The Harnack inequality for -harmonic functions.
On the definition and properties of p-superharmonic functions.
A criterion of Petrowsky's kind for a degenerate quasilinear parabolic equation.
The celebrated criterion of Petrowsky for the regularity of the latest boundary point, originally formulated for the heat equation, is extended to the so-called p-parabolic equation. A barrier is constructed by the aid of the Barenblatt solution.
Note on some greatest common divisor matrices
Some quadratic forms related to "greatest common divisor matrices" are represented in terms of L²-norms of rather simple functions. Our formula is especially useful when the size of the matrix grows, and we will study the asymptotic behaviour of the smallest and largest eigenvalues. Indeed, a sharp bound in terms of the zeta function is obtained. Our leading example is a hybrid between Hilbert's matrix and Smith's matrix.
Summability of semicontinuous supersolutions to a quasilinear parabolic equation
We study the so-called -superparabolic functions, which are defined as lower semicontinuous supersolutions of a quasilinear parabolic equation. In the linear case, when , we have supercaloric functions and the heat equation. We show that the -superparabolic functions have a spatial Sobolev gradient and a sharp summability exponent is given.
Note on infinity-superharmonic functions.
Superharmonicity of nonlinear ground states.
The objective of our note is to prove that, at least for a convex domain, the ground state of the p-Laplacian operator Δpu = div (|∇u|p-2 ∇u) is a superharmonic function, provided that 2 ≤ p ≤ ∞. The ground state of Δp is the positive solution with boundary values zero of the equation div(|∇u|p-2 ∇u) + λ |u|p-2 u = 0 in the bounded domain Ω in the n-dimensional...
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