Lipschitz continuity of the Fréchet gradient in an inverse coefficient problem for a parabolic equation with Dirichlet measured output

Lemma A.1.

Let conditions (1.3) hold. Then for the trace norm ∥u⁢(0,⋅)∥L2⁢(0,T) of the weak solution of the direct problem (1.1) the following estimate holds:

Proof.

We use the identity

Applying to the right-hand side the inequality (a-b)2≤2⁢(a2+b2), then using the Hölder inequality and integrating on [0,t], t∈[0,T], we deduce that

Integrating then on [0,l] and dividing both sides by l>0, we arrive at the required estimate (A.1). ∎

Lemma A.2.

Let conditions (1.3) hold. Then for the weak solution u∈L2⁢(0,T;H1⁢(0,l)) of the direct problem (1.1) with ut∈L2⁢(0,T;H-1⁢(0,l)) the following estimates hold:

where

and c0>0 is the constant defined in (1.3).

Proof.

Multiply both sides of equation (1.1) by u⁢(x,t), integrate on [0,l] and use the integration by parts formula. Taking into account the boundary conditions in (1.1), integrating then both sides on [0,t], t∈[0,T], and using the initial condition u⁢(x,0)=0, we obtain the following energy identity:

for a.e. t∈[0,T]. We apply the ε-inequality 2⁢a⁢b≤(1/ε)⁢a2+ε⁢b2 in the right-hand-side integral to get

for a.e. t∈[0,T]. To estimate the last integral in (A.5) we use estimate (A.1). Then we conclude that

for a.e. t∈[0,T] and ε∈(0,2⁢c0/l). Taking for convenience this parameter as ε=c0/l, we obtain

The first consequence of (A.7) is the inequality

for a.e. t∈[0,T]. Applying the Gronwall–Bellman inequality [1], we deduce that

where C1>0 is the constant defined in (A.4). Remember that according to [5, Section 5.9, Theorem 3] the weak solution u∈L2⁢(0,T;H1⁢(0,l)) with ut∈L2⁢(0,T;H-1⁢(0,l)) is in C⁢([0,T];L2⁢(0,l)). This implies estimate (A.2).

The second consequence of (A.7) is the inequality

By estimate (A.8), this inequality implies the second required estimate (A.3). ∎

Corollary A.3.

Let the conditions of Lemma A.2 hold. Then

where C1>0 and C2>0 are the constants defined in (A.4).

Corollary A.4.

Let the conditions of Lemma A.2 hold. Then for the weak solution of the adjoint problem (5.3) the following estimates hold:

where C1,C2,C3>0 are the constants defined in (A.4) and (A.9).

Proof.

Using the transformation for the time variable τ=T-t in the adjoint (backward) problem (5.3), we deduce that it becomes the well-posed problem

for the function ψ⁢(x,τ):=φ⁢(x,T-t), where q^⁢(τ)=-2⁢[u⁢(0,τ;k)-f⁢(τ)] and q^⁢(0)=0. Applying Lemma A.1 to the well-posed problem (A.12), we conclude that

Using here the inequality (a-b)≤2⁢(a2+b2), applying estimate (A.9) to the norm ∥u⁢(0,⋅)∥L2⁢(0,T) and then returning to the original function φ⁢(x,t), we arrive at the required estimates (A.10) and (A.11). ∎

Lemma A.5.

Let k1,k2∈L∞⁢(0,l) and v⁢(x,t):=u⁢(x,t;k1)-u⁢(x,t;k2). Assume that conditions (1.3) hold. Then for the weak solution v∈L2⁢(0,T;H1⁢(0,l)) of problem (2.7) the following estimates hold:

with the constants defined in Lemma A.1.

Proof.

As in the proof of Lemma A.2, we multiply both sides of equation (2.7) by v⁢(x,t), integrate on Ωt, t∈(0,T], and use the integration by parts formula. Then we obtain the following integral identity:

for all t∈[0,T], where δ⁢k⁢(x)=k1⁢(x)-k2⁢(x). The terms under the last integrals on the left- and right-hand side are zero at x=l due to the homogeneous boundary condition vx⁢(l,t)=0 in (2.7). Taking into account the Neumann boundary condition -k1⁢(0)⁢vx⁢(0,t)=δ⁢k⁢(0)⁢u2⁢x⁢(0,t;k2) in the last left-hand-side integral, we deduce that this term and the term at x=0 under the last right-hand-side integral are mutually exclusive. Then applying the ε-inequality to the first right-hand-side integral, after elementary transformations we obtain the following inequality:

for t∈[0,T], with ε∈(0,2⁢c0). Choosing this parameter as ε=c0, we get

for a.e. t∈[0,T]. Using here estimate (A.3), we obtain from this inequality the desired estimates (A.13) and (A.14). ∎

Lemma A.6.

Let the conditions of Lemma A.2 hold. Then for the weak solution

of problem (5.4) the following estimate holds:

where δ⁢k⁢(x)=k1⁢(x)-k2⁢(x),

and c0,C2,C3>0 are the constants defined in Lemma A.1 and Corollary A.3.

Proof.

We employ the same transformation τ=T-t used in Corollary A.4, then use the proof scheme of Lemma A.2 for the well-posed problem

for the function ψ⁢(x,τ):=φ⁢(x,T-t), where δ⁢w⁢(0,τ):=δ⁢u⁢(0,T-t). Specifically, multiplying both sides of (A.17) by δ⁢ψ⁢(x,τ), integrating it over (0,l)×(0,τ) and taking into account the boundary conditions in (A.17), we obtain the following integral identity:

for a.e. τ∈(0,T]. We apply the ε-inequality 2⁢a⁢b≤(1/ε)⁢a2+ε⁢b2 to the right-hand-side integrals with subsequent use of inequality (A.1) in the second right-hand-side integral. After elementary transformations, above integral identity becomes the inequality

Choosing the arbitrary parameter ε∈(0,2⁢c0/l) as ε=c0/l, we obtain

for a.e. τ∈(0,T]. Evidently, at τ=T the second, third and fourth right-hand-side integrals for the functions ψ2⁢x2⁢(x,ζ), δ⁢w2⁢(x,ζ) and δ⁢wx2⁢(x,ζ) are equal to the integrals for the functions φ2⁢x2⁢(x,t), δ⁢u2⁢(x,t) and δ⁢ux2⁢(x,t), respectively, and for these integrals we can use estimates (A.11), (A.13) and (A.14). Substituting these estimates into (A.18), we conclude that

where C42,C52>0 are defined in (A.16).

As in the proof of Lemma A.2, we first estimate from inequality (A.19) the norm ∥ψ∥L2⁢(0,T;L2⁢(0,l)), then, substituting it in the inequality

we estimate the norm ∥ψx∥L2⁢(0,T;L2⁢(0,l)). Returning then to the function φ⁢(x,t), by the above reason we arrive at the required estimate (A.15). ∎

Lemma A.7.

Let conditions (2.1) hold. Then for the regular weak solution with improved regularity defined by (2.2) the following estimates hold:

where C1,C2>0 are the constants defined in (A.4).

Proof.

Differentiate formally equation (1.1) with respect to t∈(0,T), multiply both sides by ut⁢(x,t), integrate over [0,l] and use the integration by parts formula. Taking then into account the boundary conditions, we obtain the following integral identity:

Integrating this identity on [0,t], t∈[0,T], and using the limit equation

with the initial condition u⁢(x,0)=0, we deduce that

for all t∈[0,T]. Applying now the ε-inequality 2⁢a⁢b≤(1/ε)⁢a2+ε⁢b2 to the last right-hand-side integral, we obtain the inequality

Using the same argument as in the proof of inequality (A.1), we estimate the first right-hand-side integral as follows:

for all t∈[0,T]. Substituting this in the above inequality, we conclude

This is the same inequality (A.6) with u⁢(x,t) replaced by ut⁢(x,t), and g⁢(t) replaced by g′⁢(t). Repeating the same procedure as in the proof of Lemma A.1, we deduce from this inequality the required estimates (A.20) and (A.21). ∎

Corollary A.8.

Let the conditions of Lemma A.6 hold. Then

where C3>0 is the constant defined in (A.9).

Corollary A.9.

Let the conditions of Lemma A.6 hold. Assume, in addition, that the measured output f⁢(t) satisfies the regularity condition (5.10), that is, f∈V⁢(0,T). Then for the regular weak solution with improved regularity of the adjoint problem (5.3) the following estimates hold:

Proof.

Using the transformation τ=T-t, we can prove in the same way as in Lemma A.5 that estimates similar to (A.20) and (A.21) hold also for the regular weak solution of the adjoint problem (5.3). That is,

with the same constants C1,C2>0 defined in (A.4). Using here the inequality (a-b)2≤a2+b2 and estimate (A.23), we arrive at the required estimates (A.24) and (A.25). ∎

Lemma A.10.

Let conditions (2.1) and (5.9) hold. Then for the regular weak solution with improved regularity of problem (1.1) the following estimate holds: