A doubly nonlinear evolution problem related to a model for microwave heating
aa r X i v : . [ m a t h . A P ] N ov A doubly nonlinear evolution problemrelated to a model for microwave heating ∗ Luca Scarpa
Dipartimento di Matematica "F. Casorati"Università di PaviaVia Ferrata 1, 27100 Pavia, ItalyE-mail: [email protected]
Abstract
This paper is concerned with the existence and uniqueness of the solution toa doubly nonlinear parabolic problem which arises directly from a circuit model ofmicrowave heating. Beyond the relevance from a physical point of view, the problemis very interesting also in a mathematical approach: in fact, it consists of a nonlinearpartial differential equation with a further nonlinearity in the boundary condition.Actually, we are going to prove a general result: the two nonlinearities are allowedto be maximal monotone operators and then an existence result will be shown forthe resulting problem.
AMS Subject Classification:
Key words and phrases: nonlinear parabolic equation, nonlinear boundary con-dition, existence of solutions.
In this paper, we deal with a problem that stems from a circuit model of microwaveheating (see [4] for details); in particular, we aim at proving the existence of a solution fora nonlinear partial differential equation with appropriate initial and boundary conditions.More specifically, we consider a model of a RLC circuit in which a thermistor has beeninserted: this one has a cylindrical shape and takes into account the temperature’s effects.A system of three equations is obtained in [4]; this system involves the voltage V acrossthe capacitor, resistor and inductance, the potential Φ associated to the electrostatic fieldin the thermistor and the temperature ϑ . In order to prove the existence of a solution ofthe entire system, the first step is showing the existence of a solution for the equation for ∗ Acknowledgment.
The author is grateful to Pierluigi Colli for suggesting the problem and for hisexpert advice and fundamental support throughout this project.
A doubly nonlinear evolution problem for microwave heating ϑ , complemented by a special boundary condition and initial condition. This is the aimof our work.Firstofall, the thermistor is modelled by a cylinder C l ⊆ R : D := B R (0) ⊆ R , C l := D × (0 , l ) , l, R > (1.1) B := ∂D × (0 , l ) , D I := D × { } , D F := D × { l } . (1.2)The equation in ϑ which arises from the model is the following (see [4]) c ϑ t − ∆ K ( ϑ ) = g in C l × (0 , T ) , (1.3)where T > , c > , g ∈ L ( C l × (0 , T )) and K : R → R is a Lipschitz-continuousincreasing function such that K (0) = 0 . Please note that if we write (1.3) using thevariable u = K ( ϑ ) , we obtain c ( γ ( u )) t − ∆ u = g in C l × (0 , T ) , (1.4)where γ := K − : R → R (1.5)is the inverse graph of K . In the application we are considering, γ is still a Lipschitz-continuous increasing function with γ (0) = 0 .Finally, the appropriate boundary conditions arising from the model are − ∂u∂n = ( β ( u ) − β ( u B ) in B × (0 , T ) , in D I ∪ D F × (0 , T ) , (1.6)where u B ∈ R is a datum and β : R → R , β ( r ) = hγ ( r ) + s | γ ( r ) | γ ( r ) , h, s > . (1.7)Note that β is a continuous increasing function such that β (0) = 0 .In this paper, we are interested to discuss the initial and boundary value problem c ( γ ( u )) t − ∆ u = g in C l × (0 , T ) (1.8) − ∂u∂n = β ( u ) − β ( u B ) in B × (0 , T ) (1.9) − ∂u∂n = 0 in D I ∪ D F × (0 , T ) (1.10) ( γ ( u )) (0) = v in C l . (1.11)The system (1.8)–(1.11) is indeed very interesting from a physical point of view. As amatter of fact, the variable u is strictly correlated to the temperature ϑ (we recall that u = K ( ϑ ) ): more specifically, the boundary condition (1.10) tells that the flux acrossthe top and the bottom of the cylindrical thermistor is null, while (1.9) specifies therelation between the flux across the lateral surface of the thermistor and the environment’stemperature. uca Scarpa t , and the second appears in the boundary condition (1.9). Inthis work, we are going to prove the existence of a solution of a more general problem,in which (1.8)–(1.11) appears as a special case. More precisely, we will consider an openset Ω ⊆ R n instead of the three-dimensional cylinder C l , and we will write ∂ Ω = Γ ∪ Γ ,where Γ ∩ Γ = ∅ : Γ will be the generalization of D I ∪ D F , while Γ replaces B . Fur-thermore, we will deal with two maximal monotone operators γ and β instead of thesingle-valued increasing functions (1.5) and (1.7) respectively.In the second section, we will illustrate the main results and the strategy of the proofs:relying on an important result by Di Benedetto - Showalter (see [3]), we will obtain a firstlemma, which proves the existence for a simplified problem, and then we will use this toprove the theorem ensuring the existence of a solution to (1.8)–(1.11). Finally, we willpresent a continuous dependence and uniqueness result.The third section is devoted to the proof of the lemma: it consists essentially inchecking the hypotheses of the result contained in [3] for our specific framework. In thissense, the most interesting point is the use of a fixed point argument in order to controlthe maximal monotonicity of a particular operator.The fourth section contains the proof of the existence of a solution to our problem: wewill show some uniform estimates in order to pass to the limit accurately and thus obtainthe desired result.The fifth section contains the proof of the continuous dependence and uniquenessresult: this one is provided in a simplified setting, in which a linearity assumption for γ is in order. In this section, we will focus our attention on the problem (1.8)–(1.11) and we will il-lustrate the two main results which prove the existence of a solution and the continuousdependence theorem. As we have anticipated, we abandon the specific framework of theprevious section and consider a more abstract setting: in this sense, the most importantgeneralization consists in dealing with two general maximal monotone operators on R ,instead of considering the given single-valued increasing functions (1.5) and (1.7). Fur-thermore, we also relax the hypothesis on the cylindrical shape of domain (1.1).More precisely, we now work in the following setting: Ω ⊆ R n smooth bounded domain , Γ := ∂ Ω , Γ = Γ ∪ Γ , Γ ∩ Γ = ∅ ; (2.1) γ : R → R maximal monotone ; (2.2) β : R → R maximal monotone , ∈ D ( β ) . (2.3)As we have anticipated, (2.1) is the natural generalization of environment (1.1)-(1.2): Γ here plays the role of D I ∪ D F , while γ and β are the extensions of (1.5) and (1.7), A doubly nonlinear evolution problem for microwave heating respectively. Hence, it will be natural to assume good properties on γ (such as linearboundedness), while β should be treated as generally as possible, at least at the beginning.The problem we are dealing with is the following: ∂∂t v − ∆ u = g in Ω × (0 , T ) (2.4) v ∈ γ ( u ) in Ω × (0 , T ) (2.5) − ∂u∂n ∈ β ( u ) − h in Γ × (0 , T ) , ∂u∂n = 0 in Γ × (0 , T ) (2.6) v (0) = v in Ω , (2.7)where g ∈ L (0 , T ; L (Ω)) and h ∈ L (0 , T ; L (Γ )) . As a remark, note that ∂∂n indicatesthe outward normal derivative on Γ and that we have used no particular notation for thetraces on Γ or Γ (the contest is quite clear).We would like to write a variational formulation of problem (2.4)–(2.7). Note that forall z ∈ H (Ω) , integrating by parts and taking into account condition (2.6), we have − Z Ω ∆ u ( t ) z dx = Z Ω ∇ u ( t ) · ∇ z dx + Z Γ ( ξ ( t ) − h ( t )) z ds , for some ξ ( t ) ∈ β ( u ( t )) a.e. on Γ . Hence, testing equation (2.4) by an arbitrary function z ∈ H (Ω) and integrating, we reach the following variational formulation: (cid:28) ∂v∂t ( t ) , z (cid:29) + Z Ω ∇ u ( t ) · ∇ z dx + Z Γ ξ ( t ) z ds = Z Ω g ( t ) z dx + Z Γ h ( t ) z ds for almost all t ∈ (0 , T ) and ∀ z ∈ H (Ω) , (2.8)where v ∈ γ ( u ) a.e. in Ω × (0 , T ) , (2.9) ξ ∈ β ( u ) a.e. in Γ × (0 , T ) . (2.10)In order to prove the existence of a solution for problem (2.4)–(2.7), the idea is toaccurately use an abstract result by Di Benedetto - Showalter, which we briefly remind(see [3] for further details). In particular, in this sense, we will be forced to require moreregularity on γ ; however, as we have anticipated, our assumptions will be acceptable ifwe keep in mind our application. Theorem 2.1.
Let W be a reflexive Banach space and V a Hilbert space which is denseand embedded compactly in W . Denote the injection by i : V → W and the dual operator(restriction) by i ∗ : W ∗ → V ∗ . Define A := i ∗ ◦ ∂φ ◦ i . Assume the following: • [A1] The real-valued function φ is proper, convex and lower semicontinous on W ,continuous at some point of V , and ∂φ ◦ i : V → W ∗ is bounded. • [A2] The operator ∂φ ◦ i : L (0 , T ; V ) → L (0 , T ; W ∗ ) is bounded. • [B1] The operator B : V → V ∗ is maximal monotone and bounded. uca Scarpa • [B2] The operator B : L (0 , T ; V ) → L (0 , T ; V ∗ ) is bounded and coercive, i.e. lim k u k L ,T ; V ) → + ∞ [ u,v ] ∈ B R T h v ( t ) , u ( t ) i dt k u k L (0 ,T ; V ) = + ∞ . (2.11) Then, for each f ∈ L (0 , T ; V ∗ ) and v ∈ Rg ( A ) , there exists a triple u ∈ L (0 , T ; V ) , v ∈ H (0 , T ; V ∗ ) , w ∈ L (0 , T ; V ∗ ) such that ddt v ( t ) + w ( t ) = f ( t ) , v ( t ) ∈ A ( u ( t )) , w ( t ) ∈ B ( u ( t )) a.e. t ∈ [0 , T ] (2.12) v (0) = v . (2.13)In our specific framework, we want to apply Theorem 2.1 to the spaces V = H (Ω) and W = L (Ω) : to be precise, in the development of the work we will make the identification L (Ω) ∼ = L (Ω) ′ . In the notation of the theorem, the clear choice of f is the following: h f ( t ) , z i = Z Ω g ( t ) z dx + Z Γ h ( t ) z ds , ∀ z ∈ H (Ω) , ∀ t ∈ [0 , T ] . (2.14)Furthermore, as far as A is concerned, it is natural to introduce ¯ A ( u ) = { v ∈ L (Ω) : v ( x ) ∈ γ ( u ( x )) a.e. x ∈ Ω } , (2.15) D ( ¯ A ) = { u ∈ H (Ω) : ∃ v ∈ L (Ω) such that v ( x ) ∈ γ ( u ( x )) a.e. x ∈ Ω } ; (2.16)thus, our choice of the operator A is A := i ∗ ◦ ¯ A , D ( A ) = D ( ¯ A ) . (2.17)However, while most of the assumptions of the theorem are satisfied by these particularchoices of V , W , f and A , on the other side, the intuitive way of considering B , i.e. h B ( u ) , z i = Z Ω ∇ u · ∇ z dx + Z Γ ξz ds , ξ ∈ β ( u ) a.e. on Γ , ∀ z ∈ H (Ω) , (2.18) D ( B ) = { u ∈ H (Ω) : ∃ ξ ∈ L (Γ ) such that ξ ∈ β ( u ) a.e. on Γ } , (2.19)gives us many problems: in particular, the coercivity condition (2.11) is not evident at thislevel. So, we modify expression (2.18) by adding a "correction term", which we hope willgive us the required coercivity: more precisely, we consider the following regularizationof B , h B λ ( u ) , z i = λ Z Ω uz + Z Ω ∇ u · ∇ z dx + Z Γ β λ ( u ) z ds , ∀ z ∈ H (Ω) , (2.20)where λ > is fixed and β λ indicates the Yosida approximation of β (please note that weare not using a specific notation for the traces on Γ ). The idea is to apply Theorem 2.1to B λ instead of B , for λ > fixed. As a consequence, the theorem itself will notdirectly give us a solution u for problem (2.8)–(2.10), but we will obtain a solution u λ satisfying the respective modified problem. The second step will be to prove estimates on u λ independent of λ , and then passing to the limit as λ → and finding a solution of theoriginal problem. A doubly nonlinear evolution problem for microwave heating
Remark 2.1.
Let us recall some properties of the Yosida approximation which will beuseful in the following sections: for further details about these remarks, the reader canrefer to [1, 2].Let ( H, k·k ) be a Hilbert space and A a maximal monotone operator on H : then, forall λ > , the Yosida approximation A λ of A is defined as A λ = I − J λ λ , where J λ = ( I + λA ) − is the resolvent , (2.21)while for all x ∈ D ( A ) , we will write A x for the minimum-norm element of Ax . Withthese notations, the following properties hold: • A λ is a monotone and single-valued operator such that for all x ∈ H , A λ x ∈ AJ λ x ; • A λ is Lipschitz continuous (thus maximal monotone) with Lipschitz constant ≤ λ ; • for all µ, λ > , ( A µ ) λ = A µ + λ • for all x ∈ D ( A ) , | A λ x | ≤ | A x | and λ
7→ | A λ x | is non increasing as λ ց ; • as λ ց , { A λ x } is bounded if and only if x ∈ D ( A ) , and in this case A λ x → A x .Furthermore, we will use the following result: if ϕ is a convex, proper and lower semicon-tinuous function on H such that A = ∂ϕ , then D ( A ) ⊆ D ( ϕ ) ⊆ D ( ϕ ) ⊆ D ( A ) . (2.22)Finally, if we define ϕ λ ( x ) = min y ∈ H (cid:26) λ | y − x | + ϕ ( y ) (cid:27) , x ∈ H , λ > , (2.23)then ϕ λ is convex, Fréchet-differentiable with differential A λ and we have ϕ λ ( x ) ր ϕ ( x ) as λ ց ∀ x ∈ H . (2.24)To summarize, the results we are going to present in this section could be brieflydescribed as follow: the first theorem tells us that Theorem 2.1 can be applied with theparticular choices (2.14), (2.17) and (2.20), the second theorem gives us a solution forthe original problem (2.8)–(2.10), and the third ensures that a continuos dependence anduniqueness result holds.
Theorem 2.2.
Let Ω , γ , β be as in (2.1) – (2.3) , and also suppose that D ( γ ) = R , γ (0) ∋ , β (0) ∋ , (2.25) ∃ C , C > such that | y | ≤ C | x | + C ∀ x ∈ R , ∀ y ∈ γ ( x ) . (2.26) For g ∈ L (0 , T ; L (Ω)) and h ∈ L (0 , T ; L (Γ )) , let f , A and B λ as in (2.14) , (2.17) and (2.20) . Then, for any given pair ( u , v ) ∈ (cid:0) L (Ω) (cid:1) such that v ∈ γ ( u ) a.e. in Ω , (2.27) uca Scarpa there exist u λ ∈ L (0 , T ; H (Ω)) and v λ ∈ H (0 , T ; H (Ω) ′ ) ∩ L (0 , T ; L (Ω)) such that ∂∂t v λ ( t ) + B λ ( u λ ( t )) = f ( t ) for a.e. t ∈ [0 , T ] , (2.28) v λ ( t ) ∈ A ( u λ ( t )) for a.e. t ∈ [0 , T ] , (2.29) v λ (0) = v . (2.30) Theorem 2.3.
Let Ω , γ , β , f and A as in (2.1) – (2.3) , (2.14) and (2.17) respectively,and suppose that conditions (2.25) and (2.26) of the previous theorem hold. Furthermore,assume that γ is a bi-Lipschitz continuous function(i.e., both γ and γ − are Lipschitz continuous) , (2.31) D ( β ) = R and ∃ D , D > such that (cid:12)(cid:12) β ( r ) (cid:12)(cid:12) ≤ D ˆ β ( r ) + D , (2.32) g ∈ L (0 , T ; L (Ω)) ∩ L (0 , T ; L ∞ (Ω)) , h ∈ L (0 , T ; L (Γ )) , (2.33) where ˆ β is a proper, convex and lower semicontinuous function such that ∂ ˆ β = β , and β ( r ) is the minimum-norm element in β ( r ) . Then, for any given pair ( u , v ) ∈ (cid:0) L (Ω) (cid:1) such that v = γ ( u ) a.e. in Ω and ˆ β ( u ) ∈ L (Ω) , (2.34) there are u ∈ L (0 , T ; H (Ω)) , v ∈ H (0 , T ; H (Ω) ′ ) ∩ L (0 , T ; L (Ω)) , ξ ∈ L (0 , T ; L (Γ )) such that (cid:28) ∂v∂t ( t ) , z (cid:29) + Z Ω ∇ u ( t ) · ∇ z dx + Z Γ ξ ( t ) z ds = Z Ω g ( t ) z dx + Z Γ h ( t ) z ds for a.e. t ∈ (0 , T ) and ∀ z ∈ H (Ω) , (2.35) v ( t ) ∈ A ( u ( t )) for a.e. t ∈ (0 , T ) , ξ ∈ β ( u ) a.e. in (0 , T ) × Γ , (2.36) v (0) = v . (2.37) Remark 2.2.
We can notice that hypothesis (2.32) is not too restrictive on β itself: asa matter of fact, β is allowed to have polynomial growth or even a first order exponentialgrowth. Remark 2.3.
Thanks to the assumption (2.3) on γ , the first inclusion in (2.36) yields v ( t ) = γ ( u ( t )) for a.e. t ∈ (0 , T ) (cf. (2.15)–(2.17)), so that it is possible to deduce ahigher regularity of v , that is v ∈ L (0 , T ; H (Ω)) . (2.38)Furthermore, since v ∈ H (0 , T ; H (Ω) ′ ) ∩ L (0 , T ; H (Ω)) , we also have that v ∈ C ([0 , T ]; L (Ω)) . (2.39) A doubly nonlinear evolution problem for microwave heating
Theorem 2.4.
Under the hypotheses of the previous theorem, let us also suppose that γ is linear, i.e. there exists α > such that γ ( r ) = αr , r ∈ R . (2.40) Then, there exists a constant
C > , which depends only on α , Ω and T , such that for everysolution ( u i , v i , ξ i ) of the problem (2.35) – (2.37) corresponding to the data { u i , h i , g i } , for i = 1 , , the following continuous dependence property holds: k u − u k L (0 ,T ; H (Ω)) ∩ L ∞ (0 ,T ; L (Ω)) ≤ C h(cid:13)(cid:13) u − u (cid:13)(cid:13) L (Ω) + k g − g k L (0 ,T ; L (Ω)) + k h − h |k L (0 ,T ; L (Γ )) i (2.41) In particular, problem (2.35) – (2.37) has a unique solution. The idea is to apply Theorem 2.1, for λ > fixed, in the spaces V = H (Ω) , W = L (Ω) ,to the operators f , A and B λ defined in (2.14), (2.17) and (2.20) respectively. Note atfirst that in this case W is a reflexive Banach space and V is a Hilbert space which isdense and compactly embedded in W . So, if we are able to check hypotheses [ A , [ A , [ B and [ B , then we can apply Theorem 2.1, and Lemma 2.2 follows directly. Theobjective of this section is therefore to control the hypotheses of the theorem for f , A and B λ . First of all, γ is maximal monotone on R , so there exists ˆ γ : R → ( −∞ , + ∞ ] convex,proper and lower semicontinuous such that ∂ ˆ γ = γ : it is not restrictive to suppose that ˆ γ (0) = 0 (by adding an appropriate constant). Furthermore, the condition γ (0) ∋ implies that is a minimizer for ˆ γ , so we have ˆ γ : R → [0 , + ∞ ] .Let us consider φ : L (Ω) → [0 , + ∞ ] given by φ ( u ) = (R Ω ˆ γ ( u ( x )) dx if ˆ γ ( u ) ∈ L (Ω)+ ∞ otherwise. (3.1)We know from the general theory that φ is proper, convex and lower semicontinuous on L (Ω) , and that i ∗ ◦ ∂φ ◦ i = A , where A is defined in (2.17).We have to check that φ is continuous at some point of H (Ω) . Let’s first prove that D ( ∂φ ) = L (Ω) : let u ∈ L (Ω) and v : Ω → R be a measurable function such that v ( x ) ∈ γ ( u ( x )) almost everywhere in Ω . Then, condition (2.26) implies that | v ( x ) | ≤ C | u ( x ) | + C for a.e. x ∈ Ω ; hence, it turns out that v ∈ L (Ω) , and u ∈ D ( ∂φ ) . As a consequence, D ( φ ) = L (Ω) ,since in general D ( ∂φ ) ⊆ D ( φ ) ; furthermore, since each convex, proper and lower semi-continuous function on a Banach space is continuous at the interior of its domain (for uca Scarpa φ is continuous everywhere in L (Ω) .In particular, we have φ : L (Ω) → [0 , + ∞ ) .Finally, we have to check that ∂φ ◦ i : H (Ω) → L (Ω) is bounded. Let u ∈ H (Ω) and v ∈ ( ∂φ ◦ i )( u ) : then, v ( x ) ∈ γ ( u ( x )) for almost every x ∈ Ω and condition (2.26)implies that | v ( x ) | ≤ C | u ( x ) | + C for a.e. x ∈ Ω , from which, passing to the norms, we deduce the required boundedness. We only have to control that the operator ∂φ ◦ i : L (0 , T ; H (Ω)) → L (0 , T ; L (Ω)) isbounded (more precisely, we are considering the operator induced by ∂φ ◦ i on the time-dependent spaces by the a.e. relation). Let u ∈ L (0 , T ; H (Ω)) and v ∈ ( ∂φ ◦ i )( u ) : then, u ( t ) ∈ H (Ω) and v ( t ) ∈ ∂φ ( u ( t )) almost everywhere in (0 , T ) , so, as we have observedabove, we have | v ( t )( x ) | ≤ C | u ( t )( x ) | + C for a.e. ( x, t ) ∈ Ω × (0 , T ) , which easily implies the required boundedness. We now have to control that B λ : H (Ω) → H (Ω) ′ is maximal monotone and bounded:let us start from the last property. Recall that for all ζ ∈ H (Ω) , the trace of ζ on Γ isan element of L (Γ ) and k ζ k L (Γ ) ≤ C k ζ k H (Ω) for some constant C>0 (independent of ζ ) ; (3.2)furthermore, β λ is λ -Lipschitz continuous on L (Γ ) and β λ (0) = 0 , then the followingholds: k β λ ( ζ ) k L (Γ ) ≤ λ k ζ k L (Γ ) . Hence, if u ∈ H (Ω) , then for all z ∈ H (Ω) , taking (2.20) an (3.2) into account, we have |h B λ ( u ) , z i| ≤ λ k u k L (Ω) k z k L (Ω) + k∇ u k L (Ω) k∇ z k L (Ω) + k β λ ( u ) k L (Γ ) k z k L (Γ ) ≤ (cid:20) max { λ, } + C λ (cid:21) k u k H (Ω) k z k H (Ω) ∀ z ∈ H (Ω) , from which we obtain k B λ ( u ) k H (Ω) ′ ≤ (cid:20) max { λ, } + C λ (cid:21) k u k H (Ω) ∀ u ∈ H (Ω) , (3.3)that is our required boundedness on B λ .We now have to control that B λ is maximal monotone: in this sense, we check it usinga characterization of the maximal monotonicity, i.e. we show that Rg ( R + B λ ) = H (Ω) ′ ,0 A doubly nonlinear evolution problem for microwave heating where R : H (Ω) → H (Ω) ′ is the usual Riesz operator. For any given F ∈ H (Ω) ′ , wehave to find u ∈ H (Ω) (which will depend a posteriori on λ , of course) such that Z Ω uz dx + Z Ω ∇ u · ∇ z dx + h B λ ( u ) , z i = h F, z i ∀ z ∈ H (Ω) , or in other words that (1 + λ ) Z Ω uz dx + 2 Z Ω ∇ u · ∇ z dx + Z Γ β λ ( u ) z ds = h F, z i ∀ z ∈ H (Ω) . (3.4)First of all, we introduce β ǫλ as β ǫλ ( r ) = β λ ( r ) if | β λ ( r ) | ≤ ǫ ǫ if β λ ( r ) > ǫ − ǫ if β λ ( r ) < − ǫ (3.5)and for a fixed ǫ > , we look for u ǫ ∈ H (Ω) such that (1 + λ ) Z Ω u ǫ z dx + 2 Z Ω ∇ u ǫ · ∇ z dx + Z Γ β ǫλ ( u ǫ ) z ds = h F, z i ∀ z ∈ H (Ω) . (3.6)The idea is to use a fixed point argument in the following sense: let δ ∈ (0 , / and ¯ u ∈ H − δ (Ω) . We now solve for a fixed ¯ u the following variational equation: (1 + λ ) Z Ω u ǫ z dx + 2 Z Ω ∇ u ǫ · ∇ z dx = − Z Γ β ǫλ (¯ u ) z ds + h F, z i ∀ z ∈ H (Ω) . (3.7)Please note that for such a choice of ¯ u , the trace of ¯ u on Γ is in L (Γ ) , and everythingis thus well defined. We would like to apply the Lax - Milgram lemma. First of all, notethat since the trace of ¯ u is in L (Γ ) and β ǫλ is λ -Lipschitz continuous (because so is β λ and thanks to (3.5)), then also β ǫλ (¯ u ) ∈ L (Γ ) , and thus z
7→ − Z Γ β ǫλ (¯ u ) z ds + h F, z i , z ∈ H (Ω) is an element of H (Ω) ′ . Furthermore, it is clear that ( z , z ) (1 + λ ) Z Ω z z dx + 2 Z Ω ∇ z · ∇ z dx , ( z , z ) ∈ H (Ω) × H (Ω) is a bilinear continuous and coercive form on H (Ω) . Thus, the Lax - Milgram lemmaimplies that there exists a unique u ǫ ∈ H (Ω) solving (3.7). At this point, note that if weuse the specific test function z = u ǫ in (3.7), owing to the Hölder inequality we obtain (1 + λ ) k u ǫ k L (Ω) + 2 k∇ u ǫ k L (Ω) ≤ Z Γ | β ǫλ (¯ u ) | | u ǫ | ds + |h F, u ǫ i|≤ k β ǫλ (¯ u ) k L (Γ ) k u ǫ k L (Γ ) + k F k H (Ω) ′ k u ǫ k H (Ω) ≤ ǫ | Γ | k u ǫ k L (Γ ) + k F k H (Ω) ′ k u ǫ k H (Ω) ≤ (cid:20) Cǫ | Γ | + k F k H (Ω) ′ (cid:21) k u ǫ k H (Ω) ≤ (cid:18) Cǫ | Γ | + k F k H (Ω) ′ (cid:19) + 12 k u ǫ k H (Ω) , uca Scarpa (cid:18)
12 + λ (cid:19) k u ǫ k L (Ω) + 32 k∇ u ǫ k L (Ω) ≤ (cid:18) Cǫ | Γ | + k F k H (Ω) ′ (cid:19) . At this point, it is suitable to introduce the convex set K ǫ := ( z ∈ H (Ω) : (cid:18)
12 + λ (cid:19) k z k L (Ω) + 32 k∇ z k L (Ω) ≤ (cid:18) Cǫ | Γ | + k F k H (Ω) ′ (cid:19) ) and the mapping Ψ ǫ : K ǫ → K ǫ , Ψ ǫ (¯ u ) = u ǫ , for ¯ u ∈ K ǫ ; (3.8)so, Ψ ǫ (¯ u ) is the unique solution u ǫ of problem (3.7), corresponding to ¯ u ∈ K ǫ . Further-more, u ǫ is a solution of problem (3.6) if and only if Φ ǫ ( u ǫ ) = u ǫ .Hence, in order to solve (3.7), we have to find a fixed point of Ψ ǫ : the idea is to usethe Schauder fixed point theorem, which we briefly recall. Theorem 3.1 (Schauder) . Let X be a Banach space, C a compact convex subset of X and f : C → C a continuous function: then, there exists x ∈ C such that f ( x ) = x . We want to apply Theorem 3.1 to Ψ ǫ : first of all, we have to choose the Banachspace X , in the notation of the result. In order to obtain the compactness property of K ,the idea is to work in X = H − δ (Ω) . In fact, it is clear that K ǫ is bounded in H (Ω) ,and since the inclusion H (Ω) ֒ → H − δ (Ω) is compact, we have that K ǫ is a compact setof H − δ (Ω) . We now have to check that Ψ ǫ is continuous with respect to the topology of H − δ (Ω) : so, let { ¯ u n } n ⊆ K ǫ , ¯ u ∈ K ǫ , and let us show that ¯ u n → ¯ u in H − δ (Ω) ⇒ Ψ ǫ (¯ u n ) → Ψ ǫ (¯ u ) in H − δ (Ω) . (3.9)If we call u ǫ,n = Ψ ǫ (¯ u n ) and u ǫ = Ψ ǫ (¯ u ) , then the definition of Ψ ǫ itself and the differenceof the corresponding equations allow us to infer that (1 + λ ) Z Ω ( u ǫ,n − u ǫ ) z dx + 2 Z Ω ∇ ( u ǫ,n − u ǫ ) · ∇ z dx = − Z Γ ( β ǫλ (¯ u n ) − β ǫλ (¯ u )) z ds for all z ∈ H (Ω) ; testing now by z = u ǫ,n − u ǫ , using Hölder inequality and (3.2), wearrive at (1 + λ ) k u ǫ,n − u ǫ k L (Ω) + 2 k∇ ( u ǫ,n − u ǫ ) k L (Ω) ≤ k β ǫλ (¯ u n ) − β ǫλ (¯ u ) k L (Γ ) k u ǫ,n − u ǫ k L (Γ ) ≤ λ k ¯ u n − ¯ u k L (Γ ) k u ǫ,n − u ǫ k L (Γ ) ≤ Cλ k ¯ u n − ¯ u k L (Γ ) k u ǫ,n − u ǫ k H (Ω) ≤ C λ k ¯ u n − ¯ u k L (Γ ) + 12 k u ǫ,n − u ǫ k H (Ω) . Hence, we have obtained that (cid:18)
12 + λ (cid:19) k u ǫ,n − u ǫ k L (Ω) + 32 k∇ ( u ǫ,n − u ǫ ) k L (Ω) ≤ C λ k ¯ u n − ¯ u k L (Γ ) ; A doubly nonlinear evolution problem for microwave heating now, since if ¯ u n → ¯ u in H − δ (Ω) then in particular ¯ u n → ¯ u in L (Γ ) for the traces, therelation above implies (3.9), and the continuity of Ψ ǫ is proven. So, we are able to applyTheorem 3.1 to Ψ ǫ : we find out that there exists u ǫ ∈ K ǫ such that Ψ ǫ ( u ǫ ) = u ǫ , i.e. thatthere exists a solution u ǫ of problem (3.6).At this point, we would like to find a solution of problem (3.4) taking the limit as ǫ → + : in order to do this, we need some estimates on u ǫ independent of ǫ . It isimmediate to check that if we test equation (3.6) by z = u ǫ (actually, this is an admissiblechoice of z ), we obtain (1 + λ ) Z Ω u ǫ dx + 2 Z Ω |∇ u ǫ | dx + Z Γ β ǫλ ( u ǫ ) u ǫ ds = h F, u ǫ i ; since β ǫλ is monotone and ∈ β (0) we deduce that (1 + λ ) k u ǫ k L (Ω) + 2 k∇ u ǫ k L (Ω) ≤ k F k H (Ω) ′ k u ǫ k H (Ω) ≤ k F k H (Ω) ′ + 12 k u ǫ k H (Ω) , from which (cid:18)
12 + λ (cid:19) k u ǫ k L (Ω) + 32 k∇ u ǫ k L (Ω) ≤ k F k H (Ω) ′ ∀ ǫ > . (3.10)Hence, { u ǫ } ǫ> is bounded in H (Ω) , and therefore there exists a sequence ǫ n ց and u ∈ H (Ω) such that u ǫ n ⇀ u in H (Ω) ; in particular, this condition implies that as n → ∞ u ǫ n → u in H − δ (Ω) ,u ǫ n → u in L (Γ ) . We now want to take the limit in equation (3.6) evaluated for u ǫ n . Thanks to the weakconvergence of u ǫ n , we have that Z Ω u ǫ n z dx → Z Ω uz dx and Z Ω ∇ u ǫ n · ∇ z dx → Z Ω ∇ u · ∇ z dx ; furthermore, the Lipshitz-continuity of β ǫλ leads to k β ǫλ ( u ǫ n ) − β λ ( u ) k L (Γ ) ≤ k β ǫλ ( u ǫ n ) − β ǫλ ( u ) k L (Γ ) + k β ǫλ ( u ) − β λ ( u ) k L (Γ ) ≤ λ k u ǫ n − u k L (Γ ) + k β ǫλ ( u ) − β λ ( u ) k L (Γ ) → since u ǫ n → u in L (Γ ) and thanks to the dominated convergence theorem. Hence, takingthe limit as n → ∞ we find exactly that u satisfies equation (3.4): this ends the proof ofthe maximal monotonicity of B λ . Remark 3.1.
In order to apply Theorem 2.1 we need B λ to be maximal monotone.Actually, we can say something more: B λ is a subdifferential, or, more precisely, thereexists ψ λ : H (Ω) → ( −∞ , + ∞ ] proper, convex and lower semicontinuous such that ∂ψ λ = B λ . In particular, ψ λ has the following expression: ψ λ ( z ) = λ Z Ω z dx + 12 Z Ω |∇ z | dx + Z Γ ˆ β λ ( z ) ds , where ˆ β λ is the proper, convex and continuous function on R such that ˆ β λ (0) = 0 and ∂ ˆ β λ = β λ . uca Scarpa We now have to control that the operator B λ : L (0 , T ; H (Ω)) → L (0 , T ; H (Ω) ′ ) isbounded and coercive. Let u ∈ L (0 , T ; H (Ω)) and v ∈ L (0 , T ; H (Ω)) : then, theestimate (3.3) implies that |h B λ ( u ( t )) , v ( t ) i| ≤ (cid:20) max { λ, } + C λ (cid:21) k u ( t ) k H (Ω) k v ( t ) k H (Ω) for a.e. t ∈ (0 , T ) ; integrating the previous expression on (0 , T ) we obtain Z (0 ,T ) |h B λ ( u ( t )) , v ( t ) i| dt ≤ (cid:20) max { λ, } + C λ (cid:21) k u k L (0 ,T ; H (Ω)) k v k L (0 ,T ; H (Ω) ′ ) . Since this is true for all v ∈ L (0 , T ; H (Ω)) , we have proved the required boundednesson B λ .We now focus on the coercivity of B λ : for each u ∈ L (0 , T ; H (Ω)) , using the mono-tonicity of β λ we have h B λ ( u ( t )) , u ( t ) i = λ Z Ω u ( t ) dx + Z Ω |∇ u ( t ) | dx + Z Γ β λ ( u ( t )) u ( t ) ds ≥≥ min { λ, } k u ( t ) k H (Ω) for a.e. t ∈ (0 , T ) ; so, integrating we deduce that Z T h B λ ( u ( t )) , u ( t ) i dt ≥ min { λ, } k u k L (0 ,T ; H (Ω)) , which implies R T h B λ ( u ( t )) , u ( t ) i dt k u k L (0 ,T ; H (Ω)) ≥ min { λ, } k u k L (0 ,T ; H (Ω)) → + ∞ if k u k L (0 ,T ; H (Ω)) → + ∞ , and also the last hypothesis is satisfied. First of all, Theorem 2.2 tells us that for each λ > there exist u λ ∈ L (0 , T ; H (Ω)) and v λ ∈ H (0 , T ; H (Ω) ′ ) ∩ L (0 , T ; L (Ω)) such that conditions (2.28)–(2.30) hold. Inparticular, (2.28) can be written as follows: (cid:28) ∂v λ ∂t ( t ) , z (cid:29) + λ Z Ω u λ ( t ) z + Z Ω ∇ u λ ( t ) · ∇ z dx + Z Γ β λ ( u λ ( t )) z ds = Z Ω g ( t ) z dx + Z Γ h ( t ) z ds ∀ z ∈ H (Ω) , for a.e. t ∈ (0 , T ) . (4.1)4 A doubly nonlinear evolution problem for microwave heating
We now want to obtain some estimates on u λ and β λ ( u λ ) independent of λ , and then wewill look for a solution of our original problem by taking the limit as λ → + . In thissense, the argument we are going to rely on needs a higher regularity of v λ , i.e. ∂v λ ∂t ∈ L (0 , T ; L (Ω)) , for λ > , (4.2)which is not generally ensured. Thus, the idea is to accurately approximate u and h withsome { u ,λ } and { h λ } , in order to gain the required regularity (4.2), and then exploit itin developing our argument. Indeed, we show uniform estimates on u λ and β λ ( u λ ) andcheck that such estimates are independent of both λ and the approximations of the data.In this perspective, we present a first result. Lemma 4.1. If u , v ∈ H (Ω) and h ∈ H (0 , T ; L (Γ )) , the solution components u λ and v λ of the problem (2.28) – (2.30) satisfy u λ , v λ ∈ H (0 , T ; L (Ω)) ∩ L ∞ (0 , T ; H (Ω)) , ∀ λ > . (4.3) Proof.
Let us proceed in a formal way, taking directly z = ∂u λ ∂t in (4.1) although theregularity of u λ does not allow so (actually, a rigorous approach would require a furtherregularization, which is not restrictive if we keep in mind our goal). Taking (2.3) intoaccount, we have that Z t Z Ω ∂v λ ∂t ( r ) ∂u λ ∂t ( r ) dxdr ≥ c γ Z t Z Ω (cid:12)(cid:12)(cid:12)(cid:12) ∂u λ ∂t ( r ) (cid:12)(cid:12)(cid:12)(cid:12) dxdr , where c γ is the Lipshitz constant of γ − , while Z t Z Ω (cid:18) λu λ ( r ) ∂u λ ∂t ( r ) + ∇ u λ ( r ) · ∇ ∂u λ ∂t ( r ) (cid:19) dxdr = λ Z Ω | u λ ( t ) | dx + 12 Z Ω |∇ u λ ( t ) | dx − λ Z Ω | u | dx − Z Ω |∇ u | dx , and Z t Z Γ β λ ( u λ ( r )) ∂u λ ∂t ( r ) dsdr = Z Γ ˆ β λ ( u λ ( t )) ds − Z Γ ˆ β λ ( u ) ds . Furthermore, thanks to the Young inequality we have that Z t Z Ω g ( r ) ∂u λ ∂t ( r ) dxdr ≤ c γ Z t Z Ω (cid:12)(cid:12)(cid:12)(cid:12) ∂u λ ∂t ( r ) (cid:12)(cid:12)(cid:12)(cid:12) dxdr + 12 c γ Z t Z Ω | g ( r ) | dxdr , while an integration by parts leads to Z t Z Γ h ( r ) ∂u λ ∂t ( r ) dsdr = Z Γ h ( t ) u λ ( t ) ds − Z Γ h (0) u ds − Z t Z Γ ∂h∂t ( r ) u λ ( r ) dsdr Taking all these considerations into account and using the fact that ˆ β λ ≥ , we obtain c γ Z t Z Ω (cid:12)(cid:12)(cid:12)(cid:12) ∂u λ ∂t ( r ) (cid:12)(cid:12)(cid:12)(cid:12) dxdr + min (cid:26) λ , (cid:27) k u λ ( t ) k H (Ω) ≤ λ Z Ω | u | dx + 12 Z Ω |∇ u | dx + Z Γ ˆ β λ ( u ) ds + 12 c γ k g k L (0 ,T ; L (Ω)) + Z Γ | h (0) u | ds + c γ Z t Z Ω (cid:12)(cid:12)(cid:12)(cid:12) ∂u λ ∂t ( r ) (cid:12)(cid:12)(cid:12)(cid:12) dxdr + Z Γ h ( t ) u λ ( t ) ds + Z t Z Γ (cid:12)(cid:12)(cid:12)(cid:12) ∂h∂t ( r ) u λ ( r ) (cid:12)(cid:12)(cid:12)(cid:12) dsdr ; uca Scarpa h ∈ C ([0 , T ]; L (Γ )) , we have Z Γ h ( t ) u λ ( t ) ds ≤ ǫ k h k C ([0 ,T ]; L (Γ )) + ǫC λ k u λ ( t ) k H (Ω) and Z t Z Ω (cid:12)(cid:12)(cid:12)(cid:12) ∂h∂t ( r ) u λ ( r ) (cid:12)(cid:12)(cid:12)(cid:12) dsdr ≤ (cid:13)(cid:13)(cid:13)(cid:13) ∂h∂t (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; L (Γ )) + C λ Z t k u λ ( r ) k H (Ω) dr for some constant C λ > and for all ǫ > . Thus, if we choose a sufficiently small ǫ suchthat δ := min { / , λ/ } − ǫC λ / > , we deduce that c γ Z t Z Ω (cid:12)(cid:12)(cid:12)(cid:12) ∂u λ ∂t ( r ) (cid:12)(cid:12)(cid:12)(cid:12) dxdr + δ k u λ ( t ) k H (Ω) ≤ [ . . . ] + C λ Z t k u λ ( r ) k H (Ω) dr , and the Gronwall lemma ensures that c γ δ Z t Z Ω (cid:12)(cid:12)(cid:12)(cid:12) ∂u λ ∂t ( r ) (cid:12)(cid:12)(cid:12)(cid:12) dxdr + k u λ ( t ) k H (Ω) ≤ δ [ . . . ] e C λ δ T for a.e. t ∈ (0 , T ) . Hence, u λ ∈ H (0 , T ; L (Ω)) ∩ L ∞ (0 , T ; H (Ω)) and thanks to (2.3) and a classical resultby Stampacchia (see [6]) we also have that v λ ∈ H (0 , T ; L (Ω)) ∩ L ∞ (0 , T ; H (Ω)) .As we have anticipated, we consider now some approximations { u ,λ } and { h λ } suchthat the following conditions hold: { u ,λ } ⊆ H (Ω) , v ,λ = γ ( u ,λ ) , u ,λ → u in L (Ω) , (4.4)there exists L > such that (cid:13)(cid:13)(cid:13) ˆ β λ ( u ,λ ) (cid:13)(cid:13)(cid:13) L (Ω) ≤ L ∀ λ > , (4.5) { h λ } ⊆ H (0 , T ; L (Γ )) , h λ → h in L (0 , T ; L (Γ )) . (4.6)Actually, an approximation { u ,λ } such that (4.4), (4.5) hold exists and a formal proof isgiven in Subsection 4.4. Now, thanks to Lemma 4.1, the corresponding solutions u λ , v λ given by Theorem 2.2 have the regularity (4.3). We are now ready to prove Theorem 2.3. u λ It is natural to let z = u λ ( t ) ∈ H (Ω) in equality (4.1): we obtain Z Ω ∂v λ ∂t ( t ) u λ ( t ) dx + λ k u λ ( t ) k L (Ω) + k∇ u λ ( t ) k L (Ω) + Z Γ β λ ( u λ ( t )) u λ ( t ) ds = Z Ω g ( t ) u λ ( t ) dx + Z Γ h λ ( t ) u λ ( t ) ds for a.e. t ∈ (0 , T ) . (4.7)Note that the duality pairing in (4.1) has become a scalar product, thanks to (4.2). Let’sanalyse the four terms on the left hand side, separately. First of all, note that since β λ ismonotone and β λ (0) = 0 we have Z Γ β λ ( u λ ( t )) u λ ( t ) ds ≥ A doubly nonlinear evolution problem for microwave heating furthermore, it is immediate to see that λ k u λ ( t ) k L (Ω) + k∇ u λ ( t ) k L (Ω) ≥ k∇ u λ ( t ) k L (Ω) . Now, we focus on the the first term of equation (4.7): in order to treat it, we recall aknown result (for details, see [2, Lemma 3.3, p. 72]).
Proposition 4.1.
Let H be a Hilbert space and ψ : H → ( −∞ . + ∞ ] a proper convex andlower semicontinuous function; then, for all v ∈ H (0 , T ; H ) and u ∈ L (0 , T ; H ) suchthat ( v ( t ) , u ( t )) ∈ ∂ψ for a.e. t ∈ [0 , T ] , the function t ψ ( v ( t )) is absolutely continuouson [0 , T ] and ddt ψ ( v ( t )) = ( w, v ′ ( t )) H ∀ w ∈ ∂ψ ( v ( t )) , for a.e. t ∈ [0 , T ] . (4.8)In our specific case, we know that v λ ( t ) ∈ ∂φ ( u λ ( t )) a.e. on (0 , T ) , or equivalentlythat u λ ( t ) ∈ ( ∂φ ) − ( v λ ( t )) . If we introduce the convex conjugate of φ , defined as φ ∗ : L (Ω) → ( −∞ , + ∞ ] , φ ∗ ( z ) = sup y ∈ L (Ω) { ( z, y ) L (Ω) − φ ( y ) } , (4.9)from the general theory we know that the following relation holds: ( ∂φ ) − = ∂φ ∗ . (4.10)Hence, we have that u λ ( t ) ∈ ∂φ ∗ ( v λ ( t )) for a.e. t ∈ [0 , T ] and it is natural to applyProposition 4.1 with the choices (in the notations of the proposition) H = L (Ω) and ψ = φ ∗ . Then, Proposition 4.1 tells us that Z Ω ∂v λ ∂t ( t ) u λ ( t ) dx = ddt φ ∗ ( v λ ( t )) for a.e. t ∈ [0 , T ] . Taking these remarks into account, from (4.7) we deduce that ddt φ ∗ ( v λ ( t )) + k∇ u λ ( t ) k L (Ω) ≤ Z Ω g ( t ) u λ ( t ) dx + Z Γ h λ ( t ) u λ ( t ) ds for a.e. t ∈ (0 , T ) ; furthermore, using the Hölder inequality and equation (3.2), the right hand side can beestimated by Z Ω g ( t ) u λ ( t ) dx + Z Γ h λ ( t ) u λ ( t ) ds ≤ k g ( t ) k L (Ω) k u λ ( t ) k L (Ω) + k h λ ( t ) k L (Γ ) k u λ ( t ) k L (Γ ) ≤ (cid:16) k g ( t ) k L (Ω) + C k h λ ( t ) k L (Γ ) (cid:17) k u λ ( t ) k H (Ω) . Hence, by integrating with respect to time, we easily obtain φ ∗ ( v λ ( t )) + k∇ u λ k L (0 ,t ; L (Ω)) ≤≤ φ ∗ ( v ,λ ) + Z (0 ,t ) (cid:16) k g ( s ) k L (Ω) + C k h λ ( s ) k L (Γ ) (cid:17) k u λ ( s ) k H (Ω) ds . (4.11) uca Scarpa φ ∗ ( v ,λ ) = Z Ω v ,λ u ,λ dx − φ ( u ,λ ) ≤ k v ,λ k L (Ω) k u ,λ k L (Ω) ≤ M ∀ λ > (4.12) k h λ k L (0 ,T ; L (Γ )) ≤ M ∀ λ > (4.13)for some positive constants M and M , independent of λ .We would like now to find an estimate from below of the term φ ∗ ( v λ ( t )) : at thispurpose, let c γ and C γ be the Lipschitz constants of γ − and γ respectively. Then, if x ∈ R , since ∂ ˆ γ = γ we have ˆ γ ( x ) + γ ( x )( z − x ) ≤ ˆ γ ( z ) ∀ z ∈ R . Making the particular choice z = 0 , taking into account that ˆ γ (0) = 0 and γ (0) = 0 ,we have ˆ γ ( x ) ≤ γ ( x ) x ≤ | γ ( x ) | | x | ≤ C γ | x | . At this point, note also that, if we call η ( t ) = C γ t for t ∈ R , then we have ˆ γ ∗ ( y ) = sup z ∈ R { zy − ˆ γ ( z ) } ≥ sup z ∈ R { zy − C γ z } = η ∗ ( y ) = y C γ , since it is easy to check (using the definition of conjugate function) that η ∗ ( y ) = y C γ :hence, we reach at ˆ γ ∗ ( γ ( x )) ≥ | γ ( x ) | C γ ≥ c γ C γ | x | . In particular, this estimate implies that there exists C > such that for all u ∈ L (Ω) and v ∈ L (Ω) such that v ∈ γ ( u ) a.e. in Ω we have φ ∗ ( v ) ≥ C k u k L (Ω) . In our specific case, v λ ( t ) ∈ γ ( u λ ( t )) almost everywhere, whence φ ∗ ( v λ ( t )) ≥ C k u λ ( t ) k L (Ω) . Then, from (4.11) it follows that C k u λ ( t ) k L (Ω) + k∇ u λ k L (0 ,t ; L (Ω)) ≤ M + 12 Z t (cid:16) k g ( s ) k L (Ω) + C k h λ ( s ) k L (Γ ) (cid:17) ds + 12 Z t k u λ ( s ) k H (Ω) ds , which easily implies that for all t ∈ (0 , T ) C k u λ ( t ) k L (Ω) + 12 k∇ u λ k L (0 ,t ; L (Ω)) ≤ h M + k g k L (0 ,T ; L (Ω)) + C k h λ k L (0 ,T ; L (Γ )) i + 12 Z t k u λ ( s ) k L (Ω) ds . Please note that condition (4.13) ensures the existence of a positive constant C , inde-pendent of λ , such that [ . . . ] ≤ C ∀ λ > . A doubly nonlinear evolution problem for microwave heating
In particular, we have C k u λ ( t ) k L (Ω) ≤ C + 12 Z t k u λ ( s ) k L (Ω) ds and the Gronwall lemma ensures that C k u λ ( t ) k L (Ω) ≤ C e t/ ≤ C e T/ ∀ t ∈ (0 , T ) . Hence, we have found that there exists a positive constant A > , independent of λ , suchthat k u λ k L ∞ (0 ,T ; L (Ω)) ≤ A ∀ λ > . (4.14)Furthermore, replacing (4.14) in our last inequality it follows that there exists also A > ,independent of λ , such that k u λ k L (0 ,T ; H (Ω)) ≤ A ∀ λ > , (4.15)which easily leads, thanks to (2.3), to k v λ k L (0 ,T ; H (Ω)) ≤ A ∀ λ > , (4.16)for a positive constant A , independent of λ (note the connection with Remark 2.3). β λ ( u λ ) The idea is now to test equation (4.1) by z = β λ ( u λ ( t )) : firstly, we have to control that thisis an admissible choice, or in other words that β λ ( u λ ( t )) ∈ H (Ω) . Since u λ ( t ) ∈ L (Ω) , β λ is λ -lipschitz continuous and β λ (0) = 0 , it follows that β λ ( u λ ( t )) ∈ L (Ω) . Furthermore,thanks to the Lipschitz continuity as well, we also have that β λ ( u λ ( t )) ∈ H (Ω) .Testing now (4.1) by z = β λ ( u λ ( t )) we obtain Z Ω ∂v λ ∂t ( t ) β λ ( u λ ( t )) dx + λ Z Ω u λ ( t ) β λ ( u λ ( t )) dx + Z Ω ∇ u λ ( t ) · ∇ β λ ( u λ ( t )) dx + k β λ ( u λ ( t )) k L (Γ ) = Z Ω g ( t ) β λ ( u λ ( t )) dx + Z Γ h λ ( t ) β λ ( u λ ( t )) ds . (4.17)Let’s handle the different terms of (4.17) separately: thanks to the monotonicity of β λ and the fact that β λ (0) = 0 , we have λ Z Ω u λ ( t ) β λ ( u λ ( t )) dx ≥ , while the monotonicity of β λ implies that Z Ω ∇ u λ ( t ) · ∇ β λ ( u λ ( t )) dx ≥ . uca Scarpa Z t Z Ω ∂v λ ∂s ( s ) β λ ( u λ ( s )) dx ds = Z t Z Ω ∂γ ( u λ ) ∂s ( s ) β λ ( u λ ( s )) dx ds = Z t Z Ω γ ′ ( u λ ( s )) u ′ λ ( s ) β λ ( u λ ( s )) dx ds = Z Ω j λ ( u λ ( t )) dx − Z Ω j λ ( u ,λ ) dx , where j λ ( r ) := Z r γ ′ ( s ) β λ ( s ) ds , r ∈ R . Now, thanks to (2.3), if we let c γ be the Lipschitz-constant of γ − , as usual, we have Z t Z Ω ∂v λ ∂s ( s ) β λ ( u λ ( s )) dx ds ≥ c γ Z Ω ˆ β λ ( u λ ( t )) dx − Z Ω j λ ( u ,λ ) dx . Taking all these remarks into account, from equation (4.17) we obtain c γ Z Ω ˆ β λ ( u λ ( t )) dx + k β λ ( u λ ) k L (0 ,t ; L (Γ )) ≤ Z Ω j λ ( u ,λ ) dx + Z t (cid:20)Z Ω g ( r ) β λ ( u λ ( r )) dx + Z Γ h λ ( r ) β λ ( u λ ( r )) ds (cid:21) dr . (4.18)Please note that hypotheses (2.32) and (2.33) imply that Z t Z Ω g ( r ) β λ ( u λ ( r )) dx dr ≤ Z t k g ( r ) k L ∞ (Ω) (cid:18) D Z Ω ˆ β λ ( u λ ( r )) dx + D | Ω | (cid:19) dr , while thanks to the Young inequality we have Z t Z Γ h λ ( r ) β λ ( u λ ( r )) ds dr ≤ k h λ k L (0 ,t ; L (Γ )) + 12 Z t Z Ω | β λ ( u λ ( r )) | dx dr ; substituting in (4.18) we obtain c γ Z Ω ˆ β λ ( u λ ( t )) dx + 12 k β λ ( u λ ) k L (0 ,t ; L (Γ )) ≤ Z Ω j λ ( u ,λ ) dx + 12 k h λ k L (0 ,T ; L (Γ )) + D | Ω | k g k L (0 ,T ; L ∞ (Ω)) + D Z t k g ( r ) k L ∞ (Ω) Z Ω ˆ β λ ( u λ ( r )) dx dr . (4.19)At this point, if C γ is the Lipschitz-constant of γ , property (2.24) ensures that j λ ( r ) = Z r γ ′ ( s ) β λ ( s ) ds ≤ C γ ˆ β λ ( r ) ≤ C γ ˆ β ( r ) , and consequently, thanks to (4.5) and (4.13), equation (4.19) implies that Z Ω ˆ β λ ( u λ ( t )) dx ≤ c γ (cid:20) C γ L + M D | Ω | k g k L (0 ,T ; L ∞ (Ω)) (cid:21) + D c γ Z t k g ( r ) k L ∞ (Ω) Z Ω ˆ β λ ( u λ ( r )) dx dr . A doubly nonlinear evolution problem for microwave heating
Thanks to the Gronwall lemma, we deduce that Z Ω ˆ β λ ( u λ ( t )) dx ≤ c γ [ . . . ] exp (cid:18) D c γ Z t k g ( r ) k L ∞ (Ω) dr (cid:19) ≤ c γ [ . . . ] e D cγ k g k L ,T ; L ∞ (Ω)) ; hence, there exists B > such that (cid:13)(cid:13)(cid:13) ˆ β λ ( u λ ) (cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; L (Ω)) ≤ B ∀ λ > . (4.20)Taking this estimate into account in (4.19), it immediately follows that there is B > such that k β λ ( u λ ) k L (0 ,T ; L (Γ )) ≤ B ∀ λ > . (4.21) Now, we are concerned with passing to the limit as λ → + in equation (4.1). We recallthe following result (see [7, Cor. 4, p. 85]), which we are going to use next. Proposition 4.2.
Let X ⊆ B ⊆ Y be Banach spaces with compact embedding X ֒ → B and let F ⊆ L p (0 , T ; X ) be a bounded set such that ∂F/∂t := { ∂f /∂t : f ∈ F } is boundedin L (0 , T ; Y ) for a given p ≥ . Then, F is relatively compact in L p (0 , T ; B ) . We would like to apply Proposition 4.2 with the choices X = H (Ω) , B = H − δ (Ω) ( δ ∈ (0 , / ), Y = H (Ω) ′ , p = 2 and F = { v λ } λ> in order to claim that { v λ } λ> isbounded in L (cid:0) , T ; H − δ (Ω) (cid:1) . In fact, F is bounded thanks to (4.16); furthermore, bycomparison in equation (4.1), using conditions (4.15) and (4.21), we find out that thereexists a constant E > , independent of λ , such that (cid:13)(cid:13)(cid:13)(cid:13) ∂v λ ∂t (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; H (Ω) ′ ) ≤ E ∀ λ > . By weak compactness, we infer that v λ n ⇀ v in H (0 , T ; H (Ω) ′ ) ∩ L (0 , T ; H (Ω)) (4.22)for a subsequence λ n ց . Moreover, Proposition 4.2 holds and it is a standard matterto obtain v λ n → v in L (cid:0) , T ; H − δ (Ω) (cid:1) , as n → ∞ . (4.23)Let now u λ n = γ − ( v λ n ) : then, since γ − is Lipschitz continuous (see (2.3)), condition(4.23) implies that u λ n → γ − ( v ) in L (cid:0) , T ; L (Ω) (cid:1) , as n → ∞ . (4.24)Furthermore, (4.15) tells us that there is a subsequence λ n k ց and u ∈ L (0 , T ; H (Ω)) such that u λ nk ⇀ u in L (cid:0) , T ; H (Ω) (cid:1) , as k → ∞ . (4.25)Conditions (4.24) and (4.25) imply that u = γ − ( v ) and v = γ ( u ) a.e. in (0 , T ) × Ω (4.26)and the convergence in (4.25) holds for the entire subsequence λ n . At this point, we recalla general result which is useful to us (see [5, Chapter 1]). uca Scarpa Proposition 4.3.
For all δ ∈ (0 , , there is α δ ∈ (0 , such that k z k H − δ (Ω) ≤ k z k α δ L (Ω) k z k − α δ H (Ω) , ∀ z ∈ H (Ω) . (4.27)Now, thanks to estimate (4.27) and conditions (4.24)-(4.25), we have that as k → ∞ Z T (cid:13)(cid:13) u λ nk ( t ) − u ( t ) (cid:13)(cid:13) H − δ (Ω) dt ≤ Z T (cid:13)(cid:13) u λ nk ( t ) − u ( t ) (cid:13)(cid:13) α δ L (Ω) (cid:13)(cid:13) u λ nk ( t ) − u ( t ) (cid:13)(cid:13) − α δ H (Ω) dt ≤ (cid:13)(cid:13) u λ nk − u (cid:13)(cid:13) α δ L (0 ,T ; L (Ω)) (cid:13)(cid:13) u λ nk − u (cid:13)(cid:13) − α δ ) L (0 ,T ; H (Ω)) → it follows that u λ nk → u in L (cid:0) , T ; H − δ (Ω) (cid:1) , as k → ∞ . (4.28)Furthermore, (4.28) implies the convergence for the traces on Γ : u λ nk → u in L (cid:0) , T ; L (Γ ) (cid:1) , as k → ∞ . (4.29)Let us focus now on β λ ( u λ ) : first of all, note that condition (4.21) tells us that thereis ξ ∈ L (0 , T ; L (Γ )) such that (possibly considering another subsequence) β λ nk (cid:0) u λ nk (cid:1) ⇀ ξ in L (cid:0) , T ; L (Γ ) (cid:1) , as k → ∞ . (4.30)Hence, since β is maximal monotone, the result stated in [1, Prop. 1.1, p. 42] and theconditions (4.29) and (4.30) ensure that u ∈ D ( β ) and ξ ∈ β ( u ) a.e. in (0 , T ) × Γ . (4.31)We are now almost ready to pass to the limit as k → ∞ and complete the proof. Let’srecall equation (4.1), evaluated for λ n k , and argue separately on the different terms as k → ∞ : Z Ω ∂v λ nk ( t ) ∂t z dx → (cid:28) ∂v∂t ( t ) , z (cid:29) thanks to (4.22) (4.32) λ n k Z Ω u λ nk ( t ) z → thanks to (4.25) (4.33) Z Ω ∇ u λ nk ( t ) · ∇ z dx → Z Ω ∇ u ( t ) · ∇ z dx thanks to (4.25) (4.34) Z Γ β λ nk ( u λ nk ( t )) z ds → Z Γ ξ ( t ) z ds thanks to (4.30) . (4.35)Hence, passing to the limit as k → ∞ in (4.1) we obtain exactly the thesis (2.35); further-more, conditions (4.31) and (4.26) yield (2.36). Finally, (2.37) easily follows from (4.4)and (2.34). This finishes the proof. u ,λ As we have anticipated, we now want to prove the existence of an approximation { u ,λ } such that conditions (4.4) and (4.5) hold.2 A doubly nonlinear evolution problem for microwave heating
For λ > , we define u λ as the solution of the following elliptic problem: ( u ,λ − λ ∆ u ,λ = u in Ω , ∂u ,λ ∂n = 0 on Γ . (4.36)Actually, a variational formulation of (4.36) is Z Ω u ,λ z dx + λ Z Ω ∇ u ,λ · ∇ z dx = Z Ω u z dx ∀ z ∈ H (Ω) (4.37)and a direct application of the Lax-Milgram lemma tells that such u ,λ exists and is uniquein H (Ω) . Now, it is easy to check that u ,λ → u in L (Ω) . (4.38)Indeed, if we test equation (4.37) by z = u ,λ , using the Young inequality we obtain k u ,λ k L (Ω) + λ k∇ u ,λ k L (Ω) ≤ k u k L (Ω) ; hence, there exists e u ∈ L (Ω) and a subsequence { u ,λ k } k ∈ N such that u ,λ k ⇀ e u in L (Ω) as k → ∞ , (4.39)and λu ,λ k → in H (Ω) as k → ∞ . (4.40)Taking (4.39) and (4.40) into account and letting k → ∞ in equation (4.37), we have Z Ω e u z dx = Z Ω u z dx ∀ z ∈ H (Ω) and we can conclude that e u = u since H (Ω) is dense in L (Ω) . Then, the identificationof the weak limit implies that the entire family { u ,λ } weakly converges to u . Moreover,as we have lim sup λ ց Z Ω | u ,λ | dx ≤ Z Ω | u | dx , it turns out that (4.38) holds. Thus, condition (4.4) is satisfied for such a choice of u ,λ (and clearly v ,λ = γ ( u ,λ ) ); we now check that also (4.5) is satisfied.Let z = β λ ( u ,λ ) in (4.37) ( z ∈ H (Ω) since β λ is Lipshitz continuous): taking intoaccount the monotonicity of β λ , we obtain ≤ λ Z Ω ∇ β λ ( u ,λ ) · ∇ u ,λ dx = Z Ω ( u − u ,λ ) β λ ( u ,λ ) dx . (4.41)Moreover, the definition of subdifferential tells us that ˆ β λ ( u ,λ ) ≤ ˆ β λ ( u ) + ( u ,λ − u ) β λ ( u ,λ ) a.e. in Ω ; then, integrating on Ω and taking (4.41) and (2.24) into account, we have Z Ω ˆ β λ ( u ,λ ) dx ≤ Z Ω ˆ β λ ( u ) dx + Z Ω ( u ,λ − u ) β λ ( u ,λ ) dx ≤ Z Ω ˆ β λ ( u ) dx ≤ Z Ω ˆ β ( u ) dx . Thus, condition (4.5) is satisfied with the choice L = (cid:13)(cid:13)(cid:13) ˆ β ( u ) (cid:13)(cid:13)(cid:13) L (Ω) . uca Scarpa In this last section, we aim at proving Theorem 2.4, which ensures the continuous de-pendence of the solutions from the data in problem (2.35)–(2.37). The most significantassumption in this case is the linearity of γ : if this is not true, a direct result of uniquenessor continuous dependence is not evident. Thus, let us assume (2.40) and consider twosets of data, { u , h , g } and { u , h , g } , where in particular u , u ∈ L (Ω) , h , h ∈ L (0 , T ; L (Γ )) , g , g ∈ L (0 , T ; L (Ω)) . (5.1)Then, Theorem 2.3 ensures the existence of u , u ∈ L (0 , T ; H (Ω)) , v , v ∈ H (0 , T ; H (Ω) ′ ) ∩ L (0 , T ; H (Ω)) (5.2)such that (2.35)–(2.37) hold for ( u , v , ξ ) and ( u , v , ξ ) . Taking the difference in (2.35)and testing by z = u ( t ) − u ( t ) , we obtain (cid:28) ∂ ( v − v ) ∂t ( t ) , ( u − u )( t ) (cid:29) + Z Ω |∇ ( u − u )( t ) | dx + Z Γ ( ξ − ξ )( t )( u − u )( t ) ds = Z Ω ( g − g )( t )( u − u )( t ) dx + Z Γ ( h − h )( t )( u − u )( t ) ds ; hence, recalling (2.40) and taking into account the monotonicity of β , this equation implies α ddt Z Ω | u ( t ) − u ( t ) | dx + k∇ ( u ( t ) − u ( t )) k L (Ω) ≤ Z Ω ( g − g )( t )( u − u )( t ) dx + Z Γ ( h − h )( t )( u − u )( t ) ds . (5.3)If we now integrate (5.3) with respect to time, thanks to (3.2) and the Young inequality,for all t ∈ [0 , T ] we have α k u ( t ) − u ( t ) k L (Ω) + k∇ ( u − u ) k L (0 ,t ; L (Ω)) ≤ α (cid:13)(cid:13) u − u (cid:13)(cid:13) L (Ω) ++ Z t (cid:16) k g ( s ) − g ( s ) k L (Ω) + C k h ( s ) − h ( s ) k L (Ω) (cid:17) k u ( s ) − u ( s ) k H (Ω) ds ≤ h α (cid:13)(cid:13) u − u (cid:13)(cid:13) L (Ω) + k g − g k L (0 ,T ; L (Ω)) + C k h − h |k L (0 ,T ; L (Γ )) i ++ 12 Z t k u ( s ) − u ( s ) k H (Ω) ds , which implies α k u ( t ) − u ( t ) k L (Ω) + 12 k∇ ( u − u ) k L (0 ,t ; L (Ω)) ≤ [ . . . ] + 12 Z t k u ( s ) − u ( s ) k L (Ω) ds . (5.4)In particular, Gronwall Lemma ensures that k u ( t ) − u ( t ) k L (Ω) + 1 α k∇ ( u − u ) k L (0 ,t ; L (Ω)) ≤ α [ . . . ] e t/α ≤ α [ . . . ] e T/α for all t ∈ (0 , T ) . (5.5)4 A doubly nonlinear evolution problem for microwave heating
Hence, we have shown that k u − u k L ∞ (0 ,T ; L (Ω)) ∩ L (0 ,T ; H (Ω)) ≤ e T/α α min { , /α }× h α (cid:13)(cid:13) u − u (cid:13)(cid:13) L (Ω) + k g − g k L (0 ,T ; L (Ω)) + C k h − h |k L (0 ,T ; L (Γ )) i (5.6)and the continuous dependence result is proved. The uniqueness is an easy consequencewhen we consider u = u , h = h and g = g . References [1] V. Barbu.
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