Akylbek Kerimbekov

Elmira Abdyldaeva

On the Solvability of a Nonlinear Tracking Problem under Boundary Control for the Elastic Oscillations Described by Fredholm Integro-Differential Equations

Keywords: Weak generalized solution, boundary control, functional, maximum principle, nonlinear integral equation, optimization

In the present paper we investigate nonlinear tracking problem under boundary control for the oscillation processes described by Fredholm integrodifferential equations. When we investigate this problem we use notion of a weak generalized solution of the boundary value problem. Based on the maximum principle for distributed systems we obtain optimality conditions from which follow the nonlinear integral equation of optimal control and the differential inequality. We have developed an algorithm to construct the optimization problem solution. This solving method of a nonlinear tracking problem is constructive and can be used in applications.

Introduction

With the emergence the theory control for the systems with distributed parameters, a lot of applied problems described by integral-partial differential equations, integral equations, differential and integral-functional equations ( [START_REF] Volterra | Theory of functionals and of integral and integro-differential equations[END_REF], chapter 5, P. 193-197), [START_REF] Vladimirov | Mathematical problems of speed transport theory of particles[END_REF], ( [START_REF] Richtmyer | Principles Of Advanced Mathematical Physics[END_REF], chapter 16, P. 410-414), ([4], chapter 1, P. 30-76), became investigate by methods of optimal control theory ( [START_REF] Egorov | Optimal control of thermal and diffusion processes[END_REF], chapter 6, P. 356-383), ([5], chapter 2, P. 45-78), ( [START_REF] Lions | Contrle optimal de systmes gouverns par des equations aux derives partielles[END_REF], chapter 4, P. 281-309), [START_REF] Kerimbekov | On solvability of the nonlinear optimal control problem for processes described by the semi-linear parabolic equations[END_REF]- [START_REF] Kerimbekov | On the Solvability of a Nonlinear Optimal Control Problem for the Thermal Processes Described by Fredholm Integro-Differential Equations. // Current Trends in Analysis and Its Applications[END_REF]. However, the control problems described by the integral-differential equations are little learned. In this paper we investigate the boundary tracking control problem for the elastic oscillations described by the partial Fredholm integral-differential equations in partial derivatives. This problem has a number of specific properties: according to the method of [START_REF] Plotnikov | The energy inequality and the over-determination property of systems of eigenfunction[END_REF] the generalized solution of the problem is built by the solving of countable number of integral equations; the optimal control simultaneously satisfies the two relations in the form of equality and inequality, where the relation in the form of equality leads to a nonlinear integral equation, and the relation in the form of inequality is a differential with regards to the function of the external source.

The sufficient conditions for the unique solvability of specific problems were found, and algorithm was indicated for constructing solutions of nonlinear optimization problems with arbitrary precision in the form of the triple u 0 (t),V 0 (t, x), J[u 0 (t)] , where u 0 (t) is the optimal control, V 0 (t, x) is the optimal process, J[u 0 (t)] is the functional's minimum value.

Formulation of the optimal control problem and optimality conditions

We consider the optimization problem where it is required to minimize the integral functional

J[u(t)] = T 0 Q [V (t, x) -ξ (t, x)] 2 dxdt + 2β T 0 M[t, u(t)]dt, β > 0 (1)
on the set of solutions of the boundary value problem

V tt -AV = λ T 0 K(t, τ)V (τ, x)dτ + g(t, x), x ∈ Q ⊂ R n , 0 < t ≤ T, (2)
V (0, x) = ψ 1 (x), V t (0, x) = ψ 2 (x), x ∈ Q, (3)
Γ V (t, x) = n ∑ i, j=1 a i, j (x)V x j (t, x)cos(δ , x i ) + a(x)V (t, x) = = b(t, x) f [t, u(t)], x ∈ γ, 0 < t < T. ( 4
)
Here A is the elliptic operator defined by the formula

AV (t, x) = n ∑ i, j=1 a i, j (x)V x j (t, x) x i -c(x)V (t, x), a i, j (x) = a j,i (x), n ∑ i, j=1 a i, j (x)α i α j ≥ c 0 n ∑ i=1 α 2 i , c 0 > 0 δ is a normal vector, emanating from the point x ∈ γ; K(t, τ) is a given function defined in the region D = {0 ≤ t ≤ T, 0 ≤ τ ≤ T } and satisfying the condition T 0 T 0 K 2 (t, τ)dtdτ < K 0 < ∞, i.e. K(t, s) ∈ H(D); ψ 1 (x) ∈ H 1 (Q), ψ 2 (x) ∈ H(Q), f u [t, u(t)] = 0, ∀t ∈ (0, T ), ξ (t, x) ∈ H(Q T ), M[t, u(t)] ∈ H(0, T ), Q T = (Q × T ), (5)
are given functions, a(x) ≥ 0, c(x) ≥ 0 are known measurable functions; H(X) is Hilbert space of functions defined on the set of X ; H 1 (X) is the first order Sobolev space; f u [t, u(t)] is the function of boundary source which nonlinearly depends on the control function u(t) ∈ H(0, T ) and it is an element of H(0, T ); λ is a parameter, T is a fixed moment of time; M u [t, u(t)] = 0 and satisfies the Lipschitz condition with respect to functional argument u(t) ∈ H(0, T ) . This problem is to find a control u 0 (t) ∈ H(0, T ), for which the appropriate solution V 0 (t, x) of the boundary value problem ( 2) -( 4) deviates little from the given trajectory ξ (t, x) ∈ H(Q T ) during the entire time t ∈ [0, T ] of the control.

At the same time u 0 (t) is called optimal control and V 0 (t, x) is the optimal process. We are looking for a solution of the boundary value problem ( 2)-( 4) in the form of the series

V (t, x) = ∞ ∑ n=1 V n (t)z n (x), (6)
where z n (x) are generalized eigenfunctions of the boundary value problem [START_REF] Plotnikov | The energy inequality and the over-determination property of systems of eigenfunction[END_REF]

D n (Φ, z n ) = Q n ∑ i, j=1 a i, j (x)Φ x j z nx i + c(x)z n (x)Φ(t, x) dx + γ a(x)z n (x)Φ(t, x)dx = = λ 2 n Q z n (x)Φ(t, x)dx; Γ z n (x) = 0, x ∈ γ, 0 < t < T, n = 1, 2, ...,
and they form complete orthonormal system in the Hilbert space H(Q), and the corresponding eigenvalues λ n satisfy the following conditions

λ n ≤ λ n+1 , ∀n = 1, 2, 3, ..., lim n→∞ λ n = ∞.
The Fourier coefficients V n (t) for each fixed n = 1, 2, 3, ..., satisfy the linear nonhomogeneous Fredholm integral equation of the second type

V n (t) = λ T 0 K n (t, s)V n (s)ds + a n (t), (7)
where

K n (t, s) = 1 λ n t 0 sinλ n (t -τ)K(τ, s)dτ, a n (t) = ψ 1n cosλ n t + ψ 2n λ n sinλ n t + 1 λ n t 0 sinλ n (t -τ)[q n (τ) + b n (τ) f [τ, u(τ)]]dτ. ( 8
)
The solution of equation ( 7) we find ( [START_REF] Krasnov | Integral equations[END_REF],chapter 2, P. 98-110) by the following formula

V n (t) = λ T 0 R n (t, s, λ )a n (s)ds + a n (t), (9)
where the resolvent R n (t, s, λ ) of the kernel K n (s,t) is given by

R n (t, s, λ ) = ∞ ∑ i=1 λ i-1 K n,i (t, s), K n,1 (t, s) = K n (t, s)
and the iterated kernels K n,i (t, s) for each n = 1, 2, ... are defined by the formulas

K n,i+1 (t, s) = T 0 K n (t, η)K n,i (η, s)dη, i = 1, 2, 3, ..., K n,1 (t, s) = K n (t, s). Resolvent R n (t, s, λ ) is a continuous function when |λ | < λ 1 T √ K 0
and satisfy the following estimate

T 0 R 2 n (t, s, λ )ds ≤ T K 0 λ n -|λ |T √ K 0 2 . (10)
Further, taking into account ( 8) and ( 9) solution of the boundary value problem ( 2)-( 5) can be written as

V (t, x) = ∞ ∑ n=1 (ψ n (t, λ ) + 1 λ n T 0 ε n (t, η, λ )b n (η) f (η, u(η))dη)z n (x), (11)
where

ψ n (t, λ ) = ψ 1n (cos λ n t + λ T 0 R n (t, s, λ ) cos λ n sds) + + ψ 2n λ n (sin λ n t + λ T 0 R n (t, s, λ ) sin λ n sds) + 1 λ n T 0 ε n (t, η, λ )g n (η)dη, (12)
ε n (t, η, λ ) =        sin λ n (t -η) + T η R n (t, s, λ )sinλ n (s -η)ds, 0 ≤ η ≤ t, λ T η R n (t, s, λ ) sin λ n (s -η)ds, t ≤ η ≤ T. (13)
The function [START_REF] Krasnov | Integral equations[END_REF] is an element of Gilbert space H(Q T ) and weak generalized solution of boundary problem (2) -( 5). According to condition (5) each control u(t) uniquely defines the controlled process V (t, x). Therefore for the solution V (t, x) + ∆V (t, x) of boundary value problem (2) -( 4) corresponds the control u(t) + ∆ u(t), where ∆V (t, x) is the increment corresponding to the increment ∆ u(t). By the method of to the maximum principle ( [START_REF] Egorov | Optimal control of thermal and diffusion processes[END_REF], chapter 6, P. 356-383), ([5], chapter 2, P. 45-78) the increment of functional (1) can be written as

∆ J[u] = J[u + ∆ u] = - T 0 ∆ Π [t,V (t, x), ω(t, x), u(t)]dt + T 0 Q ∆V 2 (t, x)]dxdt;
where

Π [t,V (t, x), ω(t, x), u(t)] = γ ω(t, x)b(t, x)dx f [t, u(t)] -2β M[t, u(t)],
and the function ω(t, x) is the solution of the adjoint boundary value problem

ω tt -Aω = λ T 0 K(t, τ)ω(τ, x)dτ -2[V (t, x) -ξ (t, x)], x ∈ Q, 0 < t < T, ω(T, x) = 0, ω t (T, x) = 0, x ∈ Q, (14) ω)(t, x) = 0, x ∈ γ, 0 < t < T.
According to the maximum principle for systems with distributed parameters ( [START_REF] Egorov | Optimal control of thermal and diffusion processes[END_REF], chapter 6, P. 356-383), ([5], chapter 2, P. 45-78), the optimal control is determined by the relations

2β M u [t, u(t)] f u [t, u(t)] = ∞ ∑ n=1 b n (t)ω n (t), (15)
f u [t, u(t)] M u [t, u(t)] f u [t, u(t)] u > 0, (16)
which are called optimality conditions.

Solution of the adjoint boundary-value problem

We are looking for a solution of the adjoint boundary value problem (14) in the form of the series

ω(t, x) = ∞ ∑ n=1 ω n (t)z n (x). (17)
The Fourier coefficients ω n (t) for each fixed n = 1, 2, 3, ..., satisfy the linear nonhomogeneous Fredholm integral equation of the second type

ω n (t) = λ T 0 B n (s,t)ω n (s)ds - 2 λ n T t sinλ n (τ -t)[V n (τ) -ξ n (τ)]dτ, (18)
where

B n (s,t) = 1 λ n T t sinλ n (τ -t)K(s, τ)dτ.
The solution of equation (18) we find ( [START_REF] Krasnov | Integral equations[END_REF],chapter 2, P. 98-110) by the following formula

ω n (t) = λ T 0 P n (s,t, λ ) - 2 λ n T s sinλ n (τ -s)[V n (τ) -ξ n (τ)]dτ ds- - 2 λ n T t sinλ n (τ -t)[V n (τ) -ξ n (τ)]dτ, (19)
where the resolvent P n (s,t, λ ) of the kernel B n (s,t) is given by

P n (s,t, λ ) = ∞ ∑ i=1 λ i-1 B n,i (s,t), B n,1 (s,t) = B n (s,t), B n,i+1 (s,t) = T 0 B n (η,t)B n,i (s, η)dη, i = 1, 2, 3, ...

and it is continuous function when

|λ | < λ 1 T √ K 0
and satisfy the following estimate

T 0 P 2 n (s,t, λ )dτ ≤ T K 0 λ n -|λ |T √ K 0 2 . ( 20
)
Further, taking into account ( 17) and ( 19) solution of the adjoint boundary value problem can be written as

ω(t, x) = -2{-h(t, x) + E(t, x)}, (21)
where

h(t, x) = ∞ ∑ n=1 1 λ n T 0 b n (t)E n (s,t, λ )l n (τ, λ )dτz n (x), E(t, x) = ∞ ∑ n=1 1 λ n T 0 T 0 b n (t)E n (t, τ, λ )ε n (τ, η, λ )b n (η)dτ f (η, u(η))dηz n (x), E n (t, τ, λ ) =        λ τ 0 1 λ n P n (s,t, λ )sinλ n (τ -s)ds, 0 ≤ τ ≤ t, 1
λ n sinλ n (τ -t) + λ τ 0 1 λ n P n (s,t, λ )sinλ n (τ -s)ds, t ≤ τ ≤ T, l n (t, λ ) = ξ n (t) -ψ n (t, λ ).
By means of the direct calculations we have proved the following lemmas:

Lemma 1. The function h(t, x) is an element of the space H(Q T ) . Lemma 2. Function E(t, x) is an element of H(Q T ).
Based on the Lemma 1 and Lemma 2 from (21) it follows that solution of adjoint boundary value problem ( 14) ω(t, x) is an element of the Hilbert space H(Q T ).

Nonlinear integral equation of optimal control

We find the optimal control according to optimality conditions (15) and ( 16). We substitute in (15) the solution of adjoint boundary-value problem (14) defined by (21). We rewrite the equality (15) in the form of

β M u (t, u) f u (t, u) + ∞ ∑ n=1 1 λ n b n (t) T 0 T 0 E n (t, τ, λ )ε n (τ, η, λ )dτb n (η) f (η, u(η))dη = = ∞ ∑ n=1 1 λ n b n (t) T 0 E n (t, τ, λ )l n (τ, λ )dτ. (22)
Thus, the optimal control is defined as the solution of a nonlinear integral equation ( 22) and at the same time ( 15) and ( 16) should be carried out. Condition (5) restricts the class of functions f [t, u(t)] of external influences. Therefore, we assume that the function f [t, u(t)] satisfies (16) for any control u(t) ∈ H(0, T ), i.e. the optimization problem is considered in class { f (t, u(t))} of functions satisfying (16). Nonlinear integral equation ( 22) is solved according to the procedure of work [START_REF] Kerimbekov | On solvability of the nonlinear optimal control problem for processes described by the semi-linear parabolic equations[END_REF], [START_REF] Kerimbekov | On the Solvability of a Nonlinear Optimal Control Problem for the Thermal Processes Described by Fredholm Integro-Differential Equations. // Current Trends in Analysis and Its Applications[END_REF]. We set

β M u (t, u) f u (t, u) = p(t). (23)
According to condition (16) control function u(t) is uniquely determined from equality (23), i.e. there is a such function ϕ that ( [START_REF] Lusternik | Elements of functional analysis[END_REF], chapter 8, P. 467-480)

u(t) = ϕ(t, p(t), β ). (24)
By ( 23) and ( 24) we rewrite the equation ( 22) in the operator form

p(t) + G[p, λ ] = h(t, λ ) (25)
where

G[p, λ ] = ∞ ∑ n=1 1 λ n b n (t) T 0 T 0 E n (t, τ, λ )ε(τ, η, λ )dτ × ×b n (η) f [η, ϕ(η, p(η), β )]dη, h(t, λ ) = ∞ ∑ n=1 1 λ n b n (t) T 0 E n (t, τ, λ )l n (τ, λ )dτ.
Now we investigate the questions of unique solvability of the operator equation ( 25).

Lemma 3. The function p(t) is an element of the space H(0, T ).

Proof. By (23) we have the estimate

sup M u (t, u) f u (t, u) ≤ N, ∀t ∈ [0, T ].
Since u(t) ∈ H(0, T ), the statement of the lemma follows by the inequality

T 0 |p(t)| 2 dt ≤ β 2 T 0 M u (t, u) f u (t, u) 2 dt ≤ β 2 N 2 T 2 < ∞.
Lemma 4. The operator G[p(t)] which defined by the formula (25) maps the space H(0,T) into itself, i.e. it is an element of the space H(0, T ) .

Proof. Taking into account the following estimations

T 0 ε 2 n (t, η, λ )dη ≤ 2T 1 + λ 2 T 2 K 0 λ 1 -|λ | T 2 K 0 2 , T 0 E 2 n (t, τ, λ )dτ ≤ 2T λ 2 n 1 + λ 2 T 2 K 0 λ 1 -|λ | T 2 K 0 2 ,
we obtain the assertion of lemma from following inequality Proof. The proof of this theorem follows from Lemma 4 by the following inequality, i.e. the following inequality is fulfilled

T 0 G 2 [p, λ ] = T 0 ∞ ∑ n=1 1 λ n b n (t) T 0 T 0 E n (t, τ, λ )ε n (τ, η, λ )dτ × ×b n (η) f [η, ϕ(η, p(η), β )]dη 2 dt ≤ T 0 ∞ ∑ n=1 1 λ 2 n b 2 n (t)× × ∞ ∑ n=1 T 0 T 0 E n (t, τ, λ )ε n (τ, η, λ )dτb n (η) f [η, ϕ(η, p(η), β )]dη 2 dt ≤ ≤ T 0 ∞ ∑ n=1 1 λ 2 n b 2 n (t) ∞ ∑ n=1 T 0 T 0 E 2 n (t, τ, λ )dτ T 0 ε 2 n (τ, η, λ )dτb 2 n (η)dη× × T 0 f 2 [η, ϕ(η, p(η), β )]dηdt ≤ T 0 ∞ ∑ n=1 1 λ 2 n b 2 n (t)× × ∞ ∑ n=1 T 0 2T λ 2 n      1 + λ 2 T 2 K 0 λ 1 -|λ | T 2 K 0 2      2T      1 + λ 2 T 2 K 0 λ n -|λ | T 2 K 0 2      b 2 n (η)dη× × T 0 f 2 [η, ϕ(η, p(η), β )]dηdt ≤ 1 λ 2 1 T 0 ∞ ∑ n=1 b 2 n (t)dt 2 (2T ) 2 λ 2 1 × ×      1 + λ 2 T 2 K 0 λ 1 -|λ | T 2 K 0 2      2 f [η, ϕ(η, p(η), β )] 2 H(0,T ) < ∞.
G[p, λ ] -G[p, p, λ ] 2 H ( 0,T ) ≤ C 2 0 f [t, u(t)] -f [t, ũ(t)] 2 H ( 0,T ) ≤ ≤ C 2 0 f 2 0 ϕ[t, p(t), β ] -ϕ[t, p(t), β ] 2 H ( 0,T ) ≤ C 2 0 f 2 0 ϕ 2 0 (β ) p(t) -p(t) 2 H ( 0,T ) .
Theorem 1. Suppose that conditions (5), lemma 5 and |λ | < λ 1 T √ K 0 are satisfied. Then operator equation (25) has a unique solution in the space H(0, T ).

Lemma 5 .λ 2 T 2 K 0 λ 1 2 b

 522012 Suppose that the conditionsf [t, u(t)]f [t, ū(t)] H(0,T ) ≤ f 0 u(t)ū(t) H(0,T ) , f 0 > 0, ϕ[t, p(t), β ]ϕ[t, p(t), β ] H(0,T ) ≤ φ 0 (β ) p(t)p(t) H(0,T ) , ϕ 0 (β ) > 0 are satisfied. Then if the condition γ = C 0 f 0 ϕ 0 (β ) < 1, is met, the operator G[p, λ ] is contractive. Here C -|λ | T 2 K 0 (t, x) 2H(γ T ) .

Proof. According to Lemmas 3 and 4, operator equation (25) can be considered in the space H(0, T ). According to Lemma 5 operator G(p) is contractive. Since the Hilbert space H(0, T ) is a complete metric space, by the theorem on principle of contracting mappings ( [START_REF] Lusternik | Elements of functional analysis[END_REF], chapter 1, P. 43-53) the operator G(p) has a unique fixed point, i.e. operator equation (25) has unique solution.

The solution of operator equation (25) can be found by the method of successive approximations, i.e. n th approximation of the solution is found by the formula

where p 0 (t) is an arbitrary element of the space H(0, T ). For the exact solution p(t) = lim n→∞ p n (t) we have the following estimate

or when h = p 0 (t)

where 0 < γ < 1 is the contraction constant. The exact solution can be found as the limit of the approximate solutions, i.e. substituting this solution in (24) we find the optimal control u 0 (t) = ϕ[t, p(t), β ].

We find the optimal process V 0 (t, x), i.e. the solution of boundary value problem ( 2)-( 4), corresponding to the optimal control u 0 (t, x), according to (6) from the formula

The minimum value of the functional (1) is calculated by the formula

The found triple (u 0 (t),V 0 (t, x), J[u 0 (t)] is a solution of the nonlinear optimization problem.


Alberto Paganini

Ralf Hiptmair

Approximate Riesz Representatives of Shape Gradients

Keywords:

We study finite element approximations of Riesz representatives of shape gradients. First, we provide a general perspective on its error analysis. Then, we focus on shape functionals constrained by elliptic boundary value problems and H 1 -representatives of shape gradients. We prove linear convergence in the energy norm for linear Lagrangian finite element approximations. This theoretical result is confirmed by several numerical experiments.

Introduction

A shape functional is a real map defined on a set of admissible shapes. The goal of shape optimization is to modify an initial shape so that a shape functional attains an extremal value. A common approach is to employ steepest descent algorithms [START_REF] Allaire | Conception optimale de structures[END_REF]Ch. 3.4]. Shapes may be parameterized by C 1 -mappings acting on reference configurations. Then the shape gradient is a linear continuous operator on the non-reflexive Banach space C 1 , and the concept of steepest descent may not be well-defined; see [START_REF] Hinze | Optimization with PDE constraints[END_REF]Page 103]. A compromise is to replace "steepest descents" with Riesz representatives of shape gradients with respect to a Hilbert space X. Henceforth, we refer to these representatives as X-representatives.

After recalling basic definitions of shape calculus, we provide a general perspective on error analysis in the energy norm for finite element approximations of X-representatives of shape gradients. Then, we zero in on shape functionals constrained to elliptic boundary value problems. For this case, insight into shape Hessians [START_REF] Eppler | Shape optimization for free boundary problemsanalysis and numerics[END_REF][START_REF] Schulz | A novel Steklov-Poincaré type metric for efficient PDE constrained optimization in shape spaces[END_REF] suggests to select representatives of shape gradients with respect to X = H 1 0 (D), where D is a hold-all domain that encloses the initial guess Ω; see Figure 1. For the choice X = H 1 0 (D), it is natural to consider discretization by means of linear Lagrangian finite elements [START_REF] Murai | Error analysis of H1 gradient method for shapeoptimization problems of continua[END_REF][START_REF] Schulz | A novel Steklov-Poincaré type metric for efficient PDE constrained optimization in shape spaces[END_REF][START_REF] Laurain | Domain expression of the shape derivative and application to electrical impedance tomography[END_REF]. We show that linear Lagrangian finite element approximations of H 1 -representatives of shape gradients converge linearly with respect to the mesh width. Additionally, this convergence rate does not deteriorate when state and adjoint variables are replaced by linear Lagrangian finite elements solutions. This is an improvement on the result presented in [START_REF] Murai | Error analysis of H1 gradient method for shapeoptimization problems of continua[END_REF], which involves approximations of state and adjoint variables with quadratic finite elements. Finally, we provide numerical evidence of the linear convergence rate predicted. Ω D Fig. 1. The hold-all domain D encloses the domain Ω.

Shape Functionals and Shape Gradients

Let Ω ⊂ R d , d = 2, 3, be an open bounded domain with piecewise smooth boundary ∂Ω, and let J (Ω) ∈ R be a real-valued quantity of interest associated to it. One is often interested in shape sensitivity, which quantifies the impact of small perturbations of ∂Ω on the value J (Ω).

We model perturbations of the domain Ω through maps of the form

T V (x) := x + V(x), x ∈ R d , (1)
where

V is a vector field in C 1 (R d ; R d ). It can easily be proved that the map (1) is a diffeomorphism for V C 1 < 1 [8, Lemma 6.13].
The value J (Ω) is interpreted as the realization of a shape functional, a real map

J : V → J (T V (Ω)) defined on the ball {V ∈ C 1 (R d ; R d ); V C 1 < 1}. Clearly, J (Ω) = J (T 0 (Ω)).
The sensitivity of J (Ω) with respect to the perturbation direction V is given by the Eulerian derivative of the shape functional J in the direction V, that is,

dJ (Ω; V) := lim s 0 J (T s•V (Ω)) -J (Ω) s . (2)
We say that the shape functional is shape differentiable if Formula (2) defines a linear and bounded operator V → dJ (Ω; V). In literature, this operator is called shape gradient [9, Ch. 9, Sect. 3.4]. As mentioned in the introduction, Xrepresentatives of shape gradients can be employed to solve shape optimization problems, that is, to find

Ω * ∈ argmin Ω∈U ad J (Ω) ,
where U ad denotes a set of admissible shapes.

Often, the quantity of interest takes the form

J (Ω) = B α∇(u -g) • ∇(u -g) + β(u -g) 2 dx , (3)
where the state function u is the solution of a boundary value problem stated on Ω, B ⊂ Ω, α and β are two real constants and g is a sufficiently smooth target function. In this work, u ∈ H 1 0 (Ω) is the (weak) solution of the elliptic boundary value problem with homogeneous Dirichlet boundary conditions

-∆u + u = f in Ω , u = 0 on ∂Ω , (4)
that is,

Ω ∇u • ∇v + uv dx = Ω f v dx for all v ∈ H 1 0 (Ω) , (5)
where f ∈ H 1 (Ω). For the sake of brevity, we set g = 0. Then, the shape gradient of the shape functional associated to (3) and constrained to (5) reads [4, Formula (2.9)]

dJ (Ω; V) = Ω (∇f • V)p + ∇u • (DV + DV T )∇p + div V(f p + χ B (α∇u • ∇u + βu 2 ) -∇u • ∇p -up) dx , (6)
where the adjoint function p ∈ H 1 0 (Ω) is the solution of

Ω ∇p • ∇v + pv dx = B α∇u • ∇v + βuv dx for all v ∈ H 1 0 (Ω) . (7)
Formula ( 3) is a prototypical PDE-constrained shape functional. In this work, Formula ( 6) is used as test case for proving convergence estimates and performing numerical experiments.

Remark 1. Formula (6) holds even if homogeneous Dirichlet boundary conditions in (4) are replaced by homogeneous Neumann boundary conditions, in which case the test and the trial spaces in (5) and [START_REF] Hinze | Optimization with PDE constraints[END_REF] are replaced with H 1 (Ω).

Remark 2. For the sake of simplicity, we restrict our considerations to homogeneous boundary conditions. However, we expect that the results of this work hold true for (sufficiently regular) inhomogeneous boundary conditions, too. Note that Formula (6) should be adjusted accordingly; see [4, Section 2].

Error Analysis for Finite Element Representatives

The General Case

Let (•, •) X denote the inner product of a Hilbert space X, and let us assume that the shape gradient dJ is well-defined on X. The X-representative V X of dJ can be computed by solving

(V X , W) X = dJ (Ω; W) for all W ∈ X .
Next, for an index set N , we introduce a family {X n } n∈N of finite-dimensional subspaces of X. Let V Xn n∈N be a sequence of approximate X-representatives of dJ defined by

(V Xn , W n ) X = dJ (Ω; W n ) for all W n ∈ X n . By Cea's Lemma [11, Thm 2.4.1], there exists a constant C > 0 independent of n such that V X -V Xn X ≤ C inf Wn∈Xn V X -W n X . (8)
By and large, the shape gradient of a PDE-constrained shape functional depends also on the state and the adjoint variables u and p. These functions are solutions of boundary value problem. Usually, only numerical approximations u h and p h are available. In that case, the approximate X-representative V Xn has to be replaced with the solution

V Xn h of (V Xn h , W n ) X = dJ h (Ω; W n ) for all W n ∈ X n , (9)
where dJ h is an approximation of the operator dJ obtained by replacing the functions u and p with their numerical approximations u h and p h . By Strang Lemma [11, Thm 4.1.1], the estimate ( 8) should be corrected by adding a consistency term, that is,

V X -V Xn h X ≤ C inf Wn∈Xn V X -W n X + sup Wn∈Xn |dJ (Ω; W n ) -dJ h (Ω; W n )| W n X (10)
for a constant C > 0 independent of n and h.

H 1 -representatives and Linear Lagrangian Finite Elements

A popular approach in shape optimization consists of replacing the initial domain Ω with a polygon/polyhedron equipped with a finite element mesh Ω h . This mesh is used to compute linear Lagrangian finite element approximations of the functions u and p. Then, the coordinates of the mesh nodes are (iteratively) updated according to the shape gradient [START_REF] Allaire | Conception optimale de structures[END_REF]Ch. 6.5]. This is equivalent to extending Ω h to a mesh D h that covers a hold-all domain D and choosing linear Lagrangian finite elements to construct the finite-dimensional subspace X n . Formula (10), standard finite element estimates, and Proposition 1 readily imply that, for this discretization, the approximate H 1 -representative of ( 6) satisfies

V X -V Xn h H 1 (D) = O(h) , (11)
which is the main result of this work.

Proposition 1. Let Ω ⊂ R d be a polyhedral domain, let f ∈ W 1,4 (Ω) in (5), and let us assume that the solution u of (5) satisfies

u W 2,4 (Ω) ≤ C f L 4 (Ω) . (12)
Let (V h ) h∈(0,1] be a family of H 1 0 (Ω)-conforming piecewise linear Lagrangian finite element spaces built on a quasi-uniform family of simplicial meshes (T h ) h∈(0,1] , that is, a family of meshes such that max{diam(T ) :

T ∈ T h } ≤ h diam(Ω) and min{diam(B T ) : T ∈ T h } ≥ ρh diam(Ω) for all h ∈ (0, 1] ,
for a ρ > 0, where B T is the largest ball contained in the simplex T [10, Def. 4.4.13]. Let u h , p h ∈ V h be solutions of

Ω ∇u h • ∇v h + u h v h dx = Ω f v h dx for all v h ∈ V h , (13)
Ω ∇p h • ∇v h + p h v h dx = B α∇u h • ∇v h + βu h v h dx for all v h ∈ V h , ( 14
)
where α, β ∈ R, B ⊂ Ω, and α = 0 or B = Ω, if d = 3. Let dJ h (Ω; W n ) denote the operator defined by Formula (6) with u and p replaced by u h and p h , respectively. Then,

sup Wn∈Xn |dJ (Ω; W n ) -dJ h (Ω; W n )| W n H 1 (D) ≤ C(Ω, f, u, p)h ( 15
)
for a constant C(Ω, f, u, p) > 0 independent of n and h.

Proof. First of all, note that

dJ (Ω; W n )-dJ h (Ω; W n ) = Ω (∇f • W n + f div W n )(p -p h ) dx + Ω ∇u • (DW n + DW T n )∇p -∇u h • (DW n + DW T n )∇p h dx + Ω div W n (∇u h • ∇p h + u h p h -∇u • ∇p -up) dx + B div W n α(∇u • ∇u -∇u h • ∇u h ) + β(u 2 -u 2 h ) dx . ( 16
)
We recall that, for generic functions q 0 ∈ L 2 (Ω) and q 1 , q 2 ∈ L 4 (Ω), the Cauchy-Schwarz inequality implies

q 0 q 1 q 2 L 1 (Ω) ≤ q 0 L 2 (Ω) q 1 q 2 L 2 (Ω) ≤ q 0 L 2 (Ω) q 1 L 4 (Ω) q 2 L 4 (Ω) . (17)
Thus, the first integral in ( 16) may be estimated as follows 1|

Ω (∇f • W n + f div W n )(p -p h ) dx| ≤ C W n H 1 (Ω) f W 1,4 (Ω) p -p h L 4 (Ω) .
The second integral in ( 16) may be estimated as follows

| Ω ∇u • (DW n + DW T n )∇p -∇u h • (DW n + DW T n )∇p h dx| = | Ω ∇(u -u h ) • (DW n + DW T n )∇p + ∇u h • (DW n + DW T n )∇(p -p h ) dx| ≤ C W n H 1 (Ω) u -u h W 1,4 (Ω) p W 1,4 (Ω) + u h W 1,4 (Ω) p -p h W 1,4 (Ω) .
The third and the fourth integral in ( 16) may be estimated similarly.

Stability of the Ritz projection with respect to

W 1,4 (Ω) [3] 2 u h W 1,4 (Ω) ≤ C u W 1,4 (Ω) (18)
implies uu h W 1,4 (Ω) = O(h). To show

p h W 1,4 (Ω) ≤ C p W 1,4 (Ω) , (19)
which in turn implies pp h W 1,4 (Ω) = O(h), it is necessary to repeat the proof of (18) given in [START_REF] Rannacher | Some optimal error estimates for piecewise linear finite element approximations[END_REF] tracking the consistency term

Ω ∇(p-p h )•∇g z h +(p-p h )g z h dx = B α∇(u-u h )•∇g z h +β(u-u h )g z h dx . (20)
The discrete Green's function g z h ∈ V h is given in [START_REF] Rannacher | Some optimal error estimates for piecewise linear finite element approximations[END_REF] and satisfies

g z h H 1 (Ω) = O(h -d/2
) . By the Cauchy-Schwarz inequality and standard finite element estimates,

| B α∇(u -u h ) • ∇g z h + β(u -u h )g z h dx| = O (|α|h + |β|h 2 )h -d/2 . ( 21
)
The stability result (19) holds if (21) is bounded independently of h. For this reason, we need to set α = 0 when d = 3, unless B = Ω. In this latter case, by Galerkin orthogonality, (20) is bounded by (βα)(uu h )g z h L 1 (Ω) .

Remark 3. In Proposition 1, we assume W2,4 -regularity of the solution u of (5). This assumption is made to achieve linear convergence with respect to h in the estimate [START_REF] Laurain | Domain expression of the shape derivative and application to electrical impedance tomography[END_REF]. However, a three-dimensional polyhedral domain must satisfy tight geometric conditions for u to be in W 2,4 [6, Thm 7.1]. Nevertheless, in [START_REF] Guzmán | Hölder estimates for Green's functions on convex polyhedral domains and their applications to finite element methods[END_REF] the authors show W 1,∞ -stability of the Ritz projection for general convex polyhedral domains. Therefore, we expect that (in the latter case) the righthand side of ( 15) can be replaced with a term of order O(h α ), where the rate α depends on the regularity of u and satisfies 0 < α ≤ 1.

Remark 4. In [START_REF] Hiptmair | Comparison of approximate shape gradients[END_REF][START_REF] Paganini | Approximate shape gradients for interface problems[END_REF], the authors show that one can expect superconvergence in the approximation of the shape gradient dJ . In particular, they show that

|dJ (Ω; W) -dJ h (Ω; W)| ≤ C W W 2,4 (Ω) h 2 . ( 22
)
However, in the right-hand side of ( 22) appears the W 2,4 (Ω)-norm of W. But to prove convergence in the approximation of a H 1 -representative of dJ , the upper bound of |dJ (Ω; W) -dJ h (Ω; W)| cannot involve a norm stronger than the H 1 -norm; see Equation [START_REF] Brenner | The mathematical theory of finite element methods[END_REF].

Remark 5. By the Hadamard structure theorem [9, Ch. 9, Thm 3.6], most shape gradients admit representatives g(Ω) in the space of distributions D k (∂Ω), that is,

dJ (Ω; V) = g(Ω), γ ∂Ω V • n D k (∂Ω) , (23)
where γ ∂Ω V • n is the normal component of V on the boundary ∂Ω. For instance, if u, p ∈ H 2 (Ω), Formula ( 6) is equivalent to [4, Formula (2.10)]

dJ (Ω; V) = ∂Ω (V • n) α∇u • ∇u + βu 2 + pu dS . ( 24
)
We advise against the use of g(Ω) (which corresponds to the L 2 (∂Ω)-representative of dJ ) to define descent directions because L 2 -representatives might bristle with undesirable oscillations [START_REF] Hiptmair | Shape optimization by pursuing diffeomorphisms[END_REF].

Numerical Experiments

We provide numerical evidence of the estimate [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF]. We employ linear Lagrangian finite elements on quasi-uniform triangular meshes. The experiments are performed in MATLAB and are partly based on the library LehrFEM developed at ETHZ. Mesh generation and uniform refinement are performed with the functions initmesh and refinemesh of the MATLAB PDE Toolbox [START_REF]MATLAB and Partial Differential Equation Toolbox (R2015a)[END_REF]. The boundary of computational domains is approximated by a polygon, which is generally believed not to affect the convergence of linear finite elements [10, Sect. 10.2]. For domains with curved boundaries, the refined mesh is always adjusted to fit the boundary. Integrals in the domain are computed with a 3-point quadrature rule of order 3 in each triangle and line integrals with a 6-point Gauss quadrature on each segment. We consider three different geometries for the domain Ω (see Figure 2):

1. A disc of radius 6/5 centered in (0.01,0.02). 2. A triangle with corners located at (-6/5, -6/5) , ( 6/5, -6/5) , (-6/5, 6/5) .

3.

A circular sector of radius 6/5 centered in (0.01, 0.02) of angle 0.9 • 2π.

The source function in (5) is f (x, y) = cos(x + π/4) .

The hold-all domain D is a square with edges of length 3 centered in the origin. The region of interest B is the whole domain Ω. We set α = 0 and β = 1 in (3).

Fig. 2. The domain Ω is chosen to be either a disc or a triangle or a sector.

The reference value V X is approximated by computing V Xn h on a mesh with an extra level of refinement. In light or Remark 5, we employ both Formula (6) and Formula (24) to evaluate the right-hand side dJ h in (9). To avoid biased results, we display convergence history of V X -V Xn h H 1 (D) both with self-and cross-comparison.

In Figure 3, we plot the convergence history when the domain Ω is either a disc (first row) or a triangle (second row). As predicted by [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF], we observe linear convergence when the right-hand side in ( 9) is evaluated according to [START_REF] Maz'ya | Weighted Lp estimates of solutions to boundary value problems for second order elliptic systems in polyhedral domains[END_REF]. Interestingly, using Formula (24) seems not to affect the convergence rate. The same behavior is observed when homogeneous Dirichlet boundary conditions are replaced by homogeneous Neumann boundary conditions. Note that, in this latter case, the boundary-integral counterpart of Formula (6) reads [START_REF] Hiptmair | Comparison of approximate shape gradients[END_REF]

dJ (Ω; V) = ∂Ω V • n (∇u • ∇(αu -p) + u(βu -p) + f p) dS . ( 25
)
For the sake of brevity, we omit these plots.

In Figure 4 (first row), we plot the convergence history when the domain Ω is a sector. This domain does not guarantee that u and p are in H 2 (Ω) because it has a re-entrant corner. We observe that the convergence rates decrease to fractional values. This is a consequence of the lower regularity of the functions u and p. Additionally, the convergence rates depend on the formula used to evaluate dJ h . In particular, in the cross-comparison, the convergence line saturates when Formula ( 6) is used. This may be due to a poor accuracy of the reference solution. However, we point out that Formulas ( 6) and (24) may not be equivalent due to the lack of regularity of the functions u and p; cf. Remark 5. Curiously, for homogeneous Neumann boundary conditions, the presence of the re-entrant corner seems to have a milder impact on convergence rates; see Figure 4 (second row). However, note that the approximate algebraic convergence rates of uu h H 1 (Ω) and pp h H 1 (Ω) Disc (Dirichlet BC)

10 -2 10 -1 10 -4 10 -3 1 1 h V X -V Xn h H 1 (D) self-comparison Bdry Vol 10 -2 10 -1 10 -4 10 -3 1 1 h V X -V Xn h H 1 (D)

cross-comparison Bdry Vol

Triangle (Dirichlet BC)

10 -2 10 -1 10 -4 10 -3 10 -2 1 1 h V X -V Xn h H 1 (D) self-comparison Bdry Vol 10 -2 10 -1 10 -4 10 -3 10 -2 1 1 h V X -V Xn h H 1 (D)
cross-comparison Bdry Vol Fig. 3. Convergence history when Ω is a disc (first row ) and a triangle (second row ). Line refers to evaluation of dJ h according to Formula (6); line to Formula (24). We observe the linear convergence rate predicted by [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF].

with respect to h drop to 0.67 and 0.62, respectively. By the Hadamard structure theorem (see Remark 5), vector fields W n associated to interior nodes of the mesh Ω h lie in the kernel of dJ . However, these vector fields are not in the kernel of dJ h because u and p are replaced by finite element approximations. Schulz et al. [START_REF] Schulz | A novel Steklov-Poincaré type metric for efficient PDE constrained optimization in shape spaces[END_REF] report that this numerical error might largely affect the computation of the Riesz representative. Although we have not experienced this issue, we have repeated the numerical experiments by setting to zero the values of dJ h (Ω; W n ) for all W n associated to interior nodes of Ω h . We have not observed any significative difference in the results. Thus, we acknowledge that computational resources might be saved by dropping the evaluation of dJ h (Ω; W n ) for vector fields associated to interior nodes of Ω h .

Circular Sector (Dirichlet BC)

10 -2 10 -1 10 -3 10 -2 1 0.3 h V X -V Xn h H 1 (D) self-comparison Bdry Vol 10 -2 10 -1 10 -2 10 -1 1 0.08 h V X -V Xn h H 1 (D)

cross-comparison

Bdry Vol

Circular Sector (Neumann BC)

10 -2 10 -1 10 -2 10 -1 1 1 h V X -V Xn h H 1 (D)
self-comparison Bdry Vol

10 -2 10 -1 10 -2 10 -1 1 1 h V X -V Xn h H 1 (D)
cross-comparison Bdry Vol Fig. 4. Convergence history when Ω is a Line refers to evaluation of dJ h according to Formula (6); line to Formula (24) (in the first row ) and to Formula (25) (in the second row ). For Dirichlet boundary conditions, convergence rates decay to fractional values.

Conclusion

Most shape optimization algorithms rely on Riesz representatives of shape gradients with respect to a chosen Hilbert space. Numerical discretization is inevitable when the shape functional is constrained to a boundary value problem. Formula (10) indicates how to estimate the discretization error when the Riesz representative is computed on a finite-dimensional trial space and the shape gradient can be evaluated only approximately.

For linear Lagrangian approximations of H 1 -representatives, Proposition 1 implies that the discretization error decays linearly with respect to the mesh width h. This convergence behavior is observed in several numerical experiments.

As a consequence of the Hadamard structure theorem, most shape gradients can be equivalently formulated as boundary or volume integrals. Although Proposition 1 relies on the volume formulation of the shape gradient, we have observed linear convergence independently of the formula employed to evaluate dJ . However, we advise to rely on the volume-based formula because it imposes lower regularity assumptions on the state and the adjoint variables [START_REF] Hiptmair | Comparison of approximate shape gradients[END_REF][START_REF] Delfour | Shapes and geometries. Metrics, analysis, differential calculus, and optimization[END_REF][START_REF] Laurain | Domain expression of the shape derivative and application to electrical impedance tomography[END_REF].

Henceforth, C denotes a positive generic constant independent of n and h.

The assumption Ω ⊂ R 2 made in[START_REF] Rannacher | Some optimal error estimates for piecewise linear finite element approximations[END_REF] can be replaced by Ω ⊂ R

; cf.[START_REF] Brenner | The mathematical theory of finite element methods[END_REF] Ch. 8].

Acknowledgments. The work of A. Paganini was partly supported by ETH Grant CH1-02 11-1.


Takeshi Fukao
email: fukao@kyokyo-u.ac.jp

Cahn-Hilliard approach to some degenerate parabolic equations with dynamic boundary conditions

Keywords: degenerate parabolic equation, dynamic boundary condition, weak solution, Cahn-Hilliard system AMS (MOS) subject classification: 35K65, 35K30, 47J35

In this paper the well-posedness of some degenerate parabolic equations with a dynamic boundary condition is considered. To characterize the target degenerate parabolic equation from the Cahn-Hilliard system, the nonlinear term coming from the convex part of the doublewell potential is chosen using a suitable maximal monotone graph. The main topic of this paper is the existence problem under an assumption for this maximal monotone graph for treating a wider class. The existence of a weak solution is proved.

Introduction

The relationship between the Allen-Cahn equation [START_REF] Allen | A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsing[END_REF] and the motion by mean curvature is interesting as the singular limit of the following form:

∂u ∂t -∆u + 1 ε 2 (u 3 -u) = 0 in Q := (0, T ) × Ω,
as ε 0, where 0 < T < +∞ and Ω ⊂ R d for d = 2, 3, which is a bounded domain with smooth boundary Γ . For example, Bronsard and Kohn presented a pioneering result in [START_REF] Bronsard | Motion by mean curvature as the singular limit of Ginzbrug-Landau dynamics[END_REF], and subsequently many related results have been obtained. A similar concept in this framework, the Cahn-Hilliard system [START_REF] Cahn | Free energy of a nonuniform system I. Interfacial free energy[END_REF], is connected to motion by the Mullins-Sekerka law [START_REF] Mullins | Morphological stability of a particle growing by diffusion or heat flow[END_REF] in the limit of

∂u ∂t -∆µ = 0 in Q, µ = -ε∆u + 1 ε (u 3 -u) in Q (1.1)
as ε 0. For both of these, the target problems are sharp interface models in a classical sense and a powerful analysis tool seems to be the method of matched asymptotic expansions (see [1, ?,?] and the references in these papers).

In this paper, we discuss this relation from a different view point. To do so, we begin with the following degenerate parabolic equation:

∂u ∂t -∆β(u) = g in Q, (1.2)
where g : Ω → R is a given source. This equation is characterized by the choice of β : R → R. For example, if we choose β to be a piecewise linear function of the form

β(r) :=      k s r r < 0, 0 0 ≤ r ≤ L, k (r -L) r > L;
(1. [START_REF] Andreu | A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions[END_REF] where k s and k > 0 represent the heat conductivities of the solid and liquid regions, respectively, and L > 0 is the latent heat constant, then (1.2) is the weak formulation of the Stefan problem, or the "enthalpy formulation," where the unknown u denotes the enthalpy and β(u) denotes the temperature. The informant of the sharp interface, in other words the Stefan condition, is hidden in the weak formulation. Another example is the weak formulation of the Hele-Shaw problem. If we choose β to be the inverse of the Heaviside function

H(r) :=          0 if r < 0, [0, 1] if r = 0, 1 if r > 0 for all r ∈ R, so that β is the multivalued function β(r) := H -1 (r) = ∂I [0,1] (r) for all r ∈ [0, 1], then (1.
2) can be stated as

ξ ∈ β(u), ∂u ∂t -∆ξ = g in Q,
where ∂I [0,1] is the subdifferential of the indicator function I [0,1] on the interval [0, 1], the unknown u denotes the order parameter. Details about weak formulations may be found in Visintin [START_REF] Visintin | Models of phase transitions[END_REF]. Weak formulations for this kind of sharp interface model are the focus of this paper. Therefore, we use the terms "Stefan problem" and "Hele-Shaw problem" in the sense of weak formulations throughout this paper.

Recently, the author considered the approach to the following Cahn-Hilliard system for the Stefan problem in [START_REF] Fukao | Convergence of Cahn-Hilliard systems to degenerate parabolic equations with dynamic boundary conditions[END_REF]:

∂u ∂t -∆µ = 0 in Q, (1.4) µ = -ε∆u + β(u) + επ(u) -f in Q, (1.5)
with a dynamic boundary condition of the form

∂u ∂t + ∂ ν µ -∆ Γ µ = 0 on Σ := (0, T ) × Γ, (1.6)
µ = ε∂ ν u -ε∆ Γ u + β(u) + επ(u) -f Γ on Σ, (1.7)
where the symbol ∂ ν denotes the normal derivative on the boundary Γ outward from Ω, the symbol ∆ Γ stands for the Laplace-Beltrami operator on Γ (see, e.g., [START_REF] Grigor'yan | Heat kernel and analysis on manifolds[END_REF]Chapter 3]), β is defined by (1.3), and π : R → R is a piecewise linear function defined by π(r)

:= L/2 if r < 0, π(r) := L/2 -r if 0 ≤ r ≤ L and π(r) := -L/2 if r > L.
Thanks to this choice, system (1.4)-(1.7) has the structure of a Cahn-Hilliard system. This problem originally comes from [START_REF] Goldstein | A Cahn-Hilliard model in a domain with non-permeable walls[END_REF].

Formally, if we let ε 0 in (1.4)-(1.7), then we can see that the Cahn-Hilliard system (1.4)-(1.7) converges in a suitable sense to the following Stefan problem with a dynamic boundary condition:

∂u ∂t -∆β(u) = -∆f in Q, ∂u ∂t + ∂ ν β(u) -∆ Γ β(u) = ∂ ν f Γ -∆ Γ f Γ on Σ.
Here, we should take care of the difference between the order and position of ε in (1.1) and (1.5) even when β(u) = u 3 and π(u) = -u. In [START_REF] Fukao | Convergence of Cahn-Hilliard systems to degenerate parabolic equations with dynamic boundary conditions[END_REF], β is assumed to satisfy the following condition:

β is a maximal monotone graph in R × R, and is a subdifferential β = ∂ β of some proper, lower semicontinuous, and convex function β : R → [0, +∞] satisfying β(0) = 0 with some effective domain D(β). This implies β(0) = 0. Moreover, there exist two constants c, c > 0 such that β(r) ≥ c|r| 2 -c for all r ∈ R.

(1.8)

It is easy to see that (1.2) represents a large number of problems, including the porous media equation, the nonlinear diffusion equation of Penrose-Fife type, the fast diffusion equation, and so on. However, to apply this approach from the Cahn-Hilliard system to these wider classes of the degenerate parabolic equation, the growth condition (1.8) is too strong (see, e.g. [START_REF] Damlamian | Evolution equations generated by subdifferentials in the dual space of (H 1 (Ω))[END_REF]). Therefore, in this paper based on the essential idea from [START_REF] Colli | Nonlinear diffusion equations as asymptotic limit of Cahn-Hilliard system[END_REF], we relax the assumption in (1.8). This is the different point from the previous work [START_REF] Fukao | Convergence of Cahn-Hilliard systems to degenerate parabolic equations with dynamic boundary conditions[END_REF]. See also [3, ?] for related problems of interest.

Notation. Let H := L 2 (Ω), V := H 1 (Ω), H Γ := L 2 (Γ ) and V Γ := H 1 (Γ ) with the usual norms | • | H , | • | V , | • | H Γ , | • | V Γ and inner products (•, •) H , (•, •) V , (•, •) H Γ , (•, •) V Γ ,

respectively, and let

H := H × H Γ , V := {(z, z Γ ) ∈ V × V Γ : z Γ = z | Γ a.e. on Γ } and W := H 2 (Ω) × H 2 (Γ )
. Then H, V and W are Hilbert spaces with the inner product

(u, z) H := (u, z) H + (u Γ , z Γ ) H Γ for all u, z ∈ H,
and the related norm is analogously defined as one of

V or W . Define m : H → R by m(z) := 1 |Ω| + |Γ | Ω zdx + Γ z Γ dΓ for all z ∈ H,
where |Ω| := Ω 1dx and |Γ | := Γ 1dΓ . The symbol V * denotes the dual space of V , and the pair •, • V * ,V denotes the duality pairing between V * and V . Moreover, define the bilinear form a(•,

•) : V × V → R by a(u, z) := Ω ∇u • ∇zdx + Γ ∇ Γ u Γ • ∇ Γ z Γ dΓ for all u, z ∈ V ,
where ∇ Γ denotes the surface gradient on Γ (see, e.g., [START_REF] Grigor'yan | Heat kernel and analysis on manifolds[END_REF]Chapter 3]). We introduce the subspace

H 0 := {z ∈ H : m(z) = 0} of H and V 0 := V ∩ H 0 , with their norms |z| H0 := |z| H for all z ∈ H 0 and |z| V 0 := a(z, z) 1/2 for all z ∈ V 0 . Then the duality mapping F : V 0 → V * 0 is defined by F z, z V * 0 ,V 0 := a(z, z) for all z, z ∈ V 0 and the inner product in V * 0 is defined by (z * 1 , z * 2 ) V * 0 := z * 1 , F -1 z * 2 V * 0 ,V 0 for all z * 1 , z * 2 ∈ V * 0 .
Moreover, define P : H → H 0 by P z := z -m(z)1 for all z ∈ H, where 1 := (1, 1). Thus we obtain the dense and compact embeddings V 0 → → H 0 → → V * 0 . See [8, ?] for further details.

Existence of the weak solution

In this section, we state an existence theorem for the weak solution of a degenerate parabolic equation with a dynamic boundary condition of the following form:

ξ ∈ β(u), ∂u ∂t -∆ξ = g a.e. in Q, ξ Γ ∈ β(u Γ ), ξ Γ = ξ | Γ , ∂u Γ ∂t + ∂ ν ξ -∆ Γ ξ Γ = g Γ a.e. on Σ, u(0) = u 0 a.e. in Ω, u Γ (0) = u 0Γ a.e. on Γ,
where β, g, g Γ , u 0 and u 0Γ satisfy the following assumptions:

(A1) β is a maximal monotone graph in R × R, and is a subdifferential β = ∂ β of some proper, lower semicontinuous, and convex function β : R → [0, +∞] satisfying β(0) = 0 in some effective domain D(β). This implies that β(0

) = 0; (A2) g ∈ L 2 (0, T ; H 0 ); (A3) u 0 := (u 0 , u 0Γ ) ∈ H with m 0 ∈ intD(β)
, and the compatibility conditions

β(u 0 ) ∈ L 1 (Ω), β(u 0Γ ) ∈ L 1 (Γ ) hold.
We remark that the growth condition of β in (A1) and the regularity of u 0 in (A3) are relaxations from a previous related result [START_REF] Fukao | Convergence of Cahn-Hilliard systems to degenerate parabolic equations with dynamic boundary conditions[END_REF] (cf. (1.8)).

Theorem 2.1. Under assumptions (A1)-(A3), there exists at least one pair (u, ξ)

of functions u ∈ H 1 (0, T ; V * ) ∩ L 2 (0, T ; H) and ξ ∈ L 2 (0, T ; V ) such that ξ ∈ β(u) a.e. in Q, ξ Γ ∈ β(u Γ ) and ξ Γ = ξ | Γ a.
e. on Σ, and that satisfy

u (t), z V * ,V + u Γ (t), z Γ V * Γ ,V Γ + Ω ∇ξ(t) • ∇zdx + Γ ∇ Γ ξ Γ (t) • ∇ Γ z Γ dΓ = Ω g(t)zdx + Γ g Γ (t)z Γ dΓ for all z := (z, z Γ ) ∈ V (2.1)
for a.a. t ∈ (0, T ) with u(0) = u 0 a.e. in Ω and u Γ (0) = u 0Γ a.e. on Γ .

The continuous dependence is completely the same as in a previous result [START_REF] Fukao | Convergence of Cahn-Hilliard systems to degenerate parabolic equations with dynamic boundary conditions[END_REF]Theorem 2.2]. Therefore, we devolve the uniqueness problem on [START_REF] Fukao | Convergence of Cahn-Hilliard systems to degenerate parabolic equations with dynamic boundary conditions[END_REF].

Proof of the main theorem

In this section, we prove the main theorem. The strategy of the proof is similar to that of [START_REF] Fukao | Convergence of Cahn-Hilliard systems to degenerate parabolic equations with dynamic boundary conditions[END_REF]Theorem 2.1]. However, to relax the assumption we use a different uniform estimate. Let us start with an approximate problem. Recall the Yosida approximation β λ : R → R and the related Moreau-Yosida regularization β λ of β : R → R (see, e.g., [START_REF] Barbu | Nonlinear differential equations of monotone types in Banach spaces[END_REF]). We see that 0 ≤ β λ (r) ≤ β(r) for all r ∈ R. Moreover, we define the following proper, lower semicontinuous, and convex functional ϕ : H 0 → [0, +∞]:

ϕ(z) :=      1 2 Ω |∇z| 2 dx + 1 2 Γ |∇ Γ z Γ | 2 dΓ if z ∈ V 0 , +∞ otherwise.
The subdifferential ∂ϕ on H 0 is characterized by ∂ϕ(z) = (-∆z, ∂ ν z -∆ Γ z Γ ) with z ∈ D(∂ϕ) = W ∩ V 0 (see, e.g., [9, Lemma C]). By virtue of the wellknown theory of evolution equations (see, e.g., [8, ?,?,?]), for each ε ∈ (0, 1] and λ ∈ (0, 1], there exist

v ε,λ ∈ H 1 (0, T ; H 0 ) ∩ L ∞ (0, T ; V 0 ) ∩ L 2 (0, T ; W ) and µ ε,λ ∈ L 2 (0, T ; V ) such that λv ε,λ (t) + F -1 v ε,λ (t) + ε∂ϕ v ε,λ (t) = P -β λ u ε,λ (t) -επ u ε,λ (t) + f (t) in H 0 (3.1)
for a.a. s ∈ (0, T ) with v ε,λ (0) = v 0ε in H 0 , where v 0ε ∈ V 0 solves the auxiliary problem v 0ε + ε∂ϕ(v 0ε ) = v 0 in H 0 so that there exists a constant C > 0 such that

|v 0ε | 2 H0 ≤ C, ε|v 0ε | 2 V 0 ≤ C, (3.2)
Ω β(v 0ε + m 0 )dx ≤ C, Γ β(v 0ε + m 0 )dΓ ≤ C.
Moreover, u ε,λ := v ε,λ + m 0 1, m 0 := m(u 0 ) and 1 := (1, 1), and β λ (z) := (β λ (z), β λ (z Γ )) and π(z) := (π(z), π(z Γ )) for all z ∈ H, where π : D(π) = R → R is a Lipschitz continuous function with a Lipschitz constant L π that breaks the monotonicity in β + επ; f ∈ L 2 (0, T ; D(∂ϕ)) is the solution of g(t) = ∂ϕ(f (t)) in H 0 for a.a. t ∈ (0, T ). Namely, from [9, Lemma C], we can choose f (t) := (f (t), f Γ (t)) to satisfy

   -∆f (t) = g(t) a.e. in Ω, ∂ ν f (t) -∆ Γ f Γ (t) = g Γ (t) a.e. on Γ,
for a.a. t ∈ (0, T ).

(3.3)

Uniform estimates for approximate solutions

The key strategy in the proof is to obtain uniform estimates independent of ε > 0 and λ > 0, after which we consider the limiting procedures λ 0 and ε 0. Recall (3.1) in the equivalent form

v ε,λ (s) + F P µ ε,λ (s) = 0 in V * 0 , (3.4)
µ ε,λ (s) = λv ε,λ (s) + ε∂ϕ v ε,λ (s) + β λ u ε,λ (s) + επ u ε,λ (s) -f (s) in H (3.5)
for a.a. s ∈ (0, T ). Moreover, if we put ε 0 := min{1, 1/(4L 2 π )}, then we have:

Lemma 3.1. There exist positive constants M 1 , M 2 independent of ε ∈ (0, ε 0 ] and λ ∈ (0, 1] such that λ v ε,λ (t) 2 H0 + v ε,λ (t) 2 V * 0 ≤ M 1 , ε 2 t 0 v ε,λ (s) 2 V 0 ds + 2 t 0 β λ u ε,λ (s) L 1 (Ω) ds + 2 t 0 β λ u Γ,ε,λ (s) L 1 (Γ ) ds ≤ M 2 for all t ∈ [0, T ].
Proof. Multiplying (3.1) by v ε,λ (s) ∈ V 0 , we have

λ v ε,λ (s), v ε,λ (s) H0 + v ε,λ (s), v ε,λ (s) V * 0 + ε ∂ϕ v ε,λ (s) , v ε,λ (s) H0 + P β λ v ε,λ (s) + m 0 1 , v ε,λ (s) H0 = f (s) -εP π v ε,λ (s) + m 0 1 , v ε,λ (s) H0
for a.a. s ∈ (0, T ). Using the definition of the subdifferential, we see that

λ 2 d ds v ε,λ (s) 2 H0 + 1 2 d ds v ε,λ (s) 2 V * 0 + ε 2 v ε,λ (s) 2 V 0 + β λ u ε,λ (s) L 1 (Ω) + β λ u Γ,ε,λ (s) L 1 (Γ ) ≤ |Ω| + |Γ | β(m 0 ) + 1 2 v ε,λ (s) 2 V * 0 + f (s) 2 V 0 + L 2 π ε 2 v ε,λ (s) 2 V 0
for a.a. s ∈ (0, T ).

Taking ε ∈ (0, ε 0 ] and using the Gronwall inequality, we obtain the existence of M 1 and M 2 independent of ε ∈ (0, ε 0 ] and λ ∈ (0, 1] satisfying the conclusion.

Lemma 3.2. There exists a positive constant M 3 , independent of ε ∈ (0, ε 0 ] and λ ∈ (0, 1], such that

2λ t 0 v ε,λ (s) 2 H0 ds + t 0 v ε,λ (s) 2 V * 0 ds + ε v ε,λ (t) 2 V 0 + 2 β λ u ε,λ (t) L 1 (Ω) + 2 β λ u Γ,ε,λ (t) L 1 (Γ ) ≤ M 3 , t 0 P µ ε,λ (s) 2 V 0 ds ≤ M 3 for all t ∈ [0, T ].
Proof. Multiplying (3.1) by v ε,λ (s) ∈ H 0 , we have

λ v ε,λ (s) 2 H0 + 1 2 v ε,λ (s) 2 V * 0 + ε d ds ϕ v ε,λ (s) + d ds Ω β λ u ε,λ (s) dx + d ds Γ β λ u Γ,ε,λ (s) dΓ ≤ L 2 π ε 2 v ε,λ (s) 2 V 0 + f (s) 2 V 0
for a.a. s ∈ (0, T ).

Integrating this over (0, t) with respect to s, we see that there exists a positive constant M 3 , independent of ε ∈ (0, ε 0 ] and λ ∈ (0, 1], such that the first estimate holds. Next, multiplying (3.4) by P µ ε,λ (s) ∈ V 0 and integrating the resultant over (0, t) with respect to s, we obtain the second estimate.

The previous two lemmas are essentially the same as [13, Lemmas 3.1 and 3.2]. The next uniform estimate is the point of emphasis in this paper. Lemma 3.3. There exists positive constant M 4 , independent of ε ∈ (0, 1] and λ ∈ (0, 1], such that

u ε,λ (t) 2 H ≤ M 4 1 + λ ε , v ε,λ (t) 2 H0 ≤ M 4 1 + λ ε , λ v ε,λ (t) 2 V 0 + ε t 0 ∂ϕ v ε,λ (s) 2 H0 ds ≤ M 4 1 + λ ε for all t ∈ [0, T ].
Proof. Multiplying (3.4) by v ε,λ (s) ∈ V 0 and using the fact (d/ds)m(u ε,λ (s)) = 0, we have

u ε,λ (s), u ε,λ (s) H + a µ ε,λ (s), u ε,λ (s) = 0
for a.a. s ∈ (0, T ) (see [START_REF] Fukao | Convergence of Cahn-Hilliard systems to degenerate parabolic equations with dynamic boundary conditions[END_REF]Remark 3]). On the other hand, multiplying (3.5) by ∂ϕ(v ε,λ (s)) ∈ H 0 and integrating by parts, we have

a µ ε,λ (s), u ε,λ (s) = λ 2 d ds a u ε,λ (s), u ε,λ (s) + ε ∂ϕ v ε,λ (s) 2 H0
+

Ω β λ u ε,λ (s) ∇u ε,λ (s) 2 dx + Γ β λ u Γ,ε,λ (s) ∇ Γ u Γ,ε,λ (s) 2 dΓ + ε π u ε,λ (s) , ∂ϕ u ε,λ (s) H + ∂ϕ f (s) , u ε,λ (s) H
for a.a. s ∈ (0, T ). Using the Lipschitz continuity of π and (3.3), we see that there exists a positive constant C π such that

d ds u ε,λ (s) 2 H +λ d ds v ε,λ (s) 2 V 0 +ε ∂ϕ v ε,λ (s) 2 H0 ≤ C π u ε,λ (s) 2 H +1 + g(s) 2 H0
for a.a. s ∈ (0, T ). Then, using (3.2) and the Gronwall inequality, we deduce that

u ε,λ (t) 2 H + λ v ε,λ (t) 2 V 0 ≤ |v 0ε + m 0 1| 2 H + λ|v 0ε | 2 V 0 + C π T + |g| 2 L 2 (0,T ;H0) e CπT ≤ 2C + 2|m 0 | 2 |Ω| + |Γ | + λ ε C + C π T + |g| 2 L 2 (0,T ;H0) e CπT
for all t ∈ [0, T ]. That is, there exists a positive constant M 4 independent of ε ∈ (0, 1] and λ ∈ (0, 1] such that the uniform estimates hold.

Lemma 3.4. There exists positive constant M 5 , independent of ε ∈ (0, 1] and λ ∈ (0, 1], such that

t 0 µ ε,λ (s) 2 V 0 ds ≤ M 5 1 + λ ε , t 0 β λ u ε,λ (s) 2 H ds ≤ M 5 1 + λ ε for all t ∈ [0, T ].
Using Lemmas 3.1 to 3.3, the proofs of these uniform estimates are completely the same as those for [START_REF] Colli | Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials[END_REF]Lemmas 4.3 and 4.4]. Therefore, we omit the proof.

Limiting procedure

From the previous uniform estimates, we can consider the limit as λ 0. More precisely, for each ε ∈ (0, ε 0 ], there exists a subsequence {λ k } k∈N with λ k 0 as k → +∞ and a quadruplet (

v ε , v * ε , µ ε , ξ ε ) of v ε ∈ H 1 (0, T ; V * 0 )∩L ∞ (0, T ; V 0 )∩ L 2 (0, T ; W ), v * ε ∈ L 2 (0, T ; H 0 ), µ ε ∈ L 2 (0, T ; V ), ξ ε ∈ L 2 (0, T ; H), such that v ε,λ k → v ε weakly star in L ∞ (0, T ; H 0 ), v ε,λ k → v ε weakly in L 2 (0, T ; V 0 ), λ k v ε,λ k → 0 in L 2 (0, T ; H 0 ), v ε,λ k → v ε weakly in L 2 (0, T ; V * 0 ), u ε,λ k → u ε := v ε + m 0 1 weakly star in L ∞ (0, T ; V ), ∂ϕ(v ε,λ k ) → v * ε weakly in L 2 (0, T ; H 0 ), µ ε,λ k → µ ε weakly in L 2 (0, T ; V ), β λ k (u ε,λ k ) → ξ ε weakly in L 2 (0, T ; H) as k → +∞.
From the compactness theorem (see, e.g., [START_REF] Simon | Compact sets in the spaces L p (0, T ; B)[END_REF]Section 8,Corollary 4]), this gives

v ε,λ k → v ε in C [0, T ]; H 0 , u ε,λ k → u ε in C [0, T ]; H , π(u ε,λ k ) → π(u ε ) in C [0, T ]; H as k → +∞.
Moreover, from the demi-closedness of ∂ϕ and [4, Proposition 2.2], we see that v * ε = ∂ϕ(v ε ) in L 2 (0, T ; H 0 ) and ξ ε ∈ β(u ε ) in L 2 (0, T ; H). From these facts, we deduce from (3.4) and (3.5) that

v ε (t) + F P µ ε (t) = 0 in V * 0 , (3.6)
ξ ε (t) ∈ β u ε (t) , µ ε (t) = ε∂ϕ v ε (t) + ξ ε (t) + επ u ε (t) -f (t) in H (3.7)
for a.a. t ∈ (0, T ), with v ε (0) = v 0ε in H. We also have the regularity u ε ∈ H 1 (0, T ; V * )∩L ∞ (0, T ; V )∩L 2 (0, T ; W ). Now, taking the limit inferior as λ 0 on the uniform estimates, λ/ε 0 for all ε ∈ (0, ε 0 ], and we therefore obtain the same kind of uniform estimates as in the previous lemmas independent of ε ∈ (0, ε 0 ].

Proof of Theorem 2.1. By using the estimates for v ε , u ε , µ ε and ξ ε , there exist a subsequence {ε k } k∈N with ε k 0 as k → +∞ and functions v ∈

H 1 (0, T ; V * 0 ) ∩ L ∞ (0, T ; H 0 ), u ∈ H 1 (0, T ; V * ) ∩ L ∞ (0, T ; H), µ ∈ L 2 (0, T ; V ) and ξ ∈ L 2 (0, T ; H) such that v ε k → v weakly star in L ∞ (0, T ; H 0 ), u ε k → u = v + m 0 1 weakly star in L ∞ (0, T ; H), ε k v ε k → 0 in L ∞ (0, T ; V 0 ), v ε k → v weakly in L 2 (0, T ; V * 0 ), u ε k → u weakly in L 2 (0, T ; V * ), µ ε k → µ weakly in L 2 (0, T ; V ), ξ ε k → ξ weakly in L 2 (0, T ; H), ε k π(u ε k ) → 0 in L ∞ (0, T ; H) as k → +∞.
From the Ascoli-Arzelà theorem, we also have

v ε k → v in C [

0 fξ 0 f

 00 0, T ]; V * 0 , u ε k → u in C [0, T ]; V * as k → +∞.Now, multiplying (3.7) by η ∈ L 2 (0, T ; V ) and integrating over (0, T ), we obtainT 0 µ ε k (t), η(t) H dt = ε k T 0 a v ε k (t), η(t) dt + T 0 ξ ε k (t), η(t) H dt + ε k T 0 π u ε k (t) , η(t) H dt -T (t), η(t) H dt. ) -f (t), η(t) H dt for all η ∈ L 2 (0, T ; V ), namely, µ = ξf in L 2 (0, T ; H). This implies the regularity of ξ ∈ L 2 (0, T ; V ), that is, ξ Γ = ξ | Γ a.e. on Σ. Next, we take η := u ε k ∈ L 2 (0, T ; V ) in (3ε k (t), u ε k (t) H dt ≤ T 0 u(t), µ(t) V * ,V dt + T (t), u(t) H dt = T 0 ξ(t), u(t) H dt.Thus, applying [4, Proposition 2.2] we have ξ ∈ β(u) in L 2 (0, T ; H), and so we obtain ξ ∈ β(u) a.e. in Ω. ξ Γ ∈ β(u Γ ) a.e. on Γ . Finally, letting k → +∞ and applying Hahn-Banach extension theorem of bounded linear functional on V to V × V Γ , then we see that (3.6) gives (2.1) for a.a. t ∈ (0, T ), with u(0) = u 0 a.e.in Ω and u Γ (0) = u 0Γ a.e. on Γ .

Acknowledgments

The author is indebted to professor Pierluigi Colli, who kindly gave him the opportunity for fruitful discussions. The author is supported by JSPS KAKENHI Grant-in-Aid for Scientific Research(C), Grant Number 26400164. Last but not least, the author is also grateful to the referee for the careful reading of the manuscript.


Nina Subbotina

Lyubov Shagalova

Generalized Solutions of Hamilton -Jacobi Equation to a Molecular Genetic Model

Keywords: Hamilton -Jacobi equation, method of characteristics, generalized solutions, viscosity solutions, state constraints

A boundary value problem with state constraints is under consideration for a nonlinear noncoercive Hamilton-Jacobi equation. The problem arises in molecular biology for the Crow -Kimura model of genetic evolution. A new notion of continuous generalized solution to the problem is suggested. Connections with viscosity and minimax generalized solutions are discussed. In this paper the problem is studied for the case of additional requirements to structure of solutions. Constructions of the solutions with prescribed properties are provided and justified via dynamic programming and calculus of variations. Results of simulations are exposed.

Introduction

In [START_REF] Saakian | Dynamics of the Eigen and the Crow-Kimura models for molecular evolution[END_REF] a new way to study molecular evolution has been proposed. According to this way dynamics of the Crow -Kimura model for molecular evolution can be analyzed via the following HJE ∂u/∂t + H(x, ∂u/∂x) = 0, [START_REF] Saakian | Dynamics of the Eigen and the Crow-Kimura models for molecular evolution[END_REF] where the Hamiltonian H(•) has the form

H(x, p) = -f (x) + 1 - 1 + x 2 e 2p - 1 -x 2 e -2p . (2)
The function f (•) in ( 2) is given and called fitness. Equation ( 1) is considered for t ≥ 0, -1 ≤ x ≤ 1. It is also assumed that an initial function u 0 : R → R is given such that u(0, x) = u 0 (x), x ∈ [-1; 1].

In [START_REF] Saakian | Dynamics of the Eigen and the Crow-Kimura models for molecular evolution[END_REF] problem (1)-( 3) was studied for input data u 0 (x) = -a(x -x 0 ) 2 , a > 0, f (x) = x 2 and physical interpretations were used.

The classical method for solving PDE of the first order in Cauchy problem is the method of characteristics (see, e.g. [START_REF] Courant | Methods of Mathematical Physics[END_REF]). This method reduces integration of PDEs to integration of the characteristic system of ODEs.

The characteristic system for problem

Here H x (x, p) = ∂H(x, p)/∂x, H p (x, p) = ∂H(x, p)/∂p, f (x) = ∂f (x)/∂x. Solutions of the system (4)-( 5) are called characteristics. Components x(•, y), p(•, y) and z(•, y) of the solution are called state, conjugate, and value characteristics, respectively. The method of characteristics can be applied to constructions of solutions for problem (1)-( 3) in such a neighborhood of the initial manifold [START_REF] Subbotina | On the continuous extension of a generalized solution of the Hamilton-Jacobi equation by characteristics that form a central field of extremals[END_REF] where state characteristics don't cross. As a rule, characteristics for problem (1)-( 3) are nonextendable to the whole time axis and can cross each other. Moreover, there are points in strip t ≥ 0, -1 ≤ x ≤ 1. where solution of ( 1)-(3) should be found, and where the state characteristics do not pass. An example of such a behavior of state characteristics is presented on Fig. 1.

So, one can see that solutions of the problem (1)-( 3) should be understood in a generalized sense.

In [START_REF] Subbotina | On a solution to the Cauchy problem for the HamiltonJacobi equation with state constraints[END_REF], we introduced a concept of continuous generalized solutions (see Definition 1 below) and proved it's existence in problem (1)-(3) using tools of Nonsmooth Analysis and results of the Optimal Control Theory. It was also shown that the generalized solution is not unique.

In this paper, we consider problem (1)-( 3) with additional requirements to the structure of solutions, see [START_REF] Subbotina | Construction of a Continuous Minimax/Viscosity Solution of the Hamilton JacobiBellman Equation with Nonextendable Characteristics[END_REF] and [START_REF] Subbotina | On the continuous extension of a generalized solution of the Hamilton-Jacobi equation by characteristics that form a central field of extremals[END_REF]. Namely, we need to construct a continuous solution in the strip t ≥ 0, -1 ≤ x ≤ 1 in such a way that it coincides with a solution obtained by the method of characteristics in a domain part where the characteristics defined by ( 4) and ( 5) pass.

The paper is organized as follows. In Section 2, the definition of a continuous generalized solution is introduced, and the results on its existence are presented. In Section 3 we state the problem of constructing the generalized solution with prescribed properties, give sufficient conditions under which the problem can be solved, and formulate auxiliary results on which solving is based. A scheme for constructing the generalized solution and results of a simulation are presented in Section 4 and Section 5 respectively. And, in Section 6, we compare our generalized solution with viscosity solutions. Let T > 0 be such an instant that characteristics (4), [START_REF] Subbotina | On the continuous extension of a generalized solution of the Hamilton-Jacobi equation by characteristics that form a central field of extremals[END_REF] are extendable up to T , and x(•, y), p(•, y), z(•, y) are continuous on [0, T ] for all y ∈ [-1; 1]. Exact estimates for intervals of extendibility are obtained in [START_REF] Subbotina | Construction of a Continuous Minimax/Viscosity Solution of the Hamilton JacobiBellman Equation with Nonextendable Characteristics[END_REF][START_REF] Shagalova | Applications of Dynamic Programming to Generalized Solutions for Hamilton -Jacobi Equations with State Constraints[END_REF].

We consider problem (1)-(3) on the restricted closed domain

Π T = [0; T ] × [-1; 1],
and also use the notations

Π T = (0; T ) × (-1; 1), Γ T = {(t, x)| t ∈ (0, T ), x = ±1}.
In the HJEs' theory various concepts of generalized solutions have been introduced (see, e.g. [START_REF] Kruzhkov | Generalized solutions of nonlinear equations of the first order with several variables[END_REF][START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF][START_REF] Subbotin | Generalized Solutions of First Order PDEs: The Dynamical Optimization Perspective[END_REF]). Note that definitions of generalized solutions to HJEs in open areas were applied to problems with state constraints as additional requirements to solutions on the border were imposed. These requirements play a role of boundary conditions. Unfortunately, results of the theories of generalized solutions are inapplicable to the problem (1)- [START_REF] Subbotina | On a solution to the Cauchy problem for the HamiltonJacobi equation with state constraints[END_REF]. In particular, one of the key conditions under which the known theorems on existense of a generalized viscosity solution [START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF][START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi Equations with State Constraints[END_REF] has been proved is the coercivity of the Hamiltonian (see (22) below). And the theory of minimax solutions [START_REF] Subbotin | Generalized Solutions of First Order PDEs: The Dynamical Optimization Perspective[END_REF] is not developed for problems with state constraints. So, below a new definition of a generalized solution is introduced [START_REF] Subbotina | On a solution to the Cauchy problem for the HamiltonJacobi equation with state constraints[END_REF]. This definition is based on the minimax and viscosity approahes and uses the following tools of nonsmooth analysis [START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi Equations with State Constraints[END_REF][START_REF] Clarke | Optimization and nonsmooth analysis[END_REF].

Let W be a set in R 2 . Denote by W the closure of this set, by C(W ) -the class of functions continuous on the set W .

Let u(•) ∈ C(W ) and (t, x) ∈ W . The subdifferential of the function u(•) at (t, x) is the set

D -u(t, x) =          (a, s) ∈ R × R lim inf (τ,y)→(t,x) (τ,y)∈W u(τ,y)-u(t,x)-a(τ -t)-s(y-x) |τ -t|+|y-x| ≥ 0         
.

The superdifferential of the function u(•) at (t, x) is the set

D + u(t, x) =            (a, s) ∈ R × R lim sup (τ,y)→(t,x) (τ,y)∈W u(τ,y)-u(t,x)-a(τ -t)-s(y-x) |τ -t|+|y-x| ≤ 0            .
Let Dif(u) be the set of points where the function u(•) ∈ C(W ) is differentiable. For a given set M ⊂ R 2 , the symbol coM means its convex hull [START_REF] Rockafellar | Convex Analysis[END_REF] 3) iff it satisfies the initial condition (3) and the following relations are true

; (t i , x i ) → (t, x) as i → ∞, (t i , x i ) ∈ Dif(u) . Definition 1. A continuous function u(•) : Π T → R 2 is called a generalized so- lution to problem (1)-(
a + H(x, s) ≤ 0, ∀(a, s) ∈ D + u(t, x), ∀(t, x) ∈ Π T , (6)
a + H(x, s) ≥ 0, ∀(a, s) ∈ D -u(t, x), ∀(t, x) ∈ Π T , (7)
a + H(x, s) ≥ 0, ∀ (a, s) ∈ D -u(t, x) ∩ ∂u(t, x), ∀ (t, x) ∈ Γ T . (8)

Existence of Generalized Solutions

The following statement was proved in [START_REF] Subbotina | Construction of a Continuous Minimax/Viscosity Solution of the Hamilton JacobiBellman Equation with Nonextendable Characteristics[END_REF] by using tools of Mathematical Theory of Optimal Control [START_REF] Pontryagin | The Mathematical Theory of Optimal Processes[END_REF] and the method of generalized characteristics [START_REF] Subbotina | The method of characteristics for Hamilton-Jacobi equation and its applications in dynamical optimization[END_REF][START_REF] Subbotina | The method of characteristics for Hamilton-Jacobi-Bellman equation[END_REF].

Theorem 1. Let input data u 0 (•) : [-1, 1] → R and f (•) : [-1, 1]
→ R be continuously differentiable functions. Let a function ϕ(t, x) : R 2 → R be also continuously differentiable and satisfy the relations

ϕ(0, x) = u 0 (x) ∀ x ∈ [-1, 1]; ∂ϕ(t, ±1) ∂t + H ± 1, ∂ϕ(t, ±1) ∂x = 0 ∀ t ≥ 0. ( 9
)
Then there exists a solution u(t, x) of problem (1)-( 3) in sense of definition 1. The solution has the form

u(t, x) = max x(t,y )=x ϕ(t , y )+ + t t p(τ, y )H p (x(τ, y ), p(τ, y )) -H(x(τ, y ), p(τ, y ))dτ , (10)
for all (t, x) ∈ Π T , where t ∈ [0, T ]. If t = 0 then y = y ∈ [-1, 1]; if t > 0, then y = ±1. The functions (x(•, y ), p(•, y )): [t , t] → R 2 are solutions for the system composed of the first two equations of characteristic system (4) with initial conditions

x(t , y ) = y , p(t , y ) = ∂ϕ(t , y ) ∂y = p 0 (t , y ).

To obtain u(t, x) in accordance with [START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi Equations with State Constraints[END_REF], one should consider the set of all state characteristics x(•, y ) passing through the point (t, x), namely, x(t, y ) = x. Note that the generalized solution to problem (1)-( 3) is not unique because of wide choice of functions ϕ(•) in Theorem 1.

Solutions with Prescribed Properties

Here, we consider a problem to construct the generalized solution of some particular structure.

Let x -(t) = x(t, -1) and x + (t) = x(t, +1), t ∈ [0, T ] be the state characteristics started at t = 0 from the points x = -1 and x = 1, respectively. Below, we assume that the following condition is satisfied.

A. For the state characteristics x(•, y) with initial conditions (5) at t = 0 the inequalities are valid

-1 ≤ x -(t) ≤ x(t, y) ≤ x + (t) ≤ 1, ∀ y ∈ [-1, 1], ∀ t ∈ [0, T ].

Define the subdomains

G 0 = (t, x)|t ∈ [0, T ], x -(t) ≤ x ≤ x + (t) . G + = {(t, x)| t ∈ [0, T ], x + (t) ≤ x ≤ 1}, (11)
G -= {(t, x)| t ∈ [0, T ], -1 ≤ x ≤ x -(t)}.
So, under the assumption A, we get

Π T = G + ∪ G 0 ∪ G -.
The goal of the work is to construct the generalized solution to problem (1)-( 3) such that it has the following form in G 0 :

u(t, x) = max x(t,y)=x [u 0 (y) + t 0 p(τ )H p (x(τ ), p(τ )) -H(x(τ ), p(τ ))dτ ], (12)
where x(t) = x(t, y), p(t) = p(t, y), t ≥ 0, are state and conjugate characteristics, respectively, which satisfy at t = 0 the initial conditions

x(0, y) = y, p(0, y) = u 0 (y), y ∈ [-1, 1].

Sufficient Conditions

To solve the problem (1)-( 3) with the requirement [START_REF] Crandall | Viscosity Solutions of Hamilton-Jacobi Equations at the Boundary[END_REF] we introduce the following additional assumptions on input data. B1. The derivative u 0 (•) : [-1, 1] → R is continuous and satisfies the inequalities u 0 (1) < 0, u 0 (-1) > 0.

B2. The derivative f (•): [-1, 1] → R is continuous and monotone nondecreasing. It satisfies the inequalities 2f (1) + e 2u 0 (1) < e -2u 0 (1) , -2f (-1) + e -2u 0 (-1) < e 2u 0 (-1) .

Auxiliary Problems of Calculus of Variations

Consider the following two problems of Calculus of Variations over the set of all continuously differentiable functions x(•)

: [0, T ] → R I(x(•)) = t 0 H * (x(τ ), ẋ(τ ))dτ → max (13) x 1 (0) = 1, x 1 ( t) = x, ( t, x) ∈ G + ; (14)
x 2 (0) = -1, x 2 ( t) = x, ( t, x) ∈ G - (15)
where

H * (x(t), ẋ(t)) = inf p∈R [p ẋ(t) -H(x(t), p)]. (16)
The following assertions are proven in [START_REF] Subbotina | Construction of a Continuous Minimax/Viscosity Solution of the Hamilton JacobiBellman Equation with Nonextendable Characteristics[END_REF][START_REF] Subbotina | On the continuous extension of a generalized solution of the Hamilton-Jacobi equation by characteristics that form a central field of extremals[END_REF], where conditions B1-B2 are essential.

Theorem 2. For any interior point ( t, x) ∈ G + there exists a unique extremal x = x(t) of the problem (13),( 14), [START_REF] Subbotina | The method of characteristics for Hamilton-Jacobi-Bellman equation[END_REF]. The extremal coinsides with a state characteristic x(•; 0, 1, p 0 ) satisfying the initial conditions x(0) = 1, p(0) = p 0 ∈ (-∞, u 0 (1)) where initial value p 0 can be defined uniquely from the condition x( t) = x. The functional (13) attains its strong maximum at this extremal. Theorem 3. For any interior point ( t, x) ∈ G -there exists a unique extremal x = x(t) of the problem (13), [START_REF] Subbotina | The method of characteristics for Hamilton-Jacobi equation and its applications in dynamical optimization[END_REF], [START_REF] Subbotina | The method of characteristics for Hamilton-Jacobi-Bellman equation[END_REF]. The extremal coinsides with a state characteristic x(•; 0, -1, p 0 ) satisfying the initial conditions x(0) = -1, p(0) = p 0 ∈ (u 0 (-1), ∞) where initial value p 0 can be defined uniquely from the condition x( t) = x. The functional (13) attains its strong maximum at this extremal. Theorem 4. For any boundary point ( t, 1), 0 < t ≤ T in G + , the maximum of functional (13), ( 16) is attained at a state characteristic x(•; 0, 1, p 0 ), such that the characteristics x(•; 0, 1, p 0 ), p(•; 0, 1, p 0 ) are nonextendable from the interval [0, t * (p 0 ) = t) to the right. The maximum of functional (13), ( 16) is equal to I(x(•; 0, 1, p 0 )) = (f (1) -1) t.

Theorem 5. For any boundary point ( t, -1), 0 < t ≤ T in G -, the maximum of the functional (13)-( 16) is attained at a state characteristic x(•; 0, -1, p 0 ), such that characteristics x(•; 0, -1, p 0 ), p(•; 0, -1, p 0 ) are nonextendable from the interval [0, t * (p 0 ) = t) to the right. The maximum of functional (13),( 16) is equal to I(x(•; 0, -1, p 0 )) = (f (-1) -1) t.

Construction of the generalized solution

The generalized solution of the problem (1)-( 3) with the prescribed property [START_REF] Crandall | Viscosity Solutions of Hamilton-Jacobi Equations at the Boundary[END_REF] has the following form in G 0 :

u(t, x) = max x(t,y)=x [ t 0 p(τ )H p (x(τ ), p(τ )) -H(x(τ ), p(τ ))dτ + u 0 (y)], (17)
where x(t) = x(t, y), p(t) = p(t, y), t ≥ 0, are state and conjugate characteristics satisfied initial conditions

x(0, y) = y, p(0, y) = ∂u 0 (y)/∂x, y ∈ [-1, 1]. Let (t * , x * ) ∈ G + and x * < 1. We assign u(t * , x * ) = u 0 (1) + t * 0 p(τ )H p (x(τ ), p(τ )) -H(x(τ ), p(τ )) dτ, (18)
where x(t) = x + (t, p 0 (t * , x * )), p(t) = p + (t, p 0 (t * , x * )) is the solution of the problem of Calculus Variations ( 13),( 14), [START_REF] Subbotina | The method of characteristics for Hamilton-Jacobi-Bellman equation[END_REF]. For x * = 1, 0 ≤ t * ≤ T, we set

u(t * , 1) = u 0 (1) + (f (1) -1)t * . (19)
Let (t * , x * ) ∈ G -and x * > -1. We assign

u(t * , x * ) = u 0 (-1) + t * 0 p(τ )H p (x(τ ), p(τ )) -H(x(τ ), p(τ )) dτ, (20)
where

x(t) = x -(t, p 0 (t * , x * )), p(t) = p -(t, p 0 (t * , x * ))
is the solution of the problem of Calculus Variations ( 13),( 15), [START_REF] Subbotina | The method of characteristics for Hamilton-Jacobi-Bellman equation[END_REF].

For

x * = -1, 0 ≤ t * ≤ T, we set u(t * , -1) = u 0 (-1) + (f (-1) -1)t * . (21)
So, we have defined function u(•) for all points from Π T by relations (17)-(21). Following the Cauchy method of characteristics [START_REF] Courant | Methods of Mathematical Physics[END_REF], one can show that u(•) is continuously differentiable at any interior point (t, x) ∈ G + ∪ G -, and the gradient of u(•) is equal to (-H(x, p(t)), p(t)). Theorems 2-5 imply that u(•) is continuous in Π T , and inequalities ( 6)-( 7) are valid. Below, in section 6.2, we will show that D -u(t, x) ∩ ∂u(t, x) = ∅, (t, x) ∈ Γ T . So, u(•) is a generalized solution of problem ( 1)-( 3) in sense of Definition 1. It follows from (17) that u(•) satisfies the prescribed property [START_REF] Crandall | Viscosity Solutions of Hamilton-Jacobi Equations at the Boundary[END_REF].

Numerical example

Results of simulation for the input data u 0 (x) = -0.02x 2 + 0.001 cos 2πx, f (•) = -0.5x 2 are presented in Fig. 2. One can easily check that these input data satisfy the conditions B1, B2.

Comparison with Viscosity Solution

One can see that Definition 1 coincides with the definition of viscosity solution in the interior points of the region Π T . The difference between these definitions is evident at boundary points, namely, on the set Γ T . In condition (8), the inequality holds for such points (a, s) of the subdifferential D -u(t, x) which at the same time belong to the set ∂u(t, x). In contrast to Definition 1, the notion of viscosity solution [START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi Equations with State Constraints[END_REF] for equation (1) on the set Π T requires that this solution satisfies inequality (8) at the boundary points (t, x) ∈ Γ T for all (a, s) ∈ D -u(t, x).

On coercivity condition

Conditions ensuring existence and uniqueness for viscosity solutions on the compact domain were obtained in [START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi Equations with State Constraints[END_REF]. One of the key conditions for the existence of viscosity solutions (see [START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi Equations with State Constraints[END_REF][START_REF] Crandall | Viscosity Solutions of Hamilton-Jacobi Equations at the Boundary[END_REF]) is the coercivity of the Hamiltonian:

H(x, p) → +∞ as |p| → ∞ (22)
It can be easily checked that the Hamiltonian (2) does not satisfy condition (22), for example, at x = 1 and x = -1. Therefore, the existence theorems for viscosity solutions in the sense of paper [START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi Equations with State Constraints[END_REF] cannot be used in the problem (1)-( 3). Moreover, the notion of generalized viscosity solution is inapplicable to the initial value problem ( 1)-( 3) on the compact setΠ T : If a viscosity solution would satisfy the condition D -u(t * , x * ) = ∅ at some point (t * , x * ) ∈ Γ T , then inequality [START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF] would not hold in this case. Indeed, let 0 ≤ t * ≤ T and, for definiteness, x * = 1. Let (a, s) ∈ D -u(t * , x * ). One can use the above definition of subdifferential to get the following inclusion: (a, s + k) ∈ D -u(t * , x * ) ∀k ≥ 0. So, if condition (8) would hold then a + H(1, s + k) = a -f (x) + 1 -e 2(s+k) ≥ 0, ∀k ≥ 0, which is obviously unfair. Therefore, we use the intersection of the subdifferential with the set ∂u(t, x) (see condition [START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF]) in Definition 1 of a continuous generalized solution to the initial value problem (1)-(3) on the compact set Π T .

Structure of subdifferentials on the border

Let's consider the structure of the sets D -u(t, x) and ∂u(t, x) for the function u(•) defined by ( 17)-(20) if (t, x) ∈ Γ T .

In the case 0 < t < T, x = 1, we have D -u(t, x) = D -u(t, 1) = {(f (1) -1, s)| s ∈ R, s ≥ 0}, ∂u(t, x) = ∂u(t, 1) = {(-H(1, -∞), -∞)} = {(f (1) -1, -∞)}.

In the case 0 < t < T, x = -1, we have D -u(t, x) = D -u(t, -1) = {(f (-1) -1, s)| s ∈ R, s ≤ 0}, ∂u(t, x) = ∂u(t, -1) = {(-H(-1, ∞), ∞)} = {(f (-1) -1, ∞)}.

Thus, for generalized solution (17)-(20)

D -u(t, x) ∩ ∂u(t, x) = ∅, (t, x) ∈ Γ T .

  (1)-(3) has the formẋ = H p (x, p) = -(1 + x)e 2p + (1 -x)e -2p , ṗ = -H x (x, p) = f (x) + (e 2p -e -2p )/2,(4)ż = pH p (x, p) -H(x, p), with initial conditions x(0, y) = y, p(0, y) = u 0 (y), z(0, y) = u 0 (y), y ∈ [-1; 1].

Fig. 1 .

 1 Fig. 1. State characteristics for the case f (x) = -0.25x 2 , u0(x) = 0.25(x -0.5) 2 -0.1 cos 2πx.

Fig. 2 .

 2 Fig.2. The graph of the generalized solution and its projection on the (t, x)-plane for the input data u0(x) = -0.02x 2 + 0.001 cos 2πx, f (x) = -0.5x 2 .

Acknowledgments. This work was partially supported by the Russian Foundation for Basic Research (project no. 14-01-00168) and the Complex Program of Ural Branch of RAS (project no. 15-16-1-11).