Table Of ContentDiscrete maximum principles for nonlinear elliptic
finite element problems on Riemannian manifolds with
boundary
J. Kar´atson1,, B. Kova´cs2,∗, S. Korotov3
7
1 January 3, 2017
0
2
n
a 1 Department of Applied Analysis & MTA-ELTE Numerical Analysis and Large
J
Networks Research Group, ELTE University; Department of Analysis, Technical
2
University; Budapest, Hungary; [email protected]
A] 2 Mathematisches Institut, Universita¨t Tu¨bingen, Auf der Morgenstelle 10, 72076
N Tu¨bingen, Germany, e-mail: [email protected]
. 3 Department of Computing, Mathematics and Physics, Bergen University College,
h
t Bergen, Norway; e-mail: [email protected]
a
m
Abstract: The maximum principle forms an important qualitative property of second
[
order elliptic equations, therefore its discrete analogues, the so-called discrete maximum
1
v principles (DMPs) have drawn much attention. In this paper DMPs are established for
4 nonlinear surface finite element problems on Riemannian manifolds, corresponding to the
2
classical pointwise maximum principles on surfaces in the spirit of Pucci et al. Various
4
0 real-life examples illustrate the scope of the results.
0
.
1 Keywords: nonlinear elliptic problems on surfaces, maximum principle, discrete maxi-
0
mum principle, finite element method, simplicial mesh.
7
1
: Mathematical Subject Classification: 35B50, 35J65, 65N30, 65N50
v
i
X
r 1 Introduction
a
The maximum principle forms an important qualitative property of second order elliptic
equations [17, 24, 25]. We will refer to such results as continuous maximum principles
(CMPs). Typical maximum principles often arise in a form stating that a solution attains
a nonnegative maximum on the boundary, i.e.
maxu ≤ max{0,maxu}. (1)
S ∂S
CMPs have also been studied on Riemannian manifolds. Thereby the main subject is
often an Omori–Yau type maximum principle, but in this paper we are interested in
classical pointwise versions, such as the ones studied by Antonini, Mugnai and Pucci in
[2], which give analogous results to (1).
1
The discrete analogues of CMPs, the so-called discrete maximum principles (DMPs)
have drawn much attention. The DMP is in fact an important measure of the qualitative
reliabilityofthenumericalscheme, otherwiseonecouldgetunphysicalnumericalsolutions
like negative concentrations, etc.
We are interested for DMPs on a surface with boundary, in the context of the surface
finite element method (SFEM). Originally, the surface FEM was developed by Dziuk in
[12] for elliptic problems on closed surfaces (and further developed for evolving surfaces by
Dziuk and Elliott in [13, 14]). The theory was extended to elliptic problems on surfaces
with boundaries in the recent preprint [5]. In such a case a DMP reproduces one of the
above relations for the surface finite element solution u instead of u. Various DMPs
h
in Euclidean domains, including geometric conditions on the computational meshes for
FEM solutions, have been given, e.g. in [8, 9, 11, 26]. The authors’ previous work,
e.g. [18, 21, 22], involves various types of linear and nonlinear equations and systems,
and typical geometric conditions are nonobtuseness or acuteness in the case of simplicial
meshes.
To our knowledge, no DMP has been established for elliptic problems on Riemannian
manifolds yet. In this paper we provide such a result in the spirit of [2] and using
analogous treatment of the finite element matrix as in our earlier paper [18]. We work
in the mentioned surface finite element setting, cf. [5, 12]. The surface approximation
introduces an additional geometric error, which requires extra care during the analysis
as compared to the flat domain case. In addition, the issue of generating meshes with
suitable angle properties is addressed. The theoretical results are supported by numerical
tests.
2 Formulation and properties of the problem
Our formulation of the problem is based on [2]. We consider a nonlinear elliptic problem
which is, on the one hand, a special case of the problem studied in [2] on manifolds, and
on the other hand, an analogue of the problem studied in our earlier paper [18] in the
context of DMP on usual Euclidean domains. Due to [2], there holds a CMP for this
problem, and our goal is to adapt our approach from [18] to verify a DMP as well for
proper SFEM discretisations [5, 12].
We follow [2] and [5, 12] for the description of the manifold and of the elliptic problem.
2.1 The nonlinear elliptic problem on a Riemannian manifold
2.1.1 The manifold
Following the setting and notations of [2], let M be a smooth complete d-dimensional
Riemannian manifold, and let S ⊂ M be a closed bounded and regular subdomain of M,
so that S is a smooth manifold with boundary, with outward normal ν. Furthermore, let
δ denote the associated signed distance function, satisfying S = {x ∈ Rd+1 | δ(x) = 0}
and ∇δ = ν, cf. [12, 14]. In the sequel we shall write TS× R in place of TS× (S×R) to
S S
denote the fibered product bundle. In analogy with the Euclidean case, points of TS× R
S
willbedenotedwith(x,z,ξ), where(x,ξ) ∈ TS and(x,z) ∈ S×R. Integralswillbetaken
with respect to the natural Riemannian measure, denoted by dM. The surface gradient,
or tangential gradient (i.e. the orthographic projection of the standard gradient onto the
2
surface) of a function u on S will be simply denoted by ∇ u, i.e. ∇ u = ∇u−(∇u·ν)ν.
S S
Further, ∆ denotes the Laplace–Beltrami operator, and div the tangential divergence.
S S
For p ≥ 1, the Lebesgue and Sobolev norms are defined as
(cid:90)
(cid:16) (cid:17)1/p
(cid:107)u(cid:107) := |u|pdM and (cid:107)u(cid:107) := (cid:107)u(cid:107) +(cid:107)∇u(cid:107) ,
p,S 1,p,S p,S p,S
S
respectively, where |u| := g(u,u)1/2 using the metric tensor g, further, W1,p(S) and
W1,p(S) are defined as the closure of C∞(S) and C∞(S), respectively, in the Sobolev
0 0
norm (cid:107)·(cid:107) . We define p(cid:48) := p/(p−1) if p > 1 and p(cid:48) := ∞ if p = 1 (see e.g. [12, 14]).
1,p,S
TherearesimilarSobolevembeddingestimatesofW1,p(S)availableasintheEuclidean
case. Let p∗ := pd if d > p and p∗ := ∞ if d ≤ p. Here S is a closed bounded subdomain
d−p
of the complete Riemannian d-manifold M, hence it is a compact Riemannian d-manifold
itself. Therefore, by [3], for all 1 ≤ p < p∗ we have W1,p(S) ⊂ Lp1(S) and
1
(cid:107)v(cid:107) ≤ k (cid:107)v(cid:107) (∀v ∈ W1,p(S)) (2)
p1,S 1 1,p,S
for some constant k > 0 depending on p and p but independent of v.
1 1
2.1.2 The problem
We consider a nonlinear boundary value problem of the following type, involving a scalar
nonlinear coefficient in the principal part:
(cid:16) (cid:17)
−div b(x,u,∇ u)∇ u +q(x,u) = f(x) onS,
S S S
(3)
u = g(x) on ∂S,
with the bounded surface S ⊂ M, with boundary ∂S, under the following conditions:
Assumptions 2.1.2.
(A1) Let p ≥ 2 be a given parameter.
(A2) The scalar functions b : S ×R×Rd+1 → R, q : S ×R → R are continuous. There
holds f ∈ Lp(cid:48)(Ω), further, g = g∗| for some g∗ ∈ W1,γ(S), where γ ≥ p if p > d
∂S
and γ > d if p ≤ d .
(A3) (Uniform ellipticity.) The function b satisfies
µ +µ |ξ|p−2 ≤ b(x,z,ξ) ≤ M +M |ξ|p−2
0 1 0 1
with constants µ ,µ ,M ,M > 0 independent of (x,z,ξ) ∈ S ×R×Rd+1.
0 1 0 1
(A4) Let 2 ≤ p < +∞ if d ≤ p, or 2 ≤ p < pˆ := 2d if d > p. There exist constants
1 1 d−p
α,β ≥ 0 such that for any x ∈ S and z ∈ R the following growth conditions hold,
for every z ≥ 0,
0 ≤ q(x,0) ≡ q(x,z) (∀z ≤ 0), 0 ≤ q(x,z)−q(x,0) ≤ αz +βzp1−1.
3
The weak solution u ∈ W1,p(S) of problem (3) will be defined as follows:
(cid:90) (cid:90)
(cid:2)b(x,u,∇ u) ∇ u·∇ v +q(x,u)v(cid:3) dM = fv dM (∀v ∈ W1,p(S)) (4)
S S S 0
S S
and u − g∗ ∈ W1,p(S). (5)
0
Remark 1. As mentioned before, related problems have been studied in the context of
the DMP on usual Euclidean domains in our earlier paper [18], i.e. the above problem
(3) is a surface analogue of the problems considered therein. Now in (3) we only consider
Dirichlet boundary conditions, on the other hand, compared to [18], we impose more
general growth conditions allowing p (cid:54)= 2 as well, which involves a more general setting
using the Banach space W1,p(Ω).
0
(cid:82) (cid:82)
In the sequel we omit the sign dM in the integrals for brevity, i.e. f := f dM.
S S
2.2 Continuous maximum principle on the manifold
2.2.1 Preliminaries
We will rely on [2], where CMPs are proved for a general class of elliptic inequalities on
manifolds. We summarize the required background in a rewritten form involving elliptic
equations with right-hand side of given sign.
Let us consider an elliptic equation
−div K(x,u,∇ u)+F(x,u,∇ u) = k(x) (6)
S S S
on the domain S ⊂ M under the following conditions:
Assumptions 2.2.1.
(i) The functions K : TS × R → TS and F : TS × R → R are continuous, further,
S S
K(x,z,ξ) ∈ T M for all (x,z) ∈ S ×R and ξ ∈ T M.
x x
(ii) There exist constants a > 0 and b ,a ≥ 0 such that for all (x,z,ξ) ∈ TS × R
1 1 2 S
there holds
K(x,z,ξ)·ξ ≥ a |ξ|p −a |z|p, F(x,z,ξ) ≥ −b |ξ|p .
1 2 1
The treatment in [2] involves p-regular weak solutions of (6), defined as functions
u ∈ L1 (S) satisfying
loc
K(·,u,∇ u) ∈ Lp(cid:48) (S,TS), F(·,u,∇ u) ∈ Lp(cid:48) (S)
S loc S loc
and demanding (4) only for compactly supported v ∈ W1,p(S). The following result holds:
0
Theorem 1. Let Assumptions 2.2.1 hold with a = 0 in item (ii). Let u be a p-regular
2
weak solution of (6) such that u ∈ W1,p(S), and let the right-hand side k ≤ 0 on S. If
loc
u ≤ M on ∂S for some constant M ≥ 0, then u ≤ M in S.
In fact, this result is the reformulation of [2, Theorem 3.3.] for equations instead of
inequalities and with rearranged signs.
4
2.2.2 CMP for the studied class
We reformulate the above CMP in the vein of inequality (1).
Theorem 2. Let Assumptions 2.1.2 hold. Assume that u is a weak solution of problem
(3), defined as in (4)–(5), which satisfies u ∈ C(S). Let
f(x)−q(x,0) ≤ 0 (a.e. ∀x ∈ S). (7)
(1) Then
maxu ≤ max{0,maxg}. (8)
S ∂S
(2) In particular, if g ≥ 0 then
maxu = maxg, (9)
S ∂S
and, if g ≤ 0 then we have the nonpositivity property
maxu ≤ 0.
S
(3) If we do not assume u ∈ C(S) but only g ∈ L∞(S), then the above statements hold
with replacing each max by esssup, respectively.
Proof. We verify that problem (3) under Assumptions 2.1.2 is a special case of problem
(6) under Assumptions 2.2.1. Throughout, the notation (x,z,ξ) will stand for the general
points of S ×R×Rd+1. We first subtract q(x,0) from both sides of (3), then
(cid:16) (cid:17)
ˆ
−div b(x,u,∇ u)∇ u +qˆ(x,u) = f(x) on S,
S S S
(10)
u = g(x) on ∂S
where
ˆ
qˆ(x,z) := q(x,z)−q(x,0), f(x) := f(x)−q(x,0)
for all x ∈ S, z ∈ R. Let
K(x,z,ξ) := b(x,z,ξ)ξ, F(x,z,ξ) := qˆ(x,z).
Owing to the continuity of b and q, assumption (i) obviously holds. Further, for all
arguments, assumptions (A3) and (A4) yield
K(x,z,ξ)·ξ = b(x,z,ξ)|ξ|2 ≥ µ |ξ|2 +µ |ξ|p ≥ µ |ξ|p
0 1 1
and
F(x,z,ξ) ≥ 0,
i.e. assumption (ii) holds with a := µ > 0, a = 0 and b = 0.
1 1 2 1
We also have to check that u is a p-regular weak solution of (6) such that u ∈ W1,p(S),
loc
and the right-hand side satisfies k ≤ 0 on S. Here, first, assumption (A3) yields
|K(x,z,ξ)| ≤ b(x,z,ξ)|ξ| ≤ M |ξ|+M |ξ|p−1,
0 1
5
hence
|K(·,u,∇ u)|p(cid:48) ≤ M |∇ u|p(cid:48) +M |∇ u|(p−1)p(cid:48) = M |∇ u|p(cid:48) +M |∇ u|p.
S 3 S 4 S 3 S 4 S
Here the second term is integrable on S since the weak solution u ∈ W1,p(S), and also
the first term is integrable since p(cid:48) ≤ 2 ≤ p. Hence K(·,u,∇ u) ∈ Lp(cid:48)(S,TS). Similarly,
S
from (A4), for all arguments
0 ≤ qˆ(x,z) ≤ α|z|+β|z|p1−1,
hence
|F(·,u,∇ u)|p(cid:48) = |q(·,u)|p(cid:48) ≤ α˜|u|p(cid:48) +β˜|u|p
S
is integrable on S since u ∈ Lp(S) and p(cid:48) ≤ 2 ≤ p. Also, since u ∈ W1,p(S), all local
requirements are fulfilled even globally on S. Equation (4) holds for all v ∈ W1,p(S),
0
hence especially for compactly supported v ∈ W1,p(S). Altogether, u satisfies all criteria
0
of a p-regular weak solution. Further, for all x ∈ S, now
ˆ
k(x) := f(x) := f(x)−q(x,0) ≤ 0.
That is, we can apply Theorem 1: if u ≤ M on ∂S for some constant M ≥ 0, then u ≤ M
in S. Now we can already prove the desired statements.
(1) Let µ := maxu = maxg. If µ ≥ 0 then we can apply Theorem 1 with choice
∂S ∂S
M := µ, namely, since u ≤ µ on ∂S, therefore u ≤ µ in S too, i.e.
maxu ≤ µ = max{0,µ} = max{0,maxg}.
S ∂S
If µ < 0 then we can apply Theorem 1 with choice M := 0, namely, since u ≤ 0 on ∂S,
therefore u ≤ 0 in S too, i.e.
maxu ≤ 0 = max{0,µ} = max{0,maxg}.
S ∂S
In both cases we have obtained the desired property (8).
(2)–(3) These are obvious consequences of statement (1).
In the special case q ≡ 0, the equality (9) holds without assuming g ≥ 0:
Theorem 3. Consider problem (3) with q ≡ 0, under the assumptions of Theorem 2.
That is, let (A1)–(A3) hold, u ∈ C(S), and assumption (7) now takes the form f(x) ≤ 0
(∀x ∈ S). Then
maxu = maxg. (11)
S ∂S
Proof. If maxg ≥ 0 then (8) implies (11). Let maxg < 0, say, maxg = −K with some
∂S ∂S ∂S
K > 0. Then the function w := u+K satisfies the same type of problem with right-hand
side f and b.c. g + K, respectively, hence Theorem 2 is valid for this problem as well,
and (8) for w yields
maxw ≤ max{0, max(g +K)} = 0.
S ∂S
Then
maxu ≤ −K = maxg.
S ∂S
6
Remark 2. We note that the corresponding minimum principles and nonnegativity prop-
erty hold if the sign conditions on f are reversed.
2.3 Computational scheme: the surface finite element method
The smooth surface S with boundary ∂S is approximated by a triangulated one denoted
by S , whose vertices B ,B ,...,B , are sitting on the surface, also boundary nodes of
h 1 2 n¯
S are sitting on ∂S, and it is given as
h
(cid:91)
S = T.
h
T∈T
h
We always assume that the simplices T form an admissible triangulation (i.e. partition)
T , with h denoting the maximum diameter. Admissible triangulations were introduced
h
in [13, Section 5.1]: S is a uniform triangulation, i.e. every T ∈ T satisfies that the inner
h h
radius σ is bounded from below by ch with c > 0, and S is not a global double covering
h h
of S. The discrete tangential gradient on the discrete surface S is given by
h
∇ φ = ∇φ−(∇φ·ν ) ν ,
S h h
h
understoodinapiecewisesense, withν denotingthenormaltoS , cf.[12]. Theboundary
h h
of the approximation surface is simply ∂S , which is an analogous approximation of ∂S.
h
We define the surface finite element discretisation of our problem, following [12] and
[5]. We use simplicial elements and continuous piecewise linear basis functions. The finite
element subspace V is spanned by the continuous, piecewise linear basis functions χ ,
h j
satisfying
χ (B ) = δ for all i,j = 1,2,...,n¯,
j i ij
therefore
(cid:8) (cid:9)
V = span χ ,χ ,...,χ .
h 1 2 n¯
Now, let n < n¯ be such that
B ,B ,...,B
1 2 n
are the vertices that lie on S but not on ∂S , and let
h h
B ,...,B
n+1 n¯
be the vertices that lie only on ∂S . Then the basis functions χ ,χ ,...,χ satisfy the
h 1 2 n
homogeneous Dirichlet boundary condition on ∂S . Hence, we can define
h
V0 := span{χ ,χ ,...,χ } ⊂ W1,p(S ).
h 1 2 n 0 h
Further, let
n¯
(cid:88)
g = g χ ∈ V (12)
h j j h
j=n+1
(with g ∈ R) be the piecewise linear approximation of the function g on ∂S (and on the
j h
neighbouring elements).
7
2.3.1 Lift
In the following we recall the so-called lift operator, which was introduced in [12] and
further investigated in [13]. The lift operator projects a finite element function on the
discrete surface onto a function on the smooth surface.
Using the oriented distance function δ (cf. [13, Section 2.1]), for a continuous function
η : S → R its lift is defined as
h h
ηl(P) := η (x), x ∈ S ,
h h h
where for every x ∈ S the value P = P(x) ∈ S is uniquely defined via
h
x = P +ν(P)δ(x). (13)
By η−l we mean the function whose lift is η. Further, we have the lifted finite element
spaces
(cid:8) (cid:9)
Vl := ϕ = φl | φ ∈ V ,
h h h h h
(cid:8) (cid:9)
(V0)l := ϕ = φl | φ ∈ V0 .
h h h h h
The following equivalence between discrete and continuous norms has been shown in
[10, 12] for functions η : S → R with lift ηl : S → R.
h h h
Lemma 1 (equivalence of norms). The original and lifted norms are equivalent in both
the Lp and W1,p case, independently of sufficiently small mesh sizes h. That is, there exist
constants c,h > 0 such that for all 0 < h ≤ h ,
0 0
c−1(cid:107)η (cid:107) ≤ (cid:107)ηl(cid:107) ≤ c(cid:107)η (cid:107) (∀η ∈ Lp(S )),
h Lp(Sh) h Lp(S) h Lp(Sh) h h
c−1(cid:107)η (cid:107) ≤ (cid:107)ηl(cid:107) ≤ c(cid:107)η (cid:107) (∀η ∈ W1,p(S )).
h W1,p(Sh) h W1,p(S) h W1,p(Sh) h h
2.3.2 The discrete problem
Inordertofindtheapproximatesolutionof (3),wehavetosolvethecounterpartof (4)–(5)
in V . We note here that in order to transfer the assumptions on the coefficient functions
h
(A1)–(A4) to the discrete level we have to use negative lifts in the spatial variable. For
any x ∈ S with corresponding lift P ∈ S, cf. (13), we set
h
b−l(x,·,·) = b(P,·,·),
and similarly using the negative lift in the first argument of all coefficient functions (f,
q, r, etc.), as well. No lift is used in the other arguments. By this construction all the
properties of (A1)–(A4) are clearly transferred to these lifted functions, with S instead
h
of S.
Following (10), we first subtract q−l(x,0) from both sides and consider the following
discrete problem: find u ∈ V such that
h h
(cid:90) (cid:90)
(cid:0)b−l(x,u ,∇ u ) ∇ u ·∇ v +qˆ−l(x,u )v (cid:1) = fˆ−lv (∀v ∈ V0), (14)
h Sh h Sh h Sh h h h h h h
S S
h h
and u −g ∈ V0 (15)
h h h
8
where
qˆ−l(x,z) := q−l(x,z)−q−l(x,0), fˆ−l(x) := f−l(x)−q−l(x,0)
for all x ∈ S , z ∈ R. Further, denote
h
q−l(x,z)−q−l(x,0)
, if z > 0,
r−l(x,z) := z
0, if z ≤ 0.
Then, using also assumption (A4),
qˆ−l(x,z) = r−l(x,z)z, 0 ≤ r−l(x,z) ≤ α+β|z|p1−2 (∀x ∈ S , z ∈ R). (16)
h
In what follows, we will rewrite problem (14) as
(cid:90) (cid:90)
(cid:0)b−l(x,u ,∇ u ) ∇ u ·∇ v + r−l(x,u )u v (cid:1) = fˆ−lv (∀v ∈ V0). (17)
h Sh h Sh h Sh h h h h h h h
S S
h h
We set
n¯
(cid:88)
u = c χ , (18)
h j j
j=1
and look for the coefficients c ,c ,...,c .
1 2 n¯
Lemma 2. Let g∗ and γ be as given in Assumption 2.1.2 (A2). Then there exists a
constant c > 0 such that for all h ≤ h with a h > 0 sufficiently small, the discrete
0 0
function g , from (12), satisfies
h
(cid:107)g (cid:107) ≤ c(cid:107)g∗(cid:107) . (19)
h 1,p,S 1,γ,S
h
Proof. The assumption on γ in (A2) implies that in each case we have γ ≥ p and γ > d.
Since the compact manifold S has bounded measure, the relation γ ≥ p implies that
h
(cid:107)g (cid:107) ≤ c (cid:107)g (cid:107)
h 1,p,S 1 h 1,γ,S
h h
for some constant c > 0 independent of h. Further, the relation γ > d implies that
1
W1,γ(S ) ⊂ C(S ), see, e.g., [1]. Hence, by [7, Thm. 3.1.6], the stability of the Lagrange
h h
interpolation holds for (g∗)−l in W1,γ-norm on each triangle and hence also on all S ,
h
therefore
(cid:107)g (cid:107) ≤ c (cid:107)(g∗)−l(cid:107)
h 1,γ,S 2 1,γ,S
h h
for some constant c > 0 independent of h. Finally, Lemma 1 implies that
2
(cid:107)(g∗)−l(cid:107) ≤ c (cid:107)g∗(cid:107)
1,γ,S 3 1,γ,S
h
for some constant c > 0 independent of h. These three estimates together yield the
3
desired result.
Later on the norm-boundedness of the SFEM solutions will play an important role.
Lemma 3. The norms (cid:107)u (cid:107) are bounded independently of h.
h 1,p,S
h
9
Proof. We will use the shorthand notation
|v | := (cid:107)∇ v (cid:107)
h 1,p,S S h p,S
h h h
Since (15) shows that v := u −g ∈ V0, we use it as a test function in the discrete weak
h h h h
problem (17). Rearranging the terms, with g on the right, we obtain
h
(cid:90)
(cid:0)b−l(x,u ,∇ u ) |∇ u |2 +r−l(x,u )|u |2(cid:1)
h S h S h h h
h h
S
h
(cid:90) (cid:90)
(cid:16) (cid:17)
= b−l(x,u ,∇ u )∇ u ·∇ g + r−l(x,u )u g + fˆ−l(u −g ) .
h S h S h S h h h h h h
h h h
S S
h h
We estimate this equality appropriately from both above and below. For the lower
bound of the left-hand side we use (A3), which yields b−l(x,·,ξ)|ξ|2 ≥ µ |ξ|p (x ∈ S ),
1 h
and since r−l(x,·) ≥ 0 (x ∈ S ), we obtain the lower bound
h
(cid:90) (cid:90)
µ |u |p = µ |∇ u |p ≤ b−l(x,u ,∇ u ) |∇ u |2 +r−l(x,u )|u |2. (20)
1 h 1,p,Sh 1 Sh h h Sh h Sh h h h
S S
h h
For the first term on the right-hand side, we have from (A3) that
(cid:90) (cid:90)
b−l(x,u ,∇ u )∇ u ·∇ g ≤ (M |∇ u |+M |∇ u |p−1)|∇ g |
h S h S h S h 0 S h 1 S h S h
h h h h h h
S S
h h
≤ M |u | |g | +M |u |p−1 |g |
0 h 1,p(cid:48),Sh h 1,p,Sh 1 h 1,p,Sh h 1,p,Sh
≤ c(cid:0)M |u | +M |u |p−1 (cid:1)(cid:107)g (cid:107)
0 h 1,p,Sh 1 h 1,p,Sh h 1,p,Sh
using Ho¨lder’s inequality, with 1+ 1 = 1, p(cid:48) ≤ 2 ≤ p, and the boundedness of the surface.
p p(cid:48)
For the second term on the right-hand side, using (16), we have
(cid:90) (cid:90)
(cid:16) (cid:17) (cid:16) (cid:17)
r−l(x,u )u g +fˆ−l(u −g ) = r−l(x,v +g )(v +g )g + fˆ−lv
h h h h h h h h h h h
S S
h h
(cid:90)
(cid:16) (cid:17)
≤ (α+β|v +g |p1−2)(v +g )g +fˆ−lv
h h h h h h
S
h
(cid:0)ˆ (cid:1)
≤ k |v | +(cid:107)g (cid:107) (cid:107)g (cid:107)
1 h 1,p,S h 1,p,S h 1,p,S
h h h
+(cid:0)kˆ |v | +(cid:107)g (cid:107) (cid:1)p1−1(cid:107)g (cid:107)
1 h 1,p,S h 1,p,S h 1,p,S
h h h
ˆ ˆ
+ck (cid:107)f(cid:107) |v | ,
1 p(cid:48),S h 1,p,S
h
where the last estimates is obtained, similarly as before, using repeatedly Ho¨lder’s in-
equality (with p(cid:48) ≤ 2 ≤ p) and again the boundedness of the surface, and the equivalent
version of (2) on W1,p(S ):
0 h
(cid:107)v(cid:107) ≤ kˆ |v| , (∀v ∈ W1,p(S )),
p1,Sh 1 1,p,Sh 0 h
together with the bound (cid:107)fˆ−l(cid:107) ≤ c(cid:107)fˆ(cid:107) . This can be further estimated using
p(cid:48),S p(cid:48),S
h
|v | ≤ |u | +|g | ≤ |u | +(cid:107)g (cid:107)
h 1,p,S h 1,p,S h 1,p,S h 1,p,S h 1,p,S
h h h h h
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