Byeon, Jaeyoung

EPrint Series of Department of Mathematics, Hokkaido University

Phase transition solutions
to an Allen-Cahn model equation
Jaeyoung Byeon
KAIST, Republic of Korea
We consider the following equation
(0.1)  u+ A(x)G0(u) = 0 in Rn
where A(x) = A(x+ i) for x 2 Rn; i 2 Zn; G(u+ j) = G(u) for u 2 R; j 2 Z;
G(u) > 0 for u 2 R nZ, G(j) = 0; G00(j) > 0 for j 2 Z and G(t) = G( t) for
t 2 R:
Several authors have studied Allen-Cahn phase transition models in which
the spatial phase transition manifests itself as a heteroclinic or homoclinic
solution of the corresponding partial dierential equation. See e.g. [1]-[4],
[16]-[17] and [20]. Similar type of solutions in more general settings arise in
extension's of Moser's work [14] on developing an Aubry-Mather theory for
PDE's: [5], [6]-[7], [9]-[11], [18]-[19], [21], and [23]. In all of these sources the
transition type solutions are unidirectional in the sense that they change in
a particular direction.
In this talk, I would like to introduce some recent works with Paul Ra-
binowitz about some multidirectional solutions. For autonomous problem,
that is, A = constant, there have been many works [22], [12],[13] and ref-
erences therein which show that the bounded solutions of (0.1) are closely
related to the minimal surfaces in Rn:
We believe that there would be a generic condition for existence of multi
directional solutions like gap conditions for construction of multi-transition
(unidirectional) solutions(see [21] and references therein). We do not know
how to do this yet for (0.1), but will present a certain type of non-autonomous
term A for which such solutions can be found.
To describe our results, let x = (x1;    ; xn) 2 Rn and A 2 C1(Rn)
be a nonnegative function that is 1-periodic in xi; 1  i  n. Set 
 
1
fx 2 (0; 1)n j A(x) > 0g. We assume 2d = j@
   @[0; 1]nj > 0 and @
 is a
smooth manifold. Set 
d  fx 2 
 j jx   @
j > dg. Then for suciently
small d 2 (0; d); @
d is dieomorphic to @
. Fixing such a small d, let " > 0
and A" = 1 +
1
"
A: Our model equation is
(0.2)  u+ A"G0(u) = 0; x 2 Rn:
First, we are interested in solutions of (0.2) satisfying 0 < U < 1 and
that are near 1 on AT and near 0 on BT , where
T  Zn; AT = [i2T (i+ 
); BT = [i2ZnnT (i+ 
):
To describe our results more precisely, for d 2 (0; d) and 
d as above, we
dene
AT  [fi+ 
d j i 2 Tg
and
BT  [fi+ 
d j i 2 ZN n Tg:
Set
L"(u) =
1
2
jruj2 + A"G(u) and J"(u) =
Z
Rn
L"(u) dx:
Choose 0 < b < 1
2
< a < 1 and dene
~ (T ) = fu 2 C2(Rn; [0; 1]) j u  a > 1=2 on AT and u  b < 1=2 on BTg:
Whenever a solution, u 2 ~ (T ), of (0.2) satises
J"(u+ ')  J"(u) =
Z
supp '
(L"(u+ ')  L"(u)) dx  0
for all ' 2 C10 (Rn) such that u+ ' 2 ~ (T ), u will be said to be minimal in
~ (T ).
The solutions obtained here will have certain decay properties relative to
T . To help describe them, for x 2 Rn, dene
d(x; T )  dist(x; @(T + [0; 1]n))
and for S  Rn, let S denote the characteristic function of S. Our rst
main result [8] is:
2
Theorem 0.3. Under the above hypotheses on A and G, there is an "0 > 0
such that for any " 2 (0; "0) and any T  Zn,
1o there exists a solution U = U";T 2 ~ (T ) of (0.2);
2o U is minimal in ~ (T );
3o 0 < U";T < 1 ;
4o U";T converges uniformly to 1 on A
T and to 0 on BT as "! 0;
5o there exist constants C; c > 0; independent of T  Zn and of " 2 (0; "0);
satisfying
jU";T (x)  T+[0;1]n(x)j  C exp( cd(x; T )); x 2 Rn:
Second, we are interested in unbounded solutions of (0.2). As an Aubry-
Mather theory for PDE's, Moser [14] obtained among other things the fol-
lowing existence result.
Theorem 0.4. For each  2 Rn; there exists a solution u of (0.1) such
that for some C > 0; ju(x)    xj  C; x 2 Rn:
Let P : Zn + 
 ! Z be a function such that P is constant on i + 
 for
each i 2 Zn. We say P is Lipschitz continuous if there exists C > 0 such
that jP (x)  P (y)j  Cjx  yj for any x; y 2 Zn + 
:
As for multidirectional (unbounded) solutions generalizing the unidirec-
tional solutions obtained by Moser in Theorem 0.4, our second main result
is:
Theorem 0.5. Let A" be as above. Suppose P be Lipschitz continuous on S.
Then there is an "0 depending on the Lipschitz constant of P such that for
each " 2 (0; "0), there is a solution, U = U";P of (0.2) with U ! P uniformly
on Zn + 
 as "! 0.
References
[1] Francesca Alessio, Louis Jeanjean and Piero Montecchiari, Stationary
layered solutions in R2 for a class of non autonomous Allen-Cahn equations, Calc.
Var. Partial Dierential Equations, 11 (2000), 177{202
3
[2] Francesca Alessio, Louis Jeanjean and Piero Montecchiari, Existence of
innitely many stationary layered solutions in R2 for a class of periodic Allen-Cahn
equations, Comm. Partial Dierential Equations, 27 (2002), 1537{1574
[3] Francesca Alessio and Piero Montecchiari, Entire solutions in R2 for a class
of Allen-Cahn equations, ESAIM Control Optim. Calc. Var., 11 (2005), 633{672
[4] Francesca Alessio and Piero Montecchiari, Multiplicity of entire solutions for
a class of almost periodic Allen-Cahn type equations, Adv. Nonlinear Stud., 5 (2005),
515{549
[5] Victor Bangert, On minimal laminations of the torus, Ann. Inst. Poincare Anal.
Non Lineaire, 6 (1989), 95{138
[6] Ugo Bessi, Many solutions of elliptic problems on Rn of irrational slope, Comm.
Partial Dierential Equations, 30 (2005), 1773{1804
[7] Ugo Bessi, Slope-changing solutions of elliptic problems on Rn, Nonlinear Anal., 68
(2008), 3923{3947
[8] Jaeyoung, Paul H. Rabinowitz, On a phase transition model. Calc. Var. Partial
Dierential Equations, 47 (2013), 1{23
[9] Luis A. Caffarelli and Rafael de la Llave, Planelike minimizers in periodic
media. Comm. Pure Appl. Math., 54 (2001), 1403{1441
[10] Rafael de la Llave and Enrico Valdinoci, Multiplicity results for interfaces of
Ginzburg-Landau Allen-Cahn equations in periodic media, Adv. Math., 215 (2007),
379{426
[11] Rafael de la Llave and Enrico Valdinoci, A generalization of Aubry-Mather
theory to partial dierential equations and pseudo-dierential equations, Ann. Inst.
H. Poincare Anal. Non Lineaire, 26 (2009), 1309{1344
[12] Manuel del Pino, Michael Kowalczyk, Juncheng Wei, On De Giorgi's con-
jecture in dimension N  9 . Ann. of Math. 174 (2011), 1485{1569
[13] Manuel del Pino, Michal Kowalczyk, Juncheng Wei, Entire solutions of the
Allen-Cahn equation and complete embedded minimal surfaces of nite total curvature
in R3; J. Dierential Geom., 93 (2013), 67{131
[14] Jurgen Moser, Minimal solutions of a variational problems on a torus, Ann. Inst.
Poincare Anal. Non Lineaire, 3 (1986), 229{272
[15] Matteo Novaga and Enrico Valdinoci, Bump solutions for the mesoscopic
Allen-Cahn equation in periodic media. Calc. Var. Partial Dierential Equations, 40
(2011), no. 1-2, 37{49
4
[16] Paul H. Rabinowitz and Ed Stredulinsky, Mixed states for an Allen-Cahn
type equation, Comm. Pure Appl. Math., 56 (2003), 1078{1134
[17] Paul H. Rabinowitz and Ed Stredulinsky, Mixed states for an Allen-Cahn
type equation. II, Calc. Var. Partial Dierential Equations, 21 (2004), 157{207
[18] Paul. H. Rabinowitz and Ed Stredulinsky, On some results of Moser and of
Bangert, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 673{688
[19] Paul H. Rabinowitz and Ed Stredulinsky, On some results of Moser and of
Bangert. II, Adv. Nonlinear Stud., 4 (2004), 377{396
[20] Paul H. Rabinowitz and Ed Stredulinsky, On a class of innite transition
solutions for an Allen-Cahn model equation, Discrete Contin. Dyn. Syst., 21 (2008),
319{332
[21] Paul H. Rabinowitz and Ed Stredulinsky, Extensions of Moser-Bangert The-
ory: Locally Minimal Solutions, Progress in Nonlinear Dierential Equations and
Their Applications, 81, Birkhauser, Boston, (2011)
[22] Ovidiu Savin, Regularity of at level sets in phase transitions, Ann. of Math.,
169(2009), 41{78
[23] Enrico Valdinoci, Plane-like minimizers in periodic media: jet ows and Ginzburg-
Landau-type functionals, J. Reine Angew. Math., 574 (2004), 147{185
5


Phase transition solutions to an Allen-Cahn model equation

Abstract

Similar works

Full text

Available Versions

EPrint Series of Department of Mathematics, Hokkaido University