2. Statement of the Theorem

DOI:10.1051/cocv/2015034 www.esaim-cocv.org

STRICT CONVEXITY AND THE REGULARITY OF SOLUTIONS TO VARIATIONAL PROBLEMS

Arrigo Cellina

Abstract.We consider the problem of minimizing

Ω

[L(∇v(x)) +g(x, v(x))]dx on u⁰+W₀^1,2(Ω)

whereΩis a bounded open subset ofR^N andLis a convex function that grows quadratically outside the unit ball, while, when |∇v|< 1, it behaves like |∇v|^p with 1 < p < 2. We show that, for each ω⊂⊂Ω, there exists a constantH, depending onωbut not on p, such that both

∇u_W1,2(ω)≤H and ∇u

|∇u|^2−p_W1,2(ω)≤ H (p−1)²; in particular, for everyi= 1, ...N, we have max{_|∇u|^|uxi2−p^| ,|u_xi|} ∈W_loc^1,2(Ω).

Mathematics Subject Classification. 49K10.

Received December 12, 2014.

1. Introduction

This paper is concerned with the regularity properties of solutions to variational problems and, more precisely, with their properties of higher differentiability. We consider the problem of minimizing

Ω[L(∇v(x)) +g(x, v(x))]dx on w⁰+W₀^1,2(Ω) (1.1) whereLis a convex function andΩa bounded open subset ofR^N. We wish to explore the effect of an increase of the strict convexity of the Lagrangian, with respect to the variable gradient, on the regularity of the solution;

more precisely, we consider a problem whereLgrows quadratically outside the unit ball, while, when|∇v|<1,

Keywords and phrases.Regularity of solutions, higher differentiability, strict convexity.

∗This work was partially supported by INDAM-GNAMPA.

1 Dipartimento di Matematica e Applicazioni, Universit`a degli Studi di Milano-Bicocca, Via R. Cozzi 53, 20125 Milano, Italy.

arrigo.cellina@unimib.it

Article published by EDP Sciences c EDP Sciences, SMAI 2016

it behaves like |∇v|^p with 1< p <2; hence, near the origin, the norm of the matrix of the second derivatives of Ldiverges, making the problem very strictly convex when |∇v| is small. Our Theorem 2.2below describes how this increasing in the strict convexity ofLaffects the higher differentiability of the solution u, when |∇u| is small.

Regularity results in the sense ofubeing inC^1,αforL(ξ) =|ξ|^pwithp >1 have been proved by Uhlenbeck [8], Lewis [6] and Tolksdorf [7] for g = 0 and by Di Benedetto [2], Acerbi and Fusco [1] in the general case; very recently functionals with different conditions on{|ξ|>1}and on{|ξ|<1}, (withg=f u) have been considered by Colombo and Figalli [5] and the regularityC^1,α of the solution established; these results and techniques are different from ours.

2. Statement of the Theorem

The integrandLof (1.1) is described as follows: for some 1< p <2, we shall consider the function

l(t) =

⎧⎪

⎪⎨

⎪⎪

⎩ 1

2|t|²+ 1 for|t| ≥1 1

p|t|^p+3 2−1

p for|t| ≤1

(2.1)

and setL(ξ) =l(|ξ|). We have that, calling HL(ξ) the matrix of second derivatives ofLcomputed atξ, HL(ξ) =

(p−2)|ξ|^p−4ξ⊗ξ+|ξ|^p−2I for|ξ|<1

I for|ξ|>1 (2.2)

so thatz^THL(ξ)z≥ |z|²for allξ, while|HL(ξ)| → ∞ as|ξ| →0.

The assumptions on gare:

Assumption 2.1.

i) There exist τ ∈ L¹(Ω) and a non-negativeλg ∈ L²_loc(Ω) such that for a.e. x ∈ Ω and every u, we have g(x, u)≥τ(x)−λg|u|.

ii) There exist non-negativeλ₂∈L²_loc(Ω) andλ_∞∈L^∞_loc(Ω), such that |gu(x, u)| ≤λ₂(x) +λ_∞(x)|u|.

Functions likeg(x, u) =λ₂(x)uorg(x, u) = (sin(x₁)u)²satisfy Assumption 2.1.

The mapl, and the maplr to be defined, are not reallyC² everywhere, but their gradients are Lipschitzian, and, by a simple modification of results that go back to [3], one proves that a solution u to the problem of minimizing (1.1), withLandg described above, is such that∇u∈W_loc^1,2(Ω).

The purpose of this paper is to prove the following result:

Theorem 2.2. Let Ω be a bounded open subset of R^N, let l be as in (2.1) and let g satisfy Assumption 2.1.

Then, there exist u, a solution to the Euler−Lagrange equation, i.e. such that

Ω[ ∇L(∇u(x)),∇η(x)+gu(x, u(x))η(x)]dx= 0

for every η∈C_c¹(Ω), and, for eachω⊂⊂Ω, a constantH, depending on ω but not on p, such that both

∇uW^1,2(ω)≤H and ∇u

|∇u|^2−pW^1,2(ω)≤ H (p−1)²; in particular, for every i= 1, . . . N, we havemax{_|∇^|u_u^xi_|2−p^| ,|ux_i|} ∈W_loc^1,2(Ω).

Under some additional assumptions, mainly whengis convex in the variablev, a solution to the Euler−Lagrange equation is actually a solution to the minimization problem (1.1).

The additional regularity of the solution, provided by Theorem 1, is actually lost in the limit as p → 1, as the statement of Theorem 2.2 itself suggests. In fact, the limit problem consists in minimizing (1.1) where L(ξ) =l_∗(|ξ|) with

l_∗(t) =

⎧⎪

⎨

⎪⎩ 1

2|t|²+ 1 for|t| ≥1

|t|+1

2 for|t| ≤1;

(2.3) here we have at once that l_∗(0) =∞while l_∗(t) = 0 for 0<|t| <1. Wheng(x, u) =u, a (radial) solution to problem (1.1) is

u_∗(x) =

0 for|x| ≤1 1

2(|x|²−1) for|x| ≥1 whose gradient is

∇u∗(x) =

0 for|x|<1 x for|x|>1

so that the gradient is discontinuous along|x|= 1, preventing∇u∗ from being a Sobolev function.

3. Proof of Theorem 2.2

We shall use the following notations. The measure of A ⊂R^N is |A|; a^T is the transpose of a; for a fixed coordinate direction es, we set δhe_su to be the difference quotient of the function u, defined by δhe_su(x) =

u(x+he_s)−u(x)

h . For a variationη to be defined,Dη is such that|∇η(x)| ≤Dη.

For the proof of the main result we shall needlr, a regularization of l, defined to be lr(t) =

⎧⎨

⎩ 1

2r^p−2t²+ 1 p−1

2

r^p+3 2−1

p for|t| ≤r

l(t) otherwise,

(3.1) so that

l_r(t) =

r^p−2t for 0≤t≤r l(t) otherwise.

We have thatl_r is continuous and increasing, hencelris convex; moreover, fort /∈ {r,1},l_r(t) exists and l_r(t) =

⎧⎨

⎩

r^p−2 for|t|< r (p−1)|t|^p−2 forr <|t|<1

1 otherwise.

In particular,l_r is (globally) Lipschitzian with constantr^p−2. Set Lr(ξ) = lr(|ξ|) so that ∇Lr is Lipschitzian with Lipschitz constantr^p−2. In addition, we have that∇Lr→ ∇L uniformly asr→0.

Besides Problem1.1, we shall also consider the problem of minimizing

Ω[Lr(∇v(x)) +g(x, v(x))]dx on w⁰+W₀^1,2(Ω) (3.2) and callu^r its solution. By known regularity results, the functionu^ris inW_loc^2,2(Ω).

Lemma 3.1. LetΩandgas in Theorem2.2; letu^rbe a solution to the minimization of(3.2); letφ∈W^1,2(Ω) with support compactly contained inΩ. Then, fors= 1, . . . , N, we have

Ω

d

dxs∇Lr(∇u^r),∇φ

=

Ωgu(·, u^r)φx_s

Proof.

a) First, we claim that the map∇Lr(∇u^r) is inW^1,2(Ω); we have that∇Lr(ξ) =l_r(|ξ|)_|^ξ_ξ| and that|∇u^r|is in W^1,2(Ω), with _d^d_x

i|∇u^r|= ∇u^r

|∇u^r|,∇u^rx_i

. The map l_r(t)

t =

⎧⎨

⎩

r^p−2 for 0≤ |t| ≤r

|t|^p−2 forr≤ |t| ≤1 1 for|t| ≥1

is (uniformly) Lipschitzian and it is not differentiable only at at |t| = r and |t| = 1; then, as it is known, x→ ^lr(|∇_|∇u^ur^r(⁽x^x)|^)|) is a Sobolev function with

d dxi

l_r(|∇u^r(x)|)

|∇u^r(x)| = l_r(t) t

◦ |∇u^r(x)| ∇u^r

|∇u^r|,∇u^rx_i

=

⎧⎨

⎩

0 for|∇u^r(x)| ≤ror |∇u^r(x)| ≥1

(p−2)|∇u^r(x)|^p−3 ∇u^r

|∇u^r|,∇u^rx_i

otherwise. (3.3)

Then d

dxi

l_r(|∇u^r(x)|)

|∇u^r(x)| · ∇u^r(x)

= l_r(|∇u^r(x)|)

|∇u^r(x)| ∇u^rx_i(x) + d dxi

l_r(|∇u^r(x)|)

|∇u^r(x)|

∇u^r(x).

Both terms above are inL²_loc(Ω): in fact, ^lr(_t^t) is bounded and, from (3.3), the absolute value of the second term is at most|∇ux_i|. Hence,∇L^r(∇u^r) is inW_loc^1,2(Ω).

b) Under the assumptions of the Lemma, the Euler−Lagrange equation holds for u^r in the sense that for ψ∈W₀^1,2(Ω) we have

Ω[ ∇Lr(∇u^r),∇ψ+gu(x, u)ψ]dx= 0.

Forhsufficiently small, consider the variationψ=δ₋he_sφto obtain

Ω

∇L^r(∇u^r(x+hes))− ∇L^r(∇u^r(x))

h ,∇φ(x)

dx=

Ωgu(x, u^r(x))φ(x−hes)−φ(x)

−h dx. (3.4)

Since∇L^r(∇u^r) is inW_loc^1,2(Ω), the family

∇L^r(∇u^r(x+he_i))−∇L^r(∇u^r(x)) h

h is bounded inL²_loc(Ω) and we can assume the existence of a sequence (hn) such that

∇L^r(∇u^r(x+hnei))− ∇L^r(∇u^r(x))

hn d

dxi∇L^r(∇u^r) so that the left hand side of (3.4) converges to

Ω

d

dxs∇L^r(∇u^r),∇φ.

We also have

Ωgu(x, u^r(x))φ(x−hes)−φ(x)

−h dx=

₁

0

supp(φ)+the_sgu(x, u^r(x))φx_s(x−thes)dxdt

= ₁

0

supp(φ)gu(x+thes, u^r(x+thes))φx_s(x)dxdt

=

Ωgu(x, u^r(x))φx_s(x)dx+ ₁

0

Ω[gu(x+hes, u^r(x+hes))−gu(x, u^r(x)]φx_s(x)dxdt.

By Assumption 2.1, ii), we obtain that the mapx→gu(x, u^r(x)) is inL²_loc(Ω), so thatgu(·+hes, u^r(·+hes))−

gu(·, u^r(·))L²(supp(φ))→0, thus proving the lemma.

Lemma 3.2. There existsK, depending neither onrnor onp, such that∇u^rL²(Ω)≤K andu^rL²(Ω)≤K.

Proof. SetL⁰(ξ) = ¹₂|ξ|²+ 1, so that, for any 1< p <2 and anyr≤1, we haveL^r(ξ)≤L⁰(ξ) + 1.

Letu⁰be a solution to the problem of minimizing

Ω[L⁰(∇v(x)) + 1 +g(x, v(x))]dx on w⁰+W₀^1,2(Ω) (3.5) and setV =

Ω[L⁰(∇u⁰(x)) + 1 +g(x, u⁰(x))]dx. Then V ≥

Ω

[L^r(∇u⁰) +g(x, u⁰)]≥

Ω

[L^r(∇u^r) +g(x, u^r)]≥

Ω

1

2|∇u^r|²+g(x, u^r)

; on the other hand, recalling Assumption 2.1, for a constantαto be fixed, from

λg|u| ≤ ¹₂α²

(λg)²+¹_2α¹₂

|u|²

we obtain

Ω

g(x, u^r(x))dx≥

τ−1 2α²

(λg)²−1 2

1 α²

|u^r|². CallP the Poincar´e constant inW^1,2(Ω); from

|u^r|²=

|w⁰−(u^r−w⁰)|²≤2

|w⁰|²+ 2P

|∇(u^r−w⁰)|²≤ 2

|w⁰|²+ 4P

|∇u^r|²+ 4P

|∇w⁰|², we obtain

Ωg(x, u^r)≥

τ−1 2α²

(λg)²−1 2

1 α²

4P

|∇u^r|²+

|∇w⁰|²(2 + 4P)

.

Hence,

Ω

1

2|∇u^r|²≤V −

Ωg(x, u^r)≤V −

τ+1 2α²

(λg)²+2P α²

|∇u^r|²+ 1 2α²

|∇w⁰|²(2 + 4P).

Chooseαsuch that ²_α^P₂ =¹₄ to obtain

|∇u^r|²≤4[V +

(−τ+¹₂α²(λg)²+²⁺⁴_2α2^P|∇w⁰|²)] =k₁.

From this, making use of w⁰ ∈W^1,2 and of Poincar´e’s inequality, we infer that for some k₂, we also have

Ω|u^r|²≤k₂ for allr≤1.

A similar estimate was proved in [4].

Proof of Theorem 2.2.

a) Consider the function

γr(t) =l_r(t) t =

⎧⎨

⎩

r^p−2 for|t| ≤r

|t|^p−2 forr <|t|<1 1 otherwise;

then, as in the Proof of Lemma3.1, the mapx→γr(|∇u^r(x)|) is inW_loc^1,2 and d

dxsγr(|∇u^r(x)|) =

⎧⎨

⎩

0 for|∇u^r| ≤ror|∇u^r| ≥1

(p−2)|∇u^r|^p−3 ∇u^r

|∇u^r|,∇u^rx_s

forr <|∇u^r|<1.

Moreover, 1≤γr≤r^p−2 and|_d^d_xsγr(|∇u^r|)| ≤(2−p)r^p−3|∇u^r_xs|. Then, the map x→γr(|∇u^r(x)|)ux_i(x) is in W_loc^1,2(Ω) and, setting Hu^r to be the Hessian matrix ofu^r, we obtain

∇(γr(|∇u^r|)u^rx_i) =

⎧⎨

⎩

γr(|∇u^r|)∇u^r_xi for|∇u^r| ≤r or|∇u^r| ≥1 (p−2)|∇u^r|^p−2Hu^r ∇u^r

|∇u^r| u^r_xi

|∇u^r|+γr(|∇u^r|)∇u^r_xi forr≤ |∇u^r| ≤1. (3.6)

b) Let x⁰ and δ⁰ be such that B(x⁰,4δ⁰) ⊂⊂ Ω. Let η ∈ C₀^∞(B(x⁰,2δ⁰)) be such that 0 ≤ η ≤ 1 and that η(x) = 1 for x∈B(x⁰, δ⁰); we recall thatDη = sup{|∇η(x)|}. Then, the function φ= [η²γr(|∇u^r|)u^rx_i] is in W₀^1,2(B(x⁰,3δ⁰)) and from Lemma3.1we have

Ω

d dxi

l_r(|∇u^r|)

|∇u^r| ∇u^r,∇φ

=

Ωgu(·, u^r)φx_i, i.e.,

B(x⁰,3δ⁰)

l_r(|∇u^r|)

|∇u^r| ∇u^rx_i+ d dxi

l_r(|∇u^r|)

|∇u^r|

∇u^r, 2η∇ηγr(|∇u^r|)u^rx_i+η²∇(γr(|∇u^r|)ux_i)

dx

=

B(x⁰,3δ⁰)

gu(·, u^r)

2ηηx_iγr(|∇u^r|)u^rx_i+η² d

dxi(γr(|∇u^r|)u^rx_i)

dx (3.7)

We shall callGi the term at the right hand side.

Since the above equality holds for everyi, we obtain

i

B(x⁰,3δ⁰)

d dxi

l_r(|∇u^r|)

|∇u^r| ∇u^r, η²∇(γr(|∇u^r|)u^r_xi)

dx

≤

i

|

B(x⁰,3δ⁰)

d dxi

l_r(|∇u^r|)

|∇u^r| ∇u^r,2η∇ηγr(|∇u^r|)u^rx_i

|dx+

i

Gi (3.8)

c) Forj= 1, . . . , N, we have

∇x l_r(|∇u^r|)

|∇u^r| ∇u^r

i,j

= l_r(|∇u^r|)

|∇u^r| u^r_xjxi+ d dxi

l_r(|∇u^r|)

|∇u^r|

u^r_xj

= l_r

|∇u^r|u^r_xjxi+ ∇u^r

|∇u^r|,∇u^rx_i

l_r− l_r

|∇u^r| u^r_xj

|∇u^r| i.e.,

∇x l_r(|∇u^r|)

|∇u^r| ∇u^r

= l_r

|∇u^r|Hu^r+ l_r− l_r

|∇u^r|

∇u^r

|∇u^r|Hu^r

⊗ ∇u^r

|∇u^r| and we obtain

∇x l_r(|∇u^r|)

|∇u^r| ∇u^{r 2}

= l_r

|∇u^r| ₂

|Hu^r|²+ l_r− l_r

|∇u^r| ₂

∇u^r

|∇u^r|Hu^r

²+ 2 l_r− l_r

|∇u^r| l_r

|∇u^r| ∇u^r

|∇u^r|Hu^r

²

= l_r

|∇u^r| ₂

|Hu^r|²+

(l_r)²− l_r

|∇u^r| ₂

∇u^r

|∇u^r|Hu^{r 2} so that

inf

(l_r)², l_r

|∇u^r| ₂

|Hu^r|²≤ |∇x l_r(|∇u^r|)

|∇u^r| ∇u^r

|²≤sup

(l_r)², l_r

|∇u^r| ₂

|Hu^r|². (3.9) We also have, computingl_r, l_r,γrand γ_r at|∇u^r|,

i

d dxi

l_r(|∇u^r|)

|∇u^r| ∇u^r,∇(γru^r_xi)

=

i

j

l_r

|∇u^r|u^r_xjxi+

s

u^r_xs

|∇u^r|u^r_xixs

l_r− l_r

|∇u^r| u^r_xj

|∇u^r|

·

γru^r_xjxi+ (γr)

l

u^r_xjxl u^r_xl

|∇u^r|

u^r_xi

= l_r

|∇u^r|γr|Hu^r|²+ l_r− l_r

|∇u^r|

γ_r|∇u^r|

⎛

⎝

i,s

u^r_xi

|∇u^r| u^r_xs

|∇u^r|u^r_xixs

⎞

⎠

·

⎛

⎝

j,l

u^r_xj

|∇u^r| u^r_xl

|∇u^r|u^r_xjxl

⎞

⎠

+γr l_r− l_r

|∇u^r| _i

⎛

⎝

j

u^r_xj

|∇u^r|u^r_xjxi

⎞

⎠

s

u^r_xs

|∇u^r|u^r_xixs

+ l_r

|∇u^r|γ_r|∇u^r| Hu^r ∇u^r

|∇u^r| ₂

= l_r

|∇u^r|γr|Hu^r|²+ l_r− l_r

|∇u^r|

γ_r|∇u^r| ∇u^r

|∇u^r|

T

Hu^r ∇u^r

|∇u^r| ₂

+γr l_r− l_r

|∇u^r| Hu^r ∇u^r

|∇u^r|

²+ l_r

|∇u^r|γ_r|∇u^r||Hu^r ∇u^r

|∇u^r||²

= l_r

|∇u^r|γr|Hu^r|²+ l_r− l_r

|∇u^r|

γr

Hu^r ∇u^r

|∇u^r| ² + l_r

|∇u^r|γ_r|∇u^r|

⎡

⎣|Hu^r ∇u^r

|∇u^r||²− ∇u^r

|∇u^r|

T

Hu^r ∇u^r

|∇u^r| ₂⎤

⎦

+l_rγ_r|∇u^r| ∇u^r

|∇u^r|

T

Hu^r ∇u^r

|∇u^r| ₂

≥γr[ l_r

|∇u^r||Hu^r|²+ l_r− l_r

|∇u^r| Hu^r ∇u^r

|∇u^r| ²]

≥γr[inf

l_r, l_r

|∇u^r|

# ]|Hu^r|² where

γr(t) inf

l_r(t),l_r(t) t

#

=

⎧⎨

⎩

r^2(p−2) for|t| ≤r (p−1)|t|^2(p−2)forr≤ |t| ≤1

1 otherwise.

Hence we have obtained

B(x⁰,3δ⁰)

η²γr

inf

l_r, l_r

|∇u^r|

#

|Hu^r|²dx≤

i

B(x⁰,3δ⁰)

η² d

dxi

l_r(|∇u^r|)

|∇u^r|^r ∇u^r,∇(γr(|∇u^r|)ux_i)

dx

so that, from (3.8),

B(x⁰,3δ⁰)

η²γr

inf

l_r, l_r

|∇u^r|

#

|Hu^r|²dx

≤

i

B(x⁰,3δ⁰)

d dxi

l_r(|∇u^r|)

|∇u^r| ∇u^r,2η∇ηγr(|∇u^r|)u^rx_i

dx+

i

Gi (3.10) d) From (3.6) we have that

i

Gi=

B(x⁰,3δ⁰)

[gu(x, u^r)2ηγr(|∇u^r|) ∇η,∇u^r+η²gu(x, u^r)Δu^r]dx

+

B(x⁰,3δ⁰)∩{r≤|∇u^r(x)|≤1}

gu(x, u^r)η²(p−2)|∇u^r|^p−2 ∇u^r

|∇u^r| T

Hu^r ∇u^r

|∇u^r|dx

≤

B(x⁰,3δ⁰)

[η²g_u²+|∇η|²+ 4

p−1η²g_u²+p−1

4 η²|Hu^r|²]dx +

B(x⁰,3δ⁰)∩{r≤|∇u^r(x)|≤1}

4

p−1η²g²_u+η²p−1

4 |∇u^r|^2(p−2)|Hu^r|²

dx (3.11)

By Assumption 2.1,gu(x, u^r)² ≤2[(λ₂)²+ (λ_∞|u^r|)²]; hence, applying Lemma 3.2 we infer that there exists a constantK⁰, independent ofrandp, such that the right hand side of (3.11) is bounded above by

K⁰ p−1 +1

2

B(x⁰,3δ⁰)

η²(p−1)|∇u^r|^2(p−2)|Hu^r|²dx.

Then, from inf{l_r,_|∇^lr_ur|} ≥p−1, (3.10) gives 1

2

B(x⁰,3δ⁰)

η²γr

inf

l_r, l_r

|∇u^r|

#

|Hu^r|²dx

≤ 1

p−1K⁰+

i

B(x⁰,3δ⁰)

d dxi

l_r(|∇u^r|)

|∇u^r| ∇u^r,2η∇ηγr(|∇u^r|)u^rx_i

dx

≤ 1

p−1K⁰+

B(x⁰,3δ⁰)

p−1

4 |∇x(∇L(∇u^r))|²η²+ 4

p−1N|∇η|²

dx

≤ 1

p−1K⁰+

B(x⁰,3δ⁰)

1 4γrinf

l_r, l_r

|t|

#

|∇x(∇L(∇u^r))|²η²+ 4

p−1N|∇η|²

dx (3.12)

and we obtain p−1

4

B(x⁰,3δ⁰)

η²|∇x(∇Lr(∇u^r)|²dx≤1 4

B(x⁰,3δ⁰)

η²γr

inf

l_r, l_r

|∇u^r|

#

|Hu^r|²dx≤ K¹ p−1·

f) In particular,

B(x⁰,δ⁰)

|∇x∇Lr(∇u^r)|²≤ 4

(p−1)²K¹; (3.13)

hence, the family (∇x∇Lr(∇u^r))r is bounded in L²(B(x⁰, δ⁰)). The arbitrariness ofx⁰ and of δ⁰ then shows that, for everyω ⊂⊂Ω, there exists H, independent of r and p, such that (∇x∇Lr(∇u^r)L²(ω))r ≤ _(p−1)^H 2. Then, from (3.9) and since inf{(l_r)²,(_|∇^l_u^r_r|)²} ≥(p−1)², we infer that

ω|Hu^r|²≤ 1 (p−1)²

ω|∇x∇Lr(∇u^r)|²≤ 1

(p−1)⁴H². (3.14)

Then, we can assume that, fors= 1, . . . , N, there exists a sequence (rⁿ)n such that _dx^d

s∇Lr_n(∇u^rn) converges weakly in L²(ω) to some dλ, that∇Lr_n(∇u^rn) converges in L²(ω) to a functionλ, thatu^rn →uand, finally, that∇u^rn→ ∇uin L²(ω).

g) We claim that:

i)λ=∇L(∇u);dλ=_d^d_x

s∇L(∇u) and

$$$$ d

dxs∇L(∇u)$$

$$L²(ω)

≤ 1

(p−1²H.

ii)uis a solution to the Euler Lagrange equation,i.e., that, for every η∈C_c¹(Ω),

Ω[ ∇L(∇u(x)),∇η(x)+gu(x, u(x))η(x)]dx= 0.

To prove the claim, notice that, possibly passing to a subsequence, we can assume that both∇u^rn→ ∇uand

∇Lr_n(∇u^rn)→λpointwise a.e.. Fix xsuch that the above holds and fixε. By the continuity of ∇L, letδ be such that|∇u(x)−ξ| ≤δimplies|∇L(∇u(x))− ∇L(ξ)|<₂^ε; letnbe so large that both|∇u^rn(x)− ∇u(x)|< δ, and∇Lr_n− ∇LC< ^ε₂. Hence, fornlarge,

|∇Lr_n(∇u^rn(x))− ∇L(∇u(x))|

≤ |∇Lr_n(∇u^rn(x))− ∇L(∇u^rn(x))|+|∇L(∇u^rn(x))− ∇L(∇u^rn(x))|< ε, so that|λ(x)− ∇L(∇u(x))| ≤ε, and by the arbitrariness ofε, we obtain

λ(x) =∇L(∇u(x)).

Moreover,∇Lr_n(∇u^rn)→λinL²(ω) and _d^d_x

s∇Lr_n(∇u^rn) dλweakly, implydλ= _d^d_x

s∇L(∇u), so that from _dx^d_s∇Lr_n(∇u^r)L²(ω)≤ _(p−1)¹ 2H we obtain that_dx^d_s∇L(∇u)L²(ω)≤ _(p−1)¹ 2H, thus proving i).

To prove ii), fixη∈C_c¹(Ω). We have that

Ω[ ∇Lr_n(∇u^rn(x)),∇η(x)+gu(x, u^rn(x))η(x)] dx= 0;

sinceu^rn→uinL²(ω) and∇Lr_n(∇u^rn)∇L(∇u), we obtain

Ω[ ∇L(∇u(x)),∇η(x)+gu(x, u(x))η(x)]dx= 0.

Hence, we have obtained the existence of a solutionuto the Euler−Lagrange equation such that, for every s= 1, . . . , N,

d

dxs∇L(∇u) = d dxs

⎧⎨

⎩

∇u

|∇u|^2−p for|∇u| ≤1

∇u for|∇u| ≥1

belongs toL²(ω); in particular, for everyi= 1, . . . , N, bothux_i and _|∇^u_u^xi_|2−p belong toW_loc^1,2.