is to pass to the limit as $ \epsilon \rightarrow 0$. It is clear that $ v_{\epsilon} \rightarrow v$ but it is delicate to establish a sufficiently good convergence of the $ \nabla v_{\epsilon}$'s.

This is joint work with Peter Lindqvist at Trondheim (Norway).