Searching minimizers of functions on the convenient level set of the constrain function we obtain weak solutions of a non-homogenuous p-Laplace equation in border case without using the regularity results of linear elliptic equations .

Consider two perfectly conducting spheres in a homogeneous medium where the current-electric field relation is the power law. Electric field E blows up in the L∞-norm as δ, the distance between the conductors, tends to zero. We give here a concise rigorous justification of the rate of this blow-up in terms of δ. If the current-electric field relation is linear, see similar results obtained earl...

has this character. Even obvious results for this equation may require advanced estimates in the proofs. We refer to the books [DB] and [WZYL] about this equation, which is called the “evolutionary p-Laplacian equation,” the “p-parabolic equation” or even the “non-Newtonian equation of filtration.”. Our objective is to study the regularity of the viscosity supersolutions and their spatial gradi...

We establish a lower bound for the gradient of the solution to 1-Laplace equation in a strongly star-shaped annulus with capacity type boundary conditions. The proof involves properties of the radial derivative of the solution, so that starshapedness of level sets easily follows. x1. Introduction In this paper we deal with solutions to the 1-Laplace equation 1 u = n X i;j=1 u x i u x j u x i x ...