We prove existence, uniqueness, and stability of solutions of the prescribed curvature problem in , , for any given and . We also develop a linear monotone iterative scheme for approximating the solution. This equation has been proposed as a model of the corneal shape in the recent paper (Okrasiński and Płociniczak in Nonlinear Anal., Real World Appl. 13:1498-1505, 2012), where a simplified version obtained by partial linearization has been investigated.
Keywords:mean curvature equation; mixed boundary condition; positive solution; existence; uniqueness; linear stability; order stability; Lyapunov stability; lower and upper solutions; monotone approximation; topological degree
In this paper we study existence, uniqueness, stability, and approximation of classical solutions of the one-dimensional prescribed curvature problem
has been proposed in [1-4] as a mathematical model for the geometry of the human cornea. However, in these papers a simplified version of (2) has been investigated, where the mean curvature operator has been replaced by its linearization around 0. In particular, it has been proved in  that, if , then the problem
has a unique solution which is the limit of a sequence of successive approximations. The above limitations on the parameters have recently been removed in .
Unlike all these works we tackle here the fully nonlinear problem (1) and we prove the existence of a unique solution for the whole range of positive parameters a, b. The study of problem (1) requires some care because, even if pairs of constant lower and upper solutions can easily be exhibited, the presence of the curvature term rules out in general the possibility of applying the standard existence results, due to the possible occurrence of derivative blow-up phenomena (see, e.g., ). On the other hand, the non-variational structure of (1) puts the problem, as it stands, out of the scope of the methods developed in [6-8] for the curvature equation. Nevertheless, we show that an a priori bound in for all possible solutions can be obtained by an elementary, but delicate, argument which exploits the qualitative properties - positivity, monotonicity, and concavity - of the solutions themselves. These estimates eventually enable us to use a degree argument in order to prove the existence of solutions. The proof of the uniqueness is then based on suitable fixed point index calculations, which are performed via linearization. A similar approach, applied to an associated evolutionary problem, is exploited for detecting the linear stability of the solution.
Next, taking inspiration from [9,10], we develop a linear iterative scheme for approximating the solution by two monotone sequences of strict lower and upper solutions, starting from an explicit pair of constant lower and upper solutions. These two sequences, besides providing accurate two-sided bounds on the solution, yield the strict order stability and hence, according to , the (Lyapunov) asymptotic stability of the solution itself, yielding as well an explicitly computable estimate of its basin of attractivity. We finally illustrate the use of this approximation scheme in order to compute numerically the solution u of (1) for the same choice of the parameters a and b as the one considered in .
We finally mention that part of our results extends to the general N-dimensional problem (2); this topic will be discussed elsewhere.
2 Existence, qualitative properties and approximation
In this section we are concerned with the study of the existence, the qualitative properties and the approximation of classical solutions, i.e., belonging to , of problem (1), where and are fixed constants. Clearly, problem (1) can be written in the equivalent form
It is obvious that, due to the symmetry properties of the function f, the mixed problem (4) is equivalent to the Dirichlet problem
Notations As usual, for functions , we write in if for all and in if and . We also write in if for all and, if , , where , denote the left Dini derivatives; this is equivalent to requiring that there exists such that for all . Whenever no confusion occurs, we omit the indication of the interval.
Existence, uniqueness, and linear stability
We start with a preliminary result, where some properties of the possible solutions of problem (1) are highlighted.
Lemma 2.1The following assertions hold.
Step 1. Proof of (i). Let us first show that in . Assume by contradiction that . The boundary condition implies that . Suppose that . We have and . Condition (6) yields . Hence there exists such that for all and therefore for all , which is a contradiction. Now suppose that . We have and and condition (6) yields again a contradiction. Hence we conclude that in . In a completely similar way we prove that in .
Next, in order to prove that
for all , it is sufficient to note that, if for some , then (6) would yield , which is impossible. Moreover, as the constant function is a solution of the equation in (4), the uniqueness of solutions for any Cauchy problem associated with this equation implies that
Let us now prove that
in . By contradiction, assume that there exists such that . As , there exist and such that and for all . Since we have for all , the function is decreasing in . We also know that the function au is decreasing in . Hence the function is decreasing in . On the other hand, as , from the equation in (4) we get
In conclusion, setting
We are now in position to prove the existence of a unique solution of problem (1), which is linearly stable.
Proof The proof is divided into three steps.
Clearly, is completely continuous. Moreover, let be the Nemitski operator associated with f, i.e., for any . The operator is continuous and maps bounded sets into bounded sets. Introduce the open bounded subset of
By Lemma 2.1, u satisfies the conditions (i), (ii), and (iii).
for all , the maximum principle [, Appendix, Theorem 5.2] implies that . Hence the local inversion theorem applies to Φ at every point and thus any fixed point of is isolated. The compactness in of the set of all fixed points of then implies that is finite, i.e., for some positive integer N.
has no non-trivial solution w. Accordingly, for any given , the operator does not have any eigenvalue . Therefore, we infer from [, Theorem 3.20] that
Step 3. Linear stability. The solution u of (4) is an equilibrium of the parabolic problem (11), in particular, it is a 1-periodic solution of (11). In order to show that u is linearly stable, and hence locally exponentially asymptotically stable, it is enough, after a standard cut-off argument, to show that the eigenvalue problem
does not have any eigenvalue (see, e.g., [, Chapter III.23]), or [, Chapter V.22]). Indeed, if w is a solution of (18) for some , then using again condition (15), together with the interior form of the parabolic maximum principle and the Hopf boundary point lemma (see, e.g., [, Chapter III.13]), we conclude that . □
Monotone approximation and order stability
In this section we discuss approximation and stability of the solution of (1), or equivalently of (4). To this end, we define a linear iterative scheme that allows one to construct an increasing sequence of strict lower solutions and a decreasing sequence of strict upper solutions of (4) which converge in to the unique solution u of (4), that is, of (1). Then, according to [11,13], we see that u is strictly order stable from above and from below and hence it is (Lyapunov) asymptotically stable as an equilibrium of the parabolic problem (11). In addition, the converging sequences of lower and upper solutions provide explicitly computable estimates of the basin of attractivity of the solution.
Lower and upper solutions Let us consider the problem
Remark 2.1 The Lipschitz character of g implies (see [, Chapter 3, Proposition 1.7, Proposition 2.7]) that a lower solution α of (19), which is not a solution, is a strict lower solution, that is, any solution u of (19), such that , satisfies in . Similarly, an upper solution β of (19), which is not a solution, is a strict upper solution, that is, any solution u of (19), such that , satisfies in .
Remark 2.2 Any constant is a strict lower solution of (4) and any constant is a strict upper solution of (4). In particular, one can choose and . We wish to point out that, with this choice of lower and upper solutions, the existence of at least one solution u of problem (4) between α and β can be alternatively achieved by applying [, Chapter 2, Theorem 3.1]; the relevant observations being here the facts that , and f satisfies the one-sided Nagumo condition
for all such that . We point out that one-sided Nagumo conditions were introduced for the first time by Kiguradze in .
Let us consider the following modified problem:
Next we prove that in . The proof of assertion (ii) in Lemma 2.1 can be repeated verbatim in order to show that for all and in . Assume now that there exists such that . In particular, we have in and in . By definition of , u satisfies
Combining (25) and (27) yields
Let us consider the following auxiliary linear problem:
A simple computation yields
The conclusion then easily follows by direct calculations.
holds as well. This yields the validity of (37).
Theorem 2.5Letandbe given. Then there exists, given by (39), such that for anythe sequencesandrecursively defined in (41) and (42), respectively, converge into the unique solutionuof (21) and hence of (1). In addition, for eachthe following conditions hold:
Notice that in . Hence the maximum principle implies that , that is, , in . Now, let us show that is a strict lower solution of (21). Using the definition of , together with conditions (h1) and (h2), we get
In a similar way, one can prove the following conclusion.
It follows from the previous steps that the sequence is increasing and bounded in . Therefore there exists a function which is the pointwise limit of in ; in particular, in for all . Moreover, by the Arzelà-Ascoli theorem, any subsequence of admits a subsequence which is convergent in to u. Then the whole sequence converges in to u. From the equation in (41) we see that the convergence takes place in . Hence u is a solution of problem (21) and, by Lemma 2.3 and Theorem 2.2, it is in fact the unique solution of problem (1).
In a similar way, one can prove the following conclusion.
Thus the proof is completed. □
Proof Let us note that any lower, respectively upper, solution of (21) is a lower, respectively upper, solution of the parabolic problem
Arguing as in the proof of Theorem 2.2 we see that u is the unique solution of (56). Then Theorem 2.5 implies that u is strictly order stable from below and from above. Actually, since any constant is a strict lower solution and any constant is a strict upper solution of (56), the results in [, Section 2.6] imply that u is (Lyapunov) globally asymptotically stable as a solution of (56) and hence as an equilibrium of (55). □
Remark 2.4 The definition of implies that the solution u of (1) is strictly order stable from below and from above and (Lyapunov) asymptotically stable as an equilibrium of the parabolic problem (11).
We present here some experiments concerning the numerical approximation of the solution of problem (1), for the same choice of the parameters as in .
The iterative scheme in case We have computed various approximations, at different precision levels, of the unique solution u of problem (1) by implementing in MatLab the linear iterative scheme defined by (41) and (42); at each step of the iteration the resulting linear equations have been solved using the bvp4c routine with a 100-point grid. We have chosen , with given by (39), and . Theorem 2.5 guarantees that the approximating sequences and are constituted by lower and upper solutions and monotonically converge to u, in an increasing or decreasing fashion, respectively; thus, for each n, the couple , brackets the solution u, thus providing lower and upper estimates. In what follows the -norm of a given function is intended to have been computed as the -norm of its discretization on the given grid. We have denoted by the minimum number of iterations needed in order that for ; the corresponding values are , and . In Table 1 we have tabulated , , for , at the mesh points ; the graphs of , are displayed in Figure 1; whereas Figure 2 describes the rate of decay of , as well as of the errors and , plotted against the number n of iterations. Here u denotes a reference approximation of the solution of (1), calculated using the same scheme up to a precision of 10−5. Although the lower solutions converge slightly faster than the upper solutions , it is evident that the monotone iterative scheme defined by (41) and (42) turns out to be extremely slow.
Figure 1. Graphs of the approximations,(in violet),,(in green) and,(in blue), with,, defined by (41), (42) with, such thatfor.
Figure 2. Graphs of(in blue),(in green) and(in violet), for, plotted against the numbernof iterations.
Table 1. Values of the approximations,, defined by (41), (42) with, such thatfor
The iterative scheme in case We start from the obvious observation that the iterative scheme given by (41) and (42) is well defined for any fixed ; hence it is clear that, if the resulting sequences and are Cauchy sequences in , then, by the uniqueness of the solution of (1), they converge in to u. Of course, if we cannot anymore guarantee that either is a lower solution, or is an upper solution, or the sequences and enjoy any monotonicity property. Let us take in (41) and let be the sequence of iterates obtained for some given . The numerical experiments, we have performed for several different choices of , show that the sequence converges to u, but the magnitude of L strongly affects the speed of convergence; namely, as L decreases, the required number of iterations n in order that goes beneath a prescribed threshold, decreases. In particular, the speed of convergence significantly increases as L approximates 1 and, for this choice of L, it becomes comparable even with the speed of Newton’s method. Indeed, if we fix an error tolerance of 10−3, the iterative scheme defined by (41), with and , converges in 4 iterations, whereas Newton’s method, starting from too, converges in 2 iterations: these results are displayed in Tables 2 and 3. This computational remark suggests the possibility of using the iterative scheme also in case the condition fails; however, its convergence properties should be theoretically analyzed.
Table 2. Values of the approximations, defined by (41) with, for
Table 3. Values of the Newton approximationsfor
A comparison between the solutions of (1) and (3) Here we present a numerical comparison between the solution u of the fully nonlinear problem (1) and the solution of the partially linearized problem (3) investigated in . We have approximated u by the lower solution obtained by implementing the monotone iterative scheme given by (41), with , and stopping criterion . An approximation of , matching the one obtained in , has been calculated using the bvp4c routine of MatLab with a 100-point grid. Table 4 reports the values of u and at the mesh points and Figure 3 displays the graphs of u and .
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
This paper was written under the auspices of INdAM-GNAMPA. The first named author has been supported by Fundação para a Ciência e a Tecnologia (SFRH/BD/61484/2009). The last two named authors have been supported by Università di Trieste, in the frame of the FRA projects ‘Equazioni differenziali ordinarie: aspetti qualitativi e numerici’ and ‘Nonlinear Ordinary Differential Equations: Qualitative Theory, Numerics and Applications’. They also wish to thank Igor Moret for some useful discussions.
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