Finding Multiple Solutions to Elliptic PDE with
Nonlinear Boundary Conditions
An. Le∗ , Zhi-Qiang Wang∗ and Jianxin Zhou†
Abstract
In this paper, in order to solve an elliptic partial differential equation with a nonlinear boundary condition for multiple solutions, the authors combine a minimax approach
with a boundary integral-boundary element method, and identify a subspace and its
special expression so that all numerical computation and analysis can be carried out
more efficiently based on information of functions only on the boundary. Some mathematical justification of the new approach is established. An efficient and reliable local
minimax-boundary element method is developed to numerically search for solutions.
Details on implementation of the algorithm are also addressed. The existence and multiplicity of solutions to the problem are established under certain regular assumptions.
Some conditions related to convergence of the algorithm and instability of solutions
found by the algorithm are verified. To illustrated the method, numerical multiple solutions to some examples on domains with different geometry are displayed with their
profile and contour plots.
Keywords: Multiple solutions, elliptic PDE, nonlinear boundary condition, local minimaxboundary element method
AMS Mathematical Subject Classification: 35J65,35A40,35A15,58E05,58E30
1
Introduction
Many studies in convection-diffusion systems, corrosion/oxidation modeling, metal-insulator
or metal-oxide semiconductor systems [2,11,14,16,19,24,29] lead to solve a linear elliptic
∗
Department of Mathematics and Statistics, Utah State University, Logan, UT 84322. The second author’s research was supported in part by NSF DMS-0820327
†
Department of Mathematics, Texas A&M University, College Station, TX 77843. jzhou@math.tamu.edu.
This autho’s research was supported in part by NSF DMS-0713872/0820327/1115384.
1
A LOCAL MINIMAX-BEM METHOD
2
partial differential equation with a nonlinear boundary condition (BC) of the form
{
(−∆ + aI)u(x) = 0 in Ω,
∂u(x)
= g(x, u(x))
on ∂Ω,
∂ν
2
(1.1)
2
∂
∂
where Ω ⊂ Rn is bounded and open with a smooth boundary Γ = ∂Ω, ∆ = ∂x
2 + · · · + ∂x2
n
1
is the Laplace operator, I is the identity, a > 0, g satisfies certain regularity and growth
conditions and ν is the unit outward normal vector on Γ. Such a model also appears in
the study of stationary solutions to Laplace equation with dynamical nonlinear boundary
conditions [1,17,28 and references therein].
A huge literature exists on analysis and numerical computation of the case where (1.1) has
a unique solution, e.g., finite element method (FEM), boundary integral equation method
(BIE) and boundary element method (BEM) [6,18,26,27, etc.]. In this work we focus on the
case where (1.1) has multiple unstable solutions.
Multiple unstable solutions, lowly or highly, singly or multiply excited, to many nonlinear
systems have been physically observed and mathematically proved to exist with a variety
of solution configurations, instabilities/maneuverabilities. They used to be considered too
hard to catch and therefore to apply by traditional technologies. Now scientists are able to
induce, reach or control them with new advanced (synchrotronic, laser, etc.) technologies
and search for NEW applications. So far, people’s understanding of such solutions is still
quite limited and analytic solutions are too difficult to obtain. On the other hand, due to
their strong nonlinearity, multiplicity, unstable nature, such solutions are extremely difficult
to solve and very elusive to traditional numerical methods. Thus development of efficient and
reliable numerical methods to solve such problems becomes very interesting to both research
and applications. Most results in the literature of studying multiple solution problems focus
on the case where nonlinearities appear in a differential equation. While our model equation
(1.1) has nonlinearities appeared in boundary conditions. So far we have not seen any related
numerical results in the literature, to the best of our knowledge. Note that in application, Ω
can also be unbounded exterior domain with a bounded boundary Γ. We will keep this fact
in mind while dealing with interior domains in the current work. This paper is organized as
follows: in Section 2, we show how to combine a minimax approach with a boundary integralboundary element method to solve this problem for multiple solutions. Along the line, we
identify a subspace in which all numerical computation and analysis can be carried out
more efficiently based on the information of functions only on the boundary. Mathematical
jusificationnof such approach is established. Then an efficient and reliable local minimaxBEM method is developed to find multiple solutions. More details on implementation of the
algorithm are also presented in this section. With this framework, in Section 3, we establish
the existence and multiplicity of solutions to (1.1) under certain regular assumptions. Some
A LOCAL MINIMAX-BEM METHOD
3
conditions related to the convergence of the algorithm and instability of solutions found by
the algorithm are also verified there. To illustrate the method, in Section 4, we solve some
examples. Multiple numerical solutions are displayed with their profile and contour plots.
2
A Local Minimax-BEM Method
It is known that solutions to (1.1) coincide with critical points of the C 2 functional
∫
∫
1
2
2
J(u) =
(|∇u(x)| + au (x))dx − G(x, u(x))dσx , u ∈ H 1 (Ω),
Ω 2
Γ
where
= g(x, u(x)). For any u, v ∈ H 1 (Ω), we have
dJ(u + tv) (J ′ (u), v)W −1,2 ×H 1 =
∀u, v ∈ H 1 (Ω) = W 1,2 (Ω)
dt
t=0
∫
∫
=
(∇u(x)·∇v(x) + au(x)v(x))dx − g(x, u(x))v(x)dσx ,
(2.1)
∂
G(x, u(x))
∂u
Ω
(2.2)
Γ
where J ′ (u) ∈ W −1,2 (Ω) is the Frechet derivative of J at u. Denote an inner product
∫
⟨u, v⟩a = (∇u(x)·∇v(x) + au(x)v(x))dx, ∀u, v ∈ H 1 (Ω),
(2.3)
Ω
and
∥u∥2a
= ⟨u, u⟩a . Thus u∗ ∈ H 1 (Ω) is a solution to (1.1) iff u∗ satisfies (J ′ (u∗ ), v) = 0 or
∫
∗
⟨u , v⟩a = g(x, u∗ (x))v(x)dσx ∀v ∈ H,
(2.4)
Γ
where the left hand side is a volume integral and the right hand side is a boundary integral.
Such a mixture of two types of integrals cause inconvenience and inefficiency in numerical
computation. We will try to resolve this problem. Also a key issue in finding multiple critical
points is whether or not such solutions can be found in certain order in their instabilities.
2.1
A Local Minimax Characterization
The local minimax method (LMM) developed in [21,22,32,33,34,35] is a 2-level optimization
method for finding critical points of a functional in the order of their energy levels.
We briefly describe its mathematical background. Let H be a Hilbert space with norm
∥ · ∥ and J ∈ C 1 (H, R). For a closed subspace L ⊂ H, denote H = L ⊕ L⊥ and SL⊥ = {v ∈
L⊥ : ∥v∥ = 1}. For each v ∈ SL⊥ , denote [L, v] = span{L, v}.
Definition 1. The peak mapping is a set-valued mapping P : SL⊥ → 2H s.t.
P (v) = the set of all local maxima of J on [L, v], ∀v ∈ SL⊥ .
A peak selection is a single-valued mapping p : SL⊥ → H s.t. p(v) ∈ P (v), ∀v ∈ SL⊥ . If p
is locally defined, then p is called a local peak selection.
A LOCAL MINIMAX-BEM METHOD
4
Recall J is said to satisfy the Palais-Smale (PS) condition in H, if any sequence {un } ⊂ H
s.t. {J(un )} is bounded and J ′ (un ) → 0 has a convergent subsequence.
The following theorem provides a mathematical justification for LMM and also gives an
estimate for the instability of a solution approximated by LMM.
Theorem 1. If p is a local peak selection of J near v0 ∈ SL⊥ s.t. (a) p is Lipschitz continuous
at v0 , (b) p(v0 ) ̸∈ L and (c) v0 = arg local minv∈SL⊥ J(p(v)), then u0 = p(v0 ) is a saddle
point of J. If in addition, p is differentiable at v0 and denote H 0 = ker(J ′′ (u0 )), then
dim(L) + 1 = MI(u0 ) + dim(H 0 ∩ [L, v0 ]).
For a given closed nonempty subspace L, let M = {p(v) : v ∈ SL⊥ }. Theorem 1 states
that local-min J(u) yields a saddle point u∗ = p(v ∗ ) of J, which is unstable in H but stable
u∈M
on M and can be numerically approximated by, e.g., a steepest descent method. Then it
leads to the following local minimax algorithm: a two-level local optimization algorithm. A
solution characterized by Theorem 1 has an instability index equal to dim(L) + 1. When
L = {0}, M is the wellknown Nehari manifold in the literature.
2.2
A Local Minimax Algorithm
Let w1 , ..., wn−1 be n-1 previously found critical points, L = [w1 , ..., wn−1 ]. Given ε > 0, λ > 0
and v 0 ∈ SL⊥ be an ascent-descent direction at wn−1 .
Step 1: Let t00 = 1, vL0 = 0 and set k = 0;
Step 2: Using the initial guess w = tk0 v k + vLk , solve for wk ≡ p(v k ) = arg max J(u),
u∈[L,v k ]
and denote
tk0 v k
+
vLk
= w ≡ p(v );
k
k
Step 3: Compute the steepest descent vector dk := −J ′ (wk );
Step 4: If ∥dk ∥ ≤ ε then output wn = wk , stop; else goto Step 5;
v k + sdk
∈ SL⊥ and find
∥v k + sdk ∥
{
}
λ
tk0 λ k 2
k
m
k
k λ
k
s := max
: 2 > ∥d ∥, J(p(v ( m ))) − J(w ) ≤ − m+1 ∥d ∥ .
m∈N
2m
2
2
Step 5: Set v k (s) :=
Initial guess u = tk0 v k ( 2λm ) + vLk is used to find p(v k ( 2λm )). (track a peak selection)
k+1
+ vLk+1 , k = k + 1, then goto Step 3.
Step 6: Set v k+1 = v k (sk ), wk+1 = p(v k+1 ) = tk+1
0 v
A LOCAL MINIMAX-BEM METHOD
5
Remark 1. LMM starts from n = 0 with L = {0} to find a solution w1 and then with n = 1
and L = span{w1 } to find another solution w2 . Following this way, LMM continues with L
gradually expanded by previously found solutions w1 , ..., wn−1 , i.e., L = span{w1 , ..., wn−1 }.
Theorem 2. [37] Let J be C 1 and satisfy the PS condition. If (a) p is locally Lipschitz
continuous, (b) d(L, p(v k )) > α > 0 and (c) inf v∈SL⊥ J(p(v)) > −∞, then v k → v ∗ ∈ SL⊥
with ∇J(p(v ∗ )) = 0.
2.3
A BIE/BEM Approach
For now on, we set ∥ · ∥ = ∥ · ∥a if not otherwise specified. In LMM iteration at u = p(v),
a search direction d = ∇J(u) ∈ H 1 in Step 3 of LMM is the canonical identity of J ′ (u) ∈
W −1,2 (Ω) in W 1,2 (Ω) and defined by
∫
⟨d, v⟩a =
(∇d(x) · ∇v(x) + ad(x)v(x))dx
Ω
∫
∫
∂d(x)
(−∆ + aI)d(x)v(x)dx +
=
v(x)dσx
Γ ∂ν
Ω
≡ (J ′ (u), v)W −1,2 ×H 1 , ∀v ∈ H 1 (compare to (2.2).
Thus d = ∇J(u) is solved from an inhomogeneous linear elliptic equation
{
(−∆ + aI)d(x) = (−∆ + aI)u(x), x ∈ Ω,
∂d(x)
= ∂u(x)
− g(x, u(x)),
x ∈ Γ,
∂ν
∂ν
(2.5)
where the right hand side of the equation is the residue of the equation (1.1) at u and the
right hand side of BC is the residue of BC in (1.1) at u. This is where FEM, BEM or other
methods can be applied for numerical approximation. Here we discuss how BIE/BEM can
be used more efficiently in solving such a problem.
Let E be the fundamental solution defined by
(−∆x + aI)E(|ξ − x|) = δ(ξ − x)
where δ is the delta function. It is known
{
1
− 2π
ln |ξ − x|
(n = 2), (4π|ξ
− x|)−1 (n = 3), a = 0,
√
E(|ξ − x|) =
√
− a|ξ−x|
1
(n = 3), a > 0,
K ( a|ξ − x|) (n = 2), e 4π|ξ−x|
2π 0
where K0 is the modified Bessel function of order 0. Then we have a simple-layer potential
representation (SLPR) [15] for u satisfying (−∆ + aI)u = f , i.e.,
∫
∫
u(x) =
(2.6)
E(|ξ − x|)η(ξ) dσξ + E(|ξ − x|)f (ξ) dξ
Γ
≡ (Lb η)(x) + (Lv f )(x),
Ω
∀x ∈ Rn
A LOCAL MINIMAX-BEM METHOD
6
in layer density η. For such u, by a known jump-discontinuity, we have (weakly singular)
∫
∫
∂u
1
∂E(|x − ξ|)
∂E(|x − ξ|)
(x) =
η(x) +
η(ξ) dσξ +
f (ξ) dξ
∂ν
2
∂νx
∂νx
Ω
Γ
≡ (∂ν Lb η)(x) + (∂ν Lv f )(x), ∀x ∈ Γ.
(2.7)
In the above, Lb and ∂ν Lb are two linear boundary integral operators; while Lv f and ∂ν Lv f
are two volume integrals. By BIE, d has a SLPR d(x) = (Lb η)(x) + (Lv f )(x) where
f (x) = (−∆ + aI)u(x) is known. BC in (2.5) leads to solve
(∂ν Lb η)(x) =
∂u(x)
− g(x, u(x)) − (∂ν Lv f )(x) x ∈ Γ
∂ν
for layer density η. Since accurate evaluation of volume integrals Lv f, ∂ν Lv f are expensive
and even impossible when Ω is unbounded, for efficiency, we try to avoid them in all computation of d and J. Next we process to identify a subspace and its expression so that
all numerical computation and analysis can be carried out inside the subspace based on
information of functions on the boundary Γ only.
1
From [4], for the inner product ⟨·, ·⟩a defined in (2.3), H 1 (Ω) = H 2 (Γ) ⊕ H01 (Ω) where
1
H 2 (Γ) denotes the ⟨·, ·⟩a -orthogonal complement of H01 (Ω) in H 1 (Ω) and has an ⟨·, ·⟩a orthogonal basis constructed by the Steklov eigenfunctions {ek }∞
k=1 satisfying
{
(−∆ + aI)ek (x) = 0, x ∈ Ω,
(2.8)
∂ek (x)
=
λ
e
(x),
x
∈
Γ,
k
k
∂ν
where ∥ek ∥a = 1 and λ1 < λ2 ≤ λ3 ≤ · · · are the Steklov eigenvalues. Define a subspace
Ha = {u ∈ H 1 (Ω) : (−∆ + aI)u(x) = 0}.
(2.9)
1
It is clear that {ek }∞
k=1 ⊂ Ha , Ha is ⟨·, ·⟩a -orthogonal to H0 (Ω) and contains all solutions of
1
(1.1). Thus H 2 (Γ) = H̄a . So a solution to (1.1) can be approximated by using the Steklov
eigenfunctions {ek }∞
k=1 ⊂ Ha . As for LMM, we show that all numerical approximation can
be carried out with information of functions in Ha on the boundary Γ only. To see this, first
let u ∈ Ha or f (x) = (−∆ + aI)u(x) = 0 on Ω. Then (2.5) reduces to solve d ∈ Ha from
{
(−∆ + aI)d(x) = 0,
x ∈ Ω,
(2.10)
∂u(x)
∂d(x)
=
−
g(x,
u(x)),
x
∈
Γ.
∂ν
∂ν
By SLPR d(x) = (Lb η)(x) and the layer density η can be solved from the linear system
(∂ν Lb η)(x) =
according to the following result.
∂u(x)
− g(x, u(x)),
∂ν
x ∈ Γ,
(2.11)
A LOCAL MINIMAX-BEM METHOD
7
Lemma 1. (Theorem 7.8.2. in [15]) Let a ̸= 0 and Γ be smooth. For some r ≥ − 12 , consider
the interior Neumann boundary-value problem
{
(−∆ + aI)w(x) = 0, x ∈ Ω,
(2.12)
∂w(x)
r
=
b(x)
∈
H
(Γ).
∂ν
∫
1
∂E(|x − ξ|)
η(x) +
η(ξ)dσξ = b(x), x ∈ Ω,
2
∂ν
Γ
has a unique solution η ∈ H r (Γ), and the solution w of (2.12) is given by the SLPR
∫
w(x) = E(|x − ξ|)η(ξ)dσξ ∈ H r+3/2 (Ω), x ∈ Γ.
(2.13)
Then the BIE
Γ
Since in LMM, L ⊂ Ha , in the iteration, if v k ∈ Ha with ∥v k ∥a = 1, we have p(v k ) =
tk0 v k + vLk ∈ Ha for some tk0 > 0 and vLk ∈ L, and then dk = ∇J(p(v k )) ∈ Ha by (2.10).
k −sk dk
Consequently, v k+1 = ∥vvk −s
∈ Ha . Thus LMM is invariant in Ha .
k dk ∥
a
We see that all the terms needed for computation and analysis in Ha can be expressed
by the information of its functions on Γ only, as in (2.21)-(2.25). But the space Ha is not
closed. So for analysis purpose, we define a weak form of Ha by
∫
∂u(x)
1
H̃a = {u ∈ H (Ω) : ⟨u, v⟩a =
v(x)dσx , ∀v ∈ H 1 }.
(2.14)
∂ν
Γ
1
It is clear that Ha ⊂ H̃a and H̃a is ⟨·, ·⟩a -orthogonal to H01 (Ω). Thus H̃a ⊂ H 2 (Γ).
The map from u to its Neumann derivative is related to the Hilbert transformation. As
1
the classical Sobolev embedding from H 1 (Ω) to H 2 (Γ) is continuous, this would prompt
1
to consider the problem in an alternative way. For u ∈ H 2 (Γ) we may use the norm
∫
||u||2Γ = Γ ( ∂u
u + u2 )dσ induced by the inner product [5]
∂ν
∫
1
∂u(x)
⟨u, v, ⟩Γ = (
v(x) + u(x)v(x))dσx , ∀u, v ∈ H 2 (Γ).
(2.15)
∂ν
Γ
The above process is also related to the recent work on fractional Laplacian, e.g., [5,12,13].
1
Theorem 3. H̃a is closed and thus H 2 (Γ) = H̃a .
Proof. By the Sobolev trace theorem, we see that the two norms ∥u∥2a and
∫
∫
∫
2
2
2
2
2
∥u∥a′ = ∥u∥a + u (x)dσx = (|∇u(x)| + au (x))dx + u2 (x)dσx
Γ
Ω
(2.16)
Γ
are equivalent and a set is ⟨·, ·⟩a -orthogonal to H01 (Ω) iff it is ⟨·, ·⟩a′ -orthogonal to H01 (Ω)
where
∫
∫
∫
⟨u, v⟩a′ = ⟨u, v⟩a + u(x)v(x)dσx = (∇u(x) · ∇v(x) + au(x)v(x))dx + u(x)v(x)dσx .
Γ
Ω
Γ
(2.17)
A LOCAL MINIMAX-BEM METHOD
8
1
Let uk ∈ H̃a and uk → u0 ∈ H 2 (Γ) in ∥ · ∥a -norm. We have uk → u0 in ∥ · ∥a′ -norm and
Thus
⟨uk , v⟩a′ = ⟨uk , v⟩Γ ,
∀v ∈ H 1 (Ω).
(2.18)
⟨uk , v⟩a′ → ⟨u0 , v⟩a′ ,
∀v ∈ H 1 (Ω).
(2.19)
1
2
Since the classical Sobolev embedding from H 1 (Ω) to H (Γ) is continuous, we have uk → u0
1
in ∥ · ∥Γ -norm. So ⟨uk , v⟩Γ → ⟨u0 , v⟩Γ , ∀v ∈ H 2 (Γ). On the other hand, since ⟨uk , v⟩Γ = 0
and ⟨u0 , v⟩Γ = 0, ∀v ∈ H01 (Ω), it leads to ⟨uk , v⟩Γ → ⟨u0 , v⟩Γ , ∀v ∈ H 1 (Ω). Together with
(2.18) and (2.19), we obtain ⟨u0 , v⟩a′ = ⟨u0 , v⟩Γ , ∀v ∈ H 1 (Ω), which implies u0 ∈ H̃a . So H̃a
1
1
1
is closed. Since Ha ⊂ H̃a ⊂ H 2 (Γ) and H̄a = H 2 (Γ), we conclude H 2 (Γ) = H̃a .
Since ∇J(u) is used in our numerical computation and analysis, we need to verify
Lemma 2.
∇J(u) ∈ H̃a ⇔ u ∈ H̃a .
(2.20)
Proof. Let u ∈ H̃a . Note Ha ⊂ H̃a and H̃a is closed. There are uk ∈ Ha s.t. uk → u. By
(2.10), we have ∇J(uk ) ∈ Ha . Since J is C 1 , we obtain ∇J(uk ) → ∇J(u) ∈ H̃a .
On the other hand, if d = ∇J(u) ∈ H̃a , from (2.2) and (2.3), we have
∫
⟨d, v⟩a = ⟨u, v⟩a − g(x, u(x))v(x)dσx , ∀v ∈ H 1 (Ω).
Γ
But H̃a is ⟨·, ·⟩a -orthogonal to
For each v ∈ H01 (Ω) we have ⟨d, v⟩a = 0 ⇒ ⟨u, v⟩a = 0,
1
or u ∈ (H01 (Ω))⊥a = H 2 (Γ) = H̃a where ⊥a means ⟨·, ·⟩a -orthogonal complement.
H01 (Ω).
Since all solutions of (1.1) are in H̃a and for u, v ∈ H̃a ,
∫
∂u(x)
⟨u, v⟩a =
v(x)dσx ,
Γ ∂ν
∫
∂u(x)
2
2
∥u∥ := ∥u∥a =
u(x)dσx ≥ 0,
Γ ∂ν
]
∫ [
1 ∂u(x)
J(u) =
u(x) − G(x, u(x)) dσx ,
Γ 2 ∂ν
∫ [
]
∂u(x)
′
(J (u), v) =
− g(x, u(x)) v(x)dσx (compare to (2.2)),
∂ν
Γ
u ∈ H̃a , u = 0 ⇔ ⟨u, v⟩a = 0 ∀v ∈ H̃a ,
(2.21)
(2.22)
(2.23)
(2.24)
∇J(u) ∈ H̃a ⇔ u ∈ H̃a ,
we conclude that all numerical computation and analysis of (1.1) can be carried out efficiently
in H̃a based on information of functions only on the boundary Γ. With the above notations,
the condition (2.4) can be conveniently written as: u∗ ∈ Ha is a solution to (1.1) iff
∫
∫
∂u∗ (x)
v(x)dσx = g(x, u∗ (x))v(x)dσx ∀v ∈ H̃a ,
(2.25)
∂ν
Γ
Γ
or directly as in the unknown layer density η on Γ in SLPR
u = Lb η and ∂b η(·) = g(·, Lb η(·)).
A LOCAL MINIMAX-BEM METHOD
2.4
9
More on BEM to Solve Linear System (2.11)
In LMM, SLPR in BIE leads to evaluate the gradient d by
∫
d(x) = E(|x − ξ|)η(ξ)dσξ = (Lb η)(x),
x ∈ Γ,
Γ
1
∂d(x)
= η(x) +
∂νx
2
∫
Γ
∂
E(|x − ξ|)η(ξ)dσξ = (∂ν Lb η)(x),
∂νx
x ∈ Γ.
To solve for the unknown layer density η by BEM, we partition Γ = ∪nj=1 Γj and each Γj is
a line segment centered at xj . Assume η(x) = ηj for x ∈ Γj , η = (η1 , ..., ηn ). Denote the
matrices A = (aij )n×n and DA = (daij )n×n by
∫
aij =
E(|xi − ξ|)dσξ ,
Γj
{
∫
0, i ̸= j
1
∂
daij =
δij +
E(|xi − ξ|)dσξ , δij =
,
2
1, i = j
Γj ∂νx
where for E(|x − ξ|) =
√
1
K ( a|x
2π 0
− ξ|), we have
√
√
∂
− a
(x − ξ) · νx
E(|x − ξ|) =
K1 ( a|x − ξ|)
∂νx
2π
|x − ξ|
and K1 is the modified Bessel function of order 1. The integrals in aij and daij can be
efficiently evaluated by, e.g., the Gaussian quadrature [15]. In particular, when i = j and
xi , ξ are on the same line segment Γi , we have (xi − ξ) · νx = 0. So there will not be any
singularity involved in numerical evaluation of daij . Then BC
∂d(x)
∂u(x)
= b(x) ≡
− g(x, u(x))
∂νx
∂ν
leads to solve an (n × n) linear matrix system
DA η = B
(2.26)
for η where B = (b(x1 ), ..., b(xn )). Finally we can evaluate the gradient d = Aη where
d = (d(x1 ), ..., d(xn ))T . Note that in all iterations for all solutions, the matrices A, DA remain
the same. So to solve the linear matrix system (2.26) repeatedly for different B, we only
need to do LU decomposition of DA once for all. Thus such an algorithm is very efficient.
We conclude this section by state the following result.
Proposition 1. LMM-BEM is invariant in Ha and solutions to the problem (1.1) can be
solved in Ha and by using information of u ∈ Ha only on Γ.
A LOCAL MINIMAX-BEM METHOD
3
10
Solution Existence, Multiplicity and Properties
Since most existence and multiplicity results in the literature focus on cases where nonlinearity appears in equations not BCs, in this section, we establish the existence and multiplicity
of solutions to (1.1) where nonlinearity appears in Neumann type BCs. In order to check
convergence of LMM-BEM and to estimate the instability index of solutions computed by
LMM-BEM as stated in Theorems 1 and 2, we also verify some related conditions in the theorems. Those analysis can be carried out on the space H 1 (Ω). However, since the notations
in (2.21)-(2.25) show a great advantage on the convenience and efficiency in developing our
numerical method, in order to make a direct connection between our numerical method and
the analysis below, we focus on H = H̃a with the notions in (2.21)-(2.25) and ∥ · ∥ = ∥ · ∥a .
3.1
Main Theoretical Results
Assume the standard regularity and growth
condition
(p1 ) g ∈ C 1 (Γ × R, R), g(x, 0) = gξ (x, ξ)
= 0;
ξ=0
(p2 ) there are constants C1 , C2 > 0 s.t. |g(x, ξ)| ≤ C1 + C2 |ξ|α , ∀x ∈ Γ,
n
where 1 < α < p∗ ≡ n−2
for n ≥ 3 and p∗ = +∞ for n = 2;
(p3 ) There exist µ > 2, M > 0 s.t. 0 ≤ µG(x, ξ) ≤ ξg(x, ξ), ∀|ξ| > M, x ∈ Γ;
(p4 ) gξ (x, ξ) > g(x,ξ)
, ∀x ∈ Γ and ξ ̸= 0.
ξ
By the Sobolev trace inequality, for s ≥ 1, it holds
∫
∫
1
1
2
[ Γ u(x) ∂u
(x)dσ
]
[
(x)dσx ] 2
u(x) ∂u
x
∂ν
∂ν
Γ
inf ∫
≥ inf
≥ t0 > 0
(3.1)
1
u∈H [
u∈H
s+1 dσ ] s+1
C∥u∥W 1,s+1 (Ω)
|u(x)|
x
Γ
∫
for some constant C > 0, where ∥u∥s+1
=
(|∇u(x)|s+1 + |u(x)|s+1 )dx. Then following
1,s+1
W
(Ω)
Ω
a standard approach as in [25] or [30], we have
Lemma 3. Under (p1 ) and (p2 ), J is C 2 and satisfies the PS condition.
Theorem 4. If g satisfies (p1 ), (p2 ) and (p3 ), then equation (1.1) has at least three nontrivial
solutions.
Proof. The proof of the theorem is divided into three steps.
Step 1: The existence of two nontrivial solutions — a mountain pass type approach.
Define for each x ∈ Γ,
{
g(x, t), t ≥ 0,
g̃(x, t) =
(3.2)
0,
t < 0.
1
Consider the modified functional defined on H := H 2 (Γ)
∫ [
]
1 ∂u(x)
˜
J(u) =
u(x) − G̃(x, u(x)) dσx ,
Γ 2 ∂ν
(3.3)
A LOCAL MINIMAX-BEM METHOD
11
where G̃t (x, t) = g̃(x, t). Then following a standard approach as in [25] or [30], one may
check that J˜ ∈ C 2 (H, R) and satisfies the PS condition. It is clear that under (p1 ), (p2 ), (p3 ),
˜ 1 ) → −∞ as t → ∞. We can choose r > 0 large s.t. J(re
˜ 1 ) < 0. Set
we have J(te
Γc = {γ ∈ C 0 ([0, 1], H) : γ(0) = 0, γ(1) = re1 }
and
˜
c1 = inf sup J(γ(t)).
(3.4)
γ∈Γc t∈[0,1]
By Lemma 4, c1 > ε > 0 and applying the Mountain Pass lemma, we obtain that c1 is a
critical value of J˜ or there exists u1 ∈ H s.t.
∫
∫
∂u1 (x)
v(x)dσx = g̃(x, u1 (x))v(x)dσx ∀v ∈ H.
∂ν
Γ
Γ
{
Denote
u−
1 (x) =
u1 (x), u1 (x) ≤ 0
0,
u1 (x) > 0
x ∈ Γ.
It is clear that u−
1 ∈ H and
∫
∫
∫
∂u−
∂u1 (x) −
1 (x) −
− 2
∥u1 ∥ =
u1 (x)dσx =
u1 (x)dσx = g̃(x, u1 (x))u−
1 (x)dσx = 0.
∂ν
∂ν
Γ
Γ
Γ
Thus u1 ≥ 0. By the Harnack inequality, we conclude u1 > 0 in Ω. In this case, if u1 (x0 ) = 0
0)
for some x0 ∈ Γ, by the strong maximum principle, we should have ∂u(x
< 0, which
∂ν
contradicts to g(x, 0) = 0. So u1 is strictly positive on Ω̄. It follows g̃ = g, J˜ = J and hence
u1 is also a critical point of J.
By a similar procedure, we can obtain the other solution u2 < 0. Without loss of
generality, we assume that c1 = J(u1 ) ≤ J(u2 ) = c2 .
Step 2: The local behavior of J near u1 . In following the argument developed in [30] to get
another solution, the key step is to use a generalized Morse lemma to establish certain local
behavior of J around the known solution u1 , see Lemma 5. Since the rest of the argument
is standard, we refer to [30] for more details.
So let us denote N = ker J ′′ (u1 ) and apply Lemma 5 to the function J in a neighborhood
of u1 , say Bδ = {u ∈ H : ∥u − u1 ∥ < δ}, then we have
1
J ◦ Φ(u1 + z + y) = ⟨J ′′ (u1 )z, z⟩ + J(u1 + h(y) + y),
2
∀u ∈ Bδ ,
(3.5)
where Φ : Bδ (u1 ) → H is a homeomorphism preserving u1 and h : Bδ (θ) ∩ N → N ⊥ is a C 1
mapping with h(θ) = 0. Then by Lemma 6, the local behavior of J around u1 is given by
(a) h(y) ∈ C 1 (Ω̄), for ∥y∥ < δ and (b) ∥h(y)∥C 1 → 0 as ∥y∥ → 0.
A LOCAL MINIMAX-BEM METHOD
12
Step 3. This step is exactly the same as in [30] following a topological argument and we
sketch it here. Lemma 3.2 of [30] can be proved here similarly to give us that for K > 0
large enough J −K ≡ S ∞ (homotopy equivalent). Now the local information near u1 and u2
from (3.5) can be used to show that the critical groups at u1 or at u2 are δk1 F (with F
being the coefficients group) since they are both mountain pass critical points. Finally if we
assume J has exactly three critical points 0, u1 , u2 we derive a contradiction as in the proof
of Theorem 3.1 of [30] by using the above information. Thus J has another critical point
besides 0, u1 and u2 .
By following the proof of Theorem 9.12 in [25], if in addition to (p1 ) ∼ (p3 ), g(x, ξ) is odd
in ξ, we can establish the existence of infinitely many solutions to (1.1). Next by following
standard results [25, 30], if the boundary Γ is smooth, then the solutions we get are classical
ones and the BIE/BEM approach described in Section 2.3 can be applied.
Lemma 4. For (3.3), there exist constants ρ, ε > 0 s.t.
˜
J(u)
>0
˜
∀0 < ∥u∥ < ρ and J(u)
> ε,
∀∥u∥ = ρ.
Proof. The first Steklov eigenpair (λ1 , e1 ) satisfies
∫
λ1 = min
where ∥u∥2L2 (Γ) =
u∈H
∫
g̃(x, t) ≤
Γ
Γ
∂u(x)
u(x)dσx
∂ν
2
∥u∥L2 (Γ)
> 0,
(3.6)
u2 (x)dσx . By (p1 ) and (p2 ), there exists C > 0 s.t.
λ1
|t| + C|t|α
4
and G̃(x, u(x)) ≤
λ1
C
|u(x)|2 +
|u(x)|α+1 .
4
α+1
Thus with (3.6) and Sobolev embedding inequality, we have
∫ [
∫
1 ∂u(x)
λ1 2 ]
C
˜
J(u) ≥
u(x) − u (x) dσx −
|u(x)|α+1 dσx
2
∂ν
4
α
+
1
Γ
Γ
1
1
C
1
C0
0
2
≥
∥u∥2 − ∥u∥2 −
∥u∥α+1
∥u∥α+1
Lα+1 (Ω) = ∥u∥ −
Lα+1 (Ω) ,
2
4
α+1
4
α+1
∫
α+1
where ∥u∥α+1
dx. Since α + 1 > 2, the Lemma is verified.
Lα+1 (Ω) = Ω |u(x)|
Lemma 5. (Generalized Morse Lemma) [30] Suppose U is a neighborhood of the original θ
in a Hilbert space H with inner product (·, ·) and g ∈ C 2 (H, R). Assume that θ is the only
one critical point of g in U and that A = g ′′ (θ) with kernel N . If θ is at most an isolated
point of the spectrum ρ(A), then there exist a ball Bδ centered at θ, an origin preserving local
homeomorphism Φ and a C 1 mapping h : Bδ ∩ N → N ⊥ s.t.
1
g ◦ Φ(z + y) = (Az, z) + g(h(y) + y), ∀u ∈ Bδ ,
2
where y = PN u, z = PN ⊥ u and PN is the orthogonal projection onto N .
A LOCAL MINIMAX-BEM METHOD
13
Lemma 6. In (3.5), we have (a) h(y) ∈ C 1 (Ω̄) ∀∥y∥ < δ and (b) ∥h(y)∥C 1 → 0 as ∥y∥ → 0.
Proof. Conclusion (a) comes directly from the generalized Morse lemma, i.e., Lemma 5. So
we only need to prove (b). Recall that h(y) is the unique solution of the equation
L(z) =
i.e.,
∂J
(u1 + z + y) = 0,
∂z
∂J
≡ 0,
(u + 1 + z + y)
∂z
z=h(y)
∥y∥ ≤ δ,
∀∥y∥ ≤ δ,
or by (2.25) we have
∫ [
∫
]
∂
(u1 (x) + h(y)(x) + y(x)) z(x)dσx = g(x, u1 (x) + h(y)(x) + y(x))z(x)dσx ,
Γ ∂ν
Γ
Taking J ′ (u1 ) = 0, J ′′ (u1 )y = 0 into account, we have
∫
∫
∂h(y)(x)
z(x)dσx = f (y)(x)z(x)dσx ,
∂ν
Γ
Γ
∀z ∈ H.
∀z ∈ H
(3.7)
where
f (y)(x) = g(x, u1 (x) + h(y)(x) + y(x)) − g(x, u1 (x)) − gξ (x, u1 (x))y(x).
Since
∫
∫
{
∂w(x)
v(x)dσx = gξ (x, u1 (x))w(x)v(x)dσx
N = w∈H:
∂ν
Γ
Γ
∀v ∈ H
(3.8)
}
has a finite dimension, we can assume N = span{w1 , ..., wk } ⊂ H, all wi ’s are mutually
⟨·, ·⟩a -orthogonal with ∥wi ∥ = 1. We claim that for each y ∈ N , there are scalars βi =
βi (y), i = 1, ..., k s.t.
∂h(y) ∑ ∂wi
+
βi
= f (y) on Γ,
(3.9)
∂ν
∂ν
∑
where f (y) is given by (3.8). To prove the claim, since h(y) + βi wi ∈ H and by (2.25), we
see that (3.9) is equivalent to
∑
∫
∫
∂(h(y) + βi wi )(x)
v(x)dσ = f (y)(x)v(x)dσx ∀v ∈ H.
(3.10)
∂ν
Γ
Γ
∫
∑
If we write v = w + z ∈ N ⊕ N ⊥ and note that h(y) ∈ N ⊥ , βi wi ∈ N and Γ u(x)
v(x)dσ =
∂ν
⟨u, v⟩a , then (3.10) becomes
∫
∫
∫
∑ ∫ wi (x)
∂h(y)(x)
z(x)dσx +
βi
w(x)dσx = f (y)(x)z(x)dσx + f (y)(x)w(x)dσx .
∂ν
Γ ∂ν
Γ
Γ
Γ
A LOCAL MINIMAX-BEM METHOD
14
It follows from (3.7) that βi ’s must satisfy
∫
∑ ∫ ∂wi (x)
βi
w(x)dσx = f (y)(x)w(x)dσx
∂ν
Γ
Γ
which implies for each i,
∀w ∈ N,
∫
βi =
f (y)(x)wi (x)dσx .
(3.11)
Γ
One can check that βi ’s given by (3.11) solve (3.9).
By the assumption on g and the property of h, we have βi ∈ C 1 (Bδ (θ) ∩ N, R) and
βi (θ) = 0. Then by (p2 ) and Lp estimate, for any p > 2, there exists constant Cp , C1 > 0 s.t.
∑
∥h(y)∥W 2,p (Ω) ≤ Cp (∥h(y)∥ + ∥y∥ +
max |βi | + 1)
and
∥h(y)∥C 1 (Ω̄) ≤ C1 ,
∀∥y∥ ≤ δ.
From this, one may find a constant C > 0 s.t.
|g(x, u1 (x) + h(y)(x) + y(x)) − g(x, u1 (x))| ≤ C(∥h(y) + ∥y∥ +
∑
max |βi |),
which gives us the estimate (b) of the lemma.
We remark that by arguing some more we can claim the solutions obtained contain signchanging ones except the mountain pass ones (which are one sign solutions). This requires
the methods of invariant sets with gradient flows as used in [9](see [23] for more references
therein). Let P± = {u ∈ H | ± u ≥ 0} and W± = {u ∈ H | d(u, P± ) ≤ ϵ}. For u ∈ H
∂v
let A(u) be the solution of the equation ∂ν
= g(x, u) on Γ. Then the gradient of J is of the
form u − A(u) in Hilbert space H. We show that A is attractive in small neighborhoods of
W± . Namely we have
Proposition 2. For ϵ > 0 small enough, A(∂W± ) ⊂ int (W± ).
The proof of this is by now quite standard. We sketch it here.
∫
∫
∂(v − w)
∂vv−
2
2
d(v, P ) = inf ||v − w|| = inf
(v − w)dσ ≤
v− dσ
w∈P
w∈P Γ
∂ν
∂ν
Γ
∫
=
g(x, u− )v− dσ ≤ c||u− ||2 ||v− ||2 + C||u− ||αp∗ +1 ||v− || ∗p∗ +1
Γ
p +1−α
1
≤ ( d(u, P ) + d(u, P )α )d(v, P ).
4
By choosing ϵ > 0 small, if d(u, P ) = ϵ we have d(v, P ) ≤ 12 ϵ.
With this device we may employ minimax methods to construct critical values with
critical points outside W± . Then in the setting of Theorem 4 we obtain one positive, one
A LOCAL MINIMAX-BEM METHOD
15
negative and one sign-changing solutions. If the nonlinearity is odd in u we also obtain
infinitely many sign-changing solutions by a standard argument, see [8,9,10,20,23,31].
In the rest of this section, we verify some conditions posed in Theorems 1 and 2 for
solution instability index estimate and convergence of LMM-BEM.
Proposition 3. Under (p1 ) ∼ (p4 ), when L = {0}, the peak selection p(u) = tu u is uniquely
defined for each u ∈ SH and is C 1 . Furthermore we have J(p(u)) > 0.
Proof. By (p1 ) and (p3 ), G(x, t) is super-quadratic in t and nonnegative. For each given
u ∈ SH , we have J(tu) > 0 for small t > 0, J(tu) → 0 as t → 0 and J(tu) → −∞ as t → ∞.
So there exists a tu > 0 s.t.
∫
[ ∂u(x)
]
d J(tu) =
tu
u(x) − g(x, tu u(x))u(x) dσx = 0.
(3.12)
dt t=tu
∂ν
Γ
Then
d2 J(tu) =
dt2 t=tu
∫
[ ∂u(x)
]
u(x) − gξ (x, tu u(x))u2 (x) dσx
∂ν
∫Γ
[ ∂u(x)
g(x, tu u(x)) 2 ]
<
u(x) −
u (x) dσx (by (p4 ))
∂ν
tu u(x)
Γ
∫
[ ∂u(x)
]
1
=
tu
u(x) − g(x, tu u(x))u(x) dσx = 0.
tu Γ
∂ν
(3.13)
Thus the peak selection p(u) = tu u is uniquely
defined for each u ∈ SH . Since J(tu) > 0 for
d2
small t > 0, by the uniqueness and dt2 J(tu)
< 0, we must have J(p(u)) = J(tu u) > 0.
t=tu
d
By applying the implicit
function theorem to the equation F (t, u) = dt J(tu) = 0, we note
2 that Ft (t, u)
J(tu) < 0. Hence the peak selection p(u) = tu u is C 1 .
= dtd 2 t=tu
t=tu
Proposition 4. Under (p1 ) ∼ (p4 ), for any nonempty closed subspace L ⊂ H, we have
inf
u∈L⊥ ,∥u∥=1
∥p(u)∥ > 0
and
inf
u∈L⊥ ,∥u∥=1
J(p(u)) > 0.
∫
Proof. We first assume L = {0}. For each u ∈ H with ∥u∥2 = Γ ∂u(x)
u(x)dσx = 1, from
∂ν
(3.12), we have
∫
[ ∂u(x)
]
d g(x, tu u(x))
tu
0 =
J(tu) =
u(x) −
tu u(x) dσx
dt t=tu
∂ν
tu
Γ
∫
[ ∂u(x)
]
1 λ1
≥
tu
u(x) − ( |tu u(x)|2 + C|tu u(x)|α+1 ) dσx (by (p2 ))
∂ν
tu 4
Γ
∫
]
[ ∂u(x)
λ1
u(x) − |u(x)|2 − Ctαu |u(x)|α+1 dσx
= tu
∂ν
4
∫Γ
[ ∂u(x)
]
1 ∂u(x)
∂u(x)
≥ tu
u(x) −
u(x) − C1 tαu
u(x) dσx (by (3.6) and (3.1))
∂ν
4 ∂ν
∂ν
Γ
∫
3
= tu ( − C1 tαu ), (since ∥u∥2 = Γ ∂u(x)
u(x)dσx = 1),
∂ν
4
A LOCAL MINIMAX-BEM METHOD
which implies
16
( 3 ) α1
3
− C1 tαu ≤ 0 or tu ≥ t0 =
> 0.
4
4C1
Thus
∥p(u)∥ = ∥tu u∥ = tu > t0 > 0 ∀u ∈ H, ∥u∥ = 1.
By H = L⊥ , we have
inf
u∈L⊥ ,∥u∥=1
∥p(u)∥ > 0 and
inf
u∈L⊥ ,∥u∥=1
J(p(u)) > 0.
(3.14)
Denote
N = {p(u) = tu u : u ∈ H, ∥u∥ = 1, ∇J(p(u))⊥u}.
Then the two formulas in (3.14) are equivalent to
inf ∥w∥ > 0 and
w∈N
inf J(w) > 0.
w∈N
(3.15)
For any nonempty closed subspace L ⊂ H, denote
M = {p(u) ∈ [L, u] : u ∈ L⊥ , ∥u∥ = 1, ∇J(p(u))⊥[L, u]}.
(3.16)
Then the two formulas in (3.14) become
inf ∥w∥ > 0 and
w∈M
inf J(w) > 0.
w∈M
(3.17)
Since M ⊂ N , from (3.15), the two formulas in (3.17) are also true.
∑
For L = span{u1 , ..., un−1 }, let un = p(u0 ) = t0 u0 + n−1
i=1 ti ui be a solution characterized
by Theorem 1, or found by LMM-BEM. Following the argument in [35,36] or (3.16), a peak
selection p at un can be computed by solving the equations
Fj (t0 , t1 , ..., tn−1 , u0 ) = ⟨∇J(t0 u0 +
n−1
∑
ti ui ), uj ⟩ = 0, j = 0, 1, ..., n − 1,
(3.18)
i=1
for t0 , t1 , ..., tn−1 . If the n × n matrix
Q=[
where
∂F
(t0 , t1 , ..., tn−1 , u0 )] = [⟨J ′′ (un )ui , uj ⟩],
∂ti
i, j = 0, 1..., n − 1,
∫ [
]
∂ui (x)
− gξ (x, un (x))ui (x) uj (x)dσx
⟨J (un )ui , uj ⟩ =
∂ν
Γ
′′
(3.19)
(3.20)
is nonsingular, then by the implicit function theorem, the peak selection p is locally C 1 near
un and this condition can be easily numerically checked in the algorithm. It is clear that
when n = 1, (3.19) and (3.20) become (3.13), So Q is nonsingular in this case.
A LOCAL MINIMAX-BEM METHOD
4
17
Numerical Examples
In this last section we carry out numerical computation on some examples by our LMMBEM algorithm developed in the previous sections and display the numerical solutions. We
x2
x2
set g(x, u(x)) = u3 (x) and a = 1 in (1.1) and choose domains Ω1 = {(x1 , x2 ) : 41 + 12 < 1},
Ω2 = {(x1 , x2 ) : x21 + x22 < 1}, Ω3 = (−1, 1)2 , Ω4 = a dumbbell as in Figure 1.
It is a direct verification to check that assumptions (p1 ) ∼ (p4 ) are all satisfied.
Next we discuss how to choose initial guesses in LMM-BEM: when Ω is a nice convex
domain, we use the Steklov eigenfunctions uk = ek defined in (2.8) or use uk = Aηk where
either ηk (x) = cos(kθ(x)) or ηk (x) = sin(kθ(x)) to generate a k-periodic function on Γ.
On Ω1 , we use the Steklov eigenfunctions as initial guesses. Numerical solutions are
shown in Figures 2 and 3.
On Ω2 , this is a degenerate case. All the numerical Steklov eigenfunctions are doubled.
So we have to get rid of the multiplicity and choose only one from each group as an initial
guess. Note that the first solution is radial positive with 1-peak at the center of the domain.
Numerical solutions are shown in Figures 4 and 5.
On Ω3 , due to the corner affect, using the Steklov eigenfunctions as initial guesses will
obtain only a subset of solutions. So we use sin(kπ ∗ i/n) or cos(kπ ∗ i/n) to be initial
guesses and obtained much more solutions, including the first two positive solutions that are
4-rotation asymmetric and the third one is 4-rotation symmetric with 1-peak at the center
of the domain. Numerical solutions are shown in Figures 6 and 7.
On Ω4 , due to the complexity of the domain (not convex), we have used local symmetries
to create initial guesses. The domain is shown in Fig. 1 and the numerical solutions are
shown in Figures 8 and 9.
In all our numerical computations, iterations are terminated when ∥∇J(uk )∥ < 10−5 and
maxx∈Γ | ∂u
(x) − g(x, u(x))| < 10−5 . In order to plot 2D contours and 3D profile of a solution
∂ν
in one figure, we have shifted the 3D profile up or down. Although solution profiles are
plotted on a 2D domain Ω, the problem is actually solved only on the boundary Γ. Due to
this dimension reduction, the numerical algorithm is thus much more efficient.
1
Interior domain: counterclockwise
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Figure 1: Boundary points start from the left endpoint (−1.5, 0) counterclockwise.
A LOCAL MINIMAX-BEM METHOD
1.2
18
1.5
2
1.1
1.5
1
1
1
0.9
0.5
0.5
0.8
0.7
0
0
0.6
−0.5
−0.5
0.5
−1
0.4
−1
−1.5
0.3
0.2
0
1
2
3
4
5
6
−1.5
7
(1)
0
1
2
3
4
5
6
−2
7
(2)
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
(3)
1.5
2
2
1
1.5
1.5
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
−1
−1.5
−1.5
−1.5
−2
−2
0
1
2
3
4
5
6
7
(4)
0
1
2
3
4
5
6
−2
7
(5)
2
(6)
2.5
2.5
2
2
1.5
1.5
1.5
1
0.5
1
1
0.5
0.5
0
−0.5
0
0
−0.5
−0.5
−1
−1
−1
−1.5
−1.5
−2
−1.5
−2
0
1
2
3
4
5
6
−2.5
7
(7)
−2
0
1
2
3
4
5
6
−2.5
7
(8)
(9)
Figure 2: Profiles of solutions u1 ∼ u9 on Γ1 .
0
0
−0.5
0
−0.2
−1
−0.5
−0.4
−1.5
−0.6
−1
−2
−0.8
−1.5
−2.5
−1
−1.2
−2
−1.4
1
−3
−3.5
−2.5
1
2
0.5
0.5
1
0
−1
−1
−0.5
−0.5
−1
0
−2
(1)
−4
−1
0
−3
−2
0
−0.5
1
2
0
−1
(2)
0.5
1
−2
−1
0
1
2
(3)
0
0
0
−0.5
−0.5
−0.5
−1
−1
−1
−1.5
−1.5
−2
−2.5
−1.5
−2
−2
−2.5
−3
−2.5
−3.5
−3
−3
−4
−3.5
−3.5
1
−4
1
2
−4
0
0
0
−4.5
−2
−1
−0.5
−1
−4.5
1
−2
(4)
2
0.5
1
0.5
1
0
0
−0.5
−1.5
−1
−0.5
0
0.5
1
1.5
2
−1
−1
−1
(5)
−2
(6)
0
0
0
−0.5
−0.5
−1
−1
−1
−1.5
−1.5
−2
−2
−2
−2.5
−2.5
−3
−3
−4
−3.5
−3
−4
−3.5
−5
−4.5
−4
2
−4.5
−5
1
−5
1
(7)
0
−1
−0.5
−1
1
0
0
0.5
2
0.5
1
−2
2
0.5
1
0
−1
−1
(8)
−6
1
0
−0.5
0
−0.5
−1
−1
−2
−2
(9)
Figure 3: Profiles and contours of solutions u1 ∼ u9 on Ω1 with (J(u1 ), ..., J(u9 )) = (0.5750,
0.7025, 1.7940, 2.7011, 4.0194, 6.0398, 7.1387, 10.7212, 11.4944). u1 is positive.
A LOCAL MINIMAX-BEM METHOD
19
0.6766
2
2
0.6766
1.5
1.5
0.6766
1
1
0.5
0.5
0.6766
0.6766
0
0
−0.5
−0.5
0.6766
0.6766
0.6766
0.6766
0.6766
0
1
2
3
4
5
6
−1
−1
−1.5
−1.5
−2
7
(1)
0
1
2
3
4
5
6
−2
7
(2)
3
3
2
2
2
1
1
1
0
0
0
−1
−1
−1
−3
−2
0
1
2
3
4
5
6
−3
7
(4)
1
2
3
4
5
6
−3
7
(5)
4
3
3
3
2
2
2
1
1
0
0
0
−1
−1
−1
−2
−2
−2
−3
−3
1
2
3
4
5
6
−4
7
(7)
3
4
5
6
7
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
(6)
4
0
2
−2
0
4
−4
1
(3)
3
−2
0
1
−3
0
1
2
3
4
5
6
−4
7
(8)
(9)
Figure 4: Profiles of solutions u1 ∼ u9 on Γ2 .
0
0
0.2
−1
−0.5
0.15
−1
−2
−1.5
0.1
−2
−3
−2.5
0.05
−3
0
1
−4
−3.5
0.5
1
0.5
0
−4
1
0
−0.5
−1
0.5
0
−0.5
−1
−1
(1)
−5
−1
0
−0.5
0
1
−1
(2)
1
−1
−1
−1
−0.5
0
0.5
1
(3)
0
0
−1
−1
0
−2
−2
−2
−3
−3
−4
−4
−4
−5
−6
−5
−6
−6
1
−8
1
−7
1
1
0
−1
0.5
0
−0.5
−1
1
−1
(4)
0.5
0
0
−0.5
−1
0.5
0
0
1
(5)
−0.5
(6)
0
0
−2
−2
−4
−4
0
−2
−4
−6
−6
−6
−8
−8
1
−8
1
0.5
0.5
0
−0.5
−1
(7)
−10
1
−1
−0.5
0
0.5
0.5
0
1
−0.5
−1
(8)
−1
−0.5
0
0.5
1
1
0.5
0
0
−0.5
−0.5
−1
−1
(9)
Figure 5: Profiles and contours of solutions u1 ∼ u9 on Ω2 with (J(u1 ), ..., J(u9 )) = (0.3292,
1.3511, 4.3220, 9.2009, 16.1014, 25.1169, 36.3353, 49.8370, 65.6374).
A LOCAL MINIMAX-BEM METHOD
20
0.8
−0.2
1.4
−0.3
1.2
0.75
−0.4
1
−0.5
0.7
0.8
−0.6
0.6
−0.7
0.65
−0.8
0.4
−0.9
0.2
0
0.6
−1
0
1
2
3
4
5
6
−1.1
7
(1)
0
1
2
3
4
5
6
0.55
7
(2)
1.5
0
1
2
3
4
5
6
7
1
2
3
4
5
6
7
(3)
1.5
2
1
1.5
1
0.5
1
0
0.5
0.5
0
−0.5
0
−1
−0.5
−0.5
−1
−1.5
−1.5
0
1
2
3
4
5
6
−2
7
(4)
−1
0
1
2
3
4
5
6
−1.5
7
(5)
(6)
1.5
2
1.5
0
2
1.5
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
−1
−1.5
0
1
2
3
4
5
6
−1.5
7
(7)
−1.5
0
2
4
6
−2
8
(8)
0
2
4
6
8
(9)
Figure 6: Profiles of solutions u1 ∼ u9 on Γ3 . u1 ∼ u3 are one-sign solutions.
0
0
0
−0.1
−0.2
−0.2
−0.4
−0.5
−0.6
−1
−0.3
−0.8
−1
−1
−0.5
−1.5
−2
−1
−1.4
1
0.5
−0.5
0
0.5
1
−1.2
0
(1)
1
0
0
−0.5
0.5
0
1
1
−0.4
1
0.5
0
−0.5
−1
−1
−1
(2)
−1
(3)
0
0
−0.5
−0.5
−1
0
−1
−1.5
−1
−2
−2
−1.5
−2
−3
−2.5
−3.5
1
−3
−1
−3
−2.5
−1
−4
1
0.5
0
0
−0.5
0
0.5
−1
1
1
(4)
−1
−0.5
0
0.5
0
−0.5
−0.5
−1
(5)
−1
(6)
0
0
0
1
0.5
0
1
−0.5
−0.5
−1
−1
−1
−1.5
−2
−1.5
−2
−2.5
−2
−3
−3
−3.5
−4
1
(7)
−2.5
−1
−0.5
0.5
0
0
−0.5
0.5
−1
−1
−3
1
1
(8)
0.5
0
0
−0.5
−1
1
1
−4
−1
0
−0.5
0
0.5
1
−1
(9)
Figure 7: Profiles and contours of solutions u1 ∼ u9 on Ω3 with (J(u1 ), ..., J(u9 )) = (0.2277,
0.3228, 0.3799, 0.5660, 0.8069, 0.9165, 0.9834, 1.1244, 1.3722). u1 and u2 are 4-rotation asymmetric. u3 is 4-rotation symmetric with 1-peak.
A LOCAL MINIMAX-BEM METHOD
0.7
21
1
1.2
0.9
1
0.8
0.8
0.7
0.6
0.6
0.4
0.6
0.5
0.4
0.5
0.2
0.3
0.2
0.4
0
0.3
−0.2
0.2
−0.4
0.1
0.1
0
0
1
2
3
4
5
6
0
7
(1)
−0.6
0
1
2
3
4
5
6
−0.8
7
(2)
1.5
0
1
2
3
4
5
6
7
(3)
2
2
1.5
1.5
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−1
−1
−0.5
−1
−1.5
−1.5
0
1
2
3
4
5
6
−2
7
(4)
−1.5
0
1
2
3
4
5
6
−2
7
(5)
0
1
2
3
4
5
6
7
(6)
2
2
2
1.5
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
1
0.5
0
−0.5
−1
−1.5
−1.5
−1.5
−2
0
1
2
3
4
5
6
−2
7
0
1
2
3
4
5
6
7
−2
(7)
(8)
0
1
2
3
4
5
6
7
(9)
Figure 8: The profiles of solutions u1 ∼ u9 on Γ4 . u1 and u2 are positive.
1
1.5
0.8
1
0.7
0.6
0.6
0.5
0.5
0.4
0
0.4
0.2
−0.5
0.3
0
0.2
−1
1
−1
0.1
1
0
0
0
0
−2
−1
0
1
2
1
−1
3
−1.5
(1)
0.5
0
−0.5
−1
1.5
1
2
−1
3
2.5
(2)
−1.5
−0.5
−1
0
0.5
2
1.5
1
2.5
3
(3)
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0
−0.5
−1
−1.5
1
0
0
−0.5
−0.5
−1
−1
−1.5
−1
−1.5
1
0
0
0
−1
−1.5
−1
−0.5
0.5
0
1
1.5
2
2.5
1
3
−1.5
−0.5
−1
1
0.5
0
1.5
2
3
2.5
−1
(4)
(5)
2
0.5
0
−0.5
−1
−1.5
1
2.5
2
1.5
3
(6)
1.5
2
1
2
0.5
1.5
1.5
0
−0.5
1
1
0.5
−1
0.5
−1.5
0
0
−2
1
−0.5
−0.5
0.5
−1
1
0
−1.5
0
−0.5
−2
−1
−1.5
(7)
−1
−0.5
0
0.5
1
1.5
2
2.5
3
−1.5
(8)
−1
−0.5
0
0.5
1
1.5
−1
2
2.5
3
−1
−1.5
−2
−1.5
1
0
−1
−0.5
0
0.5
1
1.5
2
2.5
3
−1
(9)
Figure 9: Profiles and contours of solutions u1 ∼ u9 on Ω4 . u1 and u2 are positive.
(J(u1 ), ..., J(u9 )) = (0.0977, 0.3481, 0.4690, 1.0779, 1.3650, 1.4606, 2.1306, 2.4783, 3.3778).
A LOCAL MINIMAX-BEM METHOD
22
Acknowledgment: We would like to thank the reviewer for making helpful comments.
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A LOCAL MINIMAX-BEM METHOD
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