Perron's method for the porous medium equation

O. Perron introduced his celebrated method for the Dirichlet problem for harmonic functions in 1923. The method produces two solution candidates for given boundary values, an upper solution and a lower solution. A central issue is then to determine when the two solutions are actually the same function. The classical result in this direction is Wiener’s resolutivity theorem: the upper and lower solutions coincide for all continuous boundary values. We discuss the resolutivity theorem and the related notions for the porous medium equation ut −∆u = 0

A remark on infinite initial values for quasilinear parabolic equations

Abstract We study the possibility of prescribing infinite initial values for solutions of the Evolutionary p -Laplace Equation in the fast diffusion case p > 2 . This expository note has been extracted from our previous work. When infinite values are prescribed on the whole initial surface, such solutions can exist only if the domain is a space–time cylinder.

Irregular Time Dependent Obstacles

Abstract We study the obstacle problem for the Evolutionary p-Laplace Equation when the obstacle is discontinuous and does not have regularity in the time variable. Two quite different procedures yield the same solution.

A theorem of Radò’s type for the solutions of a quasi-linear equation

The ∞-Eigenvalue Problem

. The Euler‐Lagrange equation of the nonlinear Rayleigh quotient \( \left(\int_{\Omega}|\nabla u|^{p}\,dx\right) \bigg/ \left(\int_{\Omega}|u|^{p}\,dx\right)\) is \( -\div\left( |\nabla u|^{p-2}\nabla u \right)= \Lambda_{p}^{p} |u |^{p-2}u,\) where \(\Lambda_{p}^{p}\) is the minimum value of the quotient. The limit as \(p\to\infty\) of these equations is found to be \(\max \left\{ \Lambda_{\infty}-\frac{|\nabla u(x)|}{u(x)},\ \ \Delta_{\infty}u(x)\right\}=0,\) where the constant \(\Lambda_{\infty}=\lim_{p\to\infty}\Lambda_{p}\) is the reciprocal of the maximum of the distance to the boundary of the domain Ω.

Removability of a Level Set for Solutions of Quasilinear Equations

In this paper, we study the removability of a level set for the solutions of quasilinear elliptic and parabolic equations of the second order. We show, under rather general assumptions on the coeff...

Notes on the p-Laplace equation

2. p.