
The decomposition of the triangular set
Citation: Andrew C. Stolte, Brady R. Cox. Feasibility of in-situ evaluation of soil void ratio in clean sands using high resolution measurements of Vp and Vs from DPCH testing[J]. AIMS Geosciences, 2019, 5(4): 723-749. doi: 10.3934/geosci.2019.4.723
[1] | Yi Dong, Jinjiang Liu, Yihua Lan . A classification method for breast images based on an improved VGG16 network model. Electronic Research Archive, 2023, 31(4): 2358-2373. doi: 10.3934/era.2023120 |
[2] | Hui-Ching Wu, Yu-Chen Tu, Po-Han Chen, Ming-Hseng Tseng . An interpretable hierarchical semantic convolutional neural network to diagnose melanoma in skin lesions. Electronic Research Archive, 2023, 31(4): 1822-1839. doi: 10.3934/era.2023094 |
[3] | Dong Wu, Jiechang Li, Weijiang Yang . STD-YOLOv8: A lightweight small target detection algorithm for UAV perspectives. Electronic Research Archive, 2024, 32(7): 4563-4580. doi: 10.3934/era.2024207 |
[4] | Peng Lu, Xinpeng Hao, Wenhui Li, Congqin Yi, Ru Kong, Teng Wang . ECF-YOLO: An enhanced YOLOv8 algorithm for ship detection in SAR images. Electronic Research Archive, 2025, 33(5): 3394-3409. doi: 10.3934/era.2025150 |
[5] | Jinjiang Liu, Yuqin Li, Wentao Li, Zhenshuang Li, Yihua Lan . Multiscale lung nodule segmentation based on 3D coordinate attention and edge enhancement. Electronic Research Archive, 2024, 32(5): 3016-3037. doi: 10.3934/era.2024138 |
[6] | Jianjun Huang, Xuhong Huang, Ronghao Kang, Zhihong Chen, Junhan Peng . Improved insulator location and defect detection method based on GhostNet and YOLOv5s networks. Electronic Research Archive, 2024, 32(9): 5249-5267. doi: 10.3934/era.2024242 |
[7] | Jianting Gong, Yingwei Zhao, Xiantao Heng, Yongbing Chen, Pingping Sun, Fei He, Zhiqiang Ma, Zilin Ren . Deciphering and identifying pan-cancer RAS pathway activation based on graph autoencoder and ClassifierChain. Electronic Research Archive, 2023, 31(8): 4951-4967. doi: 10.3934/era.2023253 |
[8] | Chetan Swarup, Kamred Udham Singh, Ankit Kumar, Saroj Kumar Pandey, Neeraj varshney, Teekam Singh . Brain tumor detection using CNN, AlexNet & GoogLeNet ensembling learning approaches. Electronic Research Archive, 2023, 31(5): 2900-2924. doi: 10.3934/era.2023146 |
[9] | Yunfei Tan, Shuyu Li, Zehua Li . A privacy preserving recommendation and fraud detection method based on graph convolution. Electronic Research Archive, 2023, 31(12): 7559-7577. doi: 10.3934/era.2023382 |
[10] | Xite Yang, Ankang Zou, Jidi Cao, Yongzeng Lai, Jilin Zhang . Systemic risk prediction based on Savitzky-Golay smoothing and temporal convolutional networks. Electronic Research Archive, 2023, 31(5): 2667-2688. doi: 10.3934/era.2023135 |
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
[1] | International Organization for Standardization (2005) ISO 22476-3:2005 Geotechnical investigation and testing-Field testing-Part 3: Standard penetration test. ISO, Geneva, Switzerland. |
[2] | International Organization for Standardization (2012) ISO 22476-1:2012 Geotechnical investigation and testing-Field testing-Part 1: Electrical cone and piezocone penetration test. ISO, Geneva, Switzerland. |
[3] | Kulhawy FH, Mayne PW (1990) Estimating Soil Properties for Foundation Design. EPRI Report EL-6800, Electric Power Research Institute, Palo Alto, 306. |
[4] | Robertson PK, Cabal KL (2015) Guide to cone penetration testing 6th Edition. |
[5] | Baldi G, Bellotti R, Ghionna V, et al. (1986) Interpretation of CPTs and CPTUs, Part Ⅱ: Drained Penetration in Sands, Proc. of 4th International Geotechnical Seminar on Field Instrumentation and In Situ Measurements, Singapore. |
[6] |
Salgado R, Mitchell JK, Jamiolkowski M (1997) Cavity expansion and penetration resistance in sand. J Geotech Geoenviron Eng 123: 344-354. doi: 10.1061/(ASCE)1090-0241(1997)123:4(344)
![]() |
[7] | Jamiolkowski M, LoPresti DCF, Manassero M (2001) Evaluation of relative density and shear strength of sands from cone penetration test and flat dilatometer test. Soil Behavior and Soft Ground Construction (GSP 119), ASCE, Reston, VA, 201-238. |
[8] |
Salgado R, Prezzi M (2007) Computation of cavity expansion pressure and penetration resistance in sands. Int J Geomech 7: 251-265. doi: 10.1061/(ASCE)1532-3641(2007)7:4(251)
![]() |
[9] | ASTM International (2014) ASTM D4428/D4428M-14 Standard Test Methods for Crosshole Seismic Testing. Ann Book ASTM Stand 4. |
[10] |
Wyllie MRJ, Gregory AR, Gardner LW (1956) Elastic wave velocity in heterogeneous and porous media. Geophysics 21: 41-70. doi: 10.1190/1.1438217
![]() |
[11] | Raymer LL, Hunt ER, Gardner JS (1980) An improved sonic transit time-to-porosity transform. presented in Trans. Soc. Prof. Well Log Analysts, 21st Annual Logging Symposium. |
[12] |
Domenico SN (1984) Rock lithology and porosity determination from shear and compressional wave velocity. Geophysics 49: 1188-1195. doi: 10.1190/1.1441748
![]() |
[13] |
Castagna JP, Batzle ML, Eastwood RL (1985) Relationship between compressional-wave and shear-wave velocities in clastic silicate rocks. Geophysics 50: 571-581. doi: 10.1190/1.1441933
![]() |
[14] |
Han DH, Nur A, Morgan D (1986) Effects of porosity and clay content on wave velocity in sandstones. Geophysics 51: 2093-2107. doi: 10.1190/1.1442062
![]() |
[15] |
Ederhart-Phillips D, Han DH, Zoback MD (1989) Empirical relationships among seismic velocities, effective pressure, porosity, and clay content in sandstones. Geophysics 54: 82-89. doi: 10.1190/1.1442580
![]() |
[16] | Biot MA (1956a) Theory of propagation of elastic waves in a fluid saturated porous solid. I: Low-frequency range. J Acoust Soc Am 28: 168-178. |
[17] | Biot MA (1956b) Theory of propagation of elastic waves in a fluid saturated porous solid. Ⅱ: Higher frequency range. J Acoust Soc Am 28: 179-191. |
[18] | Krief M, Garat J, Stellingwerff J, et al. (1990) A petrophysical interpretation using the velocities of P and S waves. Log Anal 31: 355-369. |
[19] | Miura K, Yoshida N, Kim Y (2001) Frequency dependent property of waves in saturated soil. Soils Found 41: 1-19. |
[20] |
Foti S, Lai C, Lancellotta R (2002) Porosity of fluid-saturated porous media from measured seismic wave velocities. Géotechnique 52: 359-373. doi: 10.1680/geot.52.5.359.38711
![]() |
[21] |
Foti S, Lancellotta R (2004) Soil porosity from seismic velocities. Géotechnique 54: 551-554. doi: 10.1680/geot.2004.54.8.551
![]() |
[22] |
Lai CG, Crempien de la Carrera JGF (2012) Stable inversion of measured vp and vs to estimate porosity in fluid-saturated soils. Géotechnique 62: 359-364. doi: 10.1680/geot.9.P.133
![]() |
[23] | Foti S, Passeri F (2016) Reliability of soil porosity estimation from seismic wave velocities. In Isc5-International Conference on Geotechnical and Geophysical Soil Characterisation, Gold Coast, Australia 1: 425-430. |
[24] | Jamiolkowski M (2012) Role of Geophysical Testing in Geotechnical Site Characterization. Soils Rocks 35: 117-137. |
[25] | Cox BR, Stolte AC, Stokoe KH Ⅱ, et al. (2019) A Direct Push Crosshole (DPCH) Test Method for the In-Situ Evaluation of High-Resolution P- and S-wave Velocities. ASTM Geotech Test J 42: 1101-1132. |
[26] | Van Ballegooy S, Roberts JN, Stokoe KH Ⅱ, et al. (2015) Large-Scale Testing of Shallow Ground Improvements using Controlled Staged-Loading with T-Rex, 6th International Conference on Earthquake Geotechnical Engineering, Christchurch, New Zealand. |
[27] | Wentz FJ, van Ballegooy S, Rollins KM, et al. (2015) Large Scale Testing of Shallow Ground Improvements using Blast-Induced Liquefaction, 6th International Conference on Earthquake Geotechnical Engineering, Christchurch, New Zealand. |
[28] | Stokoe KH Ⅱ, Roberts JN, Hwang S, et al. (2014) Effectiveness of Inhibiting Liquefaction Triggering by Shallow Ground Improvement Methods: Initial Field Shaking Trials with T-Rex at One Site in Christchurch, New Zealand, In Orense RP, Towhata I, Chouw N (Eds.), Soil Liquefaction during Recent Large-Scale Earthquakes, CRC Press. |
[29] | Wotherspoon LM, Cox BR, Stokoe KH Ⅱ, et al. (2015) Utilizing Direct-Push Crosshole Testing to Assess the Effectiveness of Soil Stiffening Caused by Installation of Stone Columns and Rammed Aggregate Piers, 6th International Conference on Earthquake Geotechnical Engineering, Christchurch, New Zealand |
[30] | Stokoe KH Ⅱ, Roberts JN, Hwang S, et al. (2016) Effectiveness of Effectiveness of Inhibiting Liquefaction Triggering by Shallow Ground Improvement Methods: Field Shaking Trials with T-Rex at One Area in Christchurch, New Zealand, 24th Geotechnical Conference of Torino, Design; Construction & Controls of Soil Improvement Systems, Turin, Italy, 1-20. |
[31] | Wotherspoon LM, Cox BR, Stokoe KH Ⅱ, et al. (2017) Assessment of the Degree of Soil Stiffening from Stone Column Installation using Direct Push Crosshole Testing, 16th World Conference on Earthquake Engineering, Santiago, Chile. |
[32] | Hwang S, Roberts JN, Stokoe KH Ⅱ, et al. (2017) Utilizing Direct-Push Crosshole Seismic Testing to Verify the Effectiveness of Shallow Ground Improvements: A Case Study Involving Low-Mobility Grout Columns in Christchurch, New Zealand. Grouting 2017, 415-424. |
[33] | McLaughlin KA (2017) Investigation of false-positive liquefaction case history sites in Christchurch, New Zealand. M.S. Thesis. The University of Texas at Austin. |
[34] | Cox BR, McLaughlin KA, van Ballegooy S, et al. (2017) In-Situ Investigation of False-Positive Liquefaction Sites in Christchurch, New Zealand: St. Teresa's School Case History. 3rd International Conference on Performance-based Design in Earthquake Geotechnical Engineering, Vancouver, Canada. |
[35] |
Tamura S, Tokimatsu K, Abe A, et al. (2002) Effects of the air bubbles on B value and P wave velocity of a partially saturated sand. Soils Found 42: 121-129. doi: 10.3208/sandf.42.121
![]() |
[36] | Valle-Molina C (2006) Measurements of vp and vs in Dry, Unsaturated and Saturated Sand Specimens with Peizoelectric Transducers. Ph.D. Dissertation. The University of Texas at Austin |
[37] |
Valle-Molina C, Stokoe KH Ⅱ (2012) Seismic measurements in sand specimens with varying degrees of saturation using piezoelectric transducers. Can Geotech J 49: 671-685. doi: 10.1139/t2012-033
![]() |
[38] |
Bates CR (1989) Dynamic soil property measurements during triaxial testing. Géotechnique 39: 721-726. doi: 10.1680/geot.1989.39.4.721
![]() |
[39] |
Nakagawa K, Soga K, Mitchell JK (1997) Observation of Biot compressional wave of the second kind in granular soils. Géotechnique 47: 133-147. doi: 10.1680/geot.1997.47.1.133
![]() |
[40] |
Kumar J, Madhusudhan BN (2010) Effect of relative density and confining pressure on Poisson ratio from bender and extender elements tests. Géotechnique 60: 561-567. doi: 10.1680/geot.9.T.003
![]() |
[41] |
Wichtmann T, Triantafyllidis T (2010) On the influence of the grain size distribution curve on P-wave velocity, constrained elastic modulus Mmax and Poisson's ratio of quartz sands. Soil Dyn Earthq Eng 30: 757-766. doi: 10.1016/j.soildyn.2010.03.006
![]() |
[42] |
Wagner W, Prusß A (2002) The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use. J Phys Chem Ref Data 31: 387. doi: 10.1063/1.1461829
![]() |
[43] | Kusuda T, Achenbach PR (1965) Earth temperature and thermal diffusivity at selected stations in the United States. NBS Report 8972, National Bureau of Standards, Gaithersburg, MD, USA. |
[44] | Kell GS (1975) Density, thermal expansivity, and compressibility of liquid water from 0. deg. to 150. deg.. Correlations and tables for atmospheric pressure and saturation reviewed and expressed on 1968 temperature scale. J Chem Eng Data 20: 97-105. |
[45] |
Lubbers J, Graaff R (1998) A simple and accurate formula for the sound velocity in water. Ultrasoun Med Biol 24: 1065-1068. doi: 10.1016/S0301-5629(98)00091-X
![]() |
[46] | Lambe TW, Whitman RV (1967) Soil Mechanics. John Wiley & Sons. |
[47] | Hardin BO, Richart Jr FE (1963) Elastic Wave Velocities in Granular Soils. J Soil Mech Found Div ASCE 89: 33-65. |
[48] | Hardin BO, Black WL (1968) Vibration Modulus of Normally Consolidated Clay. J Soil Mech Found Div ASCE 94: 353-369. |
[49] | Hardin BO (1978) The nature of stress-strain behavior of soils. Proceedings, Geotech. Eng. Div. Specialty Conf. on Earthquake Eng. and Soil Dynamics 1, ASCE, Pasadena, 3-90. |
[50] | Menq F (2003) Dynamic properties of sandy and gravelly soils. Ph.D. Dissertation. The University of Texas at Austin. |
[51] | Beyzaei CZ (2017) Fine-Grained Soil Liquefaction Effects in Christchurch, New Zealand. PhD Thesis. The University of California, Berkeley. |
[52] | Beyzaei CZ, Bray JD, Cubrinovski M, et al. (2018) Laboratory-Based Characterization of Shallow Silty Soils in Southwest Christchurch. Soil Dyn Earthq Eng 19: 93-109. |
[53] | Taylor ML (2015) The Geotechnical Characterisation of Christchurch Sands for Advanced Soil Modelling. Ph.D. Thesis. The University of Canterbury. |
[54] |
Bray JD, Cubrinovski M, Zupan J, et al. (2014) Liquefaction Effects on Buildings in the Central Business District of Christchurch. Earthq Spectra 30: 85-109. doi: 10.1193/022113EQS043M
![]() |
1. | Maaz Bahauddin Naveed, Data Evaluation and Modeling of Billet Characteristics in the Steel Industry, 2025, 2181, 10.38124/ijisrt/25apr652 | |
2. | Renan J. S. Isneri, César E. Torres Ledesma, Saddle Solutions for Allen–Cahn Type Equations Involving the Prescribed Mean Curvature Operator, 2025, 1424-9286, 10.1007/s00032-025-00418-y |