Nonlinear Questions in Clamped Plate Models

Abstract

Abstract. The linear clamped plate boundary value problem is a classical model in mechanics. The underlying differential equation is elliptic and of fourth order. The latter is a peculiar feature with respect to which this equation differs from numerous equations in physics and engineering which are of second order. Concerning the clamped plate boundary value problem, "linear questions" may be considered as well understood. This changes completely as soon as one poses the simplest "nonlinear question": What can be said about positivity preserving? Does a plate bend upwards when being pushed upwards? It is known that the answer is "no" in general. However, there are many positivity issues as e.g. "almost positivity" to be discussed.Boundary value problems for the "Willmore equation" are nonlinear counterparts for the linear clamped plate equation. The corresponding energy functional involves curvature integrals over the unknown surface. The Willmore equation is of interest in mechanics, membrane physics and, in particular, in differential geometry. Quite far reaching results were achieved concerning closed surfaces. As for boundary value problems, by far less is known. These will be discussed in symmetric situations.

Some solutions of minimaxmax problems for the torsional displacements of rectangular plates

“…We refer to [, (2.40)] for the justification of these boundary conditions, see also [] for full details on how to derive them for the rectangular plate Ω under study. The behavior of rectangular plates subject to a variety of boundary conditions is studied in []. The solution u of –– represents the vertical displacement of the plate under the action of f and, since the boundary conditions – satisfy the complementing condition [, Lemma 4.2] so that elliptic regularity applies, u is a strong solution of whenever f belongs to suitable spaces.…”

“…We refer to [24, (2.40)] for the justification of these boundary conditions, see also [8] for full details on how to derive them for the rectangular plate Ω under study. The behavior of rectangular plates subject to a variety of boundary conditions is studied in [6,10,11,18]. The solution of (20) so that elliptic regularity applies, is a strong solution of (20) whenever belongs to suitable spaces.…”

Instability of modes in a partially hinged rectangular plate

“…Since ( 18) is linear, this is equivalent to state that all the solutions of ( 18) are bounded. Since ( 15) is nonlinear, the stability of ϕw mi depends on the initial conditions and, therefore, on the corresponding energy (17). On the contrary, the linear instability of ϕw mi occurs when the trivial solution of ( 18) is unstable: in this case, if the initial energy is almost all concentrated in (ϕ(0), ϕ (0)), the component ϕ conveys part of its energy to ψ for some t > 0.…”

“…We refer to [2,13,15] for the derivation of (1), to the recent monograph [14] for the complete updated story, and to [30] for a classical reference on models from elasticity. The behavior of rectangular plates subject to a variety of boundary conditions is studied in [7,17,18,19,25].…”

Axially Symmetric Willmore Surfaces Determined by Quadratures

“…In the past four decades, the Willmore surfaces were studied by many authors both from purely mathematical point of view (see, e.g. [1,7,10] and references therein) and in the context of applications, for instance, in structural mechanics, biophysics and mathematical biology (see, e.g. [2,7,8,11,12]).…”