Maximum principles for a fourth order equation from thin plate theory

Abstract

This paper focuses on a nonlinear equation from thin plate theory of the formWe obtain maximum principles for certain functions defined on the solution of this equation using P -functions or auxiliary functions of the types used by Payne [L.E. Payne, Some remarks on maximum principles,

Integral bounds for von Kármán's equations

“…Let w, φ ∈ C 5 (Ω)∩C 3 (Ω) be solutions to Eqs. (3) and (4). Assume that f (x, y), h(x, y) ∈ C 2 (Ω)∩C(Ω).…”

“…Since then, many others have extended Miranda's results and have derived maximum principles from auxiliary functions. For example, maximum principle results for fourth-order equations containing nonlinearities in u, Δu, or Lu = a ij u ,ij , can be found in works by Payne and Schaefer [6][7][8][9] or by Zhang [10], and for an equation from thin-plate theory in [4]. In particular, Schaefer [8] uses functions containing the square of the Laplacian of the solution, (Δu) 2 , and deduces bounds on several quantities of interest, such as the Laplacian and gradient of the solution.…”

Maximum Principles for Some Higher-Order Semilinear Elliptic Equations

“…In this paper we show that one can obtain maximum principle results for the semilinear fourth-, sixth-and eighth-order versions of the constant-coefficient, linear elliptic equations (2), (3), (4), and for the sixth-and eighth-order generalizations of equation (1), by modifying the auxiliary functions used in [6]. From these maximum principles we deduce integral bounds on certain gradients of the solutions of several partial differential equations, subject to different boundary conditions.…”

“…He shows that the maximum value, for auxiliary functions containing the terms |∇ 2 u| 2 − u ,i u ,i , and for certain restrictions on f (u), is achieved on the boundary of the domain. An application of these techniques appears in [4] where the author modifies such functions to obtain results similar to those in [6] for a semilinear equation from thin plate theory. Other results for semilinear fourth-order equations can be found in [5,8,9,12].…”

Maximum principles and bounds for a class of fourth order nonlinear elliptic equations

“…Some applications of these results to equations governing the bending of elastic plates can be found in [4,5].…”