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Chapter 8 DSM for operators satisfying a spectral assumption
Chapter 8
DSM for operators satisfying a spectral assumption In this Chapter we introduce a spectral assumption (8.1.1) and obtain some results based on this assumption.
8.1
Spectral assumption
In this Chapter we assume that the operator equation (7.2.1) is solvable, 2 y is its solution, possibly non-unique, F ∈ Cloc , and the linear operator 0 A = F (u) has the set {z : |argz − π| ≤ ϕ0 , 0 < |z| < r0 } consisting of regular points of F 0 , where ϕ0 > 0 and r0 > 0 are arbitrary small fixed numbers, and u ∈ H is arbitrary. This is a spectral assumption on F . If this condition is satisfied then c0 ||(A + εI)−1 || ≤ , 0 < ε ≤ r0 , (8.1.1) ε where c0 = sin1ϕ0 . Condition (8.1.1) is much weaker than the assumption of monotonicity 1 of F . If F is monotone, then A = F 0 (u) ≥ 0 and ||A−1 ε || ≤ ε for all ε > 0, where Aε = A + εI. If c0 = 1 and r0 = ∞ in (8.1.1), then A is a generator of a semigroup of contractions ([P]). It is known that if F is monotone, hemicontinuous, and ε > 0, then the equation F (u) + εu = f,
ε = const > 0,
(8.1.2)
is uniquely solvable for any f ∈ H. We want to prove a similar result assuming (8.1.1). Our first result is the following. 133
134
8. DSM FOR OPERATORS SATISFYING A SA
Theorem 8.1.1. Assume that F satisfies conditions (1.3.2) and (8.1.1). Then equation (8.1.2) has a solution. Proof of Theorem 8.1.1. From our assumptions it follows that the problem u˙ = −A−1 ε [F (u) + εu − f ],
u(0) = u0 ,
(8.1.3)
is locally solvable. We will prove the uniform bound sup ||u(t)|| ≤ c,
(8.1.4)
t≥0
where c = const > 0 does not depend on t and the supremum is over all t for which u(t), the local solution to (8.1.3), does exist. If (8.1.4) holds, then, as we have proved in Section 2.6, Lemma 2.6.1, the local solution is a global one, it exists on [0, ∞). We also prove that there exists u(∞), and that u(∞) solves (8.1.2). Let us prove estimate (8.1.4). Consider the function g(t) := ||F (u(t)) + εu(t) − f ||.
(8.1.5)
We have g g˙ = Re([F 0 (u(t)) + ε]u, ˙ F (u) + εu − f ) = −g 2 .
(8.1.6)
g(t) = g0 e−t ,
(8.1.7)
Thus g0 := g(0).
From (8.1.1), (8.1.3) and (8.1.7) we get c0 g0 e−t . ε This implies the existence of u(∞) and the estimates ||u|| ˙ ≤
(8.1.8)
c0 g0 c0 g0 e−t , ||u(t) − u(∞)|| ≤ . (8.1.9) ε ε For any fixed u0 and ε > 0, one can take R > 0 such that c0 g0 ≤ R, ε so that the trajectory u(t) stays in the ball B(u0 , R). Passing to the limit t → ∞ in (8.1.7) yields ||u(t) − u(0)|| ≤
F (u(∞)) + εu(∞) − f = 0.
(8.1.10)
Therefore u(∞) solves equation (8.1.2), so this equation is solvable. Moreover, the DSM (8.1.3) converges to a solution u(∞) at an exponential rate, see (8.1.9). Theorem 8.1.1 is proved. 2
8.2. SPECTRAL ASSUMPTION
135
Remark 8.1.1. Estimate (8.1.1) follows from the spectral assumption because of the known estimate of the norm of the resolvent of a linear operator. Namely, if A is a linear operator and (A − zI)−1 is its resolvent, where z is a complex number in the set of regular points of the operator A. A point z is called a regular point of A if the operator A − zI has a bounded inverse defined on the whole space. Otherwise z is called a point of spectrum σ of A. The estimate we have mentioned is: ||(A − zI)−1 || <
1 , ρ(z, σ)
(8.1.11)
where ρ(z, σ) is the distance (on the complex plane C) from the point z to the set σ of points of spectrum of A. To check this, consider the function (A − zI − µI)−1 = (A − zI)−1 [I − µ(A − zI)−1 ]−1 . If |µ| ||(A − zI)−1 || < 1, then the above function is an analytic function of µ. Therefore, if µ is smaller than the distance from z to the nearest point of spectrum of A, then the point z + µ is a regular point of A. Thus (8.1.11) follows. Remark 8.1.2. We have proved in Theorem 8.1.1 that for ε ∈ (0, r0 ) equation (8.1.2) is solvable. This does not imply that the limiting equation F (u) = f,
(8.1.12)
is solvable. For example, the equation eu + εu = 0 is solvable in R for any ε > 0, but the limiting equation eu = 0 has no solutions. Therefore it is of interest to study the following problem: If the limiting equation (8.1.12) is solvable, then what are the conditions under which the solution to equation (8.1.2), or a more general equation F (uε ) + ε(uε − z) = f,
(8.1.13)
where z ∈ H is some element, converges to a solution to (8.1.12) as ε → 0? This question is discussed in Chapter 9.
136
8.2
8. DSM FOR OPERATORS SATISFYING A SA
Existence of a solution to a nonlinear equation
Let D ⊂ R3 be a bounded domain with Lipschitz boundary S, k = const > 0 , and f : R → R be a function such that uf (u) ≥ 0,
for |u| ≥ a ≥ 0,
(8.2.1)
where a is an arbitrary nonnegative fixed number. We assume that f is continuous in the region |u| ≥ a, and bounded and piecewise-continuous with at most finitely many discontinuity points uj , such that f (uj + 0) and f (uj − 0) exist, in the region |u| ≤ a. Consider the problem (−∆ + k 2 )u + f (u) = 0
in D,
(8.2.2)
u = 0 on S.
(8.2.3)
There is a large literature on problems of this type. Usually it is assumed that f does not grow too fast or f is monotone (see, e.g., [B] and references therein). The novel point in this Section is the absence of the monotonicity restrictions on f and of the growth restrictions on f as |u| → ∞, except for the assumption (8.2.1). This assumption allows an arbitrary behavior of f inside the region |u| ≤ a, where a ≥ 0 can be arbitrary large, and an arbitrarily rapid growth of f to +∞ as u → +∞, or arbitrarily rapid decay of f to −∞ as u → −∞. Our result is: Theorem 8.2.1. Under the above assumptions problem (8.2.2)–(8.2.3) has ◦
a solution u ∈ H 2 (D) ∩ H1 (D) := H02 (D). ◦
Here H ` (D) is the usual Sobolev space, H1 (D) is the closure of C0∞ (D) in the norm H 1 (D). Uniqueness of the solution does not hold without extra assumptions. The ideas of our proof are: first, we prove that if sup |f (u)| ≤ µ, u∈R
then a solution to (8.2.2)–(8.2.3) exists by the Schauder’s fixed-point theorem (see Section 16, Theorem 16.8.1). Here µ is a constant. Secondly, we prove an a priori bound kuk∞ ≤ a.
8.2. EXISTENCE OF A SOLUTION TO A NLE If this bound is proved, f replaced by f (u), F (u) := f (a), f (−a),
137
then problem (8.2.2)–(8.2.3) with the nonlinearity |u| ≤ a u≥a u ≤ −a
(8.2.4)
has a solution, and this solution solves the original problem (8.2.2)–(8.2.3). The bound kuk∞ ≤ a is proved by using some integral inequalities. An alternative proof of this bound is also given. This proof is based on the maximum principle for elliptic equation (8.2.2). We use some ideas from [R9]. Our presentation follows [R56]. Proof of Theorem 8.2.1. If u ∈ L∞ := L∞ (D), then problem (8.2.2)–(8.2.3) is equivalent to the integral equation: Z u=− G(x, y)f (u(y))dy := T (u), (8.2.5) D
where (−∆ + k 2 )G = −δ(x − y)
in D,
g |x∈S = 0.
(8.2.6)
x, y ∈ D.
(8.2.7)
By the maximum principle, 0 ≤ G(x, y) < g(x, y) :=
e−k|x−y| , 4π|x − y|
The map T is a continuous and compact map in the space C(D) := X, and Z Z e−k|x−y| e−k|y| µ kukC(D) := kuk ≤ µ sup dy ≤ µ dy ≤ 2 . (8.2.8) 4π|x − y| 4π|y| k 3 x D R This is an a priori estimate of any bounded solution to (8.2.2)–(8.2.3) for a bounded nonlinearity f such that sup |f (u)| ≤ µ. u∈R
Thus, Schauder’s fixed-point theorem yields the existence of a solution to (8.2.5), and consequently to problem (8.2.2)–(8.2.3), for bounded f . Indeed, if B is a closed ball of radius kµ2 , then the map T maps this ball into itself by (8.2.8), and since T is compact, the Schauder principle is applicable. Thus, the following lemma is proved.
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8. DSM FOR OPERATORS SATISFYING A SA
Lemma 8.2.1. If supu∈R |f (u)| ≤ µ, then problems (8.2.5) and (8.2.2)– (8.2.3) have a solution in C(D), and this solution satisfies estimate (8.2.8). Let us now prove an a priori bound for any solution u ∈ C(D) of the problem (8.2.2)–(8.2.3) without assuming that supu∈R |f (u)| < ∞. Let u+ := max(u, 0). Multiply (8.2.2) by (u − a)+ , integrate over D and then by parts to get Z 0= [∇u · ∇(u − a)+ + k 2 u(u − a)+ + f (u)(u − a)+ ]dx, (8.2.9) D
where the boundary integral vanishes because (u − a)+ = 0
on
S
for
a ≥ 0.
Each of the terms in (8.2.9) is nonnegative, the last one due to (8.2.1). Thus (8.2.9) implies u ≤ a.
(8.2.10)
Similarly, using (8.2.1) again, and multiplying (8.2.2) by (−u − a)+ , one gets −a ≤ u.
(8.2.11)
We have proved: Lemma 8.2.2. If (8.2.1) holds, then any solution u ∈ H02 (D) to (8.2.2)– (8.2.3) satisfies the inequality |u(x)| ≤ a.
(8.2.12)
Consider now equation (8.2.5) in C(D) with an arbitrary continuous f satisfying (8.2.1). Any u ∈ C(D) which solves (8.2.5), solves (8.2.2)(8.2.3), and therefore satisfies (8.2.12) and belongs to H02 (D). This u solves problem (8.2.2)–(8.2.3) with f replaced by F , defined in (8.2.4), and vice versa. Since F is a bounded nonlinearity, equation (8.2.5) and problem (8.2.2)–(8.2.3) (with f replaced by F ) has a solution by Lemma 8.2.1. Theorem 8.2.1 is proved. 2 Let us sketch an alternative derivation of the inequality (8.2.12) using the maximum principle. Let us derive (8.2.10). The derivation of (8.2.11) is similar.
8.2. EXISTENCE OF A SOLUTION TO A NLE
139
Assume that (8.2.10) fails. Then u > a at some point in D. Therefore at a point y, at which u attains its maximum value, one has u(y) ≥ u(x) for all x ∈ D and u(y) > a. The function u attains its maximum value, which is positive, at some point in D, because u is continuous, vanishes at the boundary of D, and is positive at some point of D by the assumption u > a. At the point y, where the function u attains its maximum, one has −∆u ≥ 0 and k 2 u(y) > 0. Moreover, f (u(y)) > 0 by the assumption (8.2.1), since u(y) > a. Therefore the left-hand side of equation (8.2.2) is positive, while its left-hand side is zero. Thus, we have got a contradiction, and estimate (8.2.10) is proved. Similarly one proves estimate (8.2.11). Thus, (8.2.12) is proved. 2
DSM for operators satisfying a spectral assumption In this Chapter we introduce a spectral assumption (8.1.1) and obtain some results based on this assumption.
8.1
Spectral assumption
In this Chapter we assume that the operator equation (7.2.1) is solvable, 2 y is its solution, possibly non-unique, F ∈ Cloc , and the linear operator 0 A = F (u) has the set {z : |argz − π| ≤ ϕ0 , 0 < |z| < r0 } consisting of regular points of F 0 , where ϕ0 > 0 and r0 > 0 are arbitrary small fixed numbers, and u ∈ H is arbitrary. This is a spectral assumption on F . If this condition is satisfied then c0 ||(A + εI)−1 || ≤ , 0 < ε ≤ r0 , (8.1.1) ε where c0 = sin1ϕ0 . Condition (8.1.1) is much weaker than the assumption of monotonicity 1 of F . If F is monotone, then A = F 0 (u) ≥ 0 and ||A−1 ε || ≤ ε for all ε > 0, where Aε = A + εI. If c0 = 1 and r0 = ∞ in (8.1.1), then A is a generator of a semigroup of contractions ([P]). It is known that if F is monotone, hemicontinuous, and ε > 0, then the equation F (u) + εu = f,
ε = const > 0,
(8.1.2)
is uniquely solvable for any f ∈ H. We want to prove a similar result assuming (8.1.1). Our first result is the following. 133
134
8. DSM FOR OPERATORS SATISFYING A SA
Theorem 8.1.1. Assume that F satisfies conditions (1.3.2) and (8.1.1). Then equation (8.1.2) has a solution. Proof of Theorem 8.1.1. From our assumptions it follows that the problem u˙ = −A−1 ε [F (u) + εu − f ],
u(0) = u0 ,
(8.1.3)
is locally solvable. We will prove the uniform bound sup ||u(t)|| ≤ c,
(8.1.4)
t≥0
where c = const > 0 does not depend on t and the supremum is over all t for which u(t), the local solution to (8.1.3), does exist. If (8.1.4) holds, then, as we have proved in Section 2.6, Lemma 2.6.1, the local solution is a global one, it exists on [0, ∞). We also prove that there exists u(∞), and that u(∞) solves (8.1.2). Let us prove estimate (8.1.4). Consider the function g(t) := ||F (u(t)) + εu(t) − f ||.
(8.1.5)
We have g g˙ = Re([F 0 (u(t)) + ε]u, ˙ F (u) + εu − f ) = −g 2 .
(8.1.6)
g(t) = g0 e−t ,
(8.1.7)
Thus g0 := g(0).
From (8.1.1), (8.1.3) and (8.1.7) we get c0 g0 e−t . ε This implies the existence of u(∞) and the estimates ||u|| ˙ ≤
(8.1.8)
c0 g0 c0 g0 e−t , ||u(t) − u(∞)|| ≤ . (8.1.9) ε ε For any fixed u0 and ε > 0, one can take R > 0 such that c0 g0 ≤ R, ε so that the trajectory u(t) stays in the ball B(u0 , R). Passing to the limit t → ∞ in (8.1.7) yields ||u(t) − u(0)|| ≤
F (u(∞)) + εu(∞) − f = 0.
(8.1.10)
Therefore u(∞) solves equation (8.1.2), so this equation is solvable. Moreover, the DSM (8.1.3) converges to a solution u(∞) at an exponential rate, see (8.1.9). Theorem 8.1.1 is proved. 2
8.2. SPECTRAL ASSUMPTION
135
Remark 8.1.1. Estimate (8.1.1) follows from the spectral assumption because of the known estimate of the norm of the resolvent of a linear operator. Namely, if A is a linear operator and (A − zI)−1 is its resolvent, where z is a complex number in the set of regular points of the operator A. A point z is called a regular point of A if the operator A − zI has a bounded inverse defined on the whole space. Otherwise z is called a point of spectrum σ of A. The estimate we have mentioned is: ||(A − zI)−1 || <
1 , ρ(z, σ)
(8.1.11)
where ρ(z, σ) is the distance (on the complex plane C) from the point z to the set σ of points of spectrum of A. To check this, consider the function (A − zI − µI)−1 = (A − zI)−1 [I − µ(A − zI)−1 ]−1 . If |µ| ||(A − zI)−1 || < 1, then the above function is an analytic function of µ. Therefore, if µ is smaller than the distance from z to the nearest point of spectrum of A, then the point z + µ is a regular point of A. Thus (8.1.11) follows. Remark 8.1.2. We have proved in Theorem 8.1.1 that for ε ∈ (0, r0 ) equation (8.1.2) is solvable. This does not imply that the limiting equation F (u) = f,
(8.1.12)
is solvable. For example, the equation eu + εu = 0 is solvable in R for any ε > 0, but the limiting equation eu = 0 has no solutions. Therefore it is of interest to study the following problem: If the limiting equation (8.1.12) is solvable, then what are the conditions under which the solution to equation (8.1.2), or a more general equation F (uε ) + ε(uε − z) = f,
(8.1.13)
where z ∈ H is some element, converges to a solution to (8.1.12) as ε → 0? This question is discussed in Chapter 9.
136
8.2
8. DSM FOR OPERATORS SATISFYING A SA
Existence of a solution to a nonlinear equation
Let D ⊂ R3 be a bounded domain with Lipschitz boundary S, k = const > 0 , and f : R → R be a function such that uf (u) ≥ 0,
for |u| ≥ a ≥ 0,
(8.2.1)
where a is an arbitrary nonnegative fixed number. We assume that f is continuous in the region |u| ≥ a, and bounded and piecewise-continuous with at most finitely many discontinuity points uj , such that f (uj + 0) and f (uj − 0) exist, in the region |u| ≤ a. Consider the problem (−∆ + k 2 )u + f (u) = 0
in D,
(8.2.2)
u = 0 on S.
(8.2.3)
There is a large literature on problems of this type. Usually it is assumed that f does not grow too fast or f is monotone (see, e.g., [B] and references therein). The novel point in this Section is the absence of the monotonicity restrictions on f and of the growth restrictions on f as |u| → ∞, except for the assumption (8.2.1). This assumption allows an arbitrary behavior of f inside the region |u| ≤ a, where a ≥ 0 can be arbitrary large, and an arbitrarily rapid growth of f to +∞ as u → +∞, or arbitrarily rapid decay of f to −∞ as u → −∞. Our result is: Theorem 8.2.1. Under the above assumptions problem (8.2.2)–(8.2.3) has ◦
a solution u ∈ H 2 (D) ∩ H1 (D) := H02 (D). ◦
Here H ` (D) is the usual Sobolev space, H1 (D) is the closure of C0∞ (D) in the norm H 1 (D). Uniqueness of the solution does not hold without extra assumptions. The ideas of our proof are: first, we prove that if sup |f (u)| ≤ µ, u∈R
then a solution to (8.2.2)–(8.2.3) exists by the Schauder’s fixed-point theorem (see Section 16, Theorem 16.8.1). Here µ is a constant. Secondly, we prove an a priori bound kuk∞ ≤ a.
8.2. EXISTENCE OF A SOLUTION TO A NLE If this bound is proved, f replaced by f (u), F (u) := f (a), f (−a),
137
then problem (8.2.2)–(8.2.3) with the nonlinearity |u| ≤ a u≥a u ≤ −a
(8.2.4)
has a solution, and this solution solves the original problem (8.2.2)–(8.2.3). The bound kuk∞ ≤ a is proved by using some integral inequalities. An alternative proof of this bound is also given. This proof is based on the maximum principle for elliptic equation (8.2.2). We use some ideas from [R9]. Our presentation follows [R56]. Proof of Theorem 8.2.1. If u ∈ L∞ := L∞ (D), then problem (8.2.2)–(8.2.3) is equivalent to the integral equation: Z u=− G(x, y)f (u(y))dy := T (u), (8.2.5) D
where (−∆ + k 2 )G = −δ(x − y)
in D,
g |x∈S = 0.
(8.2.6)
x, y ∈ D.
(8.2.7)
By the maximum principle, 0 ≤ G(x, y) < g(x, y) :=
e−k|x−y| , 4π|x − y|
The map T is a continuous and compact map in the space C(D) := X, and Z Z e−k|x−y| e−k|y| µ kukC(D) := kuk ≤ µ sup dy ≤ µ dy ≤ 2 . (8.2.8) 4π|x − y| 4π|y| k 3 x D R This is an a priori estimate of any bounded solution to (8.2.2)–(8.2.3) for a bounded nonlinearity f such that sup |f (u)| ≤ µ. u∈R
Thus, Schauder’s fixed-point theorem yields the existence of a solution to (8.2.5), and consequently to problem (8.2.2)–(8.2.3), for bounded f . Indeed, if B is a closed ball of radius kµ2 , then the map T maps this ball into itself by (8.2.8), and since T is compact, the Schauder principle is applicable. Thus, the following lemma is proved.
138
8. DSM FOR OPERATORS SATISFYING A SA
Lemma 8.2.1. If supu∈R |f (u)| ≤ µ, then problems (8.2.5) and (8.2.2)– (8.2.3) have a solution in C(D), and this solution satisfies estimate (8.2.8). Let us now prove an a priori bound for any solution u ∈ C(D) of the problem (8.2.2)–(8.2.3) without assuming that supu∈R |f (u)| < ∞. Let u+ := max(u, 0). Multiply (8.2.2) by (u − a)+ , integrate over D and then by parts to get Z 0= [∇u · ∇(u − a)+ + k 2 u(u − a)+ + f (u)(u − a)+ ]dx, (8.2.9) D
where the boundary integral vanishes because (u − a)+ = 0
on
S
for
a ≥ 0.
Each of the terms in (8.2.9) is nonnegative, the last one due to (8.2.1). Thus (8.2.9) implies u ≤ a.
(8.2.10)
Similarly, using (8.2.1) again, and multiplying (8.2.2) by (−u − a)+ , one gets −a ≤ u.
(8.2.11)
We have proved: Lemma 8.2.2. If (8.2.1) holds, then any solution u ∈ H02 (D) to (8.2.2)– (8.2.3) satisfies the inequality |u(x)| ≤ a.
(8.2.12)
Consider now equation (8.2.5) in C(D) with an arbitrary continuous f satisfying (8.2.1). Any u ∈ C(D) which solves (8.2.5), solves (8.2.2)(8.2.3), and therefore satisfies (8.2.12) and belongs to H02 (D). This u solves problem (8.2.2)–(8.2.3) with f replaced by F , defined in (8.2.4), and vice versa. Since F is a bounded nonlinearity, equation (8.2.5) and problem (8.2.2)–(8.2.3) (with f replaced by F ) has a solution by Lemma 8.2.1. Theorem 8.2.1 is proved. 2 Let us sketch an alternative derivation of the inequality (8.2.12) using the maximum principle. Let us derive (8.2.10). The derivation of (8.2.11) is similar.
8.2. EXISTENCE OF A SOLUTION TO A NLE
139
Assume that (8.2.10) fails. Then u > a at some point in D. Therefore at a point y, at which u attains its maximum value, one has u(y) ≥ u(x) for all x ∈ D and u(y) > a. The function u attains its maximum value, which is positive, at some point in D, because u is continuous, vanishes at the boundary of D, and is positive at some point of D by the assumption u > a. At the point y, where the function u attains its maximum, one has −∆u ≥ 0 and k 2 u(y) > 0. Moreover, f (u(y)) > 0 by the assumption (8.2.1), since u(y) > a. Therefore the left-hand side of equation (8.2.2) is positive, while its left-hand side is zero. Thus, we have got a contradiction, and estimate (8.2.10) is proved. Similarly one proves estimate (8.2.11). Thus, (8.2.12) is proved. 2