Maximum principles and classical solutions for degenerate parabolic equations

Maximum principles, Harnack inequality for classical solutions

where the matrix (aij) is symmetric, and uniformly elliptic (aij) ≥ γI, for some γ > 0. In terms of regularity of the coefficients, let us assume the are continuous functions. As opposed to the interior and boundary regularity estimates, where we worked with integral quantities, here we work with pointwise estimates on the solution. The point is the following: assume u ∈ C2(Ω) attains a maximum...

Higher regularity for parabolic equations and maximum principles

Renormalized Entropy Solutions for Quasi-linear Anisotropic Degenerate Parabolic Equations

We prove the well-posedness (existence and uniqueness) of renormalized entropy solutions to the Cauchy problem for quasi-linear anisotropic degenerate parabolic equations with L1 data. This paper complements the work by Chen and Perthame [9], who developed a pure L1 theory based on the notion of kinetic solutions.

Strong Maximum Principles for Anisotropic Elliptic and Parabolic Equations

We investigate vanishing properties of nonnegative solutions of anisotropic elliptic and parabolic equations. We describe the optimal vanishing sets, and we establish strong maximum principles.

L1 Contraction for Bounded (Nonintegrable) Solutions of Degenerate Parabolic Equations

We will discuss new L1 contraction results for bounded entropy solutions of Cauchy problems for degenerate parablic equations. The equations we consider have possibly strongly degenerate diffusion terms. As opposed to previous results, our results apply without any integrability assumption on (the difference of) solutions. They take the form of partial Duhamel formulas and can be seen as quanti...