Lectures on parabolic equations of the p-Laplacian type
Juha Kinnunen, Aalto University, Finland
Lecture 1: Solutions to parabolic equations of the p-Laplacian type are usually taken in a weak sense and they are assumed to belong to an appropriate Sobolev space. We discuss the definition of weak solution both from the PDE and variational point of views. We review the comparison principle, energy estimates, Harnack type estimates, Holder continuity and other basic properties for weak solutions.
Lecture 2: We discuss higher integrability results which assert that weak solutions belong to a slightly higher Sobolev space than the usual one. The strategy of the proof of local higher integrability results is to combine energy estimates with Sobolev integrability of the gradient after covering arguments and estimates on the distribution function of the gradient. On the other hand, the weak formulation of the PDE makes sense under the assumption that a solution belongs to a slightly lower Sobolev space than the usual one. These functions are called very weak solutions. We shall discuss the result that very weak solutions are weak solutions and thus possess the usual regularity properties as boundedness, Holder continuity and higher integrability. In short, it is possible to pass from an exponent below to an exponent which is above the natural Sobolev exponent. The main challenge in dealing with a very weak solution is that the solution times a cutoff function cannot be used as a test function in the weak formulation of the equation. Suitable test functions are constructed by using the Whitney extension theorem to extend the function off the set where a certain maximal function of the gradient is bounded. This parabolic Lipschitz truncation method has become a standard tool in nonlinear parabolic problems.
Lecture 3: We review potential theoretic aspects of degenerate parabolic equations of the p-Laplacian type. Solutions form a similar basis for a nonlinear parabolic potential theory as the solutions of the heat equation do in the classical theory. In the parabolic potential theory, the so-called supercaloric functions are essential. Supercaloric functions arise naturally in the study of the ZKB solutions (from Zeldovich, Kompaneets and Barenblatt) and measure data problems. Supercaloric functions can be divided into two mutually exclusive classes determined by the two prototype solutions, the ZKB solution and the friendly giant. They are supercaloric functions but not weak supersolutions since they fail to belong to the usual Sobolev space. The ZKB solution is also a weak solution to the initial value problem with Dirac’s delta as a source term and it possesses the moving boundary phenomenon. Furthermore, it is an example of fundamental solution that blows up at a single point. The friendly giant is an example of solution that does not solve measure data problem for any finite Radon measure and moreover, it blows up everywhere in some instant of time. We discuss existence, structural, convergence and Sobolev space properties of supercaloric functions. We also consider the definition and properties of the nonlinear parabolic capacity and show that the infinity set of a supercaloric function is of zero capacity.