Non-coercive radially symmetric variational problems: Existence, symmetry and convexity of minimizers

We prove existence of radially symmetric solutions and validity of Euler-Lagrange necessary conditions for a class of variational problems such that neither direct methods nor indirect methods of Calculus of Variations apply. We obtain existence and qualitative properties of the solutions by means of ad-hoc superlinear perturbations of the functional having the same minimizers of the original one.


Introduction
This paper is concerned with the variational problem where B R ⊆ R N is the open ball centered at the origin and with radius R > 0. Under the sole assumptions of increasing monotonicity of the Lagrangian with respect to the gradient variable one can prove, by means of a symmetrization procedure proposed in [31], that the problem admits a one-dimensional reduction, obtained by evaluating the functional only on the set of radially symmetric functions (see Section 3). This reduction step leads to consider the minimum problem The qualitative features of the Lagrangian are that g(r, ·) is convex (in fact this assumption can be dropped in the autonomous case, see Corollary 5.4) and with, at least, linear growth, while h(r, t) is Lipschitz continuous in the t variable. These assumptions do not assure that every minimizing sequence of the functional is precompact in L 1 , and hence the direct methods of Calculus of Variations fails. For this reason indirect methods, based on the solvability of the associated Euler-Lagrange equations, have often been adopted in the literature (see [2,4,[7][8][9][10][11][12][22][23][24]32]). Specifically, if the Lagrangian is convex with respect to both variables u and |u |, then any solution of the Euler-Lagrange conditions provides a minimizer, and vice-versa.
The main feature of the present work is that we do not require convexity of the Lagrangian in the u variable, so that the above mentioned indirect methods cannot be implemented, and a brand-new approach is needed. Our starting points are an existence result and the validity of the Euler-Lagrange necessary conditions under the additional requirement that g(r, ·) has superlinear growth. These properties can be easily obtained applying well-known results (see Step 1 of the proof of Theorem 4.1). Exploiting the necessary conditions, we obtain explicit a-priori estimates on the derivative of minimizers of superlinear functionals, that depend on the Lipschitz constant of h(r, ·).
When g(r, ·) satisfies only a linear growth condition, say g(r, s) ≥ M s−C for some positive constants M and C, and the Lipschitz constant of h(r, ·) is not too large compared with M (see the compatibility relation (hgr) between g and h in the statement of Theorem 4.1), then we proceed as follows. As a first step, we construct an ad-hoc superlinear perturbation of the slow growth functional, for which we have a Lipschitz minimizer satisfying some a-priori estimates. Then, relying on these estimates, we show that this function is in fact a minimizer of the original slow-growth problem.
In some sense, our technique is reminiscent of the semiclassical approach, based on the construction of barrier functions, for the minimization of functionals of the type Ω L(∇u) dx on functions u ∈ W 1,1 (Ω) satisfying some prescribed boundary condition (see, e.g., [29,Chapter 1]).

Notation and preliminaries
In what follows | · | will denote the Euclidean norm in R N , N ≥ 1, and B R ⊂ R N is the open ball centered at the origin and with radius R > 0. We shall denote by A and int A respectively the closure and the interior of a set A, and by Dom ϕ the essential domain of an extended real-valued function ϕ : A →]−∞, +∞], i.e. Dom ϕ = {x ∈ A : ϕ(x) < +∞}. We shall always consider proper functions, that is Dom ϕ = ∅.
Given a locally Lipschitz function ϕ : A ⊂ R → R, for every x ∈ A we denote by ∂ϕ(x) its generalized gradient at x in the sense of Clarke (see [6,Chapter 2] For notational convenience, if ϕ also depends on an additional variable r ∈ R, we denote by ∂ϕ(r, x) the generalized gradient of the function x → ϕ(r, x).
We say that f :

Symmetry of minimizers
In this section we deal with the symmetry properties of minimizers in W 1,1 0 (B R ) of functionals of the form under very mild assumptions on the Lagrangian f . Our aim is to prove that the minimization problem for F in W 1,1 0 (B R ) is, in fact, equivalent to the minimization problem for the one-dimensional functional in the functional space Remark 3.1. Notice that the functional F rad is, up to a constant factor, the functional F evaluated on the radially symmetric functions belonging to W 1,1 0 (B R ). In particular, we underline that every function u ∈ W 1 rad satisfies We adopt a symmetrization procedure introduced in [31]. Given a representative of u ∈ W 1,1 0 (B R ), and θ ∈ ∂B 1 , let be the radial symmetric function obtained from the profile of u along the straight line through 0 and with direction θ.
In [31,Lemma 3.1] it is proved that u θ ∈ W 1,1 0 (B R ) for a.e. θ ∈ ∂B 1 , and Following the lines of the proof of [31, Theorem 3.4], we show that, for some θ, u θ is a better competitor than u in the minimization problem for F .
, then it admits a radially symmetric minimizer.
Proof. Let u be a function in W 1,1 0 (B R ) such that F (u) < +∞, and let u θ be the radially symmetric function defined in (3). We claim that, using (4) and the monotonicity property of the Lagrangian f , we obtain that θ ∈ ∂B 1 , and (5) implies that hence almost every u θ is a (radially symmetric) minimizer of F . Assume now that for almost every (r, t) ∈ [0, R] × R, the map s → f (r, t, s) is strictly monotone increasing, and let u be a minimizer for F . From the computation above, we deduce that (6) holds if and only if Since u θ (rθ) = u(rθ) for a.e. (r, θ), from the strict monotonicity assumption on f we deduce that |∇u θ (rθ)| = |∇u(rθ)| for L × H N −1 -a.e. (r, θ), hence, from (4), we obtain that ∇u(rθ) is parallel to θ and then u is radially symmetric (see [31,Lemma 3.3]).
As a consequence of Theorem 3.2, we obtain the following 1-dimensional reduction of the minimum problem.

Corollary 3.3. Let f be as in Theorem 3.2. Then the minimization problem
} admits a solution if and only if the one-dimensional minimization problem (8) min{F rad (u) : u ∈ W 1 rad } admits a solution, where F rad and W 1 rad are defined in (1) and (2) respectively. Proof. If problem (7) admits a solution u ∈ W 1,1 0 (B R ), then by Theorem 3.2 there exists a radially symmetric function v ∈ W 1,1 is a solution to problem (8). Assume now that problem (8) admits a solution u ∈ W p rad , and let us prove that u(x) := u(|x|) is a solution to (7). Namely, if we assume by contradiction that there exists a

Existence of minimizers and Euler-Lagrange inclusions
In this section we focus our attention to functionals of the form We prove the existence of radially symmetric Lipschitz continuous minimizers, and the validity of necessary optimality conditions of Euler-Lagrange type, when g is a convex function with possibly linear growth in the gradient variable, and h is a Lipschitz continuous function with respect to u. As usual, the Euler-Lagrange conditions involve a pair (u, p), where u is a minimizer in W 1 rad , while the function p belongs to the space We call p a momentum associated with u.
(hgr) The functions g and h are related by the condition Then the following holds true.
(i) F admits a radially symmetric minimizer in W 1,1 0 (B R ), and F rad admits a minimizer in W 1 rad . (ii) Every minimizer of F rad is Lipschitz continuous.
(iii) For every minimizer u ∈ W 1 rad of F rad there exists p ∈ W 1, * rad such that the following Euler-Lagrange inclusions hold: Remark 4.2. In (g2r) it is not restrictive to assume that ψ is a non-decreasing function, with ψ(0) = 0, and that R z → ψ(|z|) is convex and smooth (possibly replacing ψ with a suitable regularization of its convex envelope). As a consequence of these assumptions, the function s → ψ(s)/s turns out to be strictly increasing in ]s 0 , +∞[, where s 0 := max{ψ = 0}, and hence, for every m ∈]0, M [, there exists (a unique) σ > s 0 such that ψ(σ)/σ = m. In the following we shall always assume that the function ψ in (g2r) satisfies these additional properties. We recall that, if M = +∞, such a function is called a Nagumo function (see, e.g., [5,Section 10.3]).
The proof of Theorem 4.1 is divided into two steps: first we show that the result is valid in the superlinear case, i.e. when M = +∞, and then we obtain the result when M < +∞ by constructing, with the help of the a-priori estimates obtained by the Euler-Lagrange conditions, a family of superlinear functional whose radially symmetric minimizers also minimize the functional F . Since, by (h1r), it holds that then H(r, t) ≥ 0 for all r ∈ [0, R] and t ∈ R. Moreover, we have that Since (|u|, P ) ∈ W 1 rad × W 1, * rad , it holds that (see, e.g., the derivation of formula (13) in [12]). Setting C := R 0 r N −1 h(r, 0) dr, we get Observe that, by (g2r), Since ψ is a Nagumo function, then by Theorem 2.2 in [28] is convex with respect to s, and satisfies the Basic Hypotheses 4.1.2 in [6]. Moreover, the Hamiltonian of the problem, i.e., the Fenchel-Legendre transform of L with respect to the last variable: satisfies the strong Lipschitz condition near every arc, since, by (h1r), Finally, the minimization problem is calm, since it is a free-endpoint problem, hence all assumptions of Theorem 4.2.2 in [6] are satisfied.
We shall prove the theorem only in the case h + (0) ≥ 0 (since the case h − (0) ≤ 0 can be handled similarly). If 0 is a minimum point of h, then clearly parts (i)-(ii)-(iii) are satisfied choosing u ≡ 0 and p ≡ 0. Hence, it is not restrictive to prove (i)-(ii)-(iii) under the additional assumption that 0 is not a minimum point of h. Since h + (0) ≥ 0, and h is a convex function, we have that h Given . If u is a minimizer of F , then also u m is a minimizer of F ; moreover, so that u m is a minimizer of F . Hence, we have proved the following Claim 1: If u is a minimizer of F , then u m is a minimizer of both F and F .
After this preliminary reduction, let us prove (i)-(iv).
(i) Thanks to Claim 1 and Theorem 3.2, assertion (i) is a consequence of the following Claim 2: There exists a Lipschitz continuous, monotone non-decreasing minimizer u of F rad in W 1 rad satisfying m ≤ u ≤ 0. Specifically, from (hg) we have that Hence, from Theorem 4.1 the functional F rad admits a Lipschitz continuous minimizer u ∈ W 1 rad . Let us define By Riesz's Rising sun Lemma, we have that S is the union of a finite or countable family (a k , b k ), k ∈ J, of pairwise disjoint open intervals, withû m (a k ) =û m (b k ) for every k (unless a k = 0, in which case u m (0) ≤ u m (b k )). Hence, the function u(r) := u m (b k ), if r ∈ (a k , b k ) for some k ∈ J, u m (r), otherwise, is a Lipschitz continuous, monotone non-decreasing function and F rad (u) ≤ F rad ( u), i.e., u is a minimizer of F rad with the required properties, and Claim 2 is proved.
From the analysis above we can prove the following result without requiring the convexity of g. In the following, g * * denotes the bipolar function of z → g(|z|). satisfies all the assumptions of Theorem 5.1, hence there exist a radial minimizer u(x) = u(|x|) of F in W 1 µ and a momentum p ∈ W 1, * rad such that (21)-(22) hold. As in the proof of Theorem 5.1(iii), considering without loss of generality h ∈ C 2 and h − (0) > 0, we have already proved that u is convex and there exists r 0 ∈ [0, R[ such that u(r) = m for every r ∈ [0, r 0 [, and u(r) > m for every r ∈]r 0 , R]. Moreover, the function r 1−N p(r) is strictly increasing in ]r 0 , R]. Let P be the set of all z ∈ R such that (z, g * * (z)) belongs to the set of the extremal points of the epigraph of g * * . We recall that g(z) = g * * (x) for every z ∈ P (see [12,Remark 5.3]). Reasoning as in [3] (see the proof of Theorem 2), from the strict monotonicity of r 1−N p(r) in ]r 0 , R] follows that |u (r)| ∈ P for a.e. r ∈ [r 0 , R]. Since u (r) = 0 for every r ∈ [0, r 0 [, we conclude that F rad (u) = F rad (u), hence u is a minimizer of F rad .