The p-Harmonic transform beyond its natural domain of definition

Abstract

where the exponents satisfy Hölder's relation: p , q > 1 and p + q = p · q . The p-harmonic transform assigns to f the gradient of the solution.

On maximum principles for solutions to nonlinear elliptic ODEs and radial solutions to nonlinear elliptic PDE's via Opial‐type inequalities

“…By the modified computations we arrive at (15) and the remaining arguments are the same as in the proof of the case under ( ).…”

“…Singular boundary value problems involving p ‐Laplacian arise for example, in fluid dynamics 6‐8 (chapter 2 in Drabek and Wu et al 9,10 ); glaciology (Arcoya et al 11 ), stellar dynamics (Kawano et al 12 ); in the theory of electrostatic fields (Fortunato et al 13 ); in quantum physics (Benci et al 14 ); in the nonlinear elasticity theory (D'Onofrio and Iwaniec 15 ). Links of our approach with literature, focusing on Lienard and Emden‐Fowler type equations in mathematical physics, are discussed in Section 5.…”

“…We believe that the presented methods, as well as Opial-type inequalities, can be further developed and applied to the study of monotonicity properties and the a priori estimates for solutions to PDEs in the more general setting. Singular boundary value problems involving p-Laplacian arise for example, in fluid dynamics [6][7][8] (chapter 2 in Drabek and Wu et al 9,10 ); glaciology (Arcoya et al 11 ), stellar dynamics (Kawano et al ); in the theory of electrostatic fields (Fortunato et al 13 ); in quantum physics (Benci et al 14 ); in the nonlinear elasticity theory (D'Onofrio and Iwaniec 15 ). Links of our approach with literature, focusing on Lienard and Emden-Fowler type equations in mathematical physics, are discussed in Section 5.…”

An essay on the Interpolation Theorem of Józef Marcinkiewicz – Polish patriot

“…The interested reader may wish to take a note of the interpolation lemmas in [1]. In the present paper I will try to elucidate some new advances of Marcinkiewicz interpolation theorem which arise from a study of the nonlinear p-harmonic type PDEs, [12,13,14,16,18,19,20]. The principal result in this paper can be described as follows:…”

A geometric approach to accretivity

“…An A-harmonic system of partial differential equations [9,10] involves a nonlinear mapping A : R n × H → H that satisfies the assumptions of local Lipschitz continuity…”