Existence and qualitative theory for nonlinear elliptic systems with a nonlinear interface condition used in electrochemistry
aa r X i v : . [ m a t h . A P ] N ov EXISTENCE AND QUALITATIVE THEORY FOR NONLINEARELLIPTIC SYSTEMS WITH A NONLINEAR INTERFACECONDITION USED IN ELECTROCHEMISTRY
MICHAL BATHORY, MIROSLAV BUL´IˇCEK, AND ONDˇREJ SOUˇCEK
Abstract.
We study a nonlinear elliptic system with prescribed inner inter-face conditions. These models are frequently used in physical system wherethe ion transfer plays the important role for example in modelling of nano-layer growth or Li-on batteries. The key difficulty of the model consists of therapid or very slow growth of nonlinearity in the constitutive equation insidethe domain or on the interface. While on the interface, one can avoid the diffi-culty by proving a kind of maximum principle of a solution, inside the domainsuch regularity for the flux is not available in principle since the constitutivelaw is discontinuous with respect to the spatial variable. The key result ofthe paper is the existence theory for these problems, where we require thatthe leading functional satisfies either the delta-two or the nabla-two condition.This assumption is applicable in case of fast (exponential) growth as well asin the case of very slow (logarithmically superlinear) growth. Introduction
This paper focuses on the existence and uniqueness analysis of nonlinear ellipticsystems with general growth conditions that may have discontinuity on an innerinterface which describes the transfer of a certain quantity through this interface.To describe such problem mathematically, we consider a domain Ω ⊂ R d , d ≥ ∂ Ω and with an inner interface Γ. The considered domainand the interface are shown in Figure 1 and we always have in mind a similarsituation. We could also consider more interfaces inside of the domain Ω but itwould not bring any additional mathematical difficulties so we restrict ourselvesonly to the situation depicted in Fig. 1. Thus, that the domain Ω is decomposedinto two parts Ω and Ω by the interface Γ such that Ω i is also Lipschitz for i = 1 , D ⊂ ∂ Ω andthe Neumann part Γ N ⊂ ∂ Ω and we denote by n the unit normal vector on Γ,which is understood always as the unit normal outward vector to Ω at Γ (notethat then − n is the unit outward normal vector to Ω on Γ). We also use thesymbol n to denote the unit outward normal vector to Ω on ∂ Ω.The problem reads as follows: For given mappings h : Ω × R d × N → R d × N , b : Γ × R N → R N , given Dirichlet data φ : Γ D → R N and Neumann data j :Γ N → R d × N , to find φ : Ω → R N (here N ∈ N is a number of unknowns) solving Mathematics Subject Classification.
Primary 35J66; Secondary 35M32.
Key words and phrases. nonlinear elliptic systems, interface condition, Orlicz spaces, metaloxidation, nano-layer.This work was supported by the project No. 18-12719S financed by GAˇCR and by the projectSVV-2019-260455. Ω Ω Γ Γ N Γ N Γ D Γ D n Figure 1.
Prototypical domain Ω.the following system(1.1) − div h ( x, ∇ φ ( x )) = 0 in Ω , h ( x, ∇ φ ( x )) · n ( x ) = b ( x, [ φ ]( x )) on Γ , h ( x, ∇ φ ( x )) · n ( x ) = j ( x ) · n ( x ) on Γ N ,φ ( x ) = φ ( x ) on Γ D . Here, the symbol [ φ ] denotes the jump of φ on Γ. More precisely, for x ∈ Γ wedefine [ φ ]( x ) := lim h → + φ ( x + h n ( x )) − φ ( x − h n ( x )) . Consequently, we also cannot assume that φ has derivatives in the whole Ω andtherefore the symbol ∇ φ appearing in (1.1) is considered only in Ω and Ω . Fur-ther, as we shall always assume that φ is a Sobolev function on Ω as well as on Ω ,it makes sense to talk about the trace of φ on ∂ Ω and ∂ Ω and thus the definitionof [ φ ] is meaningful, see Section 2 for precise definitions and notations.The model (1.1) is frequently used when modelling the transfer of ions (or otherparticles) through the interface Γ between two different materials with possiblydifferent relevant properties represented by sets Ω and Ω . The first prototypicexample, we have in mind, is the the process of charging and discharging of lithium-ion batteries. The model of the form (1.1) with N = 1 and h being linear withrespect to ∇ φ but being discontinuous with respect to x when crossing the interfaceΓ was derived and used for modelling of this phenomenon. Note that in this set-ting, the growth or behaviour of the function b is very fast/wild, which may causeadditional difficulties. We refer to [13, 14, 15] for physical justification of such amodel and to [4, 6] for the mathematical and numerical analysis of such modelwith zero j . The second prototypic example is the modeling of porous metal oxidelayer growth in the anodization process. The unknown function φ then representsan electrochemical potential. It has been experimentally observed that under somespecial conditions, the titanium oxide forms a nanostructure which resembles pores.In the thesis [9], it is confirmed numerically that the model (1.1) (or rather its ap-propriate unsteady version) is able to capture this phenomenon if the nonlinearities h and b are chosen accordingly. For this particular application, the mapping h models the high field conduction law in Ω , while in Ω it corresponds to the stan-dard Ohm law, and b models the Butler-Volmer relation, see e.g. [5, 9, 11] and NALYSIS OF A NONLINEAR ELLIPTIC SYSTEMS WITH JUMP ON THE INTERFACE 3 references therein for more details. After some unimportant simplifications and bysetting all electrochemical constants equal to one, these electrochemical laws takethe following form:(1.2) h ( x, v ) = sinh | v || v | v for x ∈ Ω , v ∈ R d × N , v for x ∈ Ω , v ∈ R d × N ,b ( x, z ) = exp | z | − | z | z for z ∈ R N . Thus, it turns out that in some applications, the nonlinearities h and b exhibit a veryfast growth (exponential-like) with respect to the gradient of unknown and evenworse due to the discontinuity with respect to the spatial variable the growth canoscillate between linear or exponential. Therefore, our aim is to obtain a reasonablemathematical theory for (1.1) under minimal assumptions on the smoothness withrespect to the spatial variable x and on the growth with respect to the gradient ofunknown required for h and b .Without the interface condition on Γ, the system (1.1) is a nonlinear ellipticsystem (provided that h is a monotone mapping), for which the existence theorycan be obtained in a relatively standard way if h has polynomial growth and leadsto the direct application of the standard monotone operator theory. Recently, thistheory was further generalized in [1, 2] also into the framework of Orlicz spaceswith h having a general (possibly exponential) growth and being discontinuouswith respect to the spatial variable. The problem (1.1) with the interface conditionwas also recently studied in [4] for the scalar setting, i.e. with N = 1 and only for h being linear with respect to ∇ φ and having discontinuity with respect to the spatialvariable on Γ. The authors in [4] established the existence of a weak solution forrather general class of functions b describing the jump on the interface by provingthe maximum principle for φ . Note that such a procedure heavily relies on thescalar structure of the problem, the linearity of h is used in the proof and it alsorequires the zero flux j .To give the complete picture of the problem (1.1), we would like to point outthat in case that h and b are strictly monotone (and consequently invertible),we can set f := h − and g := b − . Further, we denote j := h ( ∇ φ ), which inthe electrochemical interpretation represents the current density flux. Then, thesystem (1.1) can be rewritten as(1.3) − div j ( x ) = 0 in Ω , f ( x, j ( x )) = ∇ φ ( x ) in Ω \ Γ ,g ( x, j ( x ) · n ( x )) = [ φ ]( x ) on Γ , j ( x ) · n ( x ) = j ( x ) · n ( x ) on Γ N ,φ ( x ) = φ ( x ) on Γ D . and j : Ω → R d × N can be seen as an unknown. This is the first step to the so-called mixed formulation which seems to be advantageous from the computationalviewpoint, see the numerical experiments in [9].The key result of the paper is that we provide a complete existence theory formodel (1.1) assuming very little assumption on the structure and growth of non-linearities h and b and on the data φ and j and we provide also its equivalence M. BATHORY, M. BUL´IˇCEK, AND O. SOUˇCEK to (1.3). Furthermore, we present a constructive proof based on the Galerkin ap-proximation for both formulations (1.1) and (1.3), which may serve as a startingpoint for the numerical analysis. Moreover, in case that the nonlinearities are justderivatives of some convex potentials (which is e.g. the case of (1.2)), we show thatthe solution can be sought as a minimizer to certain functional. Finally, we wouldlike to emphasize that we aim to build a robust mathematical theory for a verygeneral class of problems allowing fast/slow growths of nonlinearities, minimal as-sumptions on data and being able to cover also general systems of elliptic PDE’s,not only the scalar problem.To end this introductory part, we just formulate a meta-theorem for the proto-typic model (1.2) and refer to Section 2 for the precise statement of our result.
Theorem 1.1 (Meta-theorem) . Let the nonlinearities h and b satisfy (1.2) . Thenfor any reasonable data φ and j there exist a unique solution φ to (1.1) and aunique solution j to (1.3) . Moreover, these solutions can be found as minimizersof certain functionals. Notations & Assumptions & Results
In this part, we formulate precisely the main result of the paper. To do sorigorously, we first need to introduce certain function spaces that are capable tocapture the very general behaviour of nonlinearities h and b . Therefore, we firstshortly introduce the Musielak–Orlicz spaces, then we formulate the assumptionson nonlinearities h and b , the geometry of Ω and the data φ and j and finallystate the main results of the paper. Also we simply write the symbol “ · ” to denotethe scalar product on R d or just to say that the product has d -summands, wheneverthere is no possible confusion. Similarly, the symbol “ (cid:5) ” denotes the scalar producton R N or the fact that the product has N -summands, and finally the symbol “:” isreserved for the scalar product on R d × N , or just for emphasizing that the producthas ( d × N )-summands.2.1. The Musielak–Orlicz spaces.
We recall here basic definitions and factsabout Musielak–Orlicz spaces and the interested reader can find proofs e.g. in [12]or in a book [10].We say that Υ : Ω × R m → [0 , ∞ ) with m ∈ N , is an N -function if it isCarath´eodory , even and convex with respect to the second variable z ∈ R m andsatisfies for almost all x ∈ Ω (note that this is a general definition but in our settingthe number m will correspond either to N or to d × N depending on the context)(2.1) lim | z |→ Υ( x, z ) | z | = 0 and lim | z |→∞ Υ( x, z ) | z | = ∞ . Further, the N -function Υ is said to satisfy the ∆ condition if there exist constants c, K ∈ (0 , ∞ ) such that for almost all x ∈ Ω and all z ∈ R m fulfilling | z | > K thereholds(2.2) Υ( x, z ) ≤ c Υ( x, z ) . The complementary (convex conjugate) function to Υ is defined for all ( x, z ) ∈ Ω × R m by (within this section, the symbol “ · ” is also used for the scalar product The function g ( x, z ) is called Carath´eodory if it is for almost all x ∈ Ω continuous with respectto z and also for all z ∈ R m measurable with respect to x . NALYSIS OF A NONLINEAR ELLIPTIC SYSTEMS WITH JUMP ON THE INTERFACE 5 on R m ) Υ ∗ ( x, z ) = sup y ∈ R m ( z · y − Υ( x, y ))and it is also an N -function. This definition directly leads to the Young inequality z · z ≤ Υ( x, z ) + Υ ∗ ( x, z ) for all z , z ∈ R m and thanks to the convexity of Υ and the fact Υ( x,
0) = 0 (it follows from (2.1)),we have that for all ( x, z ) ∈ Ω × R m and 0 < ε < x, ε z ) ≤ ε Υ( x, z ) . This allows us to introduce the ε -Young inequality (with ε ∈ (0 , z · z ≤ ε Υ( x, z ) + Υ ∗ (cid:16) x, z ε (cid:17) . Having the notion of N -function, we can now define the Musielak–Orlicz spaces.Recall that Ω ⊂ R d is an open set and for arbitrary m ∈ N define the set M Υ (Ω) := (cid:26) v ∈ L (Ω; R m ) : Z Ω Υ( x, v ( x )) d x < ∞ (cid:27) . Since the set M does not form necessarily a vector space, we define the Orlicz space L Υ (Ω) as the linear hull of M Υ (Ω) and equip it with the Luxembourg norm k v k Υ;Ω := inf (cid:26) λ > Z Ω Υ (cid:18) x, v ( x ) λ (cid:19) d x ≤ (cid:27) for all v ∈ L Υ (Ω) . We will often omit writing the subscript Ω whenever it is clear from the context. Italso directly follows from the Young inequality, that we have the H¨older inequalityin the form Z Ω v ( x ) · u ( x ) d x ≤ k v k Υ k u k Υ ∗ for all v ∈ L Υ (Ω) and all u ∈ L Υ ∗ (Ω) . Note that the equality M Υ (Ω) = L Υ (Ω) holds if and only if Υ satisfies the ∆ condition (2.2). Further, by E Υ (Ω) we denote the closure of L ∞ (Ω; R m ) in thenorm k·k Υ . The purpose of this definition is that the space E Υ (Ω) is separable,since the set of all polynomials on Ω is dense in E Υ (Ω). In addition, if Υ satisfiesthe ∆ condition, we have the following identities(2.3) E Υ (Ω) = M Υ (Ω) = L Υ (Ω) , while if the ∆ condition is not satisfied, there holds(2.4) E Υ (Ω) $ M Υ (Ω) $ L Υ (Ω) . Furthermore, since E Υ (Ω) is a linear space, we have for arbitrary v ∈ E Υ (Ω) and K ∈ R that K v ∈ E Υ (Ω). Consequently, it follows from (2.3)–(2.4) that(2.5) Z Ω Υ( x, K v ( x )) d x < ∞ for all v ∈ E Υ (Ω) and all K ∈ R . Finally, for any N -function Υ, we have the following identification of dual spaces(2.6) L Υ (Ω) = ( E Υ ∗ (Ω)) ∗ . Thus, although the space L Υ (Ω) is not reflexive in general, the property (2.6) stillensures at least the weak ∗ sequential compactness of bounded sets in L Υ (Ω) by It is evident consequence of (2.3), (2.4) and (2.6) that L Υ (Ω) is reflexive if and only if bothfunctions Υ and Υ ∗ satisfy the ∆ condition. M. BATHORY, M. BUL´IˇCEK, AND O. SOUˇCEK the Banach-Alaoglu theorem. Finally, the space L Υ (Ω) coincides with the weak ∗ closure of L ∞ (Ω; R m ).The very similar definitions can be made for the spaces defined on Γ (the ( d − E Υ (Γ), M Υ (Γ) and L Υ (Γ) withexactly same characterizations as above.2.2. Assumptions on the domain and nonlinearities.
We start this part byprecise specification of the domain Ω, whose prototype is depicted in Fig. 1, whereone can see Ω with its boundary ∂ Ω = Γ D ∪ Γ N and interface Γ. Below, we stateprecisely the necessary assumptions on Ω, however the reader should always keepin mind the “topology” of the set from Fig. 1. Domain Ω : We assume the following:(O1) The set Ω ⊂ R d , d ≥
2, is open, bounded, connected and Lipschitz.(O2) The boundary ∂ Ω can be written as a union of the closures of two relatively(in ( d −
1) topology) open disjoint sets Γ N and Γ D , where Γ D consists oftwo separated components Γ and Γ of non-zero surface measure.(O3) The interface Γ is a connected component of the set Ω that separates Γ from Γ such that the set Ω is bisected by Γ into Ω and Ω and both Ω and Ω are Lipschitz sets.We recall that the outward normal vector n on Γ is chosen to point outwards Ω .Next, we introduce the assumptions on nonlinearities. We split them into twoparts. The first one deals with the standard minimal assumption on the smoothness,growth and monotonicity, and the second one is an additional assumption that willbe used for the existence theorem. Assumptions on h and b : We assume that h : Ω × R d × N → R d × N and b :Γ × R N → R N are Carath´eodory mappings and satisfy:(A1) The mappings h and b are monotone with respect to the second variableand zero at zero, i.e. for all v , v ∈ R d × N , all z , z ∈ R N and almost all x ∈ Ω there holds( h ( x, v ) − h ( x, v )) : ( h − h ) ≥ , ( b ( x, z ) − b ( x, z )) (cid:5) ( z − z ) ≥ , h ( x,
0) = b ( x,
0) = 0 . (2.7)(A2) There exist N -functions Φ and Ψ, a nonnegative constant C and positiveconstants 0 < α h , α b ≤ v ∈ R d × N , all z ∈ R N and almostall x ∈ Ω, there holds h ( x, v ) : v ≥ α h (Φ ∗ ( x, h ( x, v )) + Φ( x, v )) − C, (2.8) b ( x, z ) (cid:5) z ≥ α b (Ψ ∗ ( x, b ( z )) + Ψ( x, z )) − C. (2.9)In case, we are more interested in the formulation for fluxes, i.e. for (1.3), wehave the following assumptions on f and g . Assumptions on f and g : We assume that f : Ω × R d × N → R d × N and g :Γ × R N → R N are Carath´eodory mappings and satisfy: NALYSIS OF A NONLINEAR ELLIPTIC SYSTEMS WITH JUMP ON THE INTERFACE 7 (A1) ∗ The mappings h and b are monotone with respect to the second variableand zero at zero, i.e. for all v , v ∈ R d × N , all z , z ∈ R N and almost all x ∈ Ω there holds( f ( x, v ) − f ( x, v )) : ( v − v ) ≥ , ( g ( x, z ) − g ( x, z )) (cid:5) ( z − z ) ≥ , f ( x,
0) = g ( x,
0) = 0 . (A2) ∗ There exist N -functions Φ and Ψ, a nonnegative constant C and positiveconstants 0 < α h , α b ≤ v ∈ R d × N , all z ∈ R N and almostall x ∈ Ω, there holds f ( x, v ) : v ≥ α f (Φ ∗ ( x, v ) + Φ( x, f ( x, v ))) − C, (2.10) g ( x, z ) (cid:5) z ≥ α g (Ψ ∗ ( x, z ) + Ψ( x, g ( x, z ))) − C. (2.11)Note that if h and b are strictly monotone, i.e. (2.7) holds for all v = v with the strict inequality sign, then we can denote their inverses (with respectto the second variable) f := h − , g := b − and the assumptions (A1)–(A2) and(A1) ∗ –(A2) ∗ are equivalent. Also the assumption h (0) = b (0) = 0 in (A1) is notnecessary, it just makes the proofs more transparent. If h (0) = 0, we can alwayswrite h ( v ) = ( h ( v ) − h (0)) + h (0) and follow step by step all proofs in the paper.Finally, we specify the assumptions that will guarantee the existence (and alsothe uniqueness) of the solution to (1.1) and (1.3), respectively. Key assumptions for the existence of solution:
In what follows we assumethat at least one of the following holds:(Π) There exists F h : Ω × R d × N → R and F b : Ω × R N → R (potentials) suchthat h and b are their Fr´echet derivatives, i.e. for all v ∈ R d × N , z ∈ R N and almost all x ∈ Ω there hold ∂F h ( x, v ) ∂ v = h ( x, v ) , ∂F b ( x, z ) ∂z = b ( x, z ) . (Π) ∗ There exists F f : Ω × R d × N → R and F g : Ω × R N → R (potentials) suchthat f and g are their Fr´echet derivatives, i.e. for all v ∈ R d × N , z ∈ R N and almost all x ∈ Ω there holds ∂F f ( x, v ) ∂ v = f ( x, v ) , ∂F g ( x, z ) ∂z = g ( x, z ) . (∆) At least one of the couples (Φ , Ψ) and (Φ ∗ , Ψ ∗ ) satisfies the ∆ condition.From now, whenever we talk about Φ and Ψ, we always mean the N -functionsfrom (2.8)–(2.9) or (2.10)–(2.11), respectively. Also to shorten the notation, we willomit writing the dependence on spatial variable x ∈ Ω but it is always assumedimplicitly, e.g. h ( v ) always means h ( x, v ) or h ( x, v ( x )) depending on the contextand similarly we use the same abbreviations for other functions/mappings. We say that a couple (Φ , Ψ) satisfies the ∆ condition if both functions Φ and Ψ satisfy the∆ condition. M. BATHORY, M. BUL´IˇCEK, AND O. SOUˇCEK
Notion of a weak solution.
In this part, we define the precise notion of aweak solution to (1.1) and/or to (1.3). Since we deal with functions that may havea jump across Γ, we use a slightly nonstandard definition of a weak gradient onΩ, which however coincides with the standard definition on Ω and Ω . Thereforefor any q ∈ L (Ω; R N ), we say that w ∈ L (Ω; R d × N ) is a gradient of q if for all ϕ ∈ C ∞ (Ω \ Γ; R d × N ) we have (2.12) Z Ω w : ϕ = − Z Ω q (cid:5) (div ϕ )and we will denote ∇ q := w as usual. This will be the default meaning of thesymbol ∇ in the whole paper. It is easy to see that if ∇ q is integrable, then therestrictions q | Ω and q | Ω are Sobolev functions on Ω and Ω , respectively. Hence,since both sets are Lipschitz, we can define for such q ’s the jump of q across Γ as[ q ] := tr Ω q (cid:12)(cid:12) Γ − tr Ω q (cid:12)(cid:12) Γ , where tr Ω i , i = 1 ,
2, is the trace operator acting upon functions defined on Ω i . Function spaces related to problem (1.1) . First, we focus on the definitionof certain spaces that are related to the problem (1.1). Thus, we introduce thefollowing three spaces P := { q ∈ L (Ω; R N ) : ∇ q ∈ L Φ (Ω) , [ q ] ∈ L Ψ (Γ) , tr Ω q (cid:12)(cid:12) Γ D = 0 , tr Ω q (cid:12)(cid:12) Γ D = 0 } ,EP := n q ∈ P : ∇ q ∈ E Φ (Ω) , [ q ] ∈ E Ψ (Γ) o ,BP := n q ∈ P : ∃{ q n } ∞ n =1 ⊂ EP, ∇ q n ⇀ ∗ ∇ q in L Φ (Ω) , [ q n ] ⇀ ∗ [ q ] in L Ψ (Γ) o . We equip these spaces with the norm(2.13) k q k P := k∇ q k Φ;Ω + k [ q ] k Ψ , Γ , where the fact that it is a norm follows from the Poincar´e inequality and from | Γ D | >
0. The motivation for definition of such spaces are the properties of Musielak–Orliczspaces stated in Section 2.1. Moreover, we used the bold face to denote E Φ (Ω) and L Φ (Ω) to emphasize that the objects with values in R d × N are considered, while weused the normal font letters L Ψ (Γ) and E Ψ (Γ) to denote the space of mappingswith value in R N . Furthermore, the space P equipped with the norm (2.13) isa Banach space since it can be identified with a closed subspace of the Banachspace L Φ (Ω) × L ψ (Γ) (see section Section 2.1 for properties of underlying spaces).However, since it is not separable in general, we construct the space EP , which canbe again identified with a closed subspace of E Φ (Ω) × E Ψ (Γ), which is separable.Therefore the Banach space EP is separable as well. Finally, the fact, that thesolution will be in most cases found as a weak ∗ limit of functions from EP , motivatesthe definition of BP , which is thus nothing else than the weak ∗ closure of EP . It isalso evident that if Φ and Ψ satisfy ∆ condition then P = EP = BP . For sake of clarity, the identity (2.12) written in terms of components of w , ϕ and q has thefollowing form N X i =1 d X j =1 Z Ω w i,j ϕ i,j = − N X i =1 Z Ω q i d X j =1 ∂ ϕ i,j ∂x j . NALYSIS OF A NONLINEAR ELLIPTIC SYSTEMS WITH JUMP ON THE INTERFACE 9
Function spaces related to problem (1.3) . In case we are more interested insolving (1.3), we set X := { τ ∈ L Φ ∗ (Ω) , τ · n ∈ L Ψ ∗ (Γ) : Z Γ ( τ · n ) (cid:5) [ ϕ ] + Z Ω ∇ ϕ : τ = 0 ∀ ϕ ∈ EP } , EX := { τ ∈ X : τ ∈ E Φ ∗ (Ω) , τ · n ∈ E Ψ ∗ (Γ) } , BX := { τ ∈ X : ∃{ τ n } ∞ n =1 , τ n ⇀ ∗ τ in E Φ ∗ (Ω) , τ n · n ⇀ ∗ τ · n in E Ψ ∗ (Γ) } . Since we assume just integrability of τ : Ω → R d × N , we specify how the constraintsfrom the definition of X , EX and BX are understood. First, the meaning ofdivergence and the zero trace on the Neumann part of the boundary is usuallyformulated as follows:(2.14) τ · n = 0 on Γ N div τ = 0 in Ω (cid:27) def ⇔ Z Ω ∇ ϕ : τ = 0 ∀ ϕ ∈ C , (Ω; R N ) , ϕ | Γ D = 0 . Note that the right hand side of (2.14) is fulfilled for τ ∈ X since Lipschitz functionsvanishing on Γ D belong to EP . Furthermore, these functions do not have a jumpon Γ and therefore the corresponding integral in the definition of X vanishes. Hence,(2.14) is just the distributional form of the operator div (divergence) as well as thetrace of τ · n . We just allow a broader class of test functions in the definition of X . Second, we can specify the meaning of τ · n ∈ L Ψ ∗ (Γ) in the definition of X asfollows:(2.15) τ · n ∈ L Ψ ∗ (Γ) def ⇔ ∃ w ∈ L Ψ ∗ (Γ) , Z Γ w (cid:5) ϕ = Z Ω ∇ ϕ : τ ∀ ϕ ∈ C , (Ω; R N ) . Note that (2.14) also implies that Z Ω ∇ ϕ : τ = − Z Ω ∇ ϕ : τ . Hence, since we know that τ · n | Γ is well defined distribution because div τ = 0, itfollows from (2.15) that w can be identified with τ · n | Γ , which is the meaning weuse in the paper. However, also for the trace of τ · n , we shall require a broaderclass of test functions than Lipschitz, which correspond to the test function from EP in the definition of X . Finally, we equip X , EX and BX with the norm k τ k X := k τ k Φ ∗ ;Ω + k τ · n k Ψ ∗ ;Γ . Similarly as before, we have that X and EX are the Banach spaces and in addition,since EX can be identified with a closed subspace of E Φ ∗ (Ω) × E Ψ ∗ (Γ), which isseparable, we have that EX is separable as well. Assumptions on data φ and j . The last set of assumptions is related to thegiven boundary and volume data. To simplify the presentation, we assume that φ and j are defined on Ω and specify the assumptions on φ : Ω → R N and j : Ω → R d × N .(D1) We assume that φ ∈ W , (Ω; R N ) such that(2.16) ∇ φ ∈ E Φ (Ω) . The reason for such simplification is that we do not want to employ the trace and/or theinverse trace theorem in Musielak–Orlicz spaces. But clearly, every φ D ∈ W , ∞ (Γ D ) can beextended to the whole Ω such that it satisfies the assumption (D1). (D2) We assume that j : Ω → R d × N is measurable and satisfies(2.17) j ∈ E Φ ∗ (Ω) , j · n = 0 on Γ , div j = 0 in Ω . It is worth noticing, that we assume here better properties than we expect fromsolution. First, since φ is a Sobolev function, it does not have any jump on Γ. Sec-ond, we assume the the flux j over the surface Γ is also vanishing (since divergenceis zero, we can talk about the normal component of the flux on Γ, see (2.17)). Thereason for such setting is that we just want to simplify the presentation of mainresults and the proofs. Definition of a weak solution.
We shall define four notions of weak solution -two for each formulation (1.1) and (1.3). We start with the motivation of a notion ofweak solution to (1.1). We assume that we have a sufficiently good solution to (1.1)and we take the scalar product of the first equality (it has N component) in (1.1)by arbitrary q ∈ EP . We integrate the result over Ω and after using integration byparts, we deduce that (recall our notation for ∇ q in (2.12) and also our definitionof n and [ q ] on Γ)0 = − Z Ω div( h ( ∇ φ ) − j ) (cid:5) q − Z Ω div( h ( ∇ φ ) − j ) (cid:5) q = − Z ∂ Ω \ Γ ( h ( ∇ φ ) − j ) n (cid:5) q − Z ∂ Ω \ Γ ( h ( ∇ φ ) − j ) n (cid:5) q + Z Γ ( h ( ∇ φ ) − j ) n (cid:5) [ q ]+ Z Ω ( h ( ∇ φ ) − j ) : ∇ q (1.1) = (2.17) Z Γ b ([ φ ]) (cid:5) [ q ] + Z Ω h ( ∇ φ ) : ∇ q − Z Ω j : ∇ q, where we also used the facts that q vanishes on Γ D , that div j = 0 and that j · n = 0 on Γ. The above identity can thus be understood as a weak formulationof (1.1) and we are led to the following definition. Definition 2.1.
Let Ω satisfy (O1)–(O3), nonlinearities h and b satisfy (A1)–(A2),data φ and j satisfy (D1)–(D2). We say that the function φ is a weak solutionto (1.1) if φ − φ ∈ P, h ( ∇ φ ) ∈ L Φ ∗ (Ω) , b ([ φ ]) ∈ L Ψ ∗ (Γ)and(2.18) Z Ω h ( ∇ φ ) : ∇ q + Z Γ b ([ φ ]) (cid:5) [ q ] = Z Ω j : ∇ q for all q ∈ EP.
Using the H¨older inequality, we see that both integrals in (2.18) are well defined.In addition, we see that for sufficiently regular φ , the computation above shows thatthe φ solving (2.18) solves (1.1) as well. Further, we introduce another concept ofsolution, which a priori does not require any information on h ( ∇ φ ) and b ([ φ ]). Definition 2.2.
Let Ω satisfy (O1)–(O3), nonlinearities h and b satisfy (A1)–(A2),data φ and j satisfy (D1)–(D2). We say that the function φ is a variational weaksolution to (1.1) if φ − φ ∈ P and(2.19) Z Ω ( h ( ∇ φ ) − j ) : ∇ ( φ − φ − q ) + Z Γ b ([ φ ]) (cid:5) [ φ − q ] ≤ q ∈ EP.
NALYSIS OF A NONLINEAR ELLIPTIC SYSTEMS WITH JUMP ON THE INTERFACE 11
Although, we did not impose any assumptions on the integrability of h ( ∇ φ ) and b ([ φ ]), this information is included implicitly in (2.19) as it is shown in Lemma 3.5below.The next notion of a weak solution concerns the “dual” formulation (1.3) in termsof the flux j . Formally, it can be again derived from (1.3), (2.16) and integrationby parts as follows Z Ω ( f ( j ) − ∇ φ ) : τ = Z Ω ∇ ( φ − φ ) : τ + Z Ω ∇ ( φ − φ ) : τ = Z ∂ Ω ( φ − φ ) (cid:5) ( τ · n ) + Z ∂ Ω ( φ − φ ) (cid:5) ( τ · n )= − Z Γ [ φ ] (cid:5) ( τ · n ) = − Z Γ g ( j · n ) (cid:5) ( τ · n )for any τ ∈ EX .Thus, we are led to the following definition. Definition 2.3.
Let Ω satisfy (O1)–(O3), nonlinearities f and g satisfy (A1) ∗ –(A2) ∗ , data φ and j satisfy (D1)–(D2). We say that the function j is a weaksolution to (1.3) if j − j ∈ X , f ( j ) ∈ L Φ (Ω) , g ( j · n ) ∈ L Ψ (Γ)and(2.20) Z Ω f ( j ) : τ + Z Γ g ( j · n ) (cid:5) ( τ · n ) = Z Ω ∇ φ : τ for all τ ∈ EX . Analogously as for φ , we can define the variational weak solution also for j . Definition 2.4.
Let Ω satisfy (O1)–(O3), nonlinearities f and g satisfy (A1) ∗ –(A2) ∗ , data φ and j satisfy (D1)–(D2). We say that the function j is a variationalweak solution to (1.3) if j − j ∈ X and(2.21) Z Ω ( f ( j ) −∇ φ ) : ( j − j − τ )+ Z Γ g ( j · n ) (cid:5) (( j − τ ) · n ) ≤ τ ∈ EX . Note that in Definition 2.1 the boundary condition φ = φ D on Γ D is imposed by φ − φ ∈ P , whereas in Definition 2.3 the same boundary condition is encoded in(2.20) implicitly (it is shown later, see part ii) of Theorem 3.4). The situation isreversed for the boundary condition j · n = j · n on Γ N .3. Main results
We start this section with the first key result of the paper that focuses on theexistence and uniqueness of a solution to (1.1).
Theorem 3.1.
Let Ω satisfy (O1)–(O3) and φ D fulfil (D1) . Suppose that h and b satisfy (A1), (A2) . (i) Assume that (∆) holds. Then, there exists a weak solution φ to (1.1) . Inaddition the weak solution satisfies φ ∈ φ + BP and (2.18) and (2.19) arevalid for any function q ∈ BP . (ii) Assume that (Π) holds. Then, there exists a variational weak solution φ ∈ φ + P to (1.1) and this solution is also a weak solution. If, in addition, the mapping h is strictly monotone, then the weak solution is uniquein the class φ + BP . As a direct consequence of the above theorem, we also obtain the result statedin Meta-theorem 1.1, which is now formulated as
Corollary 3.2.
Let Ω satisfy (O1)–(O3) , let N = 1 and let φ and j fulfil (D1) and (D2) with Φ( v ) := cosh( | v | ) − and set Ψ( z ) := exp( | z | ) − | z | − . Then Φ and Ψ are N -functions and there exists a unique variational weak solution φ ∈ φ + BP to div (cid:18) sinh |∇ φ ||∇ φ | ∇ φ (cid:19) = 0 in Ω \ Γ , sinh |∇ φ ||∇ φ | ∇ φ · n − exp( | [ φ ] | ) − | [ φ ] | [ φ ] = 0 on Γ , ∇ φ · n = j · n on Γ N ,φ = φ on Γ D . To summarize, we can obtain the existence of a weak solution in two cases.Either in case that there exists a potential (in this case the solution will be soughtas a minimizer) or in case that (∆) holds. Note that (∆) is quite a weak assumptionas the N -functions Φ such that both Φ and Φ ∗ do not satisfy the ∆ condition arenot that easy to find, especially in the applications (see the example in [12, p. 28]).Moreover, we would like to point out here that in case (∆) holds, we obtained abetter solution than just φ ∈ φ + P and we even have φ ∈ φ + BP . Note thatit is trivial if Φ and Ψ satisfy the ∆ condition. However, if it is not the case,it is a piece of new information. Second, we obtained the uniqueness in the class φ + BP , which may be a smaller class than that introduced for weak solution.However, since we know that there exists a weak solution in φ + BP , this classmay be understood as a proper selector for obtaining a uniqueness of a solution.The second existence theorem uses the alternative weak formulation (1.3) interms of the flux j . Theorem 3.3.
Let Ω satisfy (O1)–(O3) and let j fulfil (D2) . Suppose that f and g satisfy (A1) ∗ and (A2) ∗ . (i) Assume that (∆) holds. Then, there exists a weak solution j (1.3) . Inaddition the weak solution fulfills j ∈ j + BX and (2.20) and (2.21) arevalid for any function τ ∈ BX . (ii) Assume that (Π ∗ ) holds. Then, there exists a variational weak solution j ∈ j + X to (1.3) and this solution is also a weak solution.If, in addition, the mapping f is strictly monotone, then the weak solution is uniquein the class j + BX . Also here, we would like to point out that in case (∆) holds, we found a solutionin BX and this is also the class of solutions in which we obtained the uniqueness.Finally, we state the result about the equivalence of Definitions 2.1 and 2.3. Theorem 3.4.
Let all assumptions of Definitions 2.1 and 2.3 be satisfied. Inaddition, assume that h , f , g and b are strictly monotone, satisfying h − = f and b − = g . Then NALYSIS OF A NONLINEAR ELLIPTIC SYSTEMS WITH JUMP ON THE INTERFACE 13 i) If φ is a weak solution in sense of Definition 2.1 then j := h ( ∇ φ ) satisfies j − j ∈ X with j · n = b ([ φ ]) on Γ and (2.20) holds for all τ ∈ EX ∩C (Ω; R d × N ) . In addition if (∆) holds and φ ∈ φ + BP then j is a weaksolution in sense of Definition 2.3. ii) If j is a weak solution in sense of Definition 2.3 then there exists φ ∈ φ + P fulfilling ∇ φ = f ( j ) in Ω and [ φ ] = g ( j · n ) on Γ and φ is a weak solutionin sense of Definition 2.1. This theorem shows the equivalence between the notions of solution if (∆) holds.Furthermore, if (∆) is not satisfied then we have at least the equivalence of solutionin class of distributional solutions of (1.1) and (1.3) respectively. Furthermore, itfollows from the above theorem, that we can choose the formulation, which is moreproper e.g. for numerical purposes, and we still construct the unique solution tothe original problem. Moreover, we see that the existence of a weak solution j automatically implies the existence of a weak solution φ even in the case when (∆)is not satisfied. Therefore also from the point of view of analysis of the problem,the dual formulation (1.3) seems to be preferable to the weak formulation (1.1).The last result states when a variational weak solution is also a weak solutionand similarly when a weak solution is also a variational weak solution. Theorem 3.5.
Let φ ∈ φ + P be a variational weak solution to (1.1) . Then φ isalso a weak solution and satisfies (3.1) Z Ω (cid:16) Φ( ∇ φ ) + Φ ∗ ( h ( ∇ φ )) (cid:17) + Z Γ (cid:16) Ψ([ φ ]) + Ψ ∗ ( b ([ φ ])) (cid:17) < ∞ . Similarly, let φ ∈ φ + BP be a weak solution to (1.1) and (∆) hold. Then φ is alsoa variational weak solution.Let j ∈ j + X be a variational weak solution to (1.3) . Then j is also a weaksolution and satisfies Z Ω (cid:16) Φ( f ( j )) + Φ ∗ ( j ) (cid:17) + Z Γ (cid:16) Ψ( g ( j · n )) + Ψ ∗ ( j · n ) (cid:17) < ∞ . Similarly, let j ∈ j + BX be a weak solution to (1.3) and (∆) hold. Then j is alsoa variational weak solution. In the rest of the paper, we prove the results stated in this section and finallygive also the proof of Meta-theorem 1.1.4.
Proofs of the main results
This key part is organized as follows. First, in Section 4.1, we show Theorem 3.5.Then in Section 4.2 we prove Theorem 3.4. Sections 4.3 and 4.4 are devoted tothe proofs of Theorem 3.1 and 3.3, respectively. Since both proofs are almostidentical, we prove Theorem 3.1 rigorously only for the case ii), i.e. if (Π) holds,and Theorem 3.3 rigorously only for the case i), i.e. when (∆) holds true. Thecorresponding counterparts of the proofs can be done in the very same way andtherefore we present here only sketch of these proofs in Sections 4.5 and 4.6. Finallythe proof of Corollary 3.2 and consequently also of Meta-theorem 1.1 is presentedin Section 4.7.
Proof of Theorem 3.5.
We start the proof by showing that variational weaksolution is also weak solution. Let φ ∈ φ + P be a variational weak solution. Thanksto the Young inequality and the assumption (2.8) (coercivity of h ), we can write( h ( ∇ φ ) − j ) · ∇ ( φ − φ ) ≥ α h Φ ∗ ( h ( ∇ φ )) + α h Φ( ∇ φ ) − D − α h Φ ∗ ( h ( ∇ φ ))2 − α h Φ( ∇ φ )2 − Φ( α h ∇ φ ) − Φ ∗ ( α h j ) − Φ( ∇ φ ) − Φ ∗ ( j ) ≥ α h Φ ∗ ( h ( ∇ φ ))2 + α h Φ( ∇ φ )2 − α h ∇ φ ) − ∗ ( α h j ) − D. Similarly, we also recall (2.9) α b Ψ([ φ ]) + α b Ψ ∗ ( b ([ φ ])) ≤ D + b ([ φ ])[ φ ] . Then, we set q := 0 in (2.19) and with the help of above estimates we deduce that Z Ω Φ ∗ ( h ( ∇ φ )) + Φ( ∇ φ ) + Z Γ Ψ([ φ ]) + Ψ ∗ ( b ([ φ ])) ≤ C (cid:18) Z Ω Φ( α h ∇ φ ) + Φ ∗ ( α h j ) (cid:19) . (4.1)Since j ∈ E Φ ∗ (Ω) and ∇ φ ∈ E Φ (Ω), we can use (2.5) and obtain that the righthand side of (4.1) is finite. Hence, we obtain (3.1).Thus, we just need to show that φ also satisfies (2.18). Note that thanks to(3.1) all integrals in (2.18) and (2.19) are well defined and finite. Let us define forarbitrary q ∈ P J ( q ) := Z Ω ( h ( ∇ φ ) − j ) : ∇ q + Z Γ b ([ φ ]) (cid:5) [ q ] . Then, because we already have (3.1), we can rewrite (2.19) as −∞ < J ( φ − φ ) ≤ J ( q ) < ∞ for all q ∈ EP, which means that J is bounded from below. But since J is linear and EP is a linearspace, this is possible if and only if J ( q ) = 0 for all q ∈ EP , which is nothing elsethan (2.18).Next, we show that if (∆) holds and a weak solution satisfies in addition φ ∈ φ + BP then it is also a variational weak solution. Let us consider first the case whenΨ and Φ satisfy ∆ condition. Then EP = P and we can simply set q := φ − φ − ˜ q in (2.18) with arbitrary ˜ q ∈ EP to obtain (2.19) (where we replace q by ˜ q ). In thesecond case, i.e. if Ψ ∗ and Φ ∗ satisfy ∆ condition, we use the fact that φ − φ ∈ BP .Thus, we can find a sequence { φ n − φ } ∞ n =1 ⊂ EP such that ∇ φ n − ∇ φ ⇀ ∗ ∇ φ − ∇ φ weakly ∗ in L Φ (Ω) , (4.2) [ φ n ] ⇀ ∗ [ φ ] weakly ∗ in L Ψ (Γ) . (4.3)Then we set q := φ n − φ − ˜ q in (2.18), which is now an admissible choice to obtain(4.4) Z Ω ( h ( ∇ φ ) − j ) : ∇ ( φ n − φ − ˜ q ) + Z Γ b ([ φ ]) (cid:5) [ φ n − ˜ q ] = 0 for all ˜ q ∈ EP.
Since Ψ ∗ and Φ ∗ satisfy ∆ condition, we see that h ( ∇ φ ) ∈ E φ ∗ (Ω) and b ([ φ ]) ∈ E Ψ (Γ). Consequently, we can use (4.2)–(4.3) and let n → ∞ in (4.4) to recover(2.19). Note that in both cases, we obtain (2.19) even with the equality sign. NALYSIS OF A NONLINEAR ELLIPTIC SYSTEMS WITH JUMP ON THE INTERFACE 15
The second part of the proof, i.e. the part for j , is done analogously and thereforeis omitted here.4.2. Proof of Theorem 3.4.
We start the proof with the claim i). If φ is a weaksolution then it directly follows from (2.18) that j − j ∈ X with j · n = b ([ φ ])on Γ. Thus, it remains to check that (2.20) is satisfied. Hence, let τ ∈ EX bearbitrary. Then, using the definition of j , we have(4.5) Z Ω f ( j ) : τ + Z Γ g ( j · n ) (cid:5) ( τ · n ) − Z Ω ∇ φ : τ = Z Ω ( ∇ φ −∇ φ ) : τ + Z Γ [ φ ] (cid:5) ( τ · n ) . Thus, if τ is in addition C , then we can directly integrate by parts and we see thatthe right hand side vanishes, which finishes the first part of i). Second, assume that(∆) holds. In the first case, i.e. if Φ and Ψ satisfy ∆ condition, then we have that φ − φ ∈ EP and the right hand side of (4.5) vanishes by using the definition ofthe space EX . In the second case, we use the fact that we can approximate φ bya proper sequence defined in (4.2)–(4.3) and we can write Z Ω ( ∇ φ − ∇ φ ) : τ + Z Γ [ φ ] (cid:5) ( τ · n ) = lim n →∞ Z Ω ( ∇ φ n − ∇ φ ) : τ + Z Γ [ φ n ] (cid:5) ( τ · n ) = 0 , where the second equality follows from the fact that for each n ∈ N , there holds φ n − φ ∈ EP and from the definition of the space EX . Hence, the integral in (4.5)vanishes, which is nothing else than (2.20).Next, we focus on the part ii). Hence, let j ∈ j + X be a weak solution. Thenwe can set τ := τ in (2.20), where τ ∈ C (Ω; R d × N ) ∩ EX is arbitrary fulfilling τ ≡ to obtain(4.6) Z Ω ( f ( j ) − ∇ φ ) : τ = 0 . Consequently, the de Rahm theorem implies that there exist φ i ∈ W , (Ω i ; R N ),such that f ( j ) = ∇ φ ⇔ j = h ( ∇ φ ) in Ω . In addition, since ∂ Ω ∩ Γ D = ∅ , we have from (4.6) that φ must be chosen suchthat φ = φ on ∂ Ω ∩ Γ D . Consequently, it is unique. Similarly, we can uniquelyconstruct φ fulfilling φ = φ on ∂ Ω ∩ Γ D and f ( j ) = ∇ φ ⇔ j = h ( ∇ φ ) in Ω . Thus, defining finally φ := φ χ Ω + φ χ Ω and using the definition of a weak solution j and the fact that h = f − , we deducethat (recall here that the notion of ∇ does not reflect the jump over Γ) Z Ω Φ( ∇ φ ) + Z Ω Φ ∗ ( h ( ∇ φ )) = Z Ω Φ( f ( j )) + Z Ω Φ ∗ ( j ) < ∞ . To identify also a jump [ φ ] on Γ, we first state the following result, which will beproven at the end of this section. Lemma 4.1.
Let Ω satisfy (O1)–(O3) and f ∈ L (Γ) be given. Assume that forall τ ∈ C (Ω ; R d ) fulfilling div τ = 0 in Ω and τ · n = 0 on Γ N ∩ ∂ Ω there holds (4.7) Z Γ f τ · n = 0 . Then f ≡ almost everywhere on Γ . The above lemma is used in the following way. We set τ ∈ EX ∩ C (Ω; R d × N )in (2.20) arbitrarily and using the definition of φ and integration by parts, we findthat 0 = Z Ω ( f ( j ) − ∇ φ ) : τ + Z Γ g ( j · n ) (cid:5) ( τ · n )= Z Ω ∇ ( φ − φ ) : τ + Z Γ g ( j · n ) (cid:5) ( τ · n )= − Z Γ [ φ − φ ] (cid:5) ( τ · n ) + Z Γ g ( j · n ) (cid:5) ( τ · n )= Z Γ ( g ( j · n ) − [ φ ]) (cid:5) ( τ · n ) . Since τ was arbitrary, can use (4.7) to conclude[ φ ] = g ( j · n ) ⇔ b ([ φ ]) = j · n on Γ . Consequently, we also have (by using of the notion of weak solution and the factthat g = b − ) Z Γ Ψ([ φ ]) + Z Ω Ψ ∗ ( b ([ φ ])) = Z Γ Ψ( g ( j · n )) + Z Γ Ψ ∗ ( j · n ) < ∞ . Finally, it directly follows from the definition of X and the identification of φ thatit satisfies (2.18) and thanks to the above estimates φ is a weak solution. It justremains to prove Lemma 4.1. Proof of Lemma 4.1.
We start the proof by considering arbitrary Γ i ⊂ Γ, where Γ i can be described as a graph of Lipschitz function depending on the first ( d − x , . . . , x d − (here we use the fact that Ω is Lipschitz) andfulfills for some cube Q R i ⊂ R d , Γ i ⊂ Q R i ⊂ Q R i ⊂ Ω, where Q R i := x +( − R i , R i ) d with some x ∈ R d . Furthermore, we can require (this also follows fromthe Lipschitz regularity of Ω and from proper orthogonal transformation) that forsome ε > n · ( 0 , . . . , | {z } ( d − , ≥ ε on Γ i . Next, let ψ ∈ C ∞ ( { x + ( − R i , R i ) d − } ) be arbitrary function depending only on x , . . . , x d − and g ∈ C ∞ ( Q R i ) be arbitrary function fulfilling g ≡ Q R i . Thenwe set τ := ( 0 , . . . , | {z } ( d − , ψ ( x , . . . , x d − ) g ( x , . . . , x d )) . Note that τ ∈ C ∞ ( R d ; R d ). Finally, since Ω is connected and Γ D has positivemeasure we can find a smooth open connected set G ⊂ R d such that { x ∈ Ω ; ψ ( x ) ∂ x d g ( x ) = 0 } ⊂ G,G ∩ ∂ Ω ⊂ Γ D ,G \ Ω = ∅ . NALYSIS OF A NONLINEAR ELLIPTIC SYSTEMS WITH JUMP ON THE INTERFACE 17
Finally, we find an arbitrary h ∈ C ∞ ( G \ Ω ) such that(4.8) Z G \ Ω h = − Z G ∩ Ω ψ ( x ) ∂ x d g ( x ) . Next, we use the Bogovskii operator and we can find τ ∈ C ∞ ( G ; R d ) satisfyingdiv τ = ψ∂ x d g + h in G. Note that such function can be found due to the compatibility assumption (4.8).Furthermore, we simply extend τ by zero outside G . Having prepared τ and τ ,we set τ := τ − τ . Then it follows from the construction that in Ω (note that h is not supported in Ω )div τ = div τ − div τ = ψ∂ x d g − ψ∂ x d g = 0and that τ = on Γ N . Consequently, τ can be used in (4.7) and we have0 = Z Γ f ( τ · n ) = Z Γ i f ψn d . Since ψ is arbitrary then f n d = 0 almost everywhere in Γ i . Further, since n d > i then f = 0 on Γ i . This statement holds true for arbitrary Γ i and therefore can be extended to thewhole Γ. The proof is complete. (cid:3) Proof of Theorem 3.1.
In this part, we assume that (Π) holds, i.e. thereexists F h and F b such that for any v ∈ R d × N and z ∈ R N ∂F h ( v ) ∂ v = h ( v ) and ∂F b ( z ) ∂z = b ( z ) . Furthermore, since h and b are coercive and monotone mappings (see (2.7)–(2.9)),it directly follows that F h and F b are N -functions (non-negative, even, convexmappings). In addition, we evidently have the following identities for the Gˆateauxderivatives of h and b :(4.9) ∂ u F h ( v ) ≡ lim λ → + λ ( F h ( v + λ u ) − F h ( v )) = h ( v ) : u , v , u ∈ R d × N , and analogously(4.10) ∂ y F b ( z ) = b ( z ) y, z, y ∈ R N . In addition, it follows from the definition of the convex conjugate function that wecan replace (2.8)–(2.9) by more sharp identities h ( v ) : v = F h ( v ) + F ∗ h ( h ( v )) , (4.11) b ( z ) (cid:5) z = F b ( z ) + F ∗ b ( b ( z ))(4.12)and with the help of (4.11)–(4.12), we can identify Φ and Ψ from (2.8)–(2.9) with F h and F b , i.e. we set in the rest of the proof Φ := F h and Ψ := F b . Finally, wedefine the following functional(4.13) I ( p ) := Z Ω F h ( ∇ ( φ + p )) − j : ∇ ( φ + p ) + Z Γ F b ([ p ]) for all p ∈ P Here in fact the function ψ depends only on the first ( d −
1) variables, but since the set Γ i isdescribed as a graph of a Lipschitz mapping depending on x , . . . , x d − , we can use the standardsubstitution and the fundamental theorem about integrable functions. and look for the minimizer, i.e. we want to find p ∈ P such that for all q ∈ P thereholds(4.14) I ( p ) ≤ I ( q ) ⇔ I ( p ) = min q ∈ P I ( q ) . To prove the existence of p fulfilling (4.14), we define m := inf q ∈ P I ( q )and find { p n } ∞ n =1 as a minimizing sequence of I . It follows from the assumptionson φ and j that such a sequence can be found and it fulfils for all n ∈ N I ( p n ) ≤ I (0) < ∞ . Hence, using the assumption on j , the property (2.5) and the Young inequality,and defining φ n := φ + p n , we find that(4.15) Z Ω F h ( ∇ φ n ) + Z Γ F b ([ φ n ]) ≤ (cid:18)Z Ω F h ( ∇ φ n ) − j : ∇ φ n + Z Γ F b ([ φ n ]) (cid:19) + 2 Z Ω F ∗ h (2 j ) ≤ I (0) + 2 Z Ω F ∗ h (2 j ) < ∞ . Having such uniform bound, we can use the Banach-Alaoglu theorem, and find φ ∈ φ + P and a subsequence, that we do not relabel, such that(4.16) ∇ φ n ⇀ ∗ ∇ φ in L Φ (Ω) , [ φ n ] ⇀ ∗ [ φ ] in L Ψ (Γ)(there is no need to identify the weak limits since the operators of trace, ∇ and [ · ]are linear). Obviously, these two convergence results hold in the weak- L topologyas well (since Φ and Ψ are superlinear). Thus, thanks to the convexity of F h and F b and by the fact that F h ( ∇ ( φ + p )) − j : ∇ ( φ + p ) ≥ − F ∗ h ( j ) ∈ L (Ω) , we can use the weak lower semicontinuity of convex functionals to observe that m = lim n →∞ I ( p n ) ≥ I ( p ) ≥ m, hence I ( p ) = I ( φ − φ ) = m is a minimum. Furthermore, it follows from (4.15)that(4.17) Z Ω F h ( ∇ φ n ) + Z Γ F b ([ φ n ]) < ∞ . Now we will prove that φ is a variational weak solution. This will be done byderiving the Euler-Lagrange equation corresponding to I . Let q ∈ EP be arbitraryand denote φ q := φ + q . We set D h ( λ ) := F h ( ∇ φ + λ ( ∇ φ q − ∇ φ )) − F h ( ∇ φ ) λD b ( λ ) := F b ([ φ ] + λ ([ φ q ] − [ φ ])) − F b ([ φ ]) λ , where λ ∈ (0 ,
1) is arbitrary. Then, we use the minimizing property (4.14) to get I ( p ) ≤ I ((1 − λ ) p + λφ q ) , NALYSIS OF A NONLINEAR ELLIPTIC SYSTEMS WITH JUMP ON THE INTERFACE 19 which in terms of D h and D b can be rewritten by using (4.13) as(4.18) − Z Ω j : ( ∇ φ − ∇ φ − ∇ q ) ≤ Z Ω D h ( λ ) + Z Γ D b ( λ ) . Next, (4.9) and (4.10) imply that (recall that [ φ ] = 0 on Γ) D h ( λ ) → h ( ∇ φ ) : ∇ ( q − φ + φ ) ,D b ( λ ) → b ([ φ ]) (cid:5) [ q − φ ]almost everywhere in Ω and Γ, respectively, as λ → + . Our goal now is to let λ → + in (4.18). Indeed, if we can justify the limit procedure in the term onthe right hand side and if we use the above point-wise result, we directly obtain(2.19), i.e. φ is a variational weak solution. Then we can use the already provenTheorem 3.5 to conclude that φ is also a weak solution. Hence, to finish the proof,we need to justify the limit procedure. Since, we need to pass to the limit with theinequality sign, we use the Fatou lemma. Therefore we need to find I ∈ L (Ω) and I ∈ L (Γ) such that for all λ ∈ (0 ,
1) we have(4.19) D h ( λ ) ≤ I in Ω and D b ( λ ) ≤ I on Γand that for all λ ∈ (0 ,
1) we have (possibly non-uniformly)(4.20) Z Ω D h ( λ ) > −∞ , Z Γ D b ( λ ) > −∞ . Thanks to nonnegativity of F h and F b , and due to (4.16) and (4.17), we get Z Ω D h ( λ ) ≥ − λ Z Ω F h ( ∇ φ ) > −∞ , Z Γ D b ( λ ) ≥ − λ Z Γ F b ([ φ ]) > −∞ for all λ ∈ (0 , F h , which yields D h ( λ ) ≤ (1 − λ ) F h ( ∇ φ ) + λF h ( ∇ φ q ) − F h ( ∇ φ ) λ ≤ F h ( ∇ q + ∇ φ )for all λ ∈ (0 , I := F h ( ∇ q + ∇ φ ) ∈ L (Ω), we use the assumption on φ and q .Since both ∇ q, ∇ φ ∈ E Φ (Ω), which is a linear space, we have that ∇ q + ∇ φ ∈ E Φ (Ω) as well. Consequently, we can use (2.5) to conclude that Z Ω I = Z Ω F h ( ∇ q + ∇ φ ) < ∞ , which leads to the first part of (4.19). The second part is however proven similarly.Hence, we are allowed to use the Fatou lemma and to let λ → + in (4.18) to obtain(2.19). This finishes the existence part of the proof.4.4. Proof of Theorem 3.3.
We assume in this part that (∆) holds. We proceedhere as follows. First, we define the Galerkin approximation, then we derive uniformestimates and pass to the limit. Finally, depending on what kind of ∆ conditionis satisfied, we finish the proof. Galerkin approximation.
We know that EX is a separable space, thereforewe can find (cid:8) w i (cid:9) ∞ i =1 ⊂ EX , whose linear hull is dense in EX . Next, we constructan approximative sequence j n in the following way. For α = ( α , . . . , α n ) ∈ R n , wedenote w α = j + P ni =1 α i w i . Then we define the i -th component, i ∈ { , . . . , n } ,of the mapping F by(4.21) F i ( α ) := Z Ω f ( w α ) : w i + Z Γ g ( w α · n ) (cid:5) ( w i · n ) − Z Ω ∇ φ : w i , α ∈ R n . Our goal is to find α ∗ ∈ R n such that F ( α ∗ ) = 0. Indeed, having such α ∗ isequivalent to have j n := j + P ni =1 α ∗ i w i such that Z Ω f ( j n ) : w i + Z Γ g ( j n · n ) (cid:5) ( w i · n ) = Z Ω ∇ φ : w i for all i ∈ { , . . . , n } . (4.22)Hence, we focus now on finding the zero point of F defined in (4.21). Since weassume that f and g are Carath´eodory mappings and j ∈ E Φ ∗ (Ω), we can use(2.5) to deduce that the mapping F is continuous on R n . Moreover, using thegrowth properties of f and g (assumption (A2) ∗ ), the Young inequality, the factthat j · n = 0 on Γ, j ∈ E Φ ∗ (Ω) and also that ∇ φ ∈ E Φ (Ω), we get(4.23) F ( α ) · α := n X i =1 F i ( α ) α i = Z Ω f ( w α ) : ( w α − j )+ Z Γ g ( w α · n ) (cid:5) ( w α · n ) − Z Ω ∇ φ : ( w α − j ) ≥ α f Z Ω (Φ ∗ ( w α ) + Φ( f ( w α )) + α g Z Γ (Ψ ∗ ( w α · n ) + Ψ( g ( w α · n )) − α f Z Ω (Φ ∗ ( w α ) + Φ( f ( w α )) − Z Ω Φ (cid:18) α α ∇ φ (cid:19) + Φ ∗ (cid:18) α α j (cid:19) − C ≥ α f Z Ω (Φ ∗ ( w α ) + Φ( f ( w α )) + α g Z Γ (Ψ ∗ ( w α · n ) + Ψ( g ( w α · n )) − C. Since the mapping α w α is linear and since Φ ∗ , Ψ ∗ satisfy (2.1), there exists R > | α | > R , then F ( α ) · α >
1. Hence, using a well knownmodification of the Brouwer fixed point theorem, there exists a point α ∗ ∈ R n with F ( α ∗ ) = 0, which we wanted to show. Consequently, we also obtained the existenceof j n solving (4.22).4.4.2. Uniform estimates and limit n → ∞ . It follows from (4.22) (see the compu-tation in (4.23)) that the identity Z Ω f ( j n ) : ( j n − j ) + Z Γ g ( j n · n ) (cid:5) ( j n · n ) = Z Ω ∇ φ : ( j n − j )(4.24)is valid for all n ∈ N . Consequently, it follows by the same procedure as in (4.23)that we have the following uniform bounds(4.25) Z Ω (Φ ∗ ( j n ) + Φ( f ( j n ))) + Z Γ (Ψ ∗ ( j n · n ) + Ψ( g ( j n · n ))) ≤ C. NALYSIS OF A NONLINEAR ELLIPTIC SYSTEMS WITH JUMP ON THE INTERFACE 21
Thus, using the Banach-Alaoglu theorem, we find weakly- ∗ converging subsequences(that we do not relabel), so that j n ⇀ ∗ j in L Φ ∗ (Ω) , (4.26) f ( j n ) ⇀ ∗ f in L Φ (Ω) , (4.27) j n · n ⇀ ∗ j · n in L Ψ ∗ (Γ) , (4.28) g ( j n · n ) ⇀ ∗ g in L Ψ (Γ)(4.29)as n → ∞ . Furthermore, since j n − j ∈ EX , we have from the above convergenceresult that j − j ∈ BX . Next, we pass to the limit also in (4.22). Since w i ∈ E Φ ∗ (Ω) and w i · n ∈ E Ψ ∗ (Γ) for all i ∈ N , we can use (4.27) and (4.29) to let n → ∞ in (4.22) for fix i ∈ N and obtain Z Ω f : w i + Z Γ g (cid:5) ( w i · n ) = Z Ω ∇ φ : w i for all i ∈ { , . . . , n } (4.30)and since the linear hull of { w i } i ∈ N is dense in EX , we obtain Z Ω f : τ + Z Γ g (cid:5) ( τ · n ) = Z Ω ∇ φ : τ for all τ ∈ EX . (4.31)4.4.3. Identification of f and g and the energy (in)equality. To finish the proof, itremains to show that(4.32) f = f ( j ) a.e. in Ω and g = g ( j · n ) a.e. on Γand also that we constructed the variational solution. We start the proof by claimingthat Z Ω f : ( j − j ) + Z Γ g (cid:5) ( j · n ) = Z Ω ∇ φ : ( j − j ) . (4.33)The importance of (4.33) is not only that it will allow us to show (4.32) but also thathaving (4.32), (4.33) and (4.31), we immediately get (2.21) even with the equalitysign.Hence, we prove (4.33) provided that (∆) holds. First, in case that Φ ∗ and Ψ ∗ satisfy the ∆ condition then EX = X and (4.31) can be tested by any τ ∈ X ,in particular by j − j and (4.33) follows. In the opposite case, i.e. if Φ and Ψsatisfy the ∆ condition, then we have from (4.27) and (4.29) that f ∈ E Φ (Ω) and g ∈ E Ψ (Γ). Furthermore, it follows from (4.30) that for all i ∈ N Z Ω f : ( j i − j ) + Z Γ g (cid:5) ( j i · n ) = Z Ω ∇ φ : ( j i − j ) . (4.34)But now, we can use the convergence results (4.26) and (4.28) (thanks to f ∈ E Φ (Ω)and g ∈ E Ψ (Γ)) and let i → ∞ in (4.34) to obtain (4.33). Next, using the factsthat ∇ φ ∈ E Φ (Ω) and j ∈ E Φ ∗ (Ω) and (4.26)–(4.29), we can let n → ∞ in (4.24)to deduce(4.35) lim n →∞ (cid:18)Z Ω f ( j n ) : j n + Z Γ g ( j n · n ) (cid:5) ( j n · n ) (cid:19) = lim n →∞ (cid:18)Z Ω ∇ φ : ( j n − j ) + Z Ω f ( j n ) : j (cid:19) = Z Ω ∇ φ : ( j − j ) + Z Ω f : j = Z Ω f : j + Z Γ g (cid:5) ( j · n ) . Now we follow [3], see also [2]. Let v ∈ L ∞ (Ω; R d × N ) and z ∈ L ∞ (Γ; R N ) bearbitrary. Using the monotonicity assumptions (A1) ∗ , we have(4.36) 0 ≤ lim n →∞ Z Ω ( f ( j n ) − f ( v )) : ( j n − v ) + Z Γ ( g ( j n · n ) − g ( z )) (cid:5) ( j n · n − z )= Z Ω ( f − f ( v )) : ( j − v ) + Z Γ ( g − g ( z )) (cid:5) ( j · n − z ) , where we used (4.26)–(4.29) and (4.35). Finally, we closely follow [2, 7, 8] (see also[3, Lemma 2.4.2.] for similar procedure for more general monotone mappings). Wedefine the setsΩ j := { x ∈ Ω; | j ( x ) | ≤ j } , Γ j := { x ∈ Γ; | j ( x ) · n ( x ) | ≤ j } . Then for arbitrary ε > v ∈ L ∞ (Ω; R d × N ), z ∈ L ∞ (Γ; R N ) and arbitrary j ≤ k < ∞ , we set v := j χ Ω k − ε v χ Ω j , z := j · n χ Γ k − εzχ Γ j in (4.36). Doing so, we obtain (using also the fact that f (0) = g (0) = 0)(4.37) 0 ≤ Z Ω ( f − f ( j χ Ω k − ε v χ Ω j )) : ( j (1 − χ Ω k ) + ε v χ Ω j )+ Z Γ ( g − g ( j · n χ Γ k − εzχ Γ j )) (cid:5) (( j · n )(1 − χ Γ k ) + εzχ Γ j )= ε Z Ω j ( f − f ( j − ε v )) : v + ε Z Γ j ( g − g ( j · n − εz )) (cid:5) z + Z Ω \ Ω k f : j + Z Γ \ Γ k g (cid:5) ( j · n ) . Thanks to (4.26) and (4.27) and since | Ω \ Ω k | → | Γ \ Γ k | → k → ∞ , we canlet k → ∞ in (4.37) to deduce0 ≤ ε Z Ω j ( f − f ( j − ε v )) : v + ε Z Γ j ( g − g ( j · n − εz )) (cid:5) z. Dividing by ε and letting ε → + , using the definition of Ω j and Γ j (leading to thefact that j and also j · n are bounded on the integration domain) and the fact that f and g are Carath´edory, we finally observe0 ≤ Z Ω j ( f − f ( j )) : v + Z Γ j ( g − g ( j · n )) (cid:5) z. Setting v := − f − f ( j )1 + | f − f ( j ) | and z := − g − g ( j · n )1 + | g − g ( j · n ) | we deduce that (4.32) is valid almost everywhere in Ω j (and Γ j , respectively) forevery j ∈ N . Since | Ω \ Ω j | → | Γ \ Γ j | → j → ∞ , it directly follows that(4.32) holds. NALYSIS OF A NONLINEAR ELLIPTIC SYSTEMS WITH JUMP ON THE INTERFACE 23
Uniqueness.
We start the proof by claiming that (2.20) holds for all τ ∈ BX .Indeed, if Ψ ∗ and Φ ∗ satisfy ∆ condition then EX = BX and there is nothingto prove. On the other hand if Ψ and Φ satisfy ∆ condition, then we use thefact f ( j ) ∈ L Φ (Ω) = E Φ (Ω) and g ( j · n ) ∈ L Ψ (Γ) = E Ψ (Γ). Hence, for arbitrary τ ∈ BX , we can find an approximating sequence { τ k } ∞ k =1 ⊂ EX such that( τ k , τ k · n ) ⇀ ∗ ( τ , τ · n ) in L Φ ∗ (Ω) × L Ψ ∗ (Γ) . We replace τ by τ k in (2.20) and let k → ∞ . Using the above weak start conver-gence result, we recover that (2.20) holds also for τ .Finally, assume that we have to solutions j , j ∈ j + BX . Subtracting (2.20)for j from that one for j we have for all τ ∈ BX Z Ω ( f ( j ) − f ( j )) : τ + Z Γ ( g ( j · n ) − g ( j · n )) (cid:5) ( τ · n ) = 0 . Setting finally τ := j − j ∈ BX and using the strict monotonicity of f , we findthat j = j in Ω, which finishes the uniqueness part.4.5. Proof of Theorem 3.1- case (∆) holds.
This proof is analogous to thepreceding proof of Theorem 3.3 (i). Again, we approximate the problem usingseparability of EP and the Galerkin method. Eventually, we construct an approxi-mation φ n satisfying Z Ω h ( ∇ φ n ) : ∇ q + Z Γ b ([ φ n ]) (cid:5) [ q ] = Z Ω j : ∇ q for all q from some n -dimensional subspace of EP . Then, using the analogousa priori estimate to (4.25) and very similar limiting procedure, we let n → ∞ andobtain (2.18).In addition, it is evident, that we obtain a weak solution φ ∈ φ + BP , which is thelast claim of Theorem 3.1. Furthermore, assume that q ∈ BP is arbitrary. Thereforeit can be approximated by a weakly star convergent sequence { q n } ∞ n =1 ⊂ EP . Since h ( ∇ φ ) ∈ E Φ ∗ (Ω) and b ([ φ ]) ∈ E Ψ ∗ (Γ), we can now use (2.18), where we replace q by q n and using the weak star convergence, we can conclude that (2.18) holds evenfor all q ∈ BP . Finally, assume that we have to solutions φ , φ ∈ φ + BP . Thenusing (2.18) and the above argument, we can deduce that Z Ω ( h ( ∇ φ ) − h ( ∇ φ )) : ∇ q + Z Γ ( b ([ φ ]) − b ([ φ ])) (cid:5) [ q ] = 0 . Hence, setting q := φ − φ ∈ BP in the above identity, we observe with the helpof the strict monotonicity of h that ∇ φ = ∇ φ in Ω . Hence, since φ = φ on the sets Γ D ⊂ ∂ Ω , Γ D ⊂ ∂ Ω of positive measure, we seethat φ = φ in Ω and also in Ω and the solution is unique in the class φ + BP .4.6. Proof of Theorem 3.3 (ii).
This proof is analogous to the proof of Theo-rem 3.1 (ii). Indeed, it is easy to see that if we define I ( τ ) := Z Ω ( F f ( j + τ ) − ∇ φ : τ ) + Z Γ F g ( j · n ) , τ ∈ X , we can proceed as before to get a minimum τ ∈ X and the corresponding j := j + τ satisfying (2.21). This minimum is a weak solution by Theorem 3.5. Proof of Corollary 3.2.
We only need to prove that the nonlinearities de-fined in (1.2) satisfy all the assumptions of Theorem 3.1. Namely, we show that(Π) holds and that (∆) is valid. We define,Φ( v ) = F h ( v ) := cosh( | v | ) − , Ψ( z ) = F b ( z ) := exp( | z | ) − | z | − . It is clear that both functions are N -functions. Moreover, by a direct computation,we have that ∂F h ( v ) ∂ v = sinh | v || v | v , ∂F b ( z ) ∂z = exp( | z | ) − | z | z and thus (Π) holds. Moreover, F h and F b are strictly convex. Hence, we useTheorem 3.1 to get the existence of a weak solution.To prove also further properties, we show that Ψ ∗ and Φ ∗ satisfy ∆ conditionand consequently (∆) holds as well and having such property, we can even proveuniqueness of a weak solution. First, one can easily observe that there exists K > K Φ( v ) ≤ Φ(2 v ) for all v ∈ R d × N , | v | ≥ , K Ψ( z ) ≤ Ψ(2 z ) for all z ∈ R N , | z | ≥ . Then, by [12, Theorem 4.2.], this implies that Φ ∗ and Ψ ∗ satisfy the ∆ condition.The proof is complete. References [1] M. Bul´ıˇcek, P. Gwiazda, M. Kalousek, and A. ´Swierczewska Gwiazda,
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Mathematical Institute of Charles University, Faculty of Mathematics and Physics,Charles University, Sokolovsk´a 83, 186 75 Praha 8, Czech Republic
E-mail address : [email protected] Mathematical Institute of Charles University, Faculty of Mathematics and Physics,Charles University, Sokolovsk´a 83, 186 75 Praha 8, Czech Republic
E-mail address : [email protected] Mathematical Institute of Charles University, Faculty of Mathematics and Physics,Charles University, Sokolovsk´a 83, 186 75 Praha 8, Czech Republic
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