# Nonlinear monotone potential operators: from nonlinear ODE and PDE to computational material sciences.

1 Introduction

Second and fourth-order nonlinear ordinary and elliptic partial differential equations form basis of mathematical models of various steady-state phenomena and processes in mechanics, physics and many other areas of science (see, for example, [4, 15, 16, 24]). One important class of these equations is related to nonlinear monotone potential operators [20-22,24]. In the presented paper we study solvability and linearization of boundary value problems related to nonlinear monotone potential operators. The onedimensional model of these problems is the boundary value problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

The weak solution u [member of] [[??].sup.1][a, b] of the nonlinear boundary value problem (1.1)-(1.2) (subsequently, the problem (NBVP)) satisfies the integral identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

for all v [member of] [H.sup.1][a, b], where [H.sup.1][a, b] = {u(x) [member of] [H.sup.1][a, b] : u(a) = 0} and [H.sup.1][a, b] is the Sobolev space . Here and below a(u; u, v) := <Au, v>.

For the linear operator [A.sub.0]u = - (k(x)u')' + q(x)u the left-hand side of the integral identity (1.3) corresponds to the symmetric bilinear form (functional)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the well-known conditions

[c.sub.1] [greater than or equal to] k(x) [greater than or equal to] [c.sub.0] > 0, [c.sub.2] [greater than or equal to] q(x) [greater than or equal to] 0, (1.4)

guarantee the existence of the unique solution u [member of] [H.sup.1][a, b], if f [member of] [H.sup.0][a, b]. The case g(u) = q(x)u + p(x) corresponds to the quasilinear equation

Au [equivalent to] - (k[((u').sup.2])u') + q(x)u= f(x) - p(x), a<x<b, (1.5)

which can be considered as one-dimensional analogue of the well-known Plateau  and Kachanov equations . Spacifically, when k([xi]) = [(1 + [xi]).sup.-1/2] , [xi] = [(u').sup.2], and q(x) = 0, the operator Au [equivalent to] - k [((u').sup.2])u')' is a one-dimensional Plateau operator. Further, the case k([xi]) = [k.sub.0][[xi].sup.0.5([kappa]-1)], [kappa] [member of] (0,1], [k.sub.0] > 0 and q(x) = 0, corresponds to the one-dimensional analogue of Kachanov's equation for engineering materials . The differential operator Au [equivalent to] - (k[((u').sup.2])u')' + q(x)u sometimes is defined to be the nonlinear Sturm-Liouville operator.

Comparing the linear equation - (k(x)u')' + q(x)u = f (x) with the nonlinear equation - (k(x)u')'+g(u) = f (x), and taking account the second condition (1.4) for the linear equation, we conclude that extension of this condition for the function g(u) = q([eta])u, [eta] = [u.sup.2], is the condition [c.sub.2] [greater than or equal to] q([eta]) [greater than or equal to] 0. Otherwise, if this condition does not hold, then the problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

may have an infinite number of solutions. For example, if

[lim over (u[right arrow][+ or -][infinity])] g(u)/u [right arrow] [infinity],

then, as it was shown in , the problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

has an infinite number of solutions u [member of] [C.sup.2] [a,b], for every f [member of] [C.sup.0][a,b].

The problems related to solvability of boundary value problems for the quasilinear and nonlinear equations of type (1.1) have been considered by various authors (see [3-5, 16,18] and references therein). Some applications to evolution problems are presented in [12, 13]. Note that the iteration scheme for the quasilinear equations arising from elastoplasticity has first been given in  and then developed in [3,6,7]. An abstract iteration scheme and convergence criteria for these type of nonlinear problems were proposed in .

In this paper we present some review related to solvability of nonlinear boundary value problems for the second and fourth-orders nonlinear monotone potential differential operators. We also analyze questions related to linearization of these problems and convergence of approximate solutions in appropriate Sobolev spaces. The main subject of the analysis is to derive explicit, from the point of view practice, sufficient conditions for the leading coefficient k = k([xi]), [xi] := [[absolute value of [nabla]u].sup.2]. In Section 2 we discuss solvability of the problem (NBVP) in [[??].sup.1][a,b] [intersection] [H.sup.2][a, b] for the Sturm-Liouville operator Au := - (k[((u').sup.2])u'(x))' + q(x)u(x). In Section 3 we extend the obtained results to the case of the nonlinear elliptic operator Au [equivalent to] -[nabla] (k([[absolute value of [nabla]u].sup.2]) [nabla]u) + q(x)u, and derive sufficient conditions for linearization and [H.sup.1]-convergence of the approximate solution. Linearization of nonlinear problems, monotonicity on iterations and convergence issues for an abstract, as well as for concrete variational problems are discussed in Section 4. As a first application, in Section 5 the mathematical model of an elastoplastic torsion of a strain hardening bar is considered within the range of [J.sub.2]-deformation theory. In the final Section 6 an elastoplastic bending problem for an incompressible thin plate is considered. Both applications show that the sufficient conditions obtained for abstract monotone potential elliptic operators are almost same with the basic conditions of [J.sub.2]-deformation theory.

2 The Problem (NBVP) in [R.sup.1]

As a sample model consider first the boundary value problem (1.1)-(1.2) with g(u) [equivalent to] q(x)u. The weak solution u [member of] [[??].sup.1][a, b] [intersection] [H.sup.2][a, b] of this problem is defined as a solution of the integral identity (or variational problem)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

The multiple coefficient q(x) and the source function f (x) are assumed to be in

[H.sup.0][a,b] [equivalent to] [L.sub.2][a,b].

To study solvability of the nonlinear problem (2.1), we shall use the variational approach and monotone operator theory (see [4,19-22]). For this aim let us introduce the functional

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

and calculate the first and the second Gateaux derivatives of this functional. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

for v, h [member of] [[??].sup.1][a, b]. It is seen from the left-hand side of (1.5) and from (2.3) that a(u,u; v) = J'(u,v) for all v [member of] [[??].sup.1][a,b] and hence the nonlinear operator A defined by (1.5) is a potential operator, with potential J(u) defined by (2.2). In this context the functional P (u) = J (u) - l(u) is defined to be the potential of the variational problem (2.1).

Theorem 2.1. Let us assume that in addition to conditions (1.4), the coefficient k = k([xi]) is piecewise differentiable and satisfies the condition

k([xi])+2k'([xi])[xi][greater than or equal to] [[gamma].sub.0]>0, [xi][member of][[[xi].sub.*], [[xi].sup.*]], (2.5)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the problem (NBVP) has a unique solution u [member of] [[??].sup.1][a, b] [intersection] [H.sup.2][a, b].

Proof. Due to the Browder-Minty theorem (see [4,19]) the operator equation Au = F, the nonlinear operator A defined by (1.5), has a unique solution, if a potential operator A is bounded, radially continuous, coercive and uniform monotone. Hence, to prove the theorem we only need to show that the operator A defined by (1.5) has the above properties. Conditions (1.5) imply the boundedness of the operator A:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Radial continuity of the operator A follows from continuity of the function

t [??] <A(u + tv),v>,

for all t [member of] R, and for fixed u,v [member of] [[??].sup.1][a, b] [intersection] [H.sup.2][a, b]. Substituting h = v in (2.4) and using condition (2.5), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all h [member of] [H.sup.1][a, b}. Applying the Poincare inequality [[parallel]v'[parallel].sup.2.sub.0] [greater than or equal to] [c.sup.2.sub.[omega]] [[parallel]v[parallel].sup.2.sub.0] ([c.sup.2.sub.[omega]] = 2/(b - a)), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which means the uniform monotonicity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

of the nonlinear operator. Since A[theta] = [theta], where [theta] [member of] [[??].sup.1][a, b] is the zero element, inequality (2.6) also implies the coercivity of the operator A, with the same coercivity constant [[gamma].sub.1] > 0. Due to the Browder-Minty theorem the problem (NBVP) has a unique solution in [H.sup.1][a, b].

Remark 2.2. Condition (2.5) has first been used in the form k'([s.sup.2])[s.sup.2] + k([s.sup.2]) [greater than or equal to] d > 0 for the function k([xi]) : [R.sub.+] [right arrow] [R.sub.+] in the classical Kachanov method for stationary conservation laws (see [24, page 544]). In the case of Dirichlet problem for the nonlinear operator Au [equivalent to] -[nabla] (k([[absolute value of [nabla]u].sup.2])[nabla]u), the boundedness [[parallel][nabla]u[parallel].sub.c] [greater than or equal to] [[xi].sup.*] of the norm [[parallel][nabla]u[parallel].sub.c] has been proved in , where [[xi].sup.*] > 0 is a positive constant. Hence condition (2.5) can also be considered as an extension of the above condition for the case [xi] [member of] [[[xi].sub.*], [[xi].sup.*]].

3 Solvability of the Problem (NBVP) in [R.sup.n] (n > 1)

Consider now the problem (NBVP) in [R.sup.n] (n > 1) for the nonlinear elliptic operator

Au [equivalent to] -[nabla] (k([[absolute value of [nabla]u].sup.2])[nabla]u) + q(x)u, x [member or] [OMEGA][subset][R.sup.n] (3.1)

Specifically, we consider the mixed boundary value problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

where [[bar.[GAMMA]].sub.1] [union] [[GAMMA].sub.2] = [partial derivative][OMEGA], [[GAMMA].sub.1] [intersection] [[GAMMA].sub.2] = [empty set] and [OMEGA] [subset] [R.sup.n] is a bounded domain with a piecewise smooth boundary [partial derivative][OMEGA]. The weak solution u [member of] [H.sup.l]([OMEGA]) [intersection] [H.sup.2] (Q) of the nonlinear problem (3.2) is defined as a solution of the variational problem

a(u, u; v) = l(v), [for all]v [member of] [[??].sup.1]([OMEGA]) [intersection] [H.sup.2]([OMEGA]), (3.3)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [[??].sup.1]([OMEGA]) = {u [member of] [H.sup.1]([OMEGA]) : u(s) = 0,s [member of] [[GAMMA].sub.1] }. It is easy to verify that the functional

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

represents the potential of the operator A, defined by (2.1), since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Calculating the second Gateaux derivative ofthis functional we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence for h = v we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

By the inequality [([n.summation over (i=1)] [a.sub.i][b.sub.i]).sup.2] [less than or equal to] [n.summation over (i=1)] [a.sup.2.sub.1] [n.summation over (i=1)] [b.sup.2.sub.1] for all [a.sub.i]; [b.sub.i] [member of] [R.sup.1], we conclude

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Assuming the condition

k'([xi])[less than or equal to]0, [xi][member of][[[xi].sub.i], [[xi].sup.*]], (3.5)

we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking into account this inequality in the right-hand of (3.4) we obtain the following upper estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This estimate with the condition (2.5) and the Poincare inequality [[parallel][nabla]v[parallel].sup.2] [greater than or equal to] [c.sup.2.sub.[OMEGA]] [[parallel]v[parallel].sup.2.sub.0], [c.sup.2.sub.[OMEGA]] > 0, implies the positivity of the second Gateaux derivative

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This means the strong convexity of the functional, which implies the uniform monotonicity of the nonlinear elliptic operator, defined by (3.1). Hence we get the following result.

Theorem 3.1. Let us assume that in addition to the conditions of Theorem 2.1 and the conditions F [member of] [H.sup.0]([OMEGA]), [phi] [member of] [H.sup.0]([GAMMA]), the coefficient k([xi]) is piecewise differentiable and satisfies condition (3.5). Then the problem (NBVP) has a unique solution u [member of] [H.sup.1]([OMEGA]) [intersection] [H.sup.2]([OMEGA]).

The above theorems show that different from the one-dimensional case, for solvability of the multi-dimensional problem (NBVP) (n > 1), one needs to impose the additional conditions (3.5).

4 Linearization of Nonlinear Problems

Consider first the abstract equation

Au = F, u [member of] H, F [member of] [H.sup.*], (4.1)

for the nonlinear strong monotone potential operator, acting from the Hilbert space H to its dual H*. Assume that a(u; .,.) is a bounded symmetric bilinear form generated by the operator A, i.e., a(u; u,v) = <Au,v>, [for all]u,v [member of] H. Suppose that A[THETA] = [THETA], i.e., A transforms zero element of H to zero element of [H.sup.*]. These mean that the operator satisfies the conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

We denote again by J(u) the potential of the operator A, and by P(u) = J(u) - l(u), l(u) = <F, u>, the potential of the corresponding variational problem

a(u; u, v) = l(v), [for all]v [member of] H. (4.3)

A monotone iteration scheme for the abstract variational problem (4.3) corresponding to the nonlinear problem (4.1) has been proposed in . For this aim, the inequality

0.5a(u; v,v) - 0.5a(u; u, u) - J(v) + J(u) [greater than or equal to] 0, [for all]u, v [member of] H (4.4)

has been introduced in , as a convexity argument for nonlinear monotone potential operators. To analyze this inequality from the point of view the leading coefficient k = k([xi]), let us consider the one-dimensional variational problem (2.1), assuming without loss of generality, that q(x) [equivalent to] 0. For this nonlinear problem we may rewrite this inequality as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.5)

for all u, v [member of] [H.sup.1][a, b] [intersection] [H.sup.2][a, b], using (2.2) and (2.3). Letting [[xi].sub.i] = [(v').sup.2] and [[xi].sub.2] = [(u').sup.2] in (4.5), we conclude that the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.6)

is a sufficient condition for fulfilment of the convexity argument, i.e., inequality (4.4), for the variational problem (2.1). Introducing the new function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we conclude that inequality (4.6) means concavity of the function K([xi]). Since K'([xi]) = k([xi]), the condition k'([xi]) < 0, [for all][xi] [member of] [[[xi].sub.*], [[xi].sup.*]*], is evidently a sufficient condition for fulfilment of the convexity argument for the variational problem (2.1).

Theorem 4.1. Let us assume that in addition to conditions of Theorem 2.1, the coefficient k([xi]) is piecewise differentiable and satisfies condition (4.6). Then the convexity argument (4.4) holds in the one-dimensional variational problem (2.1).

The same result remains true for the n-dimensional problem (3.1)-(3.2). Detailed proofs of these result are given in .

The first application of the convexity argument for nonlinear monotone potential operators, is the monotone iteration scheme

a([u.sup.(n-1)]; [u.sup.(n)],v) = l(v), [for all]v [member of] H, n = 1, 2, 3, ..., (4.7)

for the abstract variational problem (4.3). Here [u.sup.(0)] [member of] H is an initial iteration. This results asserts that the sequence of potentials {P ([u.sup.(n)])} of the linearized problem (4.7) is a monotone decreasing one, i.e., P ([u.sup.(n+1)]) [less than or equal to] P ([u.sup.(n)]), [for all]n = 1, 2,3,... (see [8, Lemma 1]). Since

P([u.sup.(n)]) = 1/2a([u.sup.(n-1)]; [u.sup.(n)], [u.sup.(n)]) - l([u.sup.(n)]), n = 1, 2, 3, ..., (4.8)

substituting in (4.7) v = [u.sup.(n)] we get a([u.sup.(n-1)]; [u.sup.(n)], [u.sup.(n)]) = l([u.sup.(n)]). This, with (4.8), implies

P([u.sup.(n)]) = -1/2a([u.sup.(n-1)] ; [u.sup.(n)], [u.sup.(n)]) < 0, n = 1, 2, 3,.... (4.9)

Thus the sequence {P([u.sup.(n)])} is bounded below, and hence it converges. Using this lemma, it is proved that (see [8, Theorem 1]), the difference [[parallel]u - [u.sup.(n)][parallel].sub.H] between the solution u [member of] H of the variational problem (4.3) and its approximation [u.sup.(n)] [member of] H obtained by the iteration scheme (4.7), can be estimated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.10)

This estimate shows that the sequence of approximate solutions {[u.sup.(n)]} [subset] H, obtained by the iteration scheme (4.7) converges to the solution u [member of] H of the variational problem (4.3) in H-norm.

Let us apply the abstract monotone iteration scheme (4.7) to the problem (NBVP) given by (4.1)-(4.2). The sequence of approximate solutions {[u.sup.(n)]} [subset] [[??].sup.1]([OMEGA]) [intersection] [H.sup.2]([OMEGA]) is defined from the linearized mixed boundary value problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](4.11)

The potential of this linearized problem is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.12)

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Evidently, if the coefficient k([xi]) is piecewise differentiable and satisfies conditions (1.4), (2.5) and (3.5), the coefficient q(x) satisfies conditions (1.4), and q(x), F(x) [member of] [H.sup.0][[[xi].sub.*], [[xi].sup.*]], then all conditions in (4.2) hold. Thus the above results obtained for the abstract monotone iteration scheme (4.7) remain true also for the problem (NBVP) given by (4.1)-(4.2).

Theorem 4.2. Let us assume that the conditions of Theorem 3.1 hold. Then

(i) the sequence of potentials {P([u.sup.(n)]} defined by (4.12) is a monotone decreasing andconvergent one;

(ii) the sequence of approximate solutions {[u.sup.(n)]} [subset] [H.sup.1]([OMEGA]) [intersection] [H.sup.2]([OMEGA]), defined by (4.11), converges to the weak solution u [member of] [H.sup.1]([OMEGA]) [intersection] [H.sup.2]([OMEGA]) of the problem (NBVP) (4.1}-(4.2) in [H.sup.1]-norm;

(iii) for the rate of convergence the following estimate holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4.13)

where [M.sub.1] = max{[c.sub.1], [c.sub.2]} > 0, and [[gamma].sub.1] > 0 is defined in (2.6).

The main distinguished feature of this theorem is that it requires the same conditions as Theorem 3.1. In other words, for the convergence of the monotone iteration (4.12) scheme, no additional conditions are required.

5 An Elastoplastic Torsion of a Strain Hardening Bar

The concept of torsional rigidity is well known in structural mechanics as one of main characteristics of a beam of uniform cross section during elastoplastic torsion. Torsional rigidity is defined as the torque required for per unit angle of twist [phi] > 0 per unit length, when the elastic modulus of the material is set equal to one [11,17]. Specifically, if u = u(x); x = ([x.sub.1], [x.sub.2]) [member of] [OMEGA] [subset] [R.sup.2] denotes the deflection function, then the torque (or torsional rigidity) is defined as to be

T[u; g, [[phi]]] = 2 [[integral].sub.[OMEGA]] u(x; g; [phi])dx, (5.1)

where [OMEGA] := (0, [l.sub.1]) x (0, [l.sub.2]), [l.sub.1],[l.sub.2] > 0, denotes the cross section of a bar, and is assumed to be in [R.sup.2], with piecewise smooth boundary. For given g = g([[xi].sup.2]) and [phi] > 0, the function u(x) := u(x; g, [phi]) is the solution of the nonlinear boundary value problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.2)

corresponding to a given function g = g([[xi].sup.2]) The boundary value problem (5 2) represents an elastoplastic torsion of a strain hardening bar, which lower end is fixed, i. e., rigid clamped. The function u(x) is the Prandtl's stress function and [xi](u) = [[[([partial derivative]u/[partial derivative][x.sub.1]).sup.2] + [([partial derivative]u/[partial derivative][x.sub.2]).sup.2]].sup.1/2] is the stress intensity. In view of [J.sub.2]-deformation theory, the function g = g([[xi].sup.2]), [[xi].sup.2] = [[absolute value of [nabla]u].sup.2], defined to be the plasticity function, describes elastoplastic properties of a homogeneous isotropic material, and satisfies the conditions (see [6,9,11,15,17,23] and references therein)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.3)

Here G > 0 is the shift modulus and [[xi].sub.0] > 0 is assumed to be the elasticity limit of a material. Note that G = E/(1 + v), where E > 0 is the elasticity modulus and v [member of] (0,0.5) is the Poisson coefficient.

Evidently the variational problem (4.7) corresponds here to the linearized problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.4)

where n = 1, 2, 3, ... and [u.sup.(0)] [member of] [H.sup.1]([OMEGA]) [intersection] [H.sup.2]([OMEGA]) is the initial iteration The potential of the linearized problem (5.4) is defined to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

With definition (5.1) of the torque, this potential has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.5.)

On the other hand, the weak solution [u.sup.(n)] [member of] [H.sup.1]([OMEGA]) [intersection] [H.sup.2]([OMEGA]) of the linearized problem (5.4) is defined by the integral identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all n = 1, 2, .... Substituting here v = [u.sup.(n)] and using the definition of the torque, we obtain the following energy identity for the linearized problem (5.4):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This, with (5.5), permits one to define the potential of the linearized problem (5.4) via the torque by

P([u.sup.(n)]) = -[??]T[[u.sup.(n)]; g,[??]], n = 1, 2, 3,....

This result agrees with (4.9), since the torque is positive.

Comparing conditions (3.3), (2.5) and (3.5), with the assumptions of [J.sub.2]-deformation theory, we conclude that all conditions (5.3) hold. Therefore based on Theorem 3.1 and Theorem 4.2, we can derive the following results for the nonlinear boundary value problem (5.2) related to the elastoplastic torsion of a strain hardening bar.

Theorem 5.1. Let assumption (5.3) of [J.sub.2]-deformation theory hold. Then

(i) the weak solution u [member of] [H.sup.1]([OMEGA]) [intersection] [H.sup.2]([OMEGA]) of the nonlinear boundary value problem (5.2) exists and unique;

(ii) the sequence of potentials {P([u.sup.(n)]} defined by (5.5) is a monotone decreasing and convergent one;

(iii) the sequence of approximate solutions {[u.sup.(n)]} [subset] [H.sup.1]([OMEGA]) [intersection] [H.sup.2]([OMEGA]), defined by (5.4), converges to the weak solution u [member of] [H.sup.1]([OMEGA]) [intersection] [H.sup.2]([OMEGA]) of problem (5.2) in [H.sup.1]-norm;

(iv) the rate of convergence can be estimated via the torque as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

6 Elastoplastic Bending of Incompressible Plates

The mathematical model of inelastic bending of an isotropic homogeneous incompressible plate under the loads normal to the middle surface of the plate, within the range of [J.sub.2]-deformation theory, is described by the following nonlinear boundary value problem (see [9,10,14]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.1)

The function u = u(x) represents deflection of a point x [member of] [OMEGA] on the middle surface of a plate, occupying the square domain [OMEGA] [subset] [R.sup.2], being in equilibrium under the action of normal loads. The coordinate plane O[x.sub.1][x.sub.2] is assumed to be the middle surface of an isotropic homogeneous incompressible plate with the thickness h > 0. F(x) = 3q(x)/[h.sup.3], and q = q(x) is the intensity (per unit area) of the loads normal to the middle surface of a plate, and n is a unit outward normal to the boundary [partial derivative][OMEGA]. It is assumed that the load q = q([x.sub.1], [x.sub.2]) acts on the upper surface only in the [x.sub.3]-axis direction, and the lower surface of the plate is free.

The coefficient g = g([[xi].sup.2](u)) is defined to be the plasticity function, and satisfies assumptions (5.3) of [J.sub.2]-deformation theory. The dependent variable [xi] = [xi](u), being referred to as effective value of the plate curvature , satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.2)

Besides the above clamped boundary conditions, simply supported and other natural boundary conditions can also be considered.

We will show here that conditions (5.3) of [J.sub.2]-deformation theory are sufficient not only for the existence and uniqueness of the weak solution u [member of] [[??].sup.2]([OMEGA]) of the nonlinear problem (6.1), but also for the convergence of the linearized problem solution in the norm of the space [[??].sup.2]([OMEGA]).

Let [H.sup.2]([OMEGA]) be the Sobolev space of functions  defined on the domain [OMEGA] with piecewise smooth boundary [partial derivative][OMEGA] and

[[??].sup.2]([OMEGA]) = {v [member of] [H.sup.2]([OMEGA]): u(x) = [partial derivative]u(x)/[partial derivative]n = 0, x [member of] [partial derivative][OMEGA]}.

Multiplying both sides of equation (6.1) by v [member of] [[??].sup.2]([OMEGA]), integrating on [OMEGA] and using the boundary conditions (6.1), we obtain the integral identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.3)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.4)

The function u [member of] [[??].sup.2]([OMEGA]) satisfying the integral identity (6 3) for all v [member of] [[??].sup.2]([OMEGA]) is said to be a weak solution of the nonlinear problem (6.1). Recalling the definition a(u; u, v) = <Au, v>, we may write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.5)

Let us introduce now the functional P(u) = J(u) - l(u), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.6)

It easy to prove that the above defined functionals J(u) and P(u) are the potentials of the nonlinear operator A and the nonlinear problem (6.1), respectively. Indeed, calculating the Gateaux derivative of the functional J(u), we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence the nonlinear bending operator A, defined by (6.1), is a potential operator with the potential J(u), defined by (6.6). To analyze monotonicity of this operator A, we will use the equivalence (see )

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of the norm [[parallel]*[parallel].sub.2] of the Sobolev space [H.sup.2]([OMEGA]) and the energy norm

[[parallel]v[parallel].sub.E] = [{[[integral].sub.[OMEGA]]H(v,v)dx}.sup.1/2]

Lemma 6.1. If the plasticity function g = g([[xi].sup.2]) satisfies conditions (5.3), then the nonlinear bending operator A, defined by (6.1), is strong monotone in [[??].sup.2]([OMEGA]), i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.7)

Proof. Calculating the second Gateaux derivative of the functional J(u), defined by (6.6), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The second condition (5.3), with the inequality [(H(u, v)).sup.2] [less than or equal to] H(u, u) H(v, v) and the formula [[xi].sup.2](v) = H(v, v) (by (6.2) and (6.4)), implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using now the third condition in (5.3) on the right-hand side and equivalence ofnorms, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The positivity of the second Gateaux derivative of the functional J(u) means that the operator A is strongly monotone.

Since A[THETA] = [THETA], where [THETA] [member of] [[??].sup.2]([OMEGA]) is zero element, monotonicity condition (6.7) for the nonlinear operator A also means its coercivity, i.e., <Av, v> [greater than or equal to] [[gamma].sub.1] [[parallel]v[parallel].sup.2.sub.2], [[gamma].sub.1] > 0. Further the operator A is radially continuous (hemicontinuous), i.e., the real-valued function t [right arrow] <A(u + tv, v>, for fixed u, v [member of] [[??].sup.2]([OMEGA]), is continuous, since both functions t [right arrow] g([[xi].sup.2] (u + tv)), t [right arrow] H (u + tv, v) are continuous, the proof of this assertion follows immediately from (6.5).

Thus, the potential operator A is radially continuous, strongly monotone and coercive. Then, by Browder-Minty theorem, we get the following.

Theorem 6.2. If conditions (5.3) hold, then the nonlinear problem (6.1) has a unique solution in [[??].sup.2]([OMEGA]), defined by the integral identity (6.3).

Now we apply the abstract iteration scheme (4.7) linearizing the variational problem (6.3) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.8)

where [u.sup.(0)]) [member of] [[??].sup.2]([OMEGA]) is an initial iteration. The solution [u.sup.(n)] [member of] [[??].sup.2]([OMEGA]) of the linearized problem (6.8) is defined to be an approximate solution of the variational problem (6.3).

To apply the abstract iteration scheme we need a sufficient condition for the fulfilment of the convexity argument (4.4) for the nonlinear bending problem.

Lemma 6.3. Let the function g = g([[xi].sup.2]) satisfy the condition g([[xi].sup.2]) [less than or equal to] 0. Then the convexity argument (4.4) holds for the nonlinear bending operator A, defined by (6.1).

Proof. Using definitions (6.5) and (6.6), on the left-hand side of inequality (4.4), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.9)

As in the proof of Theorem 4.1, introducing the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we conclude Q"(t) = g'(t) [less than or equal to] 0, which means Q(t), is a concave function. Hence inequality (4.6) holds for this function. This, with (6.9) completes the proof.

Lemma 6.3 implies that the sequence ofpotentials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.10)

defined on approximate solutions [u.sup.(n)] [member of] [H.sup.2]([OMEGA]), is a monotone decreasing one.

Next we need to show that the functional a(u; v, w), defined by (6.5), is bounded. This follows from the condition g([[xi].sup.2](u)) [less than or equal to] [c.sub.1] and the equivalence of norms [[parallel]*[parallel].sub.2] and [[parallel]*[parallel].sub.E]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore all conditions of [8, Theorem 2.1] hold, and we have the following result.

Theorem 6.4. Let u [member of] [[??].sup.2]([OMEGA]) and [u.sup.(n)] [member of] [[??].sup.2]([OMEGA]) be the solutions of the nonlinear problem (2.3), and the linearized problem (4.4), respectively. If conditions (i)--(iii) hold, then

(a) the iteration scheme defined by (4.7) is a monotone decreasing one:

[PI]([u.sup.(n)]) [less than or equal to] [PI]([u.sup.(n-1)]), [for all][u.sup.(n-1)], [u.sup.(n)] [member of] [[??].sup.2]([OMEGA]);

(b) the sequence of approximate solutions {[u.sup.(n)]} [subset] [[??].sup.2]([OMEGA]) defined by the iteration scheme (4.7) converges to the solution u [member of] [[??].sup.2]([OMEGA]) of the nonlinear problem (4.1) in the norm of the Sobolev space [[??].sup.2]([OMEGA]);

(c) for the rate of convergence the following estimate holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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Alemdar Hasanov

Izmir University

Department of Mathematics and Computer Sciences

35350 Uckuyular, Izmir, Turkey

alemdar.hasanoglu@izmir.edu.tr

Received April 9, 2010; Accepted May 6, 2010 Communicated by A. Okay Celebi
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