Existence, regularity and boundary behaviour of bounded variation

Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation Franco Obersnel and Pierpaolo Omari Dipartimento di Matematica e Informatica Universit` a degli Studi di Trieste Via A. Valerio 12/1, 34127 Trieste, Italia E-mail: [email protected], [email protected]

Dedicated to Jean Mawhin on the occasion of his 70th birthday August 29, 2011 Abstract We discuss existence and regularity of bounded variation solutions of the Dirichlet problem for the one-dimensional capillarity-type equation ”0 “ p u(−r) = a, u(r) = b. u0 / 1 + u0 2 = f (t, u) in ] − r, r[, We prove interior regularity of solutions and we obtain a precise description of their boundary behaviour. This is achieved by a direct and elementary approach that exploits the properties of the zero set of the right-hand side f of the equation. 2010 Mathematics Subject Classification: 34B15, 35J93, 76D45, 49Q20. Keywords and Phrases: quasilinear ordinary differential equation, capillarity equation, Dirichlet problem, bounded variation solution, classical solution, existence, regularity, boundary behaviour.

1

Introduction and statements

Let us consider the Dirichlet problem for the capillarity equation q (   2 div ∇u/ 1 + |∇u| = cu in Br , u=g on ∂Br ,

(1)

where Br is an open ball of radius r > 0 in RN , c > 0 and g : ∂Br → R. The following result of Serrin [25, p. 477] is classical.

1

2

Theorem 1.1. Let N ≥ 2 and c > 0 be given. Then, for each L > 0 there exists R > 0 such that for every r ∈ ]0, R[ and every g ∈ C 2 (∂Br ), with kgk∞ ≤ L, problem (1) has a ¯r ). unique solution u ∈ C 2 (B Aside from its own relevance, this result, or some extensions of its based on the same argument as in [25], has played a relevant role in the literature, because it has often been used as a technical tool to prove the regularity of solutions of problems involving the prescribed mean curvature operator (see, e.g., [8, 9, 10]). It is quite easy to see that a result similar to Theorem 1.1 does not hold in dimension N = 1. Example 1.1. For every r ∈ ]0, 2[ the Dirichlet problem 

u0 /

0 p 1 + u0 2 = u in ] − r, r[,

u(−r) = 0, u(r) = 2

(2)

has no solution u ∈ C 2 (] − r, r[) ∩ C 0 ([−r, r]). Indeed, if u were such a solution, then there should exist t0 ∈ ] − r, r[ such that u(t0 ) = 1 and u(t) > 1 in ]t0 , r]. Integrating the equation in (2) and using the boundary conditions, together with the assumption r ∈ ]0, 2[, would yield p 1 = u(r) − u(t0 ) < 2(r − t0 ) − (r − t0 )2 , which is a contradiction. On the other hand, Theorem 1 in [19] implies the existence of a solution u ∈ BV (−r, r) of the Dirichlet problem  p 0 u0 / 1 + u0 2 = cu in ] − r, r[, u(−r) = a, u(r) = b, (3) for any given a, b, c, r ∈ R, with c, r > 0. Namely, it follows from [19] that the functional Z r c 2 v dt (4) E(v) = J (v) + 2 −r has a unique global minimizer in BV (−r, r). In (4) we set, for any v ∈ BV (−r, r), Z r q 2 1 + |Dv| + |v(−r+ ) − a| + |v(r− ) − b|, J (v) = −r

where Z

r

−r

q

2

1 + |Dv| = sup

nZ

r

(vw10 + w2 ) dt | w1 , w2 ∈ C01 (] − r, r[) o and kw12 + w22 k∞ ≤ 1 , −r

v(−r+ ) is the right trace and v(r− ) is the left trace of v at the points −r and r, respectively. Any minimizer u ∈ BV (−r, r) of E in turn satisfies the variational inequality Z r J (v) − J (u) + cu(v − u) dt ≥ 0 −r

3

for all v ∈ BV (−r, r): this is what is generally meant by a bounded variation solution u of (3) (see, e.g., [14, 15, 18, 22]). In the light of Example 1.1, a bounded variation solution u cannot generally be expected to be a classical solution of (3), i.e., u ∈ C 2 (] − r, r[) ∩ C 0 ([−r, r]) and satisfies the equation in ] − r, r[ and the boundary conditions in the pointwise sense. Nevertheless, adapting to this setting an argument from [7], it is possible to prove the interior regularity of u, i.e., u ∈ C 2 (] − r, r[). Such an approach relies on some deep results proved in [16] and however requires some estimates whose obtention is quite delicate even in the one-dimensional case. It should be therefore preferable to devise a more direct and elementary proof, which might also provide more information. This is obtained here by facing the question from a rather different perspective. Indeed, we show a connection between regularity of solutions and regularity of the zero set of the right-hand side of the equation: a fact that seems to have never been explicitly pointed out before. As a consequence, we get a result which yields the interior regularity of bounded variation solutions of the Dirichlet problem for the generalized capillarity equation 0  p (5) u0 / 1 + u0 2 = f (t, u) in ] − r, r[, u(−r) = a, u(r) = b, where, e.g., f ∈ C 1 ([−r, r]×R) is strictly increasing with respect to the second variable. We also provide a precise description of the boundary behaviour of such solutions, depending on whether they attain or not the prescribed boundary conditions. Similarly to the above, a bounded variation solution of (5) is a function u ∈ BV (−r, r) such that f (·, u) ∈ L1 (−r, r) and Z r J (v) − J (u) + f (t, u)(v − u) dt ≥ 0 (6) −r

for all v ∈ BV (−r, r). Note that this is equivalent to saying that u is a global minimizer of the functional Ku : BV (−r, r) → R, defined by Z

r

Ku (v) = J (v) +

f (t, u)v dt.

(7)

−r

For later use, we also introduce the functional I : BV (−r, r) → R defined by Z

r

I(v) = J (v) +

F (t, v) dt, −r

where

Z F (t, s) =

s

f (t, ξ) dξ. 0

These definitions make sense if the following condition is assumed: (h0 ) f : [−r, r] × R → R satisfies the L1 -Carath´eodory conditions, i.e., for a.e. t ∈ [−r, r], f (t, ·) : R → R is continuous, for every s ∈ R, f (·, s) : [−r, r] → R is measurable, and, for each R > 0, there exists ρ ∈ L1 (−r, r) such that |f (t, s)| ≤ ρ(t) for a.e. t ∈ [−r, r] and every s ∈ [−R, R].

4

Note that a bounded variation solution of (5) is a subcritical point of I in the sense of convex analysis. Theorem 1.2. Let r > 0 be given. Assume that (h1 ) f : [−r, r] × R → R is continuous and locally Lipschitz with respect to the second variable, (h2 ) there exists a measurable function ϕ : [−r, r] → R, with ϕ(t) ≤ ϕ(t) < 1r in a set of positive measure, such that lim inf

|s|→+∞

F (t, s) ≥ −ϕ(t) |s|

1 r

in [−r, r] and

uniformly in [−r, r],

i.e., for every ε > 0 there exists sε > 0 such that, for a.e. t ∈ [−r, r] and s 6∈ ] − sε , sε [, F (t, s) ≥ −(ϕ(t) + ε) |s|, and (h3 ) there exists γ ∈ C 0 ([−r, r], [−∞, +∞]), which is of class C 1 in its proper domain, such that, for every t ∈ [−r, r] and s ∈ R, f (t, s) < 0, if s < γ(t), and f (t, s) > 0, if s > γ(t). Then, for any given a, b ∈ R, problem (5) has a bounded variation solution u, which is a global minimizer of the functional I in BV (−r, r). Moreover, u ∈ W 1,1 (−r, r)∩C 2 (]−r, r[), it satisfies the equation in (5) everywhere in ] − r, r[, there exist u0 (−r) = lim u0 (t) and t→−r

u0 (r) = lim u0 (t), and both t→r

either u(−r) = a,

or

u(−r) 6= a and sgn(u(−r) − a) · u0 (−r) = +∞,

(8)

and either u(r) = b,

or

u(r) 6= b and sgn(u(r) − b) · u0 (r) = −∞.

(9)

Assumption (h2 ) expresses an asymptotic control on the interaction of f with the first eigenvalue 1r of the minus 1-Laplace operator −(sgn(u0 ))0 with homogeneous Dirichlet boundary conditions in [−r, r], as it has been defined in [6]. For semilinear elliptic problems a similar assumption has been introduced in [17] as a generalization of the classical Hammerstein condition [13]. It is easy to see that, if (h2 ) fails, the solvability of (5) is not guaranteed. Example 1.2. Fix r > 0 and a, b ∈ R. Pick any λ < − 1r . Then problem (5), with f (t, s) = λ in [−r, r] × R, has no bounded variation solution. Indeed, if u ∈ BV (−r, r) were such a solution, i.e., a global minimizer of the functional Ku defined by (7), we would get Ku (k) = 2r + |k − a| + |k − b| + 2rλk ≤ 2(1 + rλ)k + 2r + |a| + |b| → −∞, as k → +∞, which is a contradiction. Assumption (h3 ) does not require any monotonicity on the function f , however it holds if some regularity and monotonicity conditions are assumed, in such a way that the implicit function theorem applies.

5

Corollary 1.3. Assume (h4 ) f ∈ C 1 ([−r, r] × R), (h5 )

∂f (t, s) > 0 ∂s

in [−r, r] × R,

and (h2 ). Then, for any given a, b ∈ R, problem (5) has a unique bounded variation solution u, which is the global minimizer of the functional I in BV (−r, r). Moreover, u ∈ W 1,1 (−r, r) ∩ C 3 (] − r, r[), it satisfies the equation in (5) everywhere in ] − r, r[, there exist u0 (−r) = lim u0 (t) and u0 (r) = lim u0 (t), and both (8) and (9) hold. t→−r

t→r

However there are cases where assumption (h3 ) can be easily checked, even by direct inspection, although (h5 ) fails. This happens for instance whenever the zero set of f can be explicitly determined, due to the special structure of f , e.g., if f (t, s) = g(t) · h(s), or f (t, s) = g(t) + h(s). Needless to say that (h3 ) holds if f has a definite sign; in this case a regularity result for positive solutions of (5) has been proved implicitly in [2, 3] (see also [12, 20] for related results). Theorem 1.2 will be obtained as the consequence of the following two statements, which may have an independent interest. The first result yields the existence and the approximability of a bounded variation solution of (5), assuming (h2 ) and replacing (h1 ) with the condition (h6 ) f : [−r, r] × R → R satisfies the Lp -Carath´eodory conditions for some p > 1, i.e., for a.e. t ∈ [−r, r], f (t, ·) : R → R is continuous, for every s ∈ R, f (·, s) : [−r, r] → R is measurable, and, for each R > 0, there exists ρ ∈ Lp (−r, r) such that |f (t, s)| ≤ ρ(t) for a.e. t ∈ [−r, r] and every s ∈ [−R, R]. Proposition 1.4. Assume (h6 ) and (h2 ). Then, for any given a, b ∈ R, problem (5) has a bounded variation solution u, which is a global minimizer of the functional I in BV (−r, r). 2,p Moreover, there exist sequences (εn )n in R+ (−r, r), satisfying, for each 0 and (un )n in W n, q 0  00 0 (10) εn un + un / 1 + u0n 2 = f (t, un ) in ] − r, r[, un (−r) = a, un (r) = b, lim εn = 0,

n→+∞

lim

n→+∞

√

(11)

εn ku0n kL2 = 0,

sup kun kW 1,1 < +∞,

(12)

n

and lim un = u

n→+∞

p

in L p−1 (−r, r).

(13)

The second statement is a regularity result for bounded variation solutions of (5), which can be approximated by solutions of the regularized problems (10). A local and weakened version of (h3 ), as expressed by (h7 ) below, suffices to reach the conclusion. Actually (h7 ) might be further weakened requiring that γ± possess finite Dini derivatives at any point of their proper domains; however, we do not discuss this topic here.

6

Proposition 1.5. Assume (h1 ). Let u ∈ BV (−r, r) be a bounded variation solution of 2,p (5) such that there exist sequences (εn )n in R+ (−r, r), for some p > 1, 0 and (un )n in W satisfying (10), (11), (12), and (13). Let M > 0 be such that sup kun k∞ < M

(14)

n

and suppose that (h7 ) there exist γ± ∈ C 0 ([−r, r], [−∞, +∞]), with γ− (t) ≤ γ+ (t) in [−r, r], which are of class C 1 in their respective proper domains, such that, for every t ∈ [−r, r] and s ∈ [−M, M ], f (t, s) < 0 if s < γ− (t), f (t, s) = 0 if γ− (t) ≤ s ≤ γ+ (t), and f (t, s) > 0 if s > γ+ (t). Then, u ∈ W 1,1 (−r, r) ∩ C 2 (] − r, r[), it satisfies the equation in (5) everywhere in ] − r, r[, there exist u0 (−r) = lim u0 (t) and u0 (r) = lim u0 (t), and both (8) and (9) hold. t→−r

t→r

In particular we get the following local regularity result. Corollary 1.6. Assume (h0 ). Let u ∈ BV (−r, r) be a non-constant bounded variation solution of (5). Suppose that (h8 ) f ∈ C 1 ([−r, r] × [ess inf u, ess sup u]) and (h9 )

∂f (t, s) > 0 ∂s

in [−r, r] × [ess inf u, ess sup u].

Then, u ∈ W 1,1 (−r, r)∩C 3 (]−r, r[) is the unique solution of (5) having range in [ess inf u, ess sup u], it satisfies the equation in (5) everywhere in ] − r, r[, there exist u0 (−r) = lim u0 (t) and u0 (r) = lim u0 (t), and both (8) and (9) hold. t→−r

t→r

Corollary 1.6, or minor variants of its, applies in particular to the study of regularity and boundary behaviour of bounded variation solutions of (5) in the presence of lower and upper solutions (see [21] for a description of the method of lower and upper solutions for (5) in the setting of bounded variation functions). Moreover, it can be used for proving the regularity of bounded variation solutions of other boundary value problems associated with the equation in (5). These topics are discussed elsewhere (see [4, 23]).

2

Proofs

Proof of Proposition 1.4. This proof is divided into three steps. Step 1. A positive definite homogeneous form. As usual we set, for v ∈ BV (−r, r), Z r nZ r o |Dv| = sup vw0 dt | w ∈ C01 (] − r, r[) and kwk∞ ≤ 1 −r

−r

and

Z

r

kvkBV =

Z

r

|Dv| + −r

|v| dt. −r

7 Claim. Let ϕ : [−r, r] → R be a measurable function such that ϕ(t) ≤ 1r a.e. in [−r, r] and ϕ(t) < 1r in a set of positive measure. Then there exists δ > 0 such that Z r Z r + + |Dv| + |v(−r )| + |v(r )| − ϕ|v| dt ≥ δkvkBV −r

−r

for all v ∈ BV (0, T ). Proof of the claim. It is clear that, replacing ϕ with max{ϕ, 0}, we can assume ϕ ∈ L∞ (−r, r). Let us first prove that Z Z r 1 r |v| dt (15) |Dv| + |v(−r+ )| + |v(r− )| ≥ r −r −r for all v ∈ BV (0, T ) and, if v ∈ BV (0, T ) is such that Z r Z 1 r |Dv| + |v(−r+ )| + |v(r− )| = |v| dt, r −r −r then v is constant a.e. in [−r, r]. Indeed, take v ∈ BV (0, T ). By the approximation property in BV (−r, r) [1, p. 491, p. 498], there exists a sequence (wn )n in W01,1 (−r, r) such that lim wn = v in L1 (−r, r), a.e. in [−r, r] and n→+∞

r

Z

|wn0 | dt

lim

n→+∞

Z

r

=

−r

|Dv| + |v(−r+ )| + |v(r− )|.

−r

Since

Z

r

|wn0 | dt

2|wn (t)| ≤ −r

a.e. in [−r, r], we deduce that r

Z

|Dv| + |v(−r+ )| + |v(r− )|

2|v(t)| ≤ −r

a.e. in [−r, r] and hence Z

r

|Dv| + |v(−r+ )| + |v(r− )|

2 kvk∞ ≤ −r

and

1 r

Z

r

Z

r

|v| dt ≤ −r

|Dv| + |v(−r+ )| + |v(r− )|.

−r

Z

r

Let now v ∈ BV (−r, r), with

|v| dt = 1, be such that −r

Z

r

−r

|Dv| + |v(−r+ )| + |v(r− )| = 1r .

Note that, as w = |v| satisfies Z r Z |Dw| + |w(−r+ )| + |w(r− )| ≤ −r

r

−r

|Dv| + |v(−r+ )| + |v(r− )|,

8

we have

Z

r

|Dw| + |w(−r+ )| + |w(r− )| =

−r

1 r

Rr 1 too. Since ess sup w ≤ and −r |w| dt = 1, we deduce that w = 2r a.e. in [−r, r], that is 1 1 1 |v| = 2r a.e. in [−r, r]. Assume by contradiction that ess sup v = 2r and ess inf v = − 2r . According to [5, Section 2.3], we can assume that v ∈ N BV (−r, r). By [5, Theorem 2.30-(ii)] we have Z 1 2r

r

|Dv|.

ess sup v − ess inf v ≤ −r

1 As |v(−r+ )| = |v(r− )| = 2r , we conclude that Z r 2 1 |Dv| + |v(−r+ )| + |v(r− )| ≥ ess sup v − ess inf v + |v(−r+ )| + |v(r− )| = , = r r −r 1 1 which is a contradiction. Therefore we have that either v = 2r a.e. in [−r, r], or v = − 2r a.e. in [−r, r]. In order to conclude the proof of the claim we suppose, by contradiction, that there exists a sequence (vn )n in BV (−r, r) such that, for each n,

kvn kBV = 1 and

Z

r

|Dvn | + |vn (−r+ )| + |vn (r− )| −

−r

Z

r

ϕ|vn | dt ≤ −r 1

1 . n

(16)

Possibly passing to a subsequence, (vn )n converges in L (−r, R r r) to some v ∈ BV (0, T ). Recall that the functional which sends v ∈ BV (−r, r) onto −r |Dv| + |v(−r+ )| + |v(r+ )| is lower semicontinuous with respect to the L1 -convergence [11]. Hence, using the properties of ϕ, in combination with (15) and (16), we deduce that Z Z r Z r Z 1 r 1 r |v| dt ≤ |Dv| + |v(−r+ )| + |v(r+ )| ≤ ϕ|v| dt ≤ |v| dt r −r r −r −r −r and hence

1 r

Z

r

Z

r

|Dv| + |v(−r+ )| + |v(r+ )| =

|v| dt = −r

−r

lim

n→+∞

Z

r

r

ϕ|v| dt.

(17)

−r

Let us observe that v 6= 0. Indeed, otherwise we would have and, by (16),

Z

lim vn = 0 in L1 (−r, r)

n→+∞

 |Dvn | + |vn (−r+ )| + |vn (r+ )| = 0,

−r

thus yielding the contradiction 1 = lim kvn kBV = 0. n→+∞

Therefore, the preceding step implies that v = K a.e. in [−r, r], for some constant K 6= 0. From (17) and the properties of ϕ, we infer the contradiction Z r 0= (ϕ − 1r )|K| dt < 0. −r

Thus the claim is proved.

9 Step 2. A regularized problem. Take a sequence (εn )n in R+ 0 with sider, for each n, the regularized problem  p 0 εn u00 + u0 / 1 + u0 2 = f (t, u)

in ] − r, r[,

lim εn = 0. Con-

n→+∞

u(−r) = a, u(r) = b

(18)

and the associated functional In : C → R defined by Z Z r q Z r εn r 0 2 2 |v | dt + In (v) = 1 + |v 0 | dt + F (t, v) dt, 2 −r −r −r where C = {v ∈ H 1 (−r, r) | v(−r) = a, v(r) = b}. Note that C is closed and convex in H 1 (−r, r) and therefore weakly closed. By (h1 ) In is weakly lower semicontinuous. Moreover, by (h2 ), using Step 1, we see that In is bounded from below. Indeed, there exists ρ ∈ L1 (−r, r) such that, for a.e. t ∈ [−r, r] and every s ∈ R, F (t, s) ≥ −(ϕ(t) + 2δ )|s| + ρ(t) and hence, for every v ∈ C, Z Z r Z r Z Z r εn r 0 2 δ r In (v) ≥ |v | dt + |v 0 | dt − ϕ|v| dt − |v| dt + ρ dt 2 −r 2 −r −r −r −r εn 0 2 δ kv kL2 + kvkW 1,1 − |a| − |b| − kρkL1 . ≥ 2 2

(19)

Fix n and let (vk )k be a minimizing sequence for In in C. By (19) we see that (vk )k is bounded in H 1 (−r, r). Accordingly, there exists a subsequence of (vk )k , we still denote by (vk )k , which is weakly convergent to some un . Note that, by the weak closedness of C, un ∈ C. The weak lower semicontinuity of In finally yields In (un ) = min In . C

Note that, setting σ(t) = a + δ 1,1 2 kun kW

b−a 2r (t

+ r) for t ∈ [−r, r], we have by (19)

− |a| − |b| − kρkL1 ≤ In (un ) Z r Z ≤ In (σ) ≤ |σ 0 |2 dt + −r

r

Z q 2 1 + |σ 0 | dt +

r

F (t, σ) dt.

−r

−r

Hence we get sup kun kW 1,1 < +∞ n

and, in particular, sup kun k∞ < +∞.

(20)

n

It is easy to verify that each un satisfies Z Z r   0 0 1 √ un w dt + εn + 0 2 −r

1+|un |

r

f (t, un )w dt = 0,

−r

for every w ∈ H01 (−r, r). Hence un ∈ W 2,p (−r, r) is a (strong) solution of (18).

(21)

10 p Step 3. Convergence of the approximation scheme. Set q = p−1 > 1. As the sequence 1,1 (un )n is bounded in W (−r, r), possibly passing to a subsequence we still denote by (un )n , it converges in Lq (−r, r) and a.e. in [−r, r] to a function u ∈ BV (−r, r). Let us prove that u is a bounded variation solution of (5). Fix n and pick z ∈ C. Take w = z − un ∈ H01 (−r, r) as a test function in (21). By convexity, we get Z Z εn r 0 2 εn r 0 2 |z | dt + J (z) − |u | dt − J (un ) ≥ 2 −r 2 −r n Z r  Z r  0 0 1 √ ≥ εn + un (z − un ) dt = − f (t, un )(z − un ) dt. (22) 0 2 1+|un |

−r

−r

Since we have, by the lower semicontinuity of J with respect to the L1 -convergence [11], lim inf J (un ) ≥ J (u) n→+∞

and, using (20) and the Lebesgue dominated convergence theorem, Z r Z r lim f (t, un )(z − un ) dt = f (t, u)(z − u) dt, n→+∞

−r

−r

we conclude that Z

r

J (z) ≥ J (u) −

f (t, u)(z − u) dt.

(23)

−r

Now, take v ∈ BV (−r, r). By the approximation property in BV (−r, r) [1, p. 491, p. 498], there exists a sequence (zk )k in C such that lim zk = v in Lq (−r, r) and lim J (zk ) = k→+∞

k→+∞

J (v). From (23) we infer Z

r

J (v) ≥ J (u) −

f (t, u)(v − u) dt, −r

i.e., u is a bounded variation solution of (5). Moreover, letting n → +∞ in (22), we get Z Z r εn r 0 2 |un | dt + J (u) − f (t, u)(z − u) dt, J (z) ≥ lim sup n→+∞ 2 −r −r for every z ∈ C, and therefore, by the same approximation procedure used before, Z Z r εn r 0 2 |un | dt + J (u) − J (v) ≥ lim sup f (t, u)(v − u) dt, n→+∞ 2 −r −r

(24)

for every v ∈ BV (−r, r). Taking v = u as a test function in (24), we conclude that Z εn r 0 2 |un | dt = 0. lim n→+∞ 2 −r On the other hand, as un is a minimizer of In in C, we have Z Z r εn r 0 2 In (un ) = |un | dt + J (un ) + F (t, un ) dt 2 −r −r Z Z r εn r 0 2 ≤ In (v) = |v | dt + J (v) + F (t, v) dt, 2 −r −r

(25)

11

for every v ∈ C. Letting n → +∞ in (25) and arguing as above, we get Z r Z r F (t, v) dt = I(v), F (t, u) dt ≤ J (v) + I(u) = J (u) + −r

−r

for every v ∈ BV (−r, r), i.e., u is a minimizer of I in BV (−r, r). Proof of Proposition 1.5. This proof is divided into two steps. Step 1. Interior regularity. Let us set

Let us denote by E± the proper domains of γ± , respectively. ˜± = {t ∈ E± | |γ± (t)| < M } E

0 ˜± are relatively open in [−r, r] and k˜ and γ˜± = γ± |E˜± . Note that E γ± k∞ < +∞. Fix ε ∈ ]0, r[ and pick any un from the sequence (un )n approximating u.

Claim. For every t ∈ [−r + ε, r − ε], |u0n (t)| ≤ max

 2M ε

0 0 , k˜ γ− k∞ , k˜ γ+ k∞ .

Proof of the claim. Fix t ∈ [−r + ε, r − ε]. We distinguish three cases. (i) Let [α, β], with α < β and t ∈ [α, β], be a maximal interval contained in [−r, r], such that u00n (τ ) < 0 in ]α, β[. Assume first that α = −r and β = r. We have, by (14), −2M ≤ un (−r) − un (−r + ε) = −u0n (ξ)ε for some ξ ∈ ] − r, −r + ε[, and hence, as u0n is decreasing in [α, β], −2M ≤ −u0n (−r + ε)ε. Similarly we see that −2M ≤ u0n (r − ε)ε. Using again the monotonicity of u0n , we obtain 0 0 0 − 2M ε ≤ un (r − ε) ≤ un (τ ) ≤ un (−r + ε) ≤

2M ε ,

that is |u0n (τ )| ≤

2M ε ,

in [−r + ε, r − ε]. Assume next that α > −r and β = r. As f (τ, un (τ )) < 0 in ]α, β[ and f (α, un (α)) = 0, ˜− and we have un (τ ) < γ− (τ ) in ]α, β[ and γ− (α) = un (α) ∈ ] − M, M [. Therefore α ∈ E 0 0 un (τ ) ≤ γ˜− (τ ) in a right neighbourhood of α. This implies that un (α) ≤ γ˜− (α). Since u0n is decreasing, arguing as above we obtain 0 γ˜− (α) ≥ u0n (α) ≥ u0n (τ ) ≥ u0n (r − ε) ≥ − 2M ε ,

that is 0 |u0n (τ )| ≤ max{ 2M γ− k∞ }, ε , k˜

12

in [α, β] ∩ [−r + ε, r − ε]. A similar argument is used to prove that 0 |u0n (τ )| ≤ max{ 2M γ+ k∞ } ε , k˜

in [α, β] ∩ [−r + ε, r − ε], if we assume that α = −r and β < r, or to prove that 0 0 |u0n (τ )| ≤ max{k˜ γ− k∞ , k˜ γ+ k∞ }

in [α, β] ∩ [−r + ε, r − ε], if we assume that α > −r and β < r. (ii) Let [α, β], with α < β and t ∈ [α, β], be a maximal interval contained in [−r, r], such that u00n (τ ) > 0 in ]α, β[. We repeat the argument used in (i) to prove that  0 0 |u0n (τ )| ≤ max 2M γ− k∞ , k˜ γ+ k∞ ε , k˜ in [α, β] ∩ [−r + ε, r − ε]. (iii) Let t ∈ [−r, r] be a limit point of zeroes of u00n . This implies that either • there exists a sequence (tk )k , with lim tk = t and tk 6= t for all k, such that un (tk ) = k→+∞

γ˜− (tk ) for all k, or un (tk ) = γ˜+ (tk ) for all k, or • there exists a maximal interval [λ, µ] ⊆ [−r, r], with t ∈ [λ, µ], such that u00n (τ ) = 0 in [λ, µ]. Assume in the former case that un (tk ) = γ˜− (tk ) for all k. A similar argument works if ˜− and un (t) = γ˜− (t). Thus we conclude un (tk ) = γ˜+ (tk ) for all k. We have, by (14), t ∈ E 0 that u0n (t) = γ˜− (t) and therefore 0 |u0n (t)| ≤ k˜ γ− k∞ .

In the latter case the following situations may occur. ? If λ = −r and µ = r, then 2M ku0n k∞ ≤ M r < ε . ? If λ = −r and µ < r, then un (µ) = γ˜− (µ), or un (µ) = γ˜+ (µ). Suppose that un (µ) = γ˜− (µ). A similar argument applies if un (µ) = γ˜+ (µ). If µ is the left endpoint of an interval where u00n (τ ) < 0, then from (i) we infer that 0 0 |u0n (t)| = |u0n (µ)| ≤ max{ 2M γ− k∞ , k˜ γ+ k∞ }. ε , k˜

If there is a sequence (tk )k , with

lim tk = µ and tk > µ for all k, such that un (tk ) =

k→+∞

γ˜− (tk ) for all k, then arguing as above we deduce that 0 |u0n (t)| = |u0n (µ)| ≤ k˜ γ− k∞ .

In any case we conclude that 0 0 γ− k∞ , k˜ γ+ k∞ }. |u0n (t)| ≤ max{ 2M ε , k˜

? If λ > −r and µ ≤ r, then the previous argument yields (26) again. This concludes the proof of the claim.

(26)

13 0 0 Set N1 = max{ 2M γ− k∞ , k˜ γ+ k∞ } and N2 = max{f (t, s) | (t, s) ∈ [−r, r]×[−M, M ]}. ε , k˜ From the equation in (18) we obtain, by the claim and by (14),

|u00n (t)|

=

1 + u0n (t)2 1 + εn 1 +


3/2


3/2 u0n (t)2

|f (t, un (t))| ≤ 1 + N12


3/2

N2

in [−r + ε, r − ε]. By Arzel` a-Ascoli theorem we infer that, possibly passing to a subsequence we still denote by (un )n , the sequence (un )n converges to u in C 1 ([−r + ε, r − ε]). From (18) we derive that u ∈ C 2 ([−r + ε, r − ε]) and fulfills the equation in (5) in [−r + ε, r − ε]. As ε ∈ ]0, r[ is arbitrary, we conclude that u ∈ C 2 (] − r, r[) and u satisfies the equation in (5) in ] − r, r[. Furthermore, as (un )n converges pointwise to u in ] − r, r[, by (14) we conclude that kuk∞ < M too. Note that u ∈ BV (−r, r) ∩ C 2 (] − r, r[) and hence u ∈ W 1,1 (−r, r) ∩ C 2 (] − r, r[). Step 2. Boundary behaviour. We know that u ∈ C 2 (] − r, r[) ∩ C 0 ([−r, r]). Claim 1. There exist, possibly infinite, u0 (−r) = lim u0 (t) t→−r

and

u0 (r) = lim u0 (t). t→r

Proof of Claim 1. We only prove the former conclusion, as the latter one follows in a similar way. Assume first that there exists ε ∈ ]0, r[ such that u00 (t) ≤ 0 in ] − r, −r + ε], or u00 (t) ≥ 0 in ] − r, −r + ε]. Then u0 is monotone in ] − r, −r + ε] and hence there exists lim u0 (t). t→−r

Assume next that there exist sequences (sk )k and (tk )k such that lim sk = lim tk = −r,

k→+∞

k→+∞

and −r < sk+1 < tk < sk < r and u00 (sk ) < 0 < u00 (tk ), ˜+ ∩ E ˜− and γ˜− (−r) = u(−r) = γ˜+ (−r). Let us prove for all k. Clearly we have −r ∈ E 0 0 that γ˜− (−r) = γ˜+ (−r) too. For each k we set 0 µk = min{t ∈ ] − r, sk [ | u(t) = γ˜− (t), u0 (t) ≤ γ˜− (t), u(τ ) ≤ γ˜+ (τ ) in [t, sk ]}, 0 λk = max{t ∈ ] − r, µk ][ | u(t) = γ˜+ (t), u0 (t) ≥ γ˜+ (t)}, 0 νk = max{t ∈ ]tk , r] | u(t) = γ˜+ (t), u0 (t) ≤ γ˜+ (t), u(τ ) ≥ γ˜− (τ ) in [tk , t]}, 0 ξk = min{t ∈ [νk , r[ | u(t) = γ˜− (t), u0 (t) ≥ γ˜− (t)}.

As u00 (t) ≥ 0 in [λk , µk ] and u00 (t) ≤ 0 in [νk , ξk ], we have 0 0 γ˜+ (λk ) ≤ u0 (λk ) ≤ u0 (µk ) ≤ γ˜− (µk )

and 0 0 γ˜− (ξk ) ≤ u0 (ξk ) ≤ u0 (νk ) ≤ γ˜+ (νk ). 0 0 Therefore we conclude that γ˜− (−r) = γ˜+ (−r). We finally prove that there exists 0 0 u0 (−r) = lim u0 (t) = γ˜− (−r) = γ˜+ (−r). t→−r

14 ˜+ ∩ E ˜− , and pick t there. If t Indeed, fix a right neighbourhood of −r, contained in E 00 belongs to a maximal interval [λ, µ] where u (τ ) ≥ 0, then 0 0 γ˜− (λ) ≤ u0 (λ) ≤ u0 (t) ≤ u0 (µ) ≤ γ˜− (µ).

If t belongs to a maximal interval [ν, ξ] where u00 (τ ) ≤ 0, then 0 0 γ˜+ (ξ) ≤ u0 (ξ) ≤ u0 (t) ≤ u0 (ν) ≤ γ˜+ (ν).

If u00 changes sign in any neighbourhood of t, then γ˜+ (t) = u(t) = γ˜− (t) and 0 0 γ˜+ (t) = u0 (t) = γ˜− (t).

From these estimates we get the conclusion. Notice that u0 (−r) is finite in this case. This concludes the proof of the claim. Claim 2. We have both either

u(−r) = a

sgn(u(−r) − a)u0 (−r) = +∞

or

and either

u(r) = b

or

sgn(b − u(r))u0 (r) = +∞.

Proof of Claim 2. We only prove the former conclusion, as the latter one follows in a similar way. Assume, by contradiction, that u(−r) 6= a and u0 (−r) ∈ R. Note that un is the solution of the Cauchy problem ( 3
−1 2 in [−r, 0], v 00 = f (t, v) εn + (1 + v 0 )− 2 v(0) = un (0), v 0 (0) = u0n (0), and u is the solution of the Cauchy problem ( 3 2 v 00 = f (t, v)(1 + v 0 )− 2 in [−r, 0], 0 0 v(0) = u(0), v (0) = u (0). By the continuous dependence on parameters and initial conditions, which follows from (h1 ), we get the contradiction a = lim un (−r) = u(−r) 6= a. n→+∞

Hence we conclude that, if u(−r) 6= a, then u0 (−r) = −∞, or u0 (−r) = +∞. We shall prove now that, in case u(−r) < a, we have u0 (−r) = −∞, while, in case u(−r) > a, we have u0 (−r) = +∞. • Assume u(−r) < a. From the proof of Claim 1 we know that there exists ε ∈ ]0, r[ such that u00 (t) ≤ 0 in ] − r, −r + ε], or u00 (t) ≥ 0 in ] − r, −r + ε]. Suppose, by contradiction, that the former condition holds. Then we have u0 (−r) = +∞ and, possibly reducing ε > 0, u00 (t) < 0

in ] − r, −r + ε].

Indeed, otherwise −r ∈ E− , the proper domain of γ− , u(−r) = γ− (−r) and u0 (−r) = 0 γ− (−r) ∈ R: a contradiction. Therefore we get u(t) < γ− (t)

in ] − r, −r + ε].

15

Possibly further reducing ε > 0, we can also assume that u0 (−r + ε) > 0,

u(−r + ε) < a, and u(−r + ε)
0

un (−r + ε) < a,

and

un (−r + ε)
a. A similar argument shows that u0 (−r) = +∞. This concludes the proof of the claim. Proof of Theorem 1.2. Theorem 1.2 immediately follows from Proposition 1.4 and Proposition 1.5. Proof of Corollary 1.3. We apply Theorem 1.2. Let us prove that (h4 ) and (h5 ) imply (h3 ). Let us set Γ = {(t, s) ∈ [−r, r] × R | f (t, s) = 0} and E = {t ∈ [−r, r] | (t, s) ∈ Γ for some s ∈ R}. Assume Γ 6= ∅. Conditions (h4 ) and (h5 ) imply that Γ is closed and E is relatively open in [−r, r]. Moreover, there exists a function γ : E → R of class C 1 whose graph is Γ. Note that E \ {−r, r} is the countable union of disjoint open intervals ]αn , βn [, such that, for each n, lim+ γ(t) = ±∞ and lim− γ(t) = ±∞, t→αn

t→βn

with the exception of the case where, for some n, αn = −r ∈ E, or βn = r ∈ E. In addition, if ]βm , αn [ ∩ E = ∅ for some m, n, then lim γ(t) = lim+ γ(t) = ±∞.

− t→βm

t→αn

16

It is then clear that we can extend the function γ to the whole of [−r, r] in such a way that the extension, we still denote by γ, belongs to C 0 ([−r, r], [−∞, +∞]). This function satisfies, for every t ∈ [−r, r] and s ∈ R, f (t, s) < 0, if s < γ(t), and f (t, s) > 0, if s > γ(t). Of course, if Γ = ∅, we set γ(t) = −∞ in [−r, r], or γ(t) = +∞ in [−r, r], according to the sign of f . Hence Theorem 1.2 applies. By (h4 ) we also have u ∈ C 3 (] − r, r[). Let us prove now the uniqueness of the solution u. Assume that u1 , u2 are solutions of (5). Testing (6), with u = u1 , against u2 and (6), with u = u2 , against u1 , we obtain Z r f (t, u1 )(u2 − u1 ) dt ≥ 0 J (u2 ) − J (u1 ) + −r

and

Z

r

f (t, u2 )(u1 − u2 ) dt ≥ 0.

J (u1 ) − J (u2 ) + −r

Summing the two inequalities we get Z r (f (t, u2 ) − f (t, u1 ))(u2 − u1 ) dt ≤ 0. −r

By (h5 ) we conclude that u1 = u2 in [−r, r]. Proof of Corollary 1.6. Set c = ess inf u and d = ess sup u. Define g : [−r, r] × [c, d] ∪ (R2 \ [−r − 1, r + 1] × [c − 1, d + 1]) → R by setting  f (t, s) in [−r, r] × [c, d], g(t, s) =  0 in R2 \ [−r − 1, r + 1] × [c − 1, d + 1]. Since f ∈ C 1 ([−r, r]×[c, d]), according to [24, Th´eor`eme 1], the function g can be extended to a function gˆ ∈ C 1 (R2 ). Let g˜ be the restriction of gˆ to [−r, r] × R. Choose a constant M > 0 so large that the function f˜ : [−r, r] × R → R, defined by f˜(t, s) = g˜(t, s) + M ((s − d)+ )2 − M ((s − c)− )2 , satisfies

∂ f˜ (t, s) > 0 ∂s

in [−r, r] × R. Hence, setting F˜ (t, s) =

Z

s

f˜(t, ξ) dξ,

0

we see that assumptions (h2 ), (h4 ), and (h5 ) are satisfied, with f and F replaced by f˜ and F˜ , respectively. According to Corollary 1.3, there exists a unique bounded variation solution z of  p 0 z 0 / 1 + z 0 2 = f˜(t, z) in ] − r, r[, z(−r) = a, z(r) = b. (28) Moreover, z ∈ W 1,1 (−r, r) ∩ C 3 (] − r, r[), it satisfies the equation in (28) everywhere in ] − r, r[, there exist z 0 (−r) = lim z 0 (t) and z 0 (r) = lim z 0 (t), and both t→−r

t→r

17

either z(−r) = a,

or z(−r) 6= a and sgn(z(−r) − a) · z 0 (−r) = +∞,

and either z(r) = b,

or z(r) 6= b and sgn(z(r) − b) · z 0 (r) = −∞.

Since, by definition of f˜, u is a solution of (28) as well, we conclude, by uniqueness, that u = z.

References [1] G. Anzellotti, The Euler equation for functionals with linear growth, Trans. Amer. Math. Soc. 290 (1985), 483–501. [2] D. Bonheure, P. Habets, F. Obersnel, and P. Omari, Classical and non-classical solutions of a prescribed curvature equation, J. Differential Equations 243 (2007), 208–237. [3] D. Bonheure, P. Habets, F. Obersnel, and P. Omari, Classical and non-classical positive solutions of a prescribed curvature equation with singularities, Rend. Ist. Mat. Univ. Trieste 39 (2007), 63–85. [4] D. Bonheure, F. Obersnel, and P. Omari, Heteroclinic solutions of the prescribed curvature equation, preprint (2011). [5] G. Buttazzo, M. Giaquinta, and S. Hildebrandt, “One-dimensional Variational Problems. An Introduction”, Clarendon Press, Oxford, 1998. [6] K.C. Chang, The spectrum of the 1-Laplace operator, Commun. Contemp. Math. 11 (2009), 865–894. [7] M. Emmer, Esistenza, unicit` a e regolarit`a nelle superfici di equilibrio nei capillari, Ann. Univ. Ferrara Sez. VII (N.S.) 18 (1973), 79–94. [8] C. Gerhardt, Existence and regularity of capillary surfaces, Boll. Un. Mat. Ital. (4) 10 (1974), 317–335. [9] C. Gerhardt, Existence, regularity, and boundary behavior of generalized surfaces of prescribed mean curvature, Math. Z. 139 (1974), 173–198. [10] M. Giaquinta, Regolarit` a delle superfici BV (Ω) con curvatura media assegnata, Boll. Un. Mat. Ital. (4) 8 (1973), 567–578. [11] E. Giusti, “Minimal Surfaces and Functions of Bounded Variations”, Birkh¨auser, Basel, 1984. [12] P. Habets and P. Omari, Multiple positive solutions of a one-dimensional prescribed mean curvature problem, Commun. Contemp. Math. 9 (2007), 701–730. [13] A. Hammerstein, Nichtlineare Integralgleichungen nebst Anwendungen, Acta Math. 54 (1930), 117–176. [14] V.K. Le, Some existence results on non-trivial solutions of the prescribed mean curvature equation, Adv. Nonlinear Stud. 5 (2005), 133–161.

18

[15] V.K. Le, Variational method based on finite dimensional approximation in a generalized prescribed mean curvature problem, J. Differential Equations 246 (2009), 3559–3578. [16] U. Massari, Esistenza e regolarit`a delle ipersuperficie di curvatura media assegnata in Rn , Arch. Rational Mech. Anal. 55 (1974), 357–382. [17] J. Mawhin, J. R. Ward Jr., and M. Willem, Variational methods and semilinear elliptic equations, Arch. Rational Mech. Anal. 95 (1986), 269–277. [18] A. Mellet and J. Vovelle, Existence and regularity of extremal solutions for a meancurvature equation, J. Differential Equations 249 (2010) 37–75. [19] M. Miranda, Dirichlet problem with L1 data for the non-homogeneous minimal surface equation, Indiana Univ. Math. J. 24 (1974/75), 227–241. [20] F. Obersnel, Classical and non-classical sign changing solutions of a one-dimensional autonomous prescribed curvature equation, Adv. Nonlinear Stud. 7 (2007), 1–13. [21] F. Obersnel and P. Omari, Existence and multiplicity results for the prescribed mean curvature equation via lower and upper solutions, Differential Integral Equations 22 (2009), 853–880. [22] F. Obersnel and P. Omari, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation, J. Differential Equations 249 (2010), 1674–1725. [23] F. Obersnel, P. Omari, and S. Rivetti, Existence, regularity and stability of periodic solutions of a capillarity equation in the presence of lower and upper solutions, preprint (2011). [24] L. Schwartz, Les th´eor`emes de Whitney sur les fonctions diff´erentiables, S´eminaire Bourbaki, Vol. 1, Exp. No. 43, 355–363, Soc. Math. France, Paris, 1995. [25] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 413– 496.