
The decomposition of the triangular set
JPEG is the most common format for storing and transmitting photographic images on social network platforms. JPEG image is widely used in people's life because of their low storage space and high visual quality. Secret image sharing (SIS) technology is important to protect image data. Traditional SIS schemes generally focus on spatial images, however there is little research on frequency domain images. In addition, the current tiny research on SIS for JPEG images only focuses on JPEG images with a compression quality factor (QF) of 100. To overcome the limitation of JPEG images in SIS, we propose a meaningful SIS for JPEG images to operate the quantized DCT coefficients of JPEG images. The random elements utilization model is applied to achieve meaningful shadow images. Our proposed scheme has a better quality of the shadow images and the recovered secret image. Experiment results and comparisons indicate the effectiveness of the scheme. The scheme can be used for JPEG images with any compression QF. Besides, the scheme has good characteristics, such as (k,n) threshold, extended shadow images.
Citation: Yue Jiang, Xuehu Yan, Jia Chen, Jingwen Cheng, Jianguo Zhang. Meaningful secret image sharing for JPEG images with arbitrary quality factors[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11544-11562. doi: 10.3934/mbe.2022538
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JPEG is the most common format for storing and transmitting photographic images on social network platforms. JPEG image is widely used in people's life because of their low storage space and high visual quality. Secret image sharing (SIS) technology is important to protect image data. Traditional SIS schemes generally focus on spatial images, however there is little research on frequency domain images. In addition, the current tiny research on SIS for JPEG images only focuses on JPEG images with a compression quality factor (QF) of 100. To overcome the limitation of JPEG images in SIS, we propose a meaningful SIS for JPEG images to operate the quantized DCT coefficients of JPEG images. The random elements utilization model is applied to achieve meaningful shadow images. Our proposed scheme has a better quality of the shadow images and the recovered secret image. Experiment results and comparisons indicate the effectiveness of the scheme. The scheme can be used for JPEG images with any compression QF. Besides, the scheme has good characteristics, such as (k,n) threshold, extended shadow images.
We consider systems of semilinear elliptic equations
−Δu(x)+Fu(x,u)=0 |
where
(
When
All the above results are based on the ordered structure of the set of minimal solutions of (PDE) in the case
The study of (PDE) when
M0={u∈E0∣J0(u)=c0:=infE0J0(u)}≠∅. |
Paul H. Rabinowitz studied the case of spatially reversible potentials
J(u)=∑p∈ZJp,0(u):=∑p∈Z(∫Tp,0L(u)dx−c0), | (1) |
(where
Γ(v−,v+)={u∈W1,2(R×Tn−1,Rm)∣‖u−v±‖L2(Tp,0,Rm)→0 as p→±∞}. |
In [30] the existence of minimal double heteroclinics was obtained assuming that the elements of
The proof of these results does not use the ordering property of the solutions and adapts to the study of (PDE) some of the ideas developed to obtain multi-transition solutions for Hamiltonian systems (see e.g. [3], [28] and the references therein). Aim of the present paper is to show how these methods, in particular a refined study of the concentrating properties of the minimal heteroclinic solutions to (PDE), can be used in a symmetric setting to obtain saddle type solutions to (PDE).
Saddle solutions were first studied by Dang, Fife and Peletier in [16]. In that paper the authors considered Allen-Cahn equations
We refer to [14,15,6,7,27] for the study of saddle solutions in higher dimensions and to [1,20,8] for the case of systems of autonomous Allen-Cahn equations. Saddle solutions can be moreover viewed as particular
In [5] the existence of saddle type solutions was studied for non autonomous Allen-Cahn type equations and this work motivated the paper [2] where solutions of saddle type for (PDE) were found in the case
In the present paper we generalize the setting considered in [2] to the case
(
(
F(x1,x2,x3,...,xn,u)=F(x2,x1,x3,...,xn,u) on Rn×Rm. |
By [29] the set
As recalled above, in [2], where
(
J″0(v)h⋅h=∫[0,1]n|∇h|2+Fu,u(x,v(x))h⋅hdx≥α0‖h‖2L2([0,1]n,Rm) |
for every
The assumption (
Γ(v)={u∈W1,2(R×Tn−1,Rm)∣u is odd in x1,limp→+∞‖u−v‖L2([p,p+1]×Tn−1,Rm)=0}. |
In §4, setting
c(v)=infu∈Γ(v)J(u) for v∈M0 |
we show that
Mmin0={v0∈M0∣c(v0)=minv∈M0c(v)}≠∅ |
and that
M(v0)={u∈Γ(v0)∣J(u)=c(v0)} |
is not empty and compact with respect to the
‖u−v0‖W1,2([p,p+1]×Tn−1,Rm)→0 as p→+∞. |
Our main result can now be stated as follows
Theorem 1.1. Assume
Moreover there exists
distW1,2(Rk,Rm)(w,M(v0))→0,ask→+∞, | (2) |
where
Note that by
The proof of Theorem 1.1 uses a variational approach similar to the one already used in previous papers like [5,2]. To adapt this approach to the case
In this section we recall some results obtained by Rabinowitz in [29], on minimal periodic solutions to (PDE). Moreover, following the argument in [2], we study some symmetry properties related to the assumptions (
(
Let us introduce the set
E0=W1,2(Tn,Rm)={u∈W1,2(Rn,Rm)∣u is 1-periodic in all its variables} |
with the norm
‖u‖W1,2([0,1]n,Rm)=(m∑i=1∫[0,1]n(|∇ui|2+|ui|2)dx)12. |
We define the functional
J0(u)=∫[0,1]n12|∇u|2+F(x,u)dx=∫[0,1]nL(u)dx. | (3) |
and consider the minimizing set
M0={u∈E0|J0(u)=c0} where c0=infu∈E0J0(u) |
Then in [29], [30] it is shown
Lemma 2.1. Assume
1.
2. if
3. For every
distW1,2([0,1]n,Rm)(u,M0):=infv∈M0‖u−v‖W1,2([0,1]n,Rm)>ρ, |
then
4. If
5. If
Assumption
Lemma 2.2. Assume
Proof. It is sufficient to observe that if
ⅰ)
ⅱ)
Property (ⅱ) gives the second part of the statement while by (i) and the unique extension property proved in [29], we obtain that the components of
On the other hand, assumption
˜u(x)={u(x),x∈T+,u(x2,x1,x3,…,xn),x∈[0,1]n∖T+. | (4) |
Then, we have
Lemma 2.3. If
Proof. Given
∫T+L(u)dx≤∫[0,1]n∖T+L(u)dx. |
Since
c0=J0(u)=∫T+L(u)dx+∫[0,1]n∖T+L(u)dx≥2∫T+L(u)dx=J0(˜u)≥c0. |
Hence, again by Lemma 2.1-(5),
As an immediate consequence, using Lemma 2.1-(5), we have the following.
Lemma 2.4. There results
minu∈W1,2(T+,Rm)∫T+L(u)dx=c02. | (5) |
Moreover, if
Remark 1. Lemma 2.3 tells us that the elements of
σ0={x∈R×[0,1]n−1|x2−1≤x1≤x2}. |
More precisely we have
Note that by Lemma 2.1-(1) and the assumption (N) we plainly derive that (
where we recall that
Note finally that by
r0:=min{‖u−v‖L2(Tn,Rm)∣u,v∈M0,u≢ | (6) |
we have
This section is devoted to introduce the variational framework to study solutions of (PDE) which are heteroclinic between minimal periodic solutions. We follow some arguments in [29], [26], introducing the renormalized functional
Let us define the set
E = \{u\in W_{loc}^{1, 2}({\mathbb{R}}^n, {\mathbb{R}}^m)\mid u\text{ is $1$-periodic in }x_2, \ldots x_n\}. |
For any
J(u) = \sum\limits_{p\in{\mathbb{Z}}}J_{p, 0}(u), |
where, denoting
J_{p, 0}(u) = \int_{T_{p, 0}}L(u)\, dx-c_{0}, \quad \forall p\in{\mathbb{Z}}. |
Denoting briefly
J_{p, 0}(u) = \int_{[0, 1]^n}L(u(\cdot+p))\, dx-c_0 = J_0(u(\cdot+p))-c_0, \quad \forall p\in{\mathbb{Z}}. |
Then, by Lemma 2.1, we have
Lemma 3.1. The functional
Proof. Consider a sequence
\liminf\limits_k J(u_k) \geq \liminf\limits_k \sum\limits_{p = -\ell}^\ell J_{p, 0}(u_k) \geq \sum\limits_{p = -\ell}^\ell J_{p, 0}(u) > J(u) -\varepsilon\, , |
thus finishing the proof.
Using the notation introduced above, note that if
First of all, let us consider the functional
\min\limits_{u\in E} J_{p, 0}(u)+J_{p+1, 0}(u) = 0 |
and the set of minima coincide with
{\rm dist}_p(u, A) = \inf \{ \| u-v \|_{W^{1, 2}(T_{p, 0}\cup T_{p+1, 0}, {\mathbb{R}}^m)} \mid v\in A \}\, . |
Remark 2. Let us fix some constants that will be used in rest of the paper. By Lemma 2.1-(3), we have that for any
\begin{equation} \hbox{if }u\in E\hbox{ satisfies }J_{p, 0}(u)+J_{p+1, 0}(u)\leq \lambda(r)\hbox{ for a }p\in{\mathbb{Z}}, \hbox{ then }{\rm dist}_p(u, {\mathcal{M}}_0)\leq r. \end{equation} | (7) |
It is not restrictive to assume that the function with
On the other hand for every
\rho(\lambda) = \sup \left\{{\rm dist}_p(u, {\mathcal{M}}_0) \mid u\in E \text{ with } J_{p, 0}(u)+J_{p+1, 0}(u) \leq \lambda, \, p\in{\mathbb{Z}} \right\}\, |
we get
\begin{equation} \Lambda(r) = \sup \left\{J_{p, 0}(u) \mid u\in E \text{ and } p\in{\mathbb{Z}}\text{ are such that } {\rm dist}_p(u, {\mathcal{M}}_0)\leq 2r \right\} \end{equation} | (8) |
which is non-decreasing and
We say that a set
Lemma 3.2. Given
Proof. Let
\| u-v_p \|_{W^{1, 2}(T_{p, 0}\cup T_{p+1, 0}, {\mathbb{R}}^m)} \leq \tfrac{r_{0}}{4} |
from which
Moreover, using the notations introduced above, we have
Lemma 3.3. If
\|u(\cdot+p)-u(\cdot+(p+1))\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}^2\le 2(J_{p, 0}(u)+J_{p+1, 0}(u)+2c_{0}). |
Proof. Setting
\|u(\cdot+p)-u(\cdot+(p+1))\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}^2 = \int_{p}^{p+1}\int_{[0, 1]^{n-1}} |u(x_{1}+1, y)-u(x_{1}, y)|^{2} dy \, dx_{1} |
and so there exists
\int_{[0, 1]^{n-1}} |u(\bar x_{1}+1, y) -u(\bar x_{1}, y)|^2 dy \geq \|u(\cdot+p)-u(\cdot+(p+1))\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}^2. |
On the other hand, by Hölder inequality,
\begin{align*} 2(J_{p, 0}(u)+J_{p+1, 0}(u)+2c_{0})&\geq\int_{p}^{p+2} \int_{[0, 1]^{n-1}} |\partial_{x_{1}}u(x_{1}, y)|^2 dy \, dx_{1}\\ &\geq \int_{[0, 1]^{n-1}} \int_{ \bar x_{1}}^{\bar x_{1} +1} |\partial_{x_{1}}u(x_{1}, y)|^2 dx_{1} \, dy\\ &\geq \int_{[0, 1]^{n-1}} |u(\bar x_{1}+1, y ) -u(\bar x_{1})|^2 dy\\ &\geq \|u(\cdot+p)-u(\cdot+(p+1))\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}^2 \end{align*} |
completing the proof.
By the previous lemmas we obtain that the elements in the sublevels of
Lemma 3.4. For every
Proof. Let
\begin{align} \|u(\cdot+p)-u(\cdot+q)\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)} &\leq l(u) \sup\limits_{k \in {\mathcal J}(u)}\|u(\cdot+k)-u(\cdot+k+1)\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}\\ &\, \, \, \, \quad + \sum\limits_{i = 1}^{\bar l (u)} \sup\limits_{p, q \in \mathcal I_i(u)}\|u(\cdot+p)-u(\cdot+q)\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}\\ &\leq l(u)(2(\Lambda +2c_{0}))^{\frac{1}{2}} + \bar l(u) \tfrac{r_{0}}{2}. \end{align} | (9) |
where the first term in (9) follows by the application of Lemma 3.3, since
2(J_{k, 0}(u)+J_{k+1, 0}(u)+2c_{0})\le 2(J(u) +2c_{0})\le 2(\Lambda+2c_0), \quad\forall k\in{\mathbb{Z}}, |
while the second one follows by the definition of
Since
The following lemma states the weak compactness of the sublevels of the functional
Lemma 3.5. Given any
Proof. First note that, by Lemma 3.4, there exists
\begin{multline*} \|u- v\|_{L^{2}(T_{p, 0}, {\mathbb{R}}^m)} = \|u(\cdot+p)-v\|_{L^2([0, 1]^n, {\mathbb{R}}^{m})}\\ \leq\|u(\cdot+p)-u(\cdot+\ell)\|_{L^2([0, 1]^n, {\mathbb{R}}^{m})}+\|u(\cdot+\ell)-v\|_{L^2([0, 1]^n, {\mathbb{R}}^{m})}\le {\mathit{R}}+\bar{\mathit{R}}. \end{multline*} |
Consider now a sequence as in the statement, setting
\|u_k -v\|^2_{L^2(Q_{L}, {\mathbb{R}}^m)} + \|\nabla u_k\|^2_{L^2(Q_{L}, {\mathbb{R}}^m)} \leq 2L(\bar{\mathit{R}}+{\mathit{R}})^{2} + 4Lc_{0} +2\Lambda. |
Hence,
By Lemma 3.2 we also deduce the following result concerning the asymptotic behaviour of the functions in the sublevels of
Lemma 3.6. If
\|u-v^\pm\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \to 0\quad\mathit{\text{as}}\quad p\to\pm\infty. |
Proof. Since
Hence the sequence
By Lemma 3.6, if
\begin{align*} \Gamma(v^-, v^+) = \big\{ u\in E \mid \|u-v^\pm\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \to 0\, \text{as}\, p\to\pm\infty \big\} \end{align*} |
where
We note that by Lemma 3.5, every sequence
In particular, given
c(v^-, v^+) = \inf\limits_{u\in\Gamma(v^-, v^+)}J(u)\, , |
as in [29], we obtain that for any
Finally, we have that
Lemma 3.7. For every
Proof. Assume that there exists
In order to prove the second part of the statement, assume the existence of two sequences
\begin{align*} \|v^+_k-v^-_k\|_{L^2([0, 1]^n, {\mathbb{R}}^m)} &\leq \|v^-_k-u_k(\cdot+ p_k)\|_{L^2([0, 1]^n, {\mathbb{R}}^m)} \\ &\phantom{\leq}+\|u_k(\cdot+ p_k)-u_k(\cdot+ q_k)\|_{L^2([0, 1]^n, {\mathbb{R}}^m)} \\ &\phantom{\leq}+ \|v^+_k-u_k(\cdot+ q_k)\|_{L^2([0, 1]^n, {\mathbb{R}}^m)} \\ &\leq \varepsilon + {\mathit{R}} +\varepsilon \end{align*} |
since, by periodicity,
We focalize now in the study of heteroclinic solutions which are odd in the first variable, hence we will consider a subset of
E^{odd} = \{u\in E \mid \text{$u$ is odd with respect to $x_1$} \}, |
In what follows, when we will consider functions
J^+(u) = \sum\limits_{p\geq 0} J_{p, 0}(u)\, . |
For any
\Gamma(v) = \{ u\in E^{odd} \mid \|u - v\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \to 0 \text{ as } p\to +\infty \} \subseteq \Gamma(-v, v)\, . |
In this setting we can rewrite Lemma 3.6 as follows.
Lemma 4.1. For every
We are going to look for minimizer of
\begin{equation} c(v) = \inf\limits_{u\in \Gamma(v)} J(u)\quad \text{ and }\quad{\mathcal{M}}(v) = \{u\in \Gamma(v) \mid J(u) = c(v)\}\, . \end{equation} | (10) |
Notice that for any
Lemma 4.2. For any
Moreover, note that, by assumption (
\begin{equation} c = \min\limits_{v\in{\mathcal{M}}_0} c(v) \end{equation} | (11) |
is well defined and the set
\begin{equation} {\mathcal{M}}_0^{min} = \{v \in{\mathcal{M}}_0 \mid c(v) = c \} \end{equation} | (12) |
is nonempty and consists of a finite number of elements. In particular, we have
\begin{equation} \min\limits_{ v\in {\mathcal{M}}_0\setminus {\mathcal{M}}_0^{min}}c(v) > c\, . \end{equation} | (13) |
The following lemma provides a concentration property for
Lemma 4.3. For any
Proof. Note that
To prove
\tilde u(x_{1}, y) = \begin{cases} u(x_{1}, y)&\hbox{if $x_{1}\in[0, p_0]$, }\\ u(x_{1}, y)(p_0+1-x_{1})+v_0(x_{1}, y)(x_{1}-p_0)& \hbox{if $x_{1}\in (p_0, p_0+1)$, }\\ v_0(x_{1}, y) &\hbox{if $x_{1}\in[p_0+1, +\infty)$} \end{cases} |
Hence,
\tfrac12 c\le\tfrac12c(v_0)\leq \tfrac12 J(\tilde u) = J^+(\tilde u) = J^+(u) -\sum\limits_{p = p_0}^{+ \infty} J_{p, 0}(u) + J_{p_{0}, 0}(\tilde u). |
By definition, on
\tfrac 12 c\leq \tfrac12 J(\tilde u) \leq J^+(u) -{\sum}_{p = p_0}^{+ \infty} J_{p, 0}(u) + \Lambda(r) \leq \tfrac 12 c -{\sum}_{p = p_0}^{+ \infty} J_{p, 0}(u) + \tfrac32 \Lambda(r). |
Then
By the previous lemma we get
Lemma 4.4. For any
Proof. Note that the existence of
As a direct consequence of Lemmas 4.3 and 4.4 we obtain the following concentration result.
Lemma 4.5. For any
Proof. The existence of
Finally,
We are now able to prove the existence of a minimum of
Theorem 4.6. Let
Proof. Let
\begin{equation} \|u_k -v\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq r_{1} \text{ for every } p\geq \tilde\ell(r_{1}). \end{equation} | (14) |
By Lemma 3.5, since
\begin{equation} \|u -v \|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)} \leq r_{1} \text{ for every } p\geq \tilde\ell(r_{1}). \end{equation} | (15) |
Therefore, by Lemma 3.6, we conclude that
By Theorem 4.6 we know that for every
Lemma 4.7. Given
\int_{{\mathbb{R}}\times[0, 1]^{n-1}} \nabla \bar u \cdot \nabla \psi + F_u(x, \bar u) \psi \, dx = 0\, . |
The proof can be adapted by the one of Lemma 3.3 of [4] or Lemma 5.2 of [6]. Therefore we get that any
Finally, we now study further compactness properties for the functional
L_{v}: W^{2, 2}([0, 1]^n, {\mathbb{R}}^m)\subset L^{2}([0, 1]^n, {\mathbb{R}}^m)\to L^{2}([0, 1]^n, {\mathbb{R}}^m)\, , |
L_{v}h = -\Delta h+F_{u, u}(\cdot, v(\cdot))h |
has spectrum which does not contain
(
J_{0}''(v)h\cdot h = \int_{[0, 1]^n}|\nabla h(x)|^{2}+F_{u, u}(x, v(x))|h(x)|^2\, dx\geq \alpha_{0}\| h\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}^{2} |
for every
As a consequence of
Lemma 4.8. There exist
\begin{equation} \omega_{0}\|u-v\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)}^{2} \leq J_{p, 0}(u) \leq \omega_1\|u-v\|_{W^{1, 2}(T_{p, 0}, {\mathbb{R}}^m)}^{2}\, . \end{equation} | (16) |
Proof. Notice that, by (
\begin{align*} \int_{[0, 1]^n} |\nabla h(x)|^{2}+F_{u, u}(x, v(x))|h(x)|^2\, dx& \geq \alpha_{0}\| h\|_{L^{2}([0, 1]^n, {\mathbb{R}}^m)}^{2}\\&\geq -\alpha_{0}f_{0} \int_{[0, 1]^n}F_{u, u}(x, v(x))|h(x)|^2\, dx, \end{align*} |
where
\int_{[0, 1]^n}\frac 1{1+\alpha_{0}f_{0}}|\nabla h(x)|^{2}\, dx+ \int_{[0, 1]^n} F_{u, u}(x, v(x))|h(x)|^2 \, dx \geq 0\, . |
We conclude that
J_0''(v)h\cdot h = \int_{[0, 1]^n} |\nabla h(x)|^{2}+ F_{u, u}(x, v(x))|h(x)|^2 \, dx \geq \frac{\alpha_{0}f_{0}}{1+\alpha_{0}f_{0}}\|\nabla h\|^{2}_{L^{2}([0, 1]^n, {\mathbb{R}}^m)} |
and so, using
\begin{equation*} \label{eq:**bis} J_{0}''(v)h\cdot h\geq 3\omega_{0}\|h\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}, \quad\forall\, h\in W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m). \end{equation*} |
Since by Taylor's formula we have
\begin{equation} J_{0}(u)-c_{0}\geq \omega_{0} \|u-v\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}. \end{equation} | (17) |
On the other hand, again Taylor's expansion gives us
\begin{align*} J_{0}(u)-c_{0}& = \tfrac12J_{0}''(v)(u-v)\cdot(u-v)+o(\|u-v\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2})\\ & = \tfrac12 \|\nabla(u-v)\|_{L^{2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}+\tfrac12\int_{[0, 1]^{n}}F_{u, u}(x, v(x))|u(x)-v(x)|^{2}\, dx\\ &\phantom{ = } + o(\|u-v\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2})\\ &\le \tfrac12 \|\nabla(u-v)\|_{L^{2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}+\tfrac1{2f_{0}}\|u-v\|_{L^{2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}\\ &\phantom{\le} +o(\|u-v\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}) \end{align*} |
and we deduce that there exists
\begin{equation} J_{0}(u)-c_{0}\le\omega_{1}\|u-v\|_{W^{1, 2}([0, 1]^{n}, {\mathbb{R}}^m)}^{2}. \end{equation} | (18) |
The lemma follows by periodicity from (17) and (18) recalling that
Remark 3. In connection with Remark 1, arguing as in Remark 3.8 of [2], we can prove that (16) holds true also for the functional
\begin{equation} \omega_{0}\|u-v\|_{W^{1, 2}(\sigma_{0}, {\mathbb{R}}^m)}^{2}\le J_{\sigma_0}(u)\le \omega_{1}\|u-v\|_{W^{1, 2}(\sigma_{0}, {\mathbb{R}}^m)}^{2}. \end{equation} | (19) |
Hence, recalling the definition (10), plainly adapting the proof of Lemma 3.10 in [2], we obtain
Lemma 4.9. Let
In this section we prove our main theorem. To this aim, following and adapting the argument in [2], we will first prove the existence of a solution of (PDE) on the unbounded triangle
{\mathcal{T}} = \{ x\in {\mathbb{R}}^n \mid x_2 \geq |x_1| \} |
satisfying Neumann boundary conditions on
{\mathcal{M}}: = \bigcup\limits_{v\in{\mathcal{M}}_0^{min}}{\mathcal{M}}(v). |
Then, by recursive reflections with respect to the hyperplanes
Let us introduce now some notations. We define the squares
T_{p, k}: = [p, p+1] \times [k, k+1] \times [0, 1]^{n-2}\, , \quad p\in{\mathbb{Z}}, \, k\in{\mathbb{N}} |
and the horizontal strips
{\mathcal{S}}_k : = {\mathbb{R}} \times [k, k+1] \times [0, 1]^{n-2} = \bigcup\limits_{p\in{\mathbb{Z}}} T_{p, k}\, , \quad k\in{\mathbb{N}} |
The intersection between the strip
{\mathcal{T}}_k : = {\mathcal{S}}_k\cap {\mathcal{T}} = \left(\bigcup\limits_{p = -k}^{k-1} T_{p, k}\right) \cup \tau_k |
where
For every
\begin{align*} E_{k} = \{u\in W^{1, 2}({\mathcal{T}}_k, {\mathbb{R}}^m) \mid u& \text{ is odd in } x_1, \, \text{1-periodic in }x_3, ..., x_n \} \end{align*} |
and the normalized functionals on the bounded strips
J_{k}(u) = \int_{{\mathcal{T}}_{k}} L(u) \, dx -(2k+1)c_0 = \sum\limits_{p = -k}^{k-1} J_{p, k}(u) + \int_{\tau_k} L(u) \, dx-c_0\, , \quad k\in{\mathbb{N}}\, , |
for every
Remark 4. Notice that
Then, we can set
c_{k} = \inf\limits_{E_{k}} J_{k}(u) \quad\text{ and }\quad {\mathcal{M}}_{k} = \{u\in E_{k} \mid J_{k}(u) = c_{k}\}\, . |
We plainly obtain that
Lemma 5.1. We have
We can now introduce on the set
{\mathcal{E}} = \{u\in W^{1, 2}_{loc}({\mathcal{T}}, {\mathbb{R}}^m) \mid u \text{ is odd in } x_1, \, u_i(x)\ge0 \text{ for } x_1\geq 0 \, , \forall i = 1, ..., m \}. |
the functional
{\mathcal{J}}(u) = \sum\limits_{k = 0}^{+\infty} \left( J_{k}(u) - c_{k}\right)\, . |
Notice that
Lemma 5.2. If
We now look for a minimum of the functional
\tilde c = \inf\limits_{{\mathcal{E}}} {\mathcal{J}}(u) \quad \text{and} \quad \widetilde{{\mathcal{M}}} = \{u\in {\mathcal{E}} \mid {\mathcal{J}}(u) = \tilde c \, \}\, . |
Lemma 5.2, gives that
Proposition 1. We have
Arguing as in [2,4,6] (see e.g. the argument in Lemma 3.3 of [4] or Lemma 5.2 of [6]), we can prove that if
In the next lemma we finally characterise the asymptotic behavior of the solution
Lemma 5.3. Let
\lim\limits_{k\to +\infty} {\rm dist}_{W^{1, 2}({\mathcal{T}}_k, {\mathbb{R}}^m)} (w, {\mathcal{M}}(\bar v)) = 0. |
Proof. Let
\begin{equation} \lim\limits_{k\to+\infty} \|w-\bar v\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)} = 0\, . \end{equation} | (20) |
We have
\|w-v_k\|_{W^{1, 2}(T_{p_k, k}, {\mathbb{R}}^m)} \to 0\, , \text{ as } k\to +\infty. |
Now, for every
w_k(x_1, x_2, y) = \begin{cases} w(x_1, x_2, y) & \text{if } 0\leq x_1 \leq p_k\\ w(x_1, x_2, y)(p_k-x_1+1)\\ \phantom{w(x_1, x_2, y)} +v_k(x_1, x_2, y)(x_1-p_k) & \text{if } p_k < x_1 \leq p_k+1\\ v_k(x_1, x_2, y) & \text{if } x_1 > p_k+1\\ \text{odd extended for } x_1 < 0 \end{cases} |
A computation gives
\lim\limits_{k\to+\infty} J_{p_k, k}(w_k) = 0\, . |
Now, consider
c \leq J(w_k^{\downarrow}) = 2 \sum\limits_{p = 0}^{p_k} J_{p, 0}(w_k^{\downarrow}) = 2 \sum\limits_{p = 0}^{p_k-1} J_{p, k}(w) + 2J_{p_k, k}(w_k) \leq J_{k}(w) + 2J_{p_k, k}(w_k)\, . |
and hence, since
J_{k}(w) -J_{k}(w_k) = 2 \sum\limits_{p = p_k}^{k-1} J_{p, k}(w) + \int_{\tau_k} L(w) \, dx - c_0 - 2 J_{p_k, k}(w_k) |
and since
\begin{equation} 2 \sum\limits_{p = p_k}^{k-1} J_{p, k}(w) + \int_{\tau_k} L(w) \, dx - c_0 \to 0\, , \text{ as } k\to +\infty\, . \end{equation} | (21) |
In particular
\begin{equation} \|w-v_k\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)}\to 0\, . \end{equation} | (22) |
Let us now consider, for every
\omega_k(x_1, x_2, y) = \begin{cases} w(x_1, x_2, y) & \text{if } 0\leq x_1 \leq k\\ w(x_1, x_2, y)(k-x_1+1)\\ \phantom{w(x_1, x_2, y)}+v_k(x_1, x_2, y)(x_1-k) & \text{if } k < x_1 \leq k+1\\ v_k(x_1, x_2, y) & \text{if } x_1 > k+1\\ \text{odd extended for } x_1 < 0 \end{cases} |
Arguing as above
c \leq J(\omega_k^{\downarrow}) \leq J_{k}(w) + 2 J_{k, 0}(\omega_k^{\downarrow}) = c + o(1) \, , |
thus giving
We now prove that the sequence
As a consequence, by definition of
\begin{align} &\|w-v_k\|_{W^{1, 2}(T_{p, k}, {\mathbb{R}}^m)} \leq r_1 < \tfrac{r_0}{4} \end{align} | (23) |
provided that
\begin{equation} \|w-\bar v\|_{W^{1, 2}(T_{ p_0, k}, {\mathbb{R}}^m)} \leq \tfrac{r_0}{4} \text{ for every } k\geq p_0+1\, . \end{equation} | (24) |
Finally, recalling (6), since both (23) and (24) holds, we must have
Moreover, we have proved that
\begin{equation*} \lim\limits_{k\to +\infty} \| \omega_k^{\downarrow} - \bar u \|_{W^{1, 2}({\mathcal{S}}_0, {\mathbb{R}}^m)} = 0\, . \end{equation*} |
Hence we obtain that
\begin{equation} {\rm dist}_{W^{1, 2}({\mathcal{S}}_0, {\mathbb{R}}^m)}( \omega_k^{\downarrow} , {\mathcal{M}}(\bar v))\to 0\quad\text{as }k\to +\infty. \end{equation} | (25) |
Finally, for every
\begin{align*} \|w-u\|_{W^{1, 2}({\mathcal{T}}_k, {\mathbb{R}}^m)}^2 & = 2\|w-u\|_{W^{1, 2}(\cup_{p = 0}^{k-1}T_{p, k}, {\mathbb{R}}^m)}^2 + \|w- u\|_{W^{1, 2}(\tau_k, {\mathbb{R}}^m)}^2\\ & = \|\omega_k^{\downarrow}- u\|_{W^{1, 2}({\mathcal{S}}_0, {\mathbb{R}}^m)}^2 - 2 \|\omega_k^{\downarrow}-u\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)}^2\\ &\phantom{ = } + \|w- u\|_{W^{1, 2}(\tau_k, {\mathbb{R}}^m)}^2\\ &\leq \|\omega_k^{\downarrow}- u\|_{W^{1, 2}({\mathcal{S}}_0, {\mathbb{R}}^m)}^2 + \|w- u\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)}^2\, . \end{align*} |
Notice that since
\begin{multline*} \|w- u\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)}^2 \leq \|w-\bar v\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)}^2 \\ + \| u-\bar v\|_{W^{1, 2}(T_{k, k}, {\mathbb{R}}^m)}^2 \to 0\, , \text{ as } k\to +\infty\, . \end{multline*} |
Hence, by (25), we conclude
\lim\limits_{k\to +\infty} {\rm dist}_{W^{1, 2}({\mathcal{T}}_k, {\mathbb{R}}^m)} (w, {\mathcal{M}}(\bar v)) = 0 \, . |
The previous lemma gives the asymptotic estimate in Theorem 1.1 since
We can conclude now the proof of Theorem 1.1 proving the sign property
Finally, for any
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