On the numerical approximation of $p$-Biharmonic and $\infty$-Biharmonic functions

2.7 MB·

by  Nikos Katzourakis

ON THE NUMERICAL APPROXIMATION OF p-BIHARMONIC AND
∞-BIHARMONIC FUNCTIONS
NIKOSKATZOURAKISANDTRISTANPRYER
7 Abstract. In [KP16b] the authors introduced a second order variational problem in L∞. The associated
1 equation,coinedthe∞-Bilaplacian,isathird order fullynonlinearPDEgivenby
0
∆2 u :=(∆u)3|D(∆u)|2=0.
2 ∞
Inthisworkwebuildanumericalmethodaimedatquantifyingthenatureofsolutionstothisproblemwhich
n
we call ∞-Biharmonic functions. For fixed p we design a mixed finite element scheme for the pre-limiting
a
equation,thep-Bilaplacian
J (cid:0) (cid:1)
∆2pu :=∆ |∆u|p−2∆u =0.
5
Weproveconvergenceofthenumericalsolutiontotheweaksolutionof∆2u=0andshowthatweareable
2 p
to pass to the limit p → ∞. We perform various tests aimed at understanding the nature of solutions of

∆2 uandin1-dweproveconvergenceofourdiscretisationtoanappropriateweaksolutionconceptofthis
] ∞
A problem,thatofD-solutions.
N
1. Introduction and the ∞-Bilaplacian
.
h
Let Ω⊂Rd be an open and bounded set. For a given function u:Ω→R we denote the gradient of u as
t
a Du:Ω→Rd and its Hessian D2u:Ω→Rd×d and Laplacian ∆u:Ω→R. The p–Bilaplacian
m Ä ä
(1.1) ∆2u :=∆ |∆u|p−2∆u =0
[ p

is a fourth order elliptic partial differential equation (PDE) which is a nonlinear generalisation of the Bi-
1
v laplacian. Such problems typically arise from areas of elasticity, in particular, the nonlinear case can be
5 used as a model for travelling waves in suspension bridges [LM90, GM10]. It is a fourth order analogue to
1 its second order sibling, the p–Laplacian, and as such it is useful as a prototypical nonlinear fourth order
4
problem.
7
The efficient numerical simulation of general fourth order problems has attracted growing interest. A
0
. conforming approach to this class of problems would require the use of C1 finite elements, the Argyris
1
element for example [Cia78, Section 6]. From a practical point of view the approach presents difficulties, in
0
7 that the C1 finite elements are difficult to design and complicated to implement, especially when working in
1 three spatial dimensions. Other possibilities include discontinuous Galerkin methods, which form a class of
: nonconformingfiniteelementmethod. Ifp=2wehavethespecialcasethatthe(2–)Bilaplacian,∆2u=0,is
v
i linear. It has been well studied in the context of both C1 finite elements [Cia78] and discontinuous Galerkin
X
methods;forexample,thepapers[LS03,GH09]studytheuseofh–k dGfiniteelements(wherek heremeans
r the local polynomial degree as opposed to the usual convention which is p) applied to the (2–)Bilaplacian.
a
Thenumericalapproximationofp-Bilaplacian(quasi-linear,fourthorder)typePDEsisrelativelyuntouched.
To the authors’ knowledge, the only known work is [Pry14] where a discontinuous Galerkin method based
on a variational principle was derived and was shown to converge under minimal regularity. However, no
rates of convergence were proven.
InthisworkweproposeamethodbasedonC0-mixedfiniteelements. Werewritetheminimisationproblem
in the spirit of a saddle point problem and prove that the method converges under minimal regularity of the
solution. In addition, using an inf-sup condition and tools from [San93, Far98, GR12] we are able to show
Date:January26,2017.
N.K.waspartiallysupportedthroughtheEPSRCgrantEP/N017412/1. T.P.waspartiallysupportedthroughtheEPSRC
grantEP/P000835/1.
Keywords: p-Bilaplacian;∞-Bilaplacian;Generalisedsolutions;CalculusofVariationsinL∞;Finiteelementmethod;Fully
nonlinearequations;Youngmeasures
1
that under additional regularity assumptions the approximation converges with specific rates that depend
on p.
Making use of these convergence results and the uniqueness of solutions in one dimension we are able
to justify that approximations of the p-Bilaplacian for large p are “good” approximations to ∞-Biharmonic
functions. These functions are solutions of the ∞-Bilaplacian which is the PDE
(1.2) ∆2 u :=(∆u)3|D(∆u)|2 =0.
∞
It was derived in [KP16b] as the formal limit of the p-Bilaplacian (1.1) as p→∞. The ∞-Bilaplacian is the
prototypical example of a PDE from 2nd order Calculus of Variations in L∞, arising as the analogue of the
Euler–Lagrange equation associated with critical points of the supremal functional
(1.3) J[u;∞]:=(cid:107)∆u(cid:107) .
L∞(Ω)
Variational problems in L∞ are notoriously challenging. The 1st order case is reasonably well understood
and was initiated in the sequence of works by Aronsson [Aro65, Aro66, Aro68, Aro69, c.f.]. In this case, the
respective Euler-Lagrange equation associated with critical points of the functional
(1.4) J[u]=(cid:107)Du(cid:107)
L∞(Ω)
is quasi-linear, 2nd order and given by
(1.5) ∆ u=(Du⊗Du):D2u=0.
∞
This equation is called the ∞-Laplacian and can be derived through a p-approximation of the underlying
W1,p energy functional, see [Pry16, KP16a].
Itcanbeeasilyseenthatsolutionsto(1.2)cannotingeneralbeC3 notevenwhend=1;inparticular,the
Dirichlet problem is not solvable in the class of classical solutions. For a more extensive discussion we refer
to [KP16b]. Hence, the development of a solution concept which can be interpreted in an appropriate weak
sense is in order. In the case of the ∞-Laplacian, the appropriate notion is that of the Crandall-Ishii-Lions
notionofviscositysolutions. Foranintroductiontothistheorywerefertothemonograph[Kat15]. Wenote
that in the framework of viscosity solutions we can obtain uniqueness of solution for the Dirichlet problem
[Jen93]. Inthecaseof2nd orderCalculusofVariationsinL∞ theviscositysolutionconceptfortheresulting
equations is no longer applicable since we do not have access to a maximum principle for 3rd order PDEs
like (1.2), from which the solution concept stems.
Onepossibilityforageneralisedsolutionconceptto(1.2)isthatofD-solutions[Kat16b,Kat16a,KP16b].
Roughly, this is a probabilistic approach where derivatives that do not exist classically are represented as
limits of difference quotients into Young measures over a compactification of the space of derivatives. This
solutionconcepthasalreadybornesubstantialfruitinthe1st ordervectorialcaseofCalculusofVariationsin
L∞,aswellasformoregeneralPDEsystems. Inthepresent2nd ordersettingitprovestobeanappropriate
notion as well, since absolute minimisers u∈W2,∞(Ω) satisfying
g
(1.6) (cid:107)∆u(cid:107) ≤(cid:107)∆v(cid:107) ∀Ω(cid:48) (cid:98)Ω and v ∈W2,∞(Ω(cid:48)),
L∞(Ω(cid:48)) L∞(Ω(cid:48)) u
areindeeduniqueD-solutionsof(1.2),atleastford=1. Notethattheappropriatespacetotakeminimisers
is not W2,∞(Ω) but rather the larger space
g
 
 (cid:92) 
(1.7) W2,∞(Ω):= u∈ W2,p(Ω): ∆u∈L∞(Ω) .
g g
 
p∈(1,∞)
Uniqueness for the Dirichlet problem amongst this solution class is, for d > 1, still an open problem at
the time of writing this1. In [KP16b] it has been shown that in one spatial dimension the problem does
indeed have a unique absolutely minimising D-solution, while we also have uniqueness in the subclass of C3
functions for all spatial dimensions.
Thedesignofnumericalschemesthatarecompatiblewiththeseduality-freesolutionconceptsisextremely
difficult. Evenforthewelldevelopedareaofviscositysolutionsmostnumericalschemesthatexistwhichare
compatiblewiththesolutionconceptarebasedontheargumentsof[BS91]whichadvocatesapproximations
1InanupcomingpaperofthefirstauthorwithR.Moseritisestablishedthatuniquenessof∞-Biharmonicfunctionswith
prescribedboundaryvaluesindeedholdsinallspatialdimensionsd>1.
2
based on differences and which satisfy a discrete monotonicity property. The only other methodology in
the design of numerical schemes for the ∞-Laplacian is to make use of the variational principle from which
the equation is derived. Galerkin approximations of the p-Laplacian can then be shown to converge to the
viscosity solution of the ∞-Laplacian [Pry16]. This method has also been used to characterise the nature of
solutions to the variational ∞-Laplace system [KP16a]. This is also the approach we use here. We build a
scheme convergent to the weak solution of the p-Bilaplacian and then justify its use as an approximation of
∞-Biharmonic functions.
The rest of the paper is set out as follows: In §2 we formalise notation and begin exploring some of the
properties of the p-Bilaplacian. In particular, we reformulate the PDE as a saddle point type problem. We
show inf-sup conditions for the underlying operators guarantee that the saddle point type problem is well
posed, motivating the discretisation of this directly. In §3 we perform the discretisation for fixed p and
show that discrete versions of the inf-sup conditions hold. A priori results for both primal and auxiliary
variables are a consequence of this. Numerical experiments are given in §4 illustrating the behaviour of
numerical approximations to this problem. In addition, we examine the solutions for large p and make
various conjectures as to the structure of solutions in multiple spatial dimensions.
2. Approximation via the p-Bilaplacian
In this section we describe how ∞-Biharmonic functions can be approximated using p-Biharmonic func-
tions. We give a brief introduction to the p–Bilaplacian problem, beginning by introducing the Sobolev
spaces
(2.1)
ß ™
(cid:90)
Lp(Ω)= φ measurable: |φ|p dx<∞ for p∈[1,∞) and L∞(Ω)={φ measurable: esssup |φ|<∞},
Ω
Ω
(2.2) Wl,p(Ω)={φ∈Lp(Ω): Dαφ∈Lp(Ω), for |α|≤l} and Hl(Ω):=Wl,2(Ω),
which are equipped with the following norms and semi-norms:
(cid:90)
(2.3) (cid:107)v(cid:107)p := |v|pdx for p∈[1,∞) and (cid:107)v(cid:107) :=esssup |v|
Lp(Ω) L∞(Ω) Ω
Ω
(2.4) (cid:107)v(cid:107)p :=(cid:107)v(cid:107)p = (cid:88) (cid:107)Dαv(cid:107)p
l,p Wl,p(Ω) Lp(Ω)
|α|≤l
(2.5) |v|p :=|v|p = (cid:88) (cid:107)Dαv(cid:107)p
l,p Wl,p(Ω) Lp(Ω)
|α|=l
where α = {α ,...,α } is a multi-index, |α| = (cid:80)d α and derivatives Dα are understood in the weak
1 d i=1 i
sense. We pay particular attention to the case l=2 and define
(2.6) W2,p(Ω):=g+W2,p(Ω)=(cid:8)φ∈W2,p(Ω): φ| =g and Dφ| =Dg(cid:9),
g 0 ∂Ω ∂Ω
for a prescribed function g ∈ W2,∞(Ω), where the boundary condition is understood in the trace sense if
∂Ω ∈ C0,1(Ω). We note that if p > d, then the boundary condition is satisfied in the pointwise sense since
W2,p(Ω)⊆C1(Ω).
0
For the p–Bilaplacian, the action functional is given as
(cid:90)
(2.7) J[u;p]= |∆u|p dx. 2
Ω
We then look to find a minimiser over the space W2,p(Ω), that is, to find u∈W2,p(Ω) such that
g g
(2.8) J[u;p]= min J[v;p].
v∈W2g,p(Ω)
2Typically J[u;p] = 1(cid:82) |∆u|p. Note here the rescaling has no effect on the resultant Euler–Lagrange equations as the
p
Lagrangianisindependentofu.
3
If we assume temporarily that we have access to a smooth minimiser, i.e., u ∈ C4(Ω), then, given that
the Lagrangian is of second order, we have that the Euler–Lagrange equations are (in general) fourth order
and read
Ä ä
(2.9) ∆ |∆u|p−2∆u =0.
Note that, for p=2, the PDE reduces to the Bilaplacian ∆2u=0. In general, the Dirichlet problem for the
p-Bilaplacian is, given g ∈W2,∞(Ω), to find u such that
Ä ä
 ∆ u:=∆ |∆u|p−2∆u =0, in Ω,
 p
(2.10) u=g, on ∂Ω,
 Du=Dg, on ∂Ω.
2.1. Definition (weak solution). The problem (2.10) has a weak formulation. Consider the semilinear form
(cid:90) Ä ä
(2.11) A(u,v):= |∆u|p−2∆u ∆vdx.
Ω
Then, u∈W2,p(Ω) is a weak solution of (2.10) if it satisfies
g
(2.12) A(u,v)=0 ∀v ∈W2,p(Ω).
0
2.2. Proposition (coercivity of J). Suppose that u∈W2,p(Ω) and f ∈Lq(Ω), where 1 + 1 =1. We have
0 p q
that the action functional J[ · ;p] is coercive over W2,p(Ω), that is,
0
(2.13) J[u;p]≥C|u|p −γ,
2,p
for some C >0 and γ ≥0. Equivalently, we have that there exists a constant C >0 such that
(2.14) A(v,v)≥C|v|p ∀v ∈W2,p(Ω).
2,p 0
2.3. Corollary (weak lower semicontinuity). The action functional J is weakly lower semi-continuous over
W2,p(Ω). That is, given a sequence of functions {u } which has a weak limit u∈W2,p(Ω), we have
g j j∈N g
(2.15) J[u;p]≤liminfJ[u ;p].
j
j→∞
Proof The proof of this fact is a straightforward extension of [Eva98, Section 8.2 Thm 1] to second order
Lagrangians, noting that J is coercive (from Proposition 2.2) and convex. We omit the full details for
brevity. (cid:3)
2.4. Corollary (existence and uniqueness). There exists a unique minimiser to the p–Dirichlet energy func-
tional. Equivalently, there exists a unique (weak) solution u ∈ W2,p(Ω) to the (weak form of the) Euler–
g
Lagrange equations:
(cid:90)
(2.16) |∆u|p−2∆u∆φdx=0 ∀φ∈W2,p(Ω).
0
Ω
Proof Again, the result can be deduced by extending the arguments in [Eva98, Section 8.2] or [Cia78, Thm
5.3.1], again, noting the results of Propositions 2.2 and convexity. The full argument is omitted for brevity.
(cid:3)
2.5. Remark (elementary properties). We will throughout this exposition use the notation p to denote the
exponent appearing in the Lagrangian and q its conjugate exponent which satisfies
1 1
(2.17) + =1.
p q
We shall now state some useful facts involving these numbers:
(1) For a given v ∈Lp(Ω) it holds that
(cid:13) (cid:13)
(2.18) (cid:13)|v|p−1(cid:13) =(cid:107)v(cid:107)p−1 .
(cid:13) (cid:13)Lq(Ω) Lp(Ω)
4
2.6. Proposition (Poincar´e inequality). Let Ω⊂Rd be a bounded domain. For any p∈[1,∞], there exists
a constant C =C(Ω,p)>0 depending only on Ω and p such that
(2.19) (cid:107)u(cid:107) ≤C(Ω,p)(cid:107)Du(cid:107) ,
Lp(Ω) Lp(Ω)
for all u∈W1,p(Ω).
0
2.7. Proposition (Calderon-Zygmund estimate [GT83, Cor 9.10]). Let Ω⊂Rd be a domain. Then, for any
p∈(1,∞), there is a constant C =C(d,p)>0 depending only on d and p such that
(2.20) (cid:13)(cid:13)D2u(cid:13)(cid:13) ≤C(d,p)(cid:107)∆u(cid:107) ,
Lp(Ω) Lp(Ω)
for all u∈W2,p(Ω).
0
AnimmediateconsequenceofPropositions2.6and2.7aboveisthatthenorm(cid:107)·(cid:107) isequivalenttoeither
2,p
of the seminorms (cid:13)(cid:13)D2(·)(cid:13)(cid:13) and (cid:107)∆(·)(cid:107) over the space W2,p(Ω).
Lp(Ω) Lp(Ω) 0
2.8. Saddle point formulation of the p-Bilaplacian. The mixed formulation we propose to analyse is
basedontheobservationthatifφ(t)=|t|p−2t,theinverseiswelldefinedasφ−1(t)=sgn(t)|t|1/(p−1) =|t|q−2t.
Using this we make the following choice of auxiliary variable
(2.21) w =|∆u|p−2∆u
from which we can infer that
(2.22) |w|q−2w =∆u.
This allows us to write the problem as the mixed system:
®
−∆u=|w|q−2w,
(2.23)
−∆w =0.
The mixed formulation can be written weakly as: Find a pair(u,w)∈W1,p(Ω)×W1,q(Ω) such that
g
®
a(w,ψ)+b(u,ψ)=f(ψ),
(2.24)
b(w,φ)=0, ∀(ψ,φ)∈W1,q(Ω)×W1,p(Ω),
0
where the semilinear form a(w,v) and bilinear form b(u,v) are given by
 (cid:90)
 a(w,ψ):= |w|q−2wψdx
(2.25) Ω
(cid:90)
 b(u,ψ):= Du·Dψdx
Ω
and the problem data
(cid:90)
(2.26) f(ψ):= Dg·nψds
∂Ω
representsthecontributionfromtheNeumannboundaryconditions. Noticethattheproblem(1.1)hasbeen
reformulated in a saddle point form. Although we already know that the problem has a unique solution as
a consequence of Corollary 2.4, we will show that the equivalent saddle point problem also admits a unique
solution since the methodology will be useful in the sequel. We begin by recalling the following result that
we will utilise.
2.9. Theorem (Generalised G¨arding inequality [Sim72, Thm 6.3]). If the boundary ∂Ω is C1, there exists a
constant C >0, such that, for all u ∈W1,p(Ω) we have
0 0
(cid:34) (cid:35)
b(u ,v)
(2.27) (cid:107)Du (cid:107) ≤C sup 0 +(cid:107)u (cid:107) .
0 Lp(Ω) (cid:107)Dv(cid:107) 0 Lp(Ω)
0(cid:54)=v∈W1,q(Ω) Lq(Ω)
0
5
2.10. Corollary (Inf-sup stability of b(·,·)). For any u ∈ W1,p(Ω), the bilinear form b(·,·) satisfies the
0 0
following inf-sup property:
b(u ,v)
(2.28) (cid:107)Du (cid:107) ≤C sup 0 .
0 Lp(Ω) (cid:107)Dv(cid:107)
0(cid:54)=v∈W1,q(Ω) Lq(Ω)
0
Proof Consider the Dirichlet problem
®
−∆z =|u |p−2u , in Ω,
0 0
(2.29)
z =0, on ∂Ω.
Then, since u ∈W1,p(Ω), we have that |u |p−2u ∈Lq(Ω) and the gradient estimate
0 0 0 0
(2.30) (cid:107)Dz(cid:107)Lq(Ω) ≤C(cid:13)(cid:13)|u0|p−1(cid:13)(cid:13)Lq(Ω).
Then
(cid:90)
(cid:107)u (cid:107)p = (cid:0)|u |p−2u (cid:1)u dx
0 Lp(Ω) 0 0 0
Ω
(cid:90)
(2.31)
=− ∆z u dx
0
Ω
=b(u ,z),
0
throughthedefinitionoftheauxiliaryproblemandanintegrationbyparts. Now,makinguseoftheestimate
(2.30) we have
b(u ,z)(cid:107)Dz(cid:107)
(cid:107)u (cid:107)p = 0 Lq(Ω)
0 Lp(Ω) (cid:107)Dz(cid:107)
Lq(Ω)
≤ (cid:107)Db(zu(cid:107)0,z) C(cid:13)(cid:13)|u0|p−1(cid:13)(cid:13)Lq(Ω)
Lq(Ω)
(2.32)
b(u ,z)
≤ 0 C(cid:107)u (cid:107)p−1
(cid:107)Dz(cid:107) 0 Lp(Ω)
Lq(Ω)
b(u ,v)
≤C sup 0 (cid:107)u (cid:107)p−1 .
(cid:107)Dv(cid:107) 0 Lp(Ω)
0(cid:54)=v∈W1,q(Ω) Lq(Ω)
0
Hence
b(u ,v)
(2.33) (cid:107)u (cid:107) ≤C sup 0 ,
0 Lp(Ω) (cid:107)Dv(cid:107)
0(cid:54)=v∈W1,q(Ω) Lq(Ω)
0
which combining with the result in Theorem 2.9 yields the desired conclusion. (cid:3)
2.11. Theorem (The mixed formulation is well posed). For every g ∈ W2,∞(Ω), there exists a unique pair
(u,w) solving (2.24) that satisfies
Ä ä
(2.34) (cid:107)Du(cid:107) +(cid:107)w(cid:107)q−1 ≤C (cid:107)∆g(cid:107) +(cid:107)Dg(cid:107) .
Lp(Ω) Lq(Ω) Lp(Ω) Lp(Ω)
Proof The results of Lemma 2.10 show that for u :=u−g ∈W1,p(Ω) we have
0 0
b(u ,v)
(cid:107)Du (cid:107) ≤ sup 0
0 Lp(Ω) (cid:107)Dv(cid:107)
0(cid:54)=v∈W1,q(Ω) Lq(Ω)
0
b(u,v) b(g,v)
(2.35) ≤ sup + sup
(cid:107)Dv(cid:107) (cid:107)Dv(cid:107)
0(cid:54)=v∈W1,q(Ω) Lq(Ω) 0(cid:54)=v∈W1,q(Ω) Lq(Ω)
0 0
−a(w,v) b(g,v)
≤ sup + sup ,
(cid:107)Dv(cid:107) (cid:107)Dv(cid:107)
0(cid:54)=v∈W1,q(Ω) Lq(Ω) 0(cid:54)=v∈W1,q(Ω) Lq(Ω)
0 0
6
in view of the compactness of the support of v. Now, by using Remark 2.5 and Proposition 2.6 we estimate
(cid:13) (cid:13)
(cid:13)wq−1(cid:13) (cid:107)v(cid:107)
Lp(Ω) Lq(Ω)
(cid:107)Du (cid:107) ≤ sup +(cid:107)Dg(cid:107)
0 Lp(Ω) (cid:107)Dv(cid:107) Lp(Ω)
0(cid:54)=v∈W1,q(Ω) Lq(Ω)
0
(2.36) (cid:107)w(cid:107)q−1 (cid:107)Dv(cid:107)
Lq(Ω) Lq(Ω)
≤C sup +(cid:107)Dg(cid:107)
(cid:107)Dv(cid:107) Lp(Ω)
0(cid:54)=v∈W1,q(Ω) Lq(Ω)
Ä 0 ä
≤C (cid:107)w(cid:107)q−1 +(cid:107)Dg(cid:107) .
Lq(Ω) Lp(Ω)
Now take ψ =w in (2.24) then
(2.37) a(w,w)+b(u,w)=f(w).
Set φ=u in (2.24) then
0
(2.38) b(w,u )=0
0
and in particular
(2.39) a(w,w)+b(u,w)−b(w,u )=f(w).
0
Noticing that b(·,·) is symmetric then
(2.40) a(w,w)+b(g,w)=f(w),
or explicitly
(cid:90) (cid:90)
(2.41) |w|q+Dg·Dwdx= Dg·nds.
Ω ∂Ω
Integrating by parts then shows that
(cid:90)
(cid:107)w(cid:107)q = ∆gwdx
(2.42) Lq(Ω)
Ω
≤(cid:107)∆g(cid:107) (cid:107)w(cid:107) .
Lp(Ω) Lq(Ω)
Hence
(2.43) (cid:107)w(cid:107)q−1 ≤(cid:107)∆g(cid:107) ,
Lq(Ω) Lp(Ω)
which yields the desired result upon noting
(2.44) (cid:107)Du(cid:107) ≤(cid:107)Du (cid:107) +(cid:107)Dg(cid:107)
Lp(Ω) 0 Lp(Ω) Lp(Ω)
and combining with (2.36). (cid:3)
2.12. Theorem (the limit as p→∞). Consider the Sobolev space (1.7) and let (u )∞ denote a sequence of
p 1
weak solutions u ∈ W2,p(Ω) to the p-Bilaplacian. Then, there exists a subsequence converging uniformly
p g
together wth their derivatives to a (candidate ∞-Biharmonic) function u ∈W2,∞(Ω). Namely,
∞ g
(2.45) u →u in C1(Ω),
pj ∞
along a subsequence as p→∞.
Proof Let u ∈ W2,p(Ω) denote the weak solution of (2.10). In view of Corollary 2.4, we know that u
p g p
minimises the energy functional
(cid:90)
(2.46) J[u ]= |∆u |p.
p p
Ω
In particular,
(2.47) J[u ]≤J[g],
p
where g ∈W2,∞(Ω) is the associated boundary data to (2.10). Using this fact, we have
(2.48) (cid:107)∆u (cid:107)p =J[u ]≤J[g]=(cid:107)∆g(cid:107)p ,
p Lp(Ω) p Lp(Ω)
7
and we may infer that
(2.49) (cid:107)∆u (cid:107) ≤(cid:107)∆g(cid:107) .
p Lp(Ω) Lp(Ω)
Now fix a k > d and take p ≥ k. Then, by using H¨older’s inequality with r = p and q = r−1 such that
k r
1 + 1 =1, we obtain
r q
Å ã Å ã
(cid:90) (cid:90) 1/q (cid:90) 1/r
(2.50) (cid:107)∆u (cid:107)k = |∆u |k ≤ 1q |∆u |p .
p Lk(Ω) p p
Ω Ω Ω
Hence
r k
(2.51) (cid:107)∆u (cid:107)k ≤|Ω|r−1 (cid:107)∆u (cid:107)k =|Ω|1−p (cid:107)∆u (cid:107)k
p Lk(Ω) p Lp(Ω) p Lp(Ω)
and we see
1 1
−
(2.52) (cid:107)∆u (cid:107) ≤|Ω|k p (cid:107)∆u (cid:107) .
p Lk(Ω) p Lp(Ω)
By using the triangle inequality, a double application of the Poincar´e inequality (since both u = g and
Du = Dg on ∂Ω) from Proposition 2.6 and the Calderon-Zygmund Lk estimates from Proposition 2.7, we
have
(cid:107)u (cid:107) ≤(cid:107)u −g(cid:107) +(cid:107)g(cid:107)
p Lk(Ω) p Lk(Ω) Lk(Ω)
(2.53) ≤C(cid:48)(k,Ω)(cid:13)(cid:13)D2up−D2g(cid:13)(cid:13)Lk(Ω)+(cid:107)g(cid:107)Lk(Ω)
≤C(k,Ω)(cid:107)∆u −∆g(cid:107) +(cid:107)g(cid:107) .
p Lk(Ω) Lk(Ω)
By utilising the triangle inequality again, we have
Ä ä
(cid:107)u (cid:107) ≤C (cid:107)∆u (cid:107) +(cid:107)g(cid:107)
p Lk(Ω) Å p Lk(Ω) W2,k(Ω) ã
(2.54) 1 1
−
≤C |Ω|k p (cid:107)∆u (cid:107) +(cid:107)g(cid:107) ,
p Lp(Ω) W2,k(Ω)
by virtue of (2.52). Similarly, one may show that
Å ã
1 1
−
(2.55) (cid:107)Du (cid:107) ≤C |Ω|k p (cid:107)∆u (cid:107) +(cid:107)g(cid:107) .
p Lk(Ω) p Lp(Ω) W2,k(Ω)
Thus, in view of (2.49) we infer that
(2.56) (cid:107)up(cid:107)W2,k(Ω) ≤C(cid:107)g(cid:107)W2,k(Ω).
This means that for any k >d we have the uniform bound
(2.57) sup(cid:107)u (cid:107) ≤C =C(k,Ω).
p W2,k(Ω)
p>k
By invoking standard weak compactness arguments, we may extract a sub-sequence {u }∞ ⊂ {u }∞
pj j=1 p p=1
and a function u ∈W2,k(Ω) such that, for any k >n,
∞
(2.58) u (cid:42)u weakly in W2,k(Ω)
pj ∞
as j →∞ and
(cid:13) (cid:13)
(cid:107)u∞(cid:107)W2,k(Ω) ≤lijm→i∞nf(cid:13)upj(cid:13)W2,k(Ω)
(2.59)
≤liminfC(cid:107)g(cid:107) .
W2,k(Ω)
j→∞
Since this is true for any fixed k, it is clear that u ∈(cid:84) W2,k(Ω). Further, by the weak lower semi-
∞ k∈(1,∞)
continuity of the Lk norm, from (2.52) we may infer ∆u ∈ L∞(Ω) and hence u ∈ W2,∞(Ω), therefore
∞ ∞ g
concluding the proof. (cid:3)
8
2.13. Remark (uniqueness of “weak” solutions to the ∞-Bilaplacian). Theorem 2.12 only guarantees con-
vergence to a candidate ∞-Harmonic function. If d>1 it is not known if the limiting problem
 (∆u)3|D(∆u)|2 =0, in Ω,

(2.60) u=g, on ∂Ω,
 Du=Dg, on ∂Ω,
has a unique solution. Hence, Theorem 2.12 only guarantees convergence to some candidate ∞-Biharmonic
function. If d=1 the problem reduces to
 (u(cid:48)(cid:48))3(u(cid:48)(cid:48)(cid:48))2 =0, in (a,b)

u(a)=g(a), u(b)=g(b),
 u(cid:48)(a)=g(cid:48)(a), u(cid:48)(b)=g(cid:48)(b).
It has been shown that for this case there is a unique D-solution [KP16b]. The solution is interpreted in
a weak sense and is the only candidate ∞-Harmonic function. This means for d = 1 that Theorem 2.12
guarantees convergence of the sequence of p-Biharmonic functions to the unique ∞-Biharmonic function.
(However, in an upcoming paper of the first author with R. Moser it will be shown that uniqueness of D-
solutions for (2.60) indeed holds in all dimensions. Notwithstanding, herein we do not attempt to study this
problem further and we leave the case d>1 of (2.60) for future work.)
3. Discretisation of the p-Bilaplacian
In this section we describe a mixed finite element discretisation of the p-Bilaplacian. Let T be a con-
forming triangulation of Ω, namely, T is a finite family of sets such that
(1) K ∈T implies K is an open simplex (segment for d=1, triangle for d=2, tetrahedron for d=3),
(2) foranyK,J ∈T wehavethatK∩J isafulllower-dimensionalsimplex(i.e.,itiseither∅,avertex,
an edge, a face, or the whole of K and J) of both K and J and
(3) (cid:83) K =Ω.
K∈T
The shape regularity constant of T is defined as the number
ρ
(3.1) µ(T ):= inf K,
K∈T hK
where ρ is the radius of the largest ball contained inside K and h is the diameter of K. An indexed
K K
family of triangulations {Tn} is called shape regular if
n
(3.2) µ:=infµ(Tn)>0.
n
Further, we define h:Ω→R to be the piecewise constant meshsize function of T given by
(3.3) h(x):=maxh .
K
K(cid:51)x
A mesh is called quasi-uniform when there exists a positive constant C such that max h ≤ Cmin h.
x∈Ω x∈Ω
In what follows we shall assume that all triangulations are shape-regular and quasi-uniform although the
results may be extendable even in the non-quasi-uniform case using techniques developed in [DK08].
We let E be the skeleton (set of common interfaces) of the triangulation T and say e ∈ E if e is on the
interior of Ω and e∈∂Ω if e lies on the boundary ∂Ω and set h to be the diameter of e.
e
We let Pk(T ) denote the space of piecewise polynomials of degree k over the triangulation T ,i.e.,
(3.4) Pk(T )={φ such that φ| ∈Pk(K)}
K
and introduce the finite element space
(3.5) V:=Pk(T )∩C0(Ω)
to be the usual space of continuous piecewise polynomial functions.
9
3.1. Definition (Ritz projection operator). The Ritz projection operator R : W1,2(Ω) → V is defined for
0
v ∈W1,2(Ω) as
0
(cid:90) (cid:90)
(3.6) D(Rv)·DΦ= Dv·DΦ ∀φ∈V.
Ω Ω
It is well known that this operator satisfies the following approximation properties: for any v ∈Wk+1,q(Ω) ,
(3.7) (cid:107)v−Rv(cid:107) =Chk+1|v| ,
Lq(Ω) k+1,q
(3.8) (cid:107)Dv−D(Rv)(cid:107) =Chk|v| .
Lq(Ω) k+1,q
3.2. Galerkin discretisation. Consider the space
(3.9) V :={φ∈V:φ| =Rg}.
g ∂Ω
Then, we consider the Galerkin discretisation of (2.10), to find(u ,w )∈V ×V such that
h h g
a(w ,ψ)+b(u ,ψ)=f(ψ),
h h
(3.10)
b(w ,φ)=0, ∀(ψ,φ)∈V×V ,
h 0
where the bilinear forms a(·,·),b(·,·) are given in (2.25) an.
3.3. Lemma. The bilinear form b(·,·) satisfies the following inf-sup property: for any Φ∈V ,
0
b(Φ,v )
(3.11) (cid:107)DΦ(cid:107) ≤C sup h .
Lp(Ω) (cid:107)Dv (cid:107)
0(cid:54)=vh∈V0 h Lq(Ω)
Proof The proof of this fact involves noticing that the Ritz projection is in fact a Fortin operator, that is
R satisfies
(3.12) b(Φ,v−Rv)=0 ∀v ∈W1,q(Ω)
and that there exists a constant such that
(3.13) (cid:107)D(Rv)(cid:107) ≤C(cid:107)Dv(cid:107) .
Lq(Ω) Lq(Ω)
The orthogonality it clear in view of the definition of the projection. For the stability bound, we refer to
[CT87]. Hence, using the inf-sup condition from Lemma 2.10 we have
b(Φ,v)
(cid:107)DΦ(cid:107) ≤ sup
Lp(Ω) (cid:107)Dv(cid:107)
0(cid:54)=v∈W1,q(Ω) Lq(Ω)
0
b(Φ,Rv)
(3.14) ≤ sup
(cid:107)Dv(cid:107)
0(cid:54)=v∈W1,q(Ω) Lq(Ω)
0
b(Φ,v )
≤ sup h ,
(cid:107)Dv (cid:107)
0(cid:54)=vh∈V0 h Lq(Ω)
as required. (cid:3)
3.4.Theorem(existenceanduniquenessofsolutionto(3.10)). Thereexistsauniquetuple (u ,w )∈V ×V
h h g
solving (3.10). They satisfy the stability bound
Ä ä
(3.15) (cid:107)Du (cid:107) +(cid:107)w (cid:107)q−1 ≤C (cid:107)∆g(cid:107) +(cid:107)Dg(cid:107) .
h Lp(Ω) h Lq(Ω) Lp(Ω) Lp(Ω)
Note that since g ∈W2,∞(Ω), the right hand side of (3.15) is finite.
Proof The proof of this mirrors that of Theorem 2.11. We begin by noting that for ψ =w we have
h
(3.16) a(w ,w )+b(u ,w )=f(w ).
h h h h h
Now for φ=u :=u −Rg we see that
h,0 h
(3.17) b(w ,u −Rg)=0,
h h
10