Renormalized solutions for nonlinear partial differential equations with variable exponents and L 1 -data

Transcription

1 Renormalized solutions for nonlinear partial differential equations with variable exponents and L 1 -data vorgelegt von Diplom-Mathematikerin Aleksandra Zimmermann geb. Zmorzynska Warschau von der Fakultät II-Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr.rer.nat. genehmigte Dissertation Promotionsausschuss: Vorsitzender: Berichter/Gutachter: Berichter/Gutachter: Prof. Dr. S. Felsner Prof. Dr. P. Wittbold PD Dr. B. Andreianov Tag der wissenschaftlichen Aussprache: Berlin 21 D 83

2 Contents Notation 3 1 Introduction 4 2 Function spaces and notation Lebesgue and Sobolev spaces with variable exponent Function spaces for the evolution problem Notation and functions The elliptic case Renormalized solutions Existence for L -data Approximate solutions for L -data A priori estimates Basic convergence results Proof of existence Weak solutions for L -data Proof of Theorem Approximate solutions for L 1 -data Conclusion of the proof of Theorem Existence in one space dimension Uniqueness of renormalized solutions Extensions and remarks The parabolic case Mild solutions of the abstract Cauchy problem Existence of mild solutions The closure of D(A β ) Solutions and function spaces for the evolution problem Renormalized solution Functional setting and weak solutions

3 CONTENTS Integration-by-parts-formula Existence of renormalized solutions Existence for L -data L 1 -contraction and uniqueness of renormalized solutions A comparison principle and weak solutions for L -data Proof of Theorem Extensions and open problems Strong L 1 -convergence A regularity result Entropy solutions Bibliography 1

4 Notation open domain in R N for N 1 topological boundary of x = (x 1,..., x n ) point in R N for N 1 dx = dx 1... dx N Lebesgue measure in (, T ) for T > t time variable Σ T (, T ) Du gradient of a function u supp f support of a function f f +, f max(f, ), min(f, ) D(), D( ),... test functions in,,... D + (), D + ( ),... positive test functions in,,... C(),C( ),... the function space of continuous functions in,,... C k (),C k ( ),... for 1 k the function space of k times continuously differentiable functions in,,... L p () {f : R f measurable, f(x) p dx < } for 1 p < L () {f : R f measurable, ess sup x f(x) < } W 1,p () {f : R f L p (), Df (L p ()) N } for 1 p + W 1,p () Closure of D() in W 1,p () for 1 p < + H 1 () {f : R f L 2 (), Df (L 2 ()) N } H 1 () dual space of H 1 () X arbitrary Banach space X dual space of the Banach space X L p (, T ; X) for 1 p <, {f : X T f(t) p Xdt < } L (, T ; X) {f : (, T ) X ess sup f(t) X < } C([, T ]; X) function space of continuous functions defined on [, T ] with values in X D((, T ); X) test functions with values in X I I : X X identity mapping on X 3

5 Chapter 1 Introduction Let be a bounded domain in R N (N 1) with Lipschitz boundary if N 2. Our aim is to prove existence and uniqueness of renormalized solutions to the nonlinear elliptic equation { β(u) div(a(x, Du) + F (u)) f in, (E, f) u = on and to the corresponding parabolic problem β(u) t div(a(x, Du) + F (u)) f in, (P, f, b ) u = on Σ T, β(u(, )) b in with right-hand side f L 1 () for (E, f) and f L 1 ( ) for (P, f, b ). Furthermore, F : R R N is locally Lipschitz continuous and β : R 2 R a set-valued, maximal monotone mapping such that β(). a : R N R N is a Carathéodory function satisfying the following assumptions: (A1) There exist a continuous function p : (1, ), 1 < min x p(x) N (the case min x p(x) > N is easy and can be solved by variational methods) and a positive constant γ such that a(x, ξ) ξ γ ξ p(x) holds for all ξ R N and almost every x. (A2) a(x, ξ) d(x)+ ξ p(x) 1 for almost every x and for every ξ R N, where d is a nonnegative function in L p ( ) () and p (x) := p(x)/(p(x) 1) for a.e. x. (A3) (a(x, ξ) a(x, η)) (ξ η) for almost every x and for every ξ, η R N. 4

6 CHAPTER 1. INTRODUCTION 5 Due to the assumptions (A1), (A2) and (A3), the functional setting involves Lebesgue and Sobolev spaces with variable exponent L p( ) () and W 1,p( ) (). The theory of Lebesgue and Sobolev spaces with variable exponent has experienced a revival of interest, shown in a substantial amount of publications over the past few years. An extensive list of references concerning the recent advances and open problems can be found in Diening et al. [38]. The equation (E, f) can be viewed as generalization of the p(x)-laplacian equation { div( Du p(x) 2 Du) = f in, (L, f) u = on. In case of a constant exponent p( ) p, 1 < p <, and f W 1,p () it follows from Minty-Browder Theorem that there exists a unique solution u W 1,p () to (L, f) in the sense of distributions. For 1 < p < N and right-hand side f L 1 () we can not expect solutions u W 1,p (). Indeed, supposing that for each f L 1 () there exists a solution u W 1,p () of (L, f), as div( Du p 2 Du) W 1,p (), it follows that L 1 () W 1,p (). By duality, this implies W 1,p () L (). From Sobolev embedding theorems we know that this is not true in general if 1 < p < N. Supposing 1 < p < 2 and that there exists a solution u W 1,1 () of (L, f), as Du p 2 Du (L 1/(p 1) ()) N we get div( Du p 2 Du) W 1,1/(p 1) () and therefore L 1 () W 1,1/(p 1) (). By duality, this implies W 1,1/(2 p) () L (). By Sobolev embedding theorems, this is true if p > 2 1. Therefore N we can not even expect solutions u W 1,1 () of (L, f) for 1 < p < 2 1 and N f L 1 (). In the case of constant p > 2 1 the existence of a distributional N solution u of (L, f) in the space W 1,q () q< N(p 1) N 1 has been shown in [25]. As it has been shown in [7] and [62], the distributional solution u is in general not unique. In order to get well-posedness for L 1 ()-data, the notion of an entropy solution for problem (L, f) was introduced by Bénilan et al. in [15] in the framework of a constant p( ) p. Moreover, existence and uniqueness of an entropy solution of (L, f) has been established for 1 < p < N. In [67] the result of [15] has been extended to nonconstant p W 1, (). An equivalent notion of solution for problem (L, f) is called renormalized solution. The concept of renormalized solutions was introduced by DiPerna and Lions in [39]. This notion was then extended to the study of various problems of partial differential equations of parabolic,

7 CHAPTER 1. INTRODUCTION 6 elliptic-parabolic and hyperbolic type, we refer to [2], [23], [56], [32], [2] and the references therein for more details. In [1], existence of renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations has been shown and in [11] the notion of renormalized solution was adapted to an anisotropic reaction-diffusion-advection system. In [12], existence and uniqueness of renormalized solutions to (L, f) has been shown for continuous functions p : (1, ), such that 2 1 < min N x p(x). (P, f, b ) can be viewed as a generalisation of the parabolic p(x)-laplacian equation u t div( Du p(x) 2 Du) = f in, (L, f, u ) u = on Σ T, u(, ) = u ( ) in. One of the motivations for studying (E, F ) and (P, f, b ) comes from applications to electro-rheological fluids (see [64], [49] for more details) as an important class of non-newtonian fluids. Other important applications are related to image processing and elasticity (see [33], [75]). Note that (L, f, u ) has a more complicated nonlinearity than the classical p-laplacian since it is nonhomogenous. In [13] existence and uniqueness of renormalized solutions to (L, f, u ) was shown for arbitrary L 1 -data. This thesis is organized as follows: In the next chapter we will present some general definitions and results concerning the necessary function spaces. Moreover, we will introduce some notation and functions which will be used frequently. In the third chapter we will study existence and uniqueness of weak and renormalized solutions to the elliptic problem (E, f). These results will serve us as a basis for the study of the evolution problem associated with the same convection-diffusion operator: More precisely, from these results we deduce that there exists a mild solution of the abstract Cauchy problem corresponding to (P, f, b ) in the sense of nonlinear semigroup theory. The nonlinear semigroup theory gives a general notion of solution, called mild solution for abstract Cauchy problems of the form du dt + Au f where A is (a possibly multivalued) operator in a Banach space X and f L 1 (, T ; X). The mild solution is, roughly speaking, the uniform limit of piecewise constant approximate solutions of time-discretized equations given by an implicit Euler scheme. This result will lead us to the appropriate energy space for weak and renormalized solutions of (P, f, b ) with variable exponent.

8 CHAPTER 1. INTRODUCTION 7 At the end of the last chapter we will show existence and uniqueness of renormalized solutions using the ideas developed in [18], [2], [68] and [22] for the case of a constant exponent. For all the basic definitions and results from nonlinear semigroup theory we mostly refer to the unpublished book of Bénilan, Crandall and Pazy (see [17]). Other references are [8], [72], [59], [58] [14], [36], [3]. Consider also [16], [34], [9], [35] and [19] for furhter reading and applications to partial differential equations. For all basic definitions and results concerning (linear and nonlinear) functional analysis and classical Lebesgue and Sobolev spaces we refer to, e.g., [29], [66]. For the theory of vector-valued integration and Sobolev spaces see, e.g., [4], [48].

9 Chapter 2 Function spaces and notation 2.1 Lebesgue and Sobolev spaces with variable exponent We recall in what follows some definitions and basic properties of Lebesgue and Sobolev spaces with variable exponent (see for example [52], [44], [43], [37], [38] for proofs and details and [6] for general theory of Orlicz spaces). For an open set R N, let p : [1, ) be a measurable function, which is called the variable exponent, such that 1 p := ess inf x p(x) p + := ess sup x p(x) <. We define the variable exponent Lebesgue space L p( ) () to consist of classes of almost everywhere equal measurable functions f : R, such that the modular ρ p (f) := f(x) p(x) dx is finite. On L p( ) (), f f L p( ) () := inf{λ > : ρ p ( f ) < 1} defines a norm, (Lp( ) (), λ L ()) is a Banach p( ) space and D() is dense in L p( ) (). If p > 1, then L p( ) () is reflexive and its dual space is isomorphic to L p ( ) (), where = 1. For any p( ) p ( ) f L p( ) () and g L p ( ) (), the Hölder type inequality ( fg dx 1 p + 1 p ) f L p( ) () g L p ( ) () (2.1.1) holds true. Convergence with respect to the modular is equivalent to convergence with respect to the norm. We have the following relation between the modular and the norm: min { f p L p( ) (), f p+ L p( ) () } { } f(x) p(x) dx max f p, L p( ) () f p+. L p( ) () (2.1.2) 8

10 CHAPTER 2. FUNCTION SPACES AND NOTATION 9 Since we always assume to be bounded, we have L q( ) () L p( ) () (2.1.3) with continuous embedding for all variable exponents q L () such that p(x) q(x) almost everywhere in. We define the Sobolev space with variable exponent W 1,p( ) () = {f L p( ) () : Df L p( ) ()}. For f W 1,p( ) (), f f W 1,p( ) () := f L p( ) () + Df L p( ) () defines a norm such that, for (W 1,p( ) (), W 1,p( )) is a Banach space and we have a continuous embedding W 1,q( ) () W 1,p( ) () (2.1.4) for all variable exponents q L () such that p(x) q(x) almost everywhere in. Moreover, if p > 1, then W 1,p( ) () is reflexive. For N = 1 and = (a, b), a, b R, a < b, it is an immediate consequence of (2.1.4) that W 1,p( ) (a, b) W 1,p (a, b) C([a, b]) (2.1.5) with continuous and dense embedding. We define also W 1,p( ) () := D() W 1,p( ) (). For exponents p C(, R + ), p 1 and f W 1,p( ) () the Poincaré inequality f L p( ) () C Df L p( ) () (2.1.6) holds true and the embedding W 1,p( ) () into L p( ) () is compact (see [52], [44]). In particular, W 1,p( ) () is a reflexive Banach space if p > 1. Its dual space will be denoted by W 1,p ( ) (). According to [44] and [37], for a bounded domain with Lipschitz boundary and p + < N, we have a compact embedding W 1,p( ) () L q( ) () for all measurable exponents q : [1, ) such that q(x) < p (x) ε almost everywhere in for some ε >, where p (x) := Np(x)/(N p(x)) almost everywhere in. Some more general Sobolev embedding results for variable exponents p L () such that p > 1 can be found in [52]. To the best of our knowledge, no general necessary and sufficient conditions for the Poincaré inequality (2.1.6) are known beyond continuity of the variable

11 CHAPTER 2. FUNCTION SPACES AND NOTATION 1 exponent. This is different in the particular case of one space dimension: If = (a, b) for a, b R, a < b, from Poincaré inequality in W 1,1 (a, b) it follows that W 1,1 (a, b) is continuously embedded into C([a, b]). Since W 1,p( ) (a, b) W 1,1 (a, b) there exists C 1 > such that f L (a,b) C 1 Df L 1 (a,b) (2.1.7) for all f W 1,p( ) (a, b). Now, from (2.1.7) and the continuous embedding of L p( ) (a, b) into L 1 (a, b) it follows that there exists C 2 > such that f L (a,b) C 2 Df L p( ) (a,b) (2.1.8) for all f W 1,p( ) (a, b). Since L (a, b) L p( ) (a, b), from (2.1.8) it follows that f L p( ) (a,b) C 3 Df L p( ) (a,b) (2.1.9) holds for all f W 1,p( ) (a, b), where C 3 > does not depend on f. Hence, the Poincaré inequality (2.1.6) holds for any p L (a, b) such that p 1. However, we will exclusively work with Lebesgue and Sobolev spaces with continuous variable exponent p : [1, ) such that 1 < p. We do not assume p( ) to be log-hölder continuous: Definition The continuous function p : [1, ) satisfies the log-hölder continuity condition iff there exists a non-decreasing function ω : (, ) R such that lim sup t + ω(t) ln(1/t) < + and holds for all x, y, x y < 1. p(x) p(y) < ω( x y ) (2.1.1) If log-hölder continuity condition (2.1.1) holds, C () is dense in W 1,p( ) () and W 1,p( ) () = W 1,p( ) () W 1,1 (). Moreover, if 1 < p < p + < N, then the Sobolev embedding holds also for q( ) = p ( ) (see [37] for more details). An additional difficulty to our setting arises from the fact that W 1,p( ) () W 1,1 () and W 1,p( ) () are in general not equal, hence different duality frameworks for (E, f) are possible and lead to different notions of solution (see [74], [73], [3] for more details). We will restrict ourselves to the W 1,p( ) ()/W 1,p ( ) () duality. We refer to [3] for some existence and uniqueness results to (E, f) in the case F and p L () such that p > 1 where different duality frameworks and notions of solution have been considered. Note that W 1,p( ) () is stable by

12 CHAPTER 2. FUNCTION SPACES AND NOTATION 11 composition with Lipschitz functions, even if for a function w W 1,p( ) () having trace zero does not guarantee that w W 1,p( ) (). Indeed, let L : R R be Lipschitz continuous such that L() = and u W 1,p( ) (). Then there exists a sequence (u n ) n D(), such that u n u in W 1,p( ) () as n. From the Lipschitz continuity of L it follows immediately that L(u n ) L(u) as n in L p( ) (). Since L is essentially bounded and D(L u n ) = L (u n )D(u n ) almost everywhere in and in D () for each n N, we have L(u n ) W 1,p+ (), hence in W 1,p( ) () (by continuous embedding of W 1,p+ () into W 1,p( ) ()). Moreover, there exists a constant C > not depending on n N such that D(L(u n )) p( ) C. By reflexivity of W 1,p( ) () it follows that there exists a (not relabeled) subsequence of (L(u n )) n converging to L(u) weak in W 1,p( ) (). Therefore, L(u) W 1,p( ) (). 2.2 Function spaces for the evolution problem If X is a Banach space, 1 q and T >, then L q (, T ; X) denotes the space of strongly measurable functions u : (, T ) X such that t u(t) X L q (, T ). Moreover, C([, T ]; X) denotes the space of continuous functions u : [, T ] X endowed with the norm u C([,T ];X) = max t [,T ] u(t) X. The following density result will be used in the study of the evolution problem: Proposition Let X = L p () or X = W 1,p () and 1 p <. Then, D((, T ) ) is dense in L q (, T ; X) for any 1 q <. Proof: From [4], Cor , p. 13 it follows that { n } Z := φ i (x)ψ i (t), n 1, φ i D(), ψ i D(, T ) D((, T ) ) i=1 is dense in L q (, T ; X) for any Banach space X such that D() is dense in X and 1 q <.

13 CHAPTER 2. FUNCTION SPACES AND NOTATION 12 For T > let := (, T ). Extending a variable exponent p : [1, ) to by setting p(t, x) := p(x) for all (t, x), we may also consider the generalized Lebesgue space { } L p( ) ( ) := u : R; u is measurable, u(t, x) p(x) d(t, x) <, endowed with the norm u L p( ) ( ) := inf µ> { u(t, x) µ p(x) d(t, x) 1 which, of course shares the same properties as L p( ) (). Moreover, if p( ) is log-hölder continuous in, so it is in. Indeed, if p( ) satisfies the log- Hölder continuity condition in, according to Definition 2.1.1, there exists a non-decreasing function ω : (, ) R such that lim sup t + ω(t) ln(1/t) < + and p(t, x) p(s, y) = p(x) p(y) < ω( x y ) ω( (t, x) (s, y) ) (2.2.1) holds for all (t, x), (s, y) such that (t, x) (s, y) < 1. Let p : [1, ) be a continuous variable exponent and T >. The abstract Bochner spaces L p+ (, T ; L p( ) ()) and L p (, T ; L p( ) ()) will be important in the study of renormalized solutions to (P, f, b ). In the following we identify an abstract function like v L p (, T ; L p( ) ()) with the realvalued function v defined by v(t, x) = v(t)(x) for almost all t (, T ) and almost all x. In the same way we associate to any function v L p( ) ( ) an abstract function v : (, T ) L p( ) () by setting v(t) := v(t, ) for almost every t (, T ). Lemma We have the following continuous dense embeddings: L p+ (, T ; L p( ) ()) d L p( ) ( ) d L p (, T ; L p( ) ()). (2.2.2) Proof: For v L p( ) ( ), the corresponding abstract function v : (, T ) L p( ) () is strongly Bochner measurable (by the Dunford-Pettis Theorem, since it is weakly measurable and L p( ) () is seperable). Moreover, using },

14 CHAPTER 2. FUNCTION SPACES AND NOTATION 13 (2.1.2) and the Jensen inequality, we find the estimate T v(t) p dt L p( ) () [ T ( max v(t, x) p(x) dx, T ( T v(t, x) p(x) dxdt + T 1 p /p + [ ] max v p, L p( ) ( ) v p+ L p( ) ( ) ] + T 1 p /p + max [ v (p ) 2 /p +, L p( ) ( ) v p L p( ) (. ) v(t, x) dx) p /p+] p(x) dt ) p /p + v(t, x) p(x) dxdt (2.2.3) Therefore, the embedding of L p( ) ( ) into L p (, T ; L p( ) ()) is continuous. If u L p+ (, T ; L p( ) ()), from L p( ) () L 1 () it follows that u L p+ (, T ; L 1 ()), hence, according to [4], Prop , p. 28, the corresponding real-valued function u : (, T ) R is measurable and using the same arguments as above we find the continuous embedding of L p+ (, T ; L p( ) ()) into L p( ) ( ). It is left to prove that both embeddings are dense. We consider the first embedding and fix u L p( ) ( ). Since D( ) is dense L p( ) ( ), we find a sequence (u n ) n D( ) converging to u in L p( ) ( ) as n. According to Proposition 2.2.1, D( ) is densely embedded into L p+ (, T ; L p+ ()), therefore u n L p+ (, T ; L p( ) ()) for all n N. To prove the denseness of the second embedding, we fix v L p (, T ; L p( ) ()). Taking a standard sequence of mollifiers (ρ n ) n D(R) and extending v by zero onto R, from [4], Proposition 1.7.1, p. 25, it follows that the regularized (in time) function (ρ n v)( ) := ρ n ( s)v(s)ds (2.2.4) R is in L p+ (R; L p( ) ()) for each n N, hence in L p( ) ( ) and converges to v in L p (, T ; L p( ) ()) (see [4], Théorème 1.7.1, p. 27). 2.3 Notation and functions Let us introduce some notation and functions that will be frequently used. If A is a Lebesgue measurable set, we will denote its Lebesgue measure by A and by χ A its characteristic function. For any u : R and k, we write { u (<, >,, =)k} for the set {x : u(x) (<, >,, =)k}. For

15 CHAPTER 2. FUNCTION SPACES AND NOTATION 14 r R, let r r + := max(r, ), r sign (r) the usual sign function which is equal to 1 on ], [, to 1 on ], [ and to for r =. r sign + (r) is the function defined by sign + (r) = 1 if r > and sign + (r) = if r. Let h l : R R be defined by h l (r) := min((l + 1 r ) +, 1) for each r R. For any given k >, we define the truncation function T k : R R by k, if r k, T k (r) := r, if r < k,. k, if r k. For δ > we define H + δ and H δ : R R by : R R by, r < H + δ (r) = 1 δ r, r δ 1, r > δ 1, r < δ 1 H δ (r) = r, δ r δ δ 1, r > δ. Clearly, H + δ is an approximation of sign + and H δ is an approximation of sign. Remark The following argument will be frequently used to treat the convection term F (u) Du in (E, f) and (P, f, b ): Observe that for F = (F 1,..., F N ) L (R, R N ) such that F () =, u W 1,p( ) () we have F (u) Du =. (2.3.1) Proof: Let us define s F (σ)dσ := ( s F 1(σ)dσ,..., s F N(σ)dσ ) and φ : R R N, φ(s) := s F (σ)dσ for s R. Observe that φ is Lipschitz continuous such that φ() = and therefore φ u is in (W 1,p( ) ()) N. Hence, (φ u) = F (u) u (2.3.2) x i x i in D () for any i = 1,..., N. Consequently, ( u ) div F (σ)dσ = F (u) Du (2.3.3) in D () and (2.3.1) follows using (2.3.3) and the Gauss-Green Theorem for Sobolev functions from u = almost everywhere on.

16 Chapter 3 The elliptic case 3.1 Renormalized solutions Definition A renormalized solution to (E, f) is a pair of functions (u, b) satisfying the following conditions: (R1) u : R is measurable, b L 1 (), u(x) D(β(x)) and b(x) β(u(x)) for a.e. x. (R2) For each k >, T k (u) W 1,p( ) () and bh(u)ϕ + (a(x, Du) + F (u)) D(h(u)ϕ) = holds for all h C 1 c (R) and all ϕ W 1,p( ) () L (). (R3) a(x, Du) Du as k. {k< u <k+1} f h(u)ϕ (3.1.1) Remark We can easily check that all the terms in (R2) make sense. We recall that a function u such that T k (u) W 1,p( ) (), for all k >, does not necessarily belong to W 1,1 (). However, it is possible to define its generalized gradient (still denoted by Du) as the unique measurable function v : R N such that DT k (u) = vχ { u <k} for a.e. x and for all k >, where χ E denotes the characteristic function of a measurable set E. Moreover, if u W 1,1 (), then v coincides with the standard distributional gradient of u. See [15], [67] for more details. The main existence result of this chapter is the following theorem: Theorem For f L 1 () there exists at least one renormalized solution (u, b) to (E, f). 15

17 CHAPTER 3. THE ELLIPTIC CASE Existence for L -data To prove Theorem 3.1.2, we will introduce and solve approximating problems. To this end, for f L 1 () and m, n N we define f m,n : R by f m,n (x) = max(min(f(x), m), n) for almost every x. Clearly, f m,n L () for each m, n N, f m,n (x) f(x) a.e. in, hence lim n lim m f m,n = f in L 1 () and almost everywhere in. The next proposition will give us existence of renormalized solutions (u m,n, b m,n ) of (E, f m,n ) for each m, n N: Proposition For f L () there exists at least one renormalized solution (u, b) to (E, f). The proof of Proposition will be divided into several steps Approximate solutions for L -data At first we approximate (E, f) for f L () by problems for which existence can be proved by standard variational arguments. For < ε 1, let β ε : R R be the Yosida approximation (see [28]) of β. We introduce the operators and A 1,ε : W 1,p( ) () (W 1,p( ) ()), u β ε (T 1/ε (u)) + ε arctan(u) div a(x, Du) A 2,ε : W 1,p( ) () (W 1,p( ) ()), u div F (T 1/ε (u)). Because of (A2) and (A3), A 1,ε is well-defined and monotone (see [55], p. 157). Since β ε T 1/ε and arctan are bounded and continuous and thanks to the growth condition (A2) on a, it follows that A 1,ε is hemicontinuous (see [55], p.157). From the continuity and boundedness of F T 1/ε it follows that A 2,ε is strongly continuous. Therefore the operator A ε := A 1,ε + A 2,ε is pseudomonotone. Using the monotonicity of β ε, the Gauss-Green Theorem for Sobolev functions and the boundary condition on the convection term F (T 1/ε(u)) Du, we show with similar arguments as in [12] that A ε is coercive and bounded. Then it follows from [55], Theorem 2.7, that A ε is surjective, i.e., for each < ε 1 and f (W 1,p( ) ()) there exists at least one solution u ε W 1,p( ) () to the problem { β ε (T 1/ε (u ε )) + ε arctan(u ε ) div(a(x, Du ε ) + F (T 1/ε (u ε ))) = f in, (E ε, f) u = on

18 CHAPTER 3. THE ELLIPTIC CASE 17 such that (β ε (T 1/ε (u ε )) + ε arctan(u ε ))ϕ + (a(x, Du ε ) + F (T 1/ε (u ε )) Dϕ = f, ϕ (3.2.1) holds for all ϕ W 1,p( ) (), where, denotes the duality pairing between W 1,p( ) () and (W 1,p( ) ()). In the next proposition, we establish uniqueness of solutions u ε of (E ε, f) with right-hand sides f L () through a comparison principle that will play an important role in the approximation of renormalized solutions to (E, f) with f L 1 (). Proposition For < ε 1 fixed and f, f L () let u ε, ũ ε W 1,p( ) () be solutions of (E ε, f) and (E ε, f), respectively. Then, the following comparison principle holds: ε (arctan(u ε ) arctan(ũ ε )) + (f f) sign + (u ε ũ ε ). (3.2.2) Proof: We use the test function ϕ = H + δ (u ε ũ ε ) in the weak formulation (3.2.1) for u ε and ũ ε. Subtracting the resulting inequalities, we obtain where I 1 ε,δ = I 2 ε,δ = I 3 ε,δ = I 4 ε,δ = I 5 ε,δ = I 1 ε,δ + I 2 ε,δ + I 3 ε,δ + I 4 ε,δ = I 5 ε,δ (β ε (T 1/ε (u ε )) β ε (T 1/ε (ũ ε )))H + δ (u ε ũ ε ), (ε arctan(u ε ) ε arctan(ũ ε ))H + δ (u ε ũ ε ), (a(x, Du ε ) a(x, Dũ ε )) DH + δ (u ε ũ ε ), (F (T 1/ε (u ε )) F (T 1/ε (ũ ε ))) DH + δ (u ε ũ ε ), (f f)h + δ (u ε ũ ε ). Passing to the limit with δ, (3.2.2) follows. Remark Let f, f L () be such that f f almost everywhere in, ε > and u ε, ũ ε W 1,p( ) () solutions to (S ε, f), (S ε, f) respectively. Then it is an immediate consequence of Propsition that u ε ũ ε almost everywhere in. Furthermore, from the monotonicity of β ε T 1/ε it follows that also β ε (T 1/ε (u ε )) β ε (T 1/ε (ũ ε )) almost everywhere in.

19 CHAPTER 3. THE ELLIPTIC CASE A priori estimates Lemma For < ε 1 and f L () let u ε solution of (E ε, f). Then, W 1,p( ) () be a i.) there exists a constant C 1 = C 1 ( f, γ, p( ), N) >, not depending on ε, such that Du ε L p( ) () C 1. (3.2.3) Moreover, ii.) iii.) holds for all < ε 1 and {l< u ε <l+k} holds for all < ε 1 and all l, k >. β ε (T 1/ε (u ε )) f L () (3.2.4) a(x, Du ε ) Du ε k f (3.2.5) { u ε >l} Proof: Taking u ε as a test function in (3.2.1), by (A1) we obtain γ Du ε p(x) C(p( ), N) f L () Du ε L (), (3.2.6) p( ) where C(p( ), N) > is a constant coming from the Hölder and Poincaré inequalities. From (2.1.2) and (3.2.6) it follows that either or Setting C 1 := max Du ε L p( ) () Du ε L p( ) () ( ) 1 1 γ f p L ()C(p( ), N) 1 ( ) 1 1 γ f p L ()C(p( ), N) + 1. ( ( ) 1 ( ) ) 1 γ f p L ()C(p( ), N) , γ f p L ()C(p( ), N) 1, we get i.). Taking 1 δ (T k+δ(β ε (T 1/ε (u ε )) T k (β ε (T 1/ε (u ε ))) as a test function in (3.2.1), passing to the limit with δ and choosing k > f we obtain ii.). For k, l > fixed we take T k (u ε T l (u ε )) as a test function in (3.2.1) to obtain iii.).

20 CHAPTER 3. THE ELLIPTIC CASE 19 Remark For k >, from Lemma 3.2.4, iii.), we deduce a(x, Du ε ) Du ε k f { u ε > l} C 2 (k)l p (3.2.7) {l< u ε <l+k} for any < ε 1 and a constant C 2 (k) > not depending on ε. Indeed, T l (u ε ) p { u ε l} { u ε l} l p C(p, N)l p ( C(p, N)l p ( { Du ε 1} Du ε p(x) + { Du ε <1} Du ε p ) ) Du ε p(x) +, (3.2.8) where C(p, N) > is a constant from the Poincaré inequality in W 1,p (). Combining (3.2.6), (3.2.3) and (3.2.8), setting ( ) C(p( ), C(p( ), p, γ, C 1 ) := C(p N) f, N) C 1 + >, γ we obtain { u ε l} l p C(p( ), p, γ, C 1 ). (3.2.9) Now, (3.2.7) follows from (3.2.9) with C 2 (k) := C(p( ), p, γ, C 1 )k f > Basic convergence results In the following it is always understood that ε takes values in a sequence in (, 1) tending to zero. The a priori estimates in Lemma and Remark imply the following basic convergences: Lemma For < ε 1 and f L () let u ε W 1,p( ) () be the solution of (E ε, f). There exist u W 1,p( ) (), b L () such that for a not relabeled subsequence of (u ε ) <ε 1 as ε : and Moreover, for any k >, and u ε u in L p( ) () and a.e. in (3.2.1) Du ε Du in (L p( ) ()) N (3.2.11) β ε (T 1/ε (u ε )) b in L (). (3.2.12) DT k (u ε ) DT k (u) in (L p( ) ()) N (3.2.13) a(x, DT k (u ε )) a(x, DT k (u)) in (L p ( ) ()) N. (3.2.14)

21 CHAPTER 3. THE ELLIPTIC CASE 2 Proof: Since (3.2.1) - (3.2.13) follow directly from Lemma and Remark 3.2.5, (3.2.14) is left to prove. To this end, we fix k > and take h l (u ε )(T k (u ε ) T k (u)) as a test function in (3.2.1). Using Gauss-Green Theorem for Sobolev functions and passing to the limit with ε and then with l we obtain lim sup a(x, DT k (u ε )) D(T k (u ε ) T k (u)). (3.2.15) ε By (A2) and (3.2.3) it follows that given any subsequence of a(x, DT k (u ε )) ε, there exists a subsequence, still denoted by a(x, DT k (u ε )) ε such that We show that a(x, DT k (u ε )) ε Φ k weakly in (L p ( ) ()) N. Φ k (x) = a(x, DT k (u)) for almost every x, (3.2.16) which allows us to conclude that the whole sequence a(x, DT k (u ε )) ε converges to a(x, DT k (u)). To this end, we define the variational operator for G (L p( ) ()) N by (AG)(H) = A : (L p( ) ()) N (L p ( ) ()) N a(x, G) H, H (L p( ) ()) N. By (A2), A ist well-defined and hemicontinuous, by (A3) it is monotone, hence A is maximal monotone (see [65], Lemma 3.4, p. 88). Using (A3) and (3.2.15), we calculate (Φ k a(x, H)) (DT k (u) H) for all H (L p( ) ()) N. (3.2.17) Since A is maximal monotone, (3.2.16) follows from (3.2.17). Remark As an immediate consequence of (3.2.15) and (A3) we obtain lim a(x, DT k (u ε ) a(x, DT k (u)) D(T k u ε T k (u)) =. (3.2.18) ε Combining (3.2.7) and (3.2.18), using the same arguments as in [6] it follows that a(x, Du) Du =. (3.2.19) lim l {l< u <l+1}

22 CHAPTER 3. THE ELLIPTIC CASE Proof of existence Now, we are able to conclude the proof of Proposition 3.2.1: Proof: Let h Cc 1 (R) and φ W 1,p( ) () L () be arbitrary. Taking h l (u ε )h(u)φ as a test function in (3.2.1), we obtain where I 1 ε,l + I 2 ε,l + I 3 ε,l + I 4 ε,l = I 5 ε,l (3.2.2) Iε,l 1 = β ε (T 1/ε (u ε ))h l (u ε )h(u)φ, Iε,l 2 = ε arctan(u ε )h l (u ε )h(u)φ, Iε,l 3 = a(x, Du ε ) D(h l (u ε )h(u)φ), Iε,l 4 = F (T 1/ε (u ε )) D(h l (u ε )h(u)φ), Iε,l 5 = fh l (u ε )h(u)φ. Step 1: Passing to the limit with ε Obviously, lim ε I 2 ε,l =. (3.2.21) Using the convergence results (3.2.1), (3.2.12) from Lemma we can immediately calculate the following limits: lim Iε,l 1 = bh l (u)h(u)φ, (3.2.22) ε lim Iε,l 5 = fh l (u)h(u)φ. (3.2.23) ε We write where I 3,1 ε,l = I 3,2 ε,l = I 3 ε,l = I 3,1 ε,l + I 3,2 ε,l, h l(u ε )a(x, Du ε ) Du ε h(u)φ, h l (u ε )a(x, Du ε ) D(h(u)φ).

23 CHAPTER 3. THE ELLIPTIC CASE 22 Using (3.2.7) we get the estimate lim ε I 3,1 ε,l h L () φ L ()C 2 (1)l p. (3.2.24) Since modular convergence is equivalent to norm convergence in L p( ) (), by Lebesgue Dominated Convergence Theorem it follows that h l (u ε ) x i (h(u)φ) h l (u) x i (h(u)φ) for any i {1,..., N} in L p( ) () as ε. Keeping in mind that I 3,2 ε,l = h l (u ε )a(x, DT l+1 (u ε )) D(h(u)φ), by (3.2.14), we get Let us write where lim I 3,2 ε,l = ε I 4,1 ε,l = I 4,2 ε,l = h l (u)a(x, DT l+1 (u)) D(h(u)φ). (3.2.25) I 4 ε,l = I 4,1 ε,l + I 4,2 ε,l, h l(u ε )F (T 1/ε (u ε )) Du ε h(u)φ, h l (u ε )F (T 1/ε (u ε )) D(h(u)φ). For any l N, there exists ε (l) such that for all ε < ε (l), I 4,1 ε,l = h l(t l+1 (u ε ))F (T l+1 (u ε )) DT l+1 (u ε )h(u)φ. (3.2.26) Using Gauss-Green Theorem for Sobolev functions in (3.2.26) we get I 4,1 ε,l = Tl+1 (u ε) h l(r)f (r)dr D(h(u)φ). (3.2.27) Now, using (3.2.1) and the Gauss-Green Theorem, after the passage to the limit with ε we get lim I 4,1 ε,l = h ε l(u)f (u) Du h(u)φ. (3.2.28)

24 CHAPTER 3. THE ELLIPTIC CASE 23 Choosing ε small enough, we can write I 4,2 ε,l = h l (u ε )F (T l+1 (u ε )) D(h(u)φ) (3.2.29) and conclude lim I 4,2 ε,l = h l (u)f (u) D(h(u)φ). (3.2.3) ε Step 2: Passage to the limit with l. Combining (3.2.2) with (3.2.21) - (3.2.3) we find where and I 1 l = I 2 l = I 1 l + I 2 l + I 3 l + I 4 l + I 5 l = I 6 l, (3.2.31) bh l (u)h(u)φ, h l (u)a(x, DT l+1 (u)) D(h(u)φ), Il 3 l p C 2 (1) h φ, Il 4 = h l (u)f (u) D(h(u)φ), Il 5 = h l(u)f (u) Du h(u)φ, Il 6 = fh l (u)h(u)φ. Obviously, we have lim l I3 l =. (3.2.32) Choosing m > such that supp h [ m, m], we can replace u by T m (u) in Il 1,..., I6 l. Therefore, it follows that lim l I1 l = bh(u)φ,, (3.2.33) lim l I2 l = a(x, Du) D(h(u)φ), (3.2.34) lim l I4 l = F (u) D(h(u)φ), (3.2.35) lim l I5 l =, (3.2.36) lim l I6 l = fh(u)φ. (3.2.37)

25 CHAPTER 3. THE ELLIPTIC CASE 24 Combining (3.2.31) with (3.2.32) - (3.2.37) we obtain bh(u)φ + (a(x, Du) + F (u)) D(h(u)φ) = for all h C 1 c (R) and all φ W 1,p( ) () L (). f h(u)φ (3.2.38) 3. Step: Subdifferential argument It is left to prove that u(x) D(β(x)) and b(x) β(u(x)) for almost all x. Since β a is maximal monotone graph, there exists a convex, l.s.c. and proper function j : R [, ], such that β(r) = j(r) for all r R. According to [28], for < ε 1, j ε : R R defined by j ε (r) = r β ε(s)ds has the following properties: i.) For any < ε 1, j ε is convex and differentiable for all r R, such that j ε(r) = β ε (r) for all r R and any < ε 1 ii.) j ε (r) j(r) for all r R as ε. From i.) it follows that for any < ε 1 j ε (r) j ε (T 1/ε (u ε )) + (r T 1/ε (u ε ))β ε (T 1/ε (u ε )) (3.2.39) holds for all r R and almost everywhere in. Let E be an arbitrary measurable set and χ E its characteristic function. We fix ε >. Multiplying (3.2.39) by h l (u ε )χ E, integrating over and using ii.), we obtain j(r) h l (u ε ) j ε (T l+1 (u ε ))h l (u ε ) + (r T l+1 (u ε ))h l (u ε )β ε (T 1/ε (u ε )) E E (3.2.4) for all r R and all < ε < min(ε, 1 ). As ε, taking into account that l E is arbitrary we obtain from (3.2.4) j(r)h l (u) j ε (T l+1 (u))h l (u) + bh l (u)(r T l+1 (u)) (3.2.41) for all r R almost everywhere in. Passing to the limit with l and then with ε in (3.2.41) finally yields j(r) j(u(x)) + b(x)(r u(x)) (3.2.42) for all r R and almost every x, hence u D(β) and b β(u) almost everywhere in. With this last step the proof of Proposition is concluded.

26 CHAPTER 3. THE ELLIPTIC CASE Weak solutions for L -data Definition A weak solution to (S, f) is a pair of functions (u, b) W 1,p( ) () L 1 loc () satisfying F (u) (L1 loc ())N, b β(u) almost everywhere in and b div(a(x, Du) + F (u)) = f (3.3.1) in D (). Remark Note that if (u, b) is a renormalized solution to (E, f) such that u W 1,p( ) (), then (u, b) in general is not a weak solution in the sense of Definition 3.3.1, since we did not assume a growth condition on F and therefore F (u) in general may fail to be locally integrable. If (u, b) is a renormalized solution of (E, f) such that u L (), it is a direct consequence of Definition that u is in W 1,p( ) () and since (3.1.1) holds with the formal choice h 1, (u, b) is a weak solution. Indeed, let us choose ϕ D() and plug h l (u)ϕ as a test function in (3.1.1). Since u L (), we can pass to the limit with l and find that u solves (E, f) in the sense of distributions. In the next proposition we will show that renormalized solutions to (E, f) for right-hand side f L () are weak solutions. In one space dimension this follows immediately from Remark since u W 1,p( ) () implies u C() (see Proposition 3.4.6). Therefore, for the rest of this section we may assume N 2. Proposition Let (u, b) be a renormalized solution to (E, f) for f L (). Then u W 1,p( ) () L () and thus, in particular, u is a weak solution to (E, f). Proof: From Lemma it follows that u W 1,p( ) (). It suffices to prove that u L (). For ε, k >, we take h l (u) 1T ε ε(u T k (u)) as a test function in (3.1.1). Neglecting positive terms and passing to the limit with l, we obtain 1 ε {k< u <k+ε} Du p(x) f N (φ(k)) (N 1)/N, (3.3.2) where φ(k) := { u > k} for k >. Now we use similar arguments as in [18]. We apply the continuous embedding of W 1,1 () into L N/(N 1) () and the Hölder inequality to get ( ) 1/(p 1 φ(k) φ(k + ε) ) ( ) 1/p 1 T ε (u T k (u)) N Du p, εc N N 1 ε ε {k< u <k+ε} (3.3.3)

27 CHAPTER 3. THE ELLIPTIC CASE 26 where C N > is the constant coming from the Sobolev embedding. Notice that 1 φ(k) φ(k + ε) Du p + 1 Du p(x), (3.3.4) ε {k< u <k+ε} ε ε {k< u <k+ε} hence from (3.3.2), (3.3.3) and (3.3.4) we deduce 1 T ε (u T k (u)) N εc N N 1 ( ) 1/(p φ(k) φ(k + ε) ) ( ) 1/p φ(k) φ(k + ε) + f N (φ(k)) (N 1)/N. ε ε (3.3.5) From (3.3.5) and Young s inequality with α > it follows that 1 C N C (φ(k + ε))(n 1)/N αp p C f N (φ(k)) (N 1)/N φ(k) φ(k + ε), ε (3.3.6) where ( ) 1 C := α (p ) (p ) + αp >. p The mapping (, ) k φ(k) is non-increasing and therefore of bounded variation, hence it is differentiable almost everywhere on (, ) with φ L 1 loc (, ). Since it is also continuous from the right, we can pass to the limit with ε in (3.3.6) to find C (φ(k)) (N 1)/N + φ (k) (3.3.7) for almost every k > and α > chosen small enough such that ( ) C C N := C αp p C f N >. Now, the conclusion of the proof follows by contradiction. We assume that φ(k) > for each k >. For k > fixed, we choose k < k. From (3.3.7) it follows that 1 N C + d ( ) (φ(s)) (1/N) (3.3.8) ds for almost all s (k, k). The left hand side of (3.3.8) is in L 1 (k, k), hence we integrate (3.3.8) over [k, k]. Moreover, since φ is non-increasing, integrating (3.3.8) over (k, k) we get (φ(k)) 1/N φ(k ) 1/N + 1 N C (k k) (3.3.9) and from (3.3.9) the contradiction follows.

28 CHAPTER 3. THE ELLIPTIC CASE Proof of Theorem Approximate solutions for L 1 -data The comparison principle from Proposition will be the main tool in the second approximation procedure. For f L 1 () and m, n N let f m,n L () be defined as in the beginning of Section 4. From Proposition it follows that for any m, n N there exists u m,n W 1,p( ) (), b m,n L (), such that (u m,n, b m,n ) is a renormalized solution of (E, f m,n ). Therefore b m,n h(u m,n )φ+ (a(x, Du m,n )+F (u m,n )) D(h(u m,n )φ) = f m,n h(u m,n )φ (3.4.1) holds for all m, n N, h Cc 1 (R), φ W 1,p( ) () L (). In the next Lemma, we give a priori estimates that will be important in the following: Lemma For m, n N let (u m,n, b m,n ) be a renormalized solution of (E, f m,n ). Then, i.) For any k > we have DT k (u m,n ) p(x) k γ f 1. (3.4.2) ii.) For k >, there exists a constant C 3 (k) >, not depending on m, n N, such that DT k (u m,n ) p( ) C 3 (k). (3.4.3) iii.) holds for all m, n N. b m,n 1 f 1 (3.4.4) Proof: Proof: For l, k >, we plug h l (u m,n )T k (u m,n ) as a test function in (3.4.1). Then i.) and ii.) follow with similar arguments as used in the proof of Lemma To prove iii.), we neglect the positive term a(x, DT k (u m,n ))DT k (u m,n ) and keep b m,n T k (u m,n ) f m,n u m,n. (3.4.5)

29 CHAPTER 3. THE ELLIPTIC CASE 28 Since b m,n β(u m,n ) a.e. in, from (3.4.5) it follows that b m,n f, (3.4.6) { u m,n >k} and we find iii.) by passing to the limit with k. By definition we have From Proposition it follows that f m,n f m+1,n and f m,n+1 f m,n (3.4.7) u ε m,n u ε m+1,n and u ε m,n+1 u ε m,n, (3.4.8) almost everywhere in for any m, n N and all ε >, hence passing to the limit with ε in (3.4.8) yields u m,n u m+1,n and u m,n+1 u m,n, (3.4.9) almost everywhere in for any m, n N. Setting b ε := β ε (T 1 (u ε )), using ε (3.4.8), Remark and the fact that b ε m,n b m,n in L () and this convergence preserves order we get b m,n b m+1,n and b m,n+1 b m,n (3.4.1) almost everywhere in for any m, n N. By (3.4.1) and (3.4.4), for any n N there exists b n L 1 () such that b m,n b n as m in L 1 () and almost everywhere and b L 1 (), such that b n b as n in L 1 () and almost everywhere in. By (3.4.9), the sequence (u m,n ) m is monotone increasing, hence, for any n N, u m,n u n almost everywhere in, where u n : R is a measurable function. Using (3.4.9) again, we conclude that the sequence (u n ) n is monotone decreasing, hence u n u almost everywhere in, where u : R is a measurable function. In order to show that that u is finite almost everywhere we will give an estimate on the level sets of u m,n in the next lemma: Lemma For m, n N let (u m,n, b m,n ) be a renormalized solution of (E, f m,n ). Then, there exist a constant C 4 >, not depending on m, n N, such that { u m,n l} C 4 l (p 1) (3.4.11) for all l 1.

30 CHAPTER 3. THE ELLIPTIC CASE 29 Proof: With the same arguments as in Remark we obtain ( ) { u m,n l} C(p, N)l p DT l (u m,n ) p(x) +, (3.4.12) for all m, n N where C(p, N) is the constant from Sobolev embedding in L p (). Now, we plug (3.4.2) into (3.4.12) to obtain (3.4.11). Note that, as (u m,n ) m is pointwise increasing with respect to m, and lim {u m,n > l} = {u n > l} (3.4.13) m lim {u m,n l} = {u n l}. (3.4.14) m Combining (3.4.11) with (3.4.13) and (3.4.14) we get {u n l} + {u n > l} C 4 l (p 1), (3.4.15) for any l 1, hence u n is finite almost everywhere for any n N. By the same arguments we get {u < l} + {u > l} C 4 l (p 1) (3.4.16) from (3.4.15), hence u is finite almost everywhere. Now, since b m,n β(u m,n ) almost everywhere in it follows by a subdifferential argument that b n β(u n ) and b β(u) almost everywhere in. Remark If (u m,n, b m,n ) is a renormalized solution of (E, f m,n ), using h ν (u m,n )T k (u m,n T l (u m,n )) as a test function in (3.4.1), neglecting positive terms and passing to the limit with ν we obtain ( ) a(x, Du m,n ) Du m,n k f + f {l< u m,n <l+k} { u m,n >l} { f <σ} { f >σ} (3.4.17) for any k, l, σ >. Now, applying (3.4.11) to (3.4.17), we find that a(x, Du m,n ) Du m,n σkc 4 l (p 1) + k f (3.4.18) {l< u m,n <l+k} holds for any k, σ >, l 1 uniformly in m, n N. { f >σ} Lemma For m, n N let (u m,n, b m,n ) be a renormalized solution of (E, f m,n ). There exists a subsequence (m(n)) n such that setting f n := f m(n),n, b n := b m(n),n, u n := u m(n),n we have u n u almost everywhere in. (3.4.19)

31 CHAPTER 3. THE ELLIPTIC CASE 3 Moreover, for any k >, as n. T k (u n ) T k (u) in L p( ) () and almost everywhere in, (3.4.2) DT k (u n ) DT k (u) in (L p( ) ()) N, (3.4.21) a(x, DT k (u n )) a(x, DT k (u)) in (L p ( ) ()) N. (3.4.22) Proof: Applying the diagonal principle in L 1 (), we construct a subsequence (m(n)) n, such that arctan(u m(n),n ) arctan(u), b n := b m(n),n b, f n := f m(n),n f as n in L 1 () and almost everywhere in. It follows that (3.4.19) and (3.4.2) hold. Combining (3.4.2) with (3.4.3) we get T k (u) W 1,p( ) (), T k (u n ) T k (u) in L p( ) () and (3.4.21) holds for any k >. From (3.4.2) and (A2) it follows, that, for fixed k >, given any subsequence of a(x, DT k (u n )) n, there exists a subsequence, still denoted by a(x, DT k (u n )) n, such that a(x, DT k (u n )) n Φ k in (L p ( ) ()) N as n. Since h l (u n )(T k (u n ) T k (u)) is an admissible test function in (3.4.1), lim sup a(x, DT k (u n )) D(T k (u n ) T k (u)) (3.4.23) n holds. Then, (3.4.22) follows with the same arguments as in the proof of Lemma Remark With the same arguments as in Remark 3.2.7, we have a(x, DT k (u n ) a(x, DT k (u))) D(T k (u n ) T k (u)) =, (3.4.24) lim n lim a(x, Du) Du =. (3.4.25) l {l< u <l+1}

32 CHAPTER 3. THE ELLIPTIC CASE Conclusion of the proof of Theorem It is left to prove that (u, b) satisfies bh(u)φ + (a(x, Du) + F (u)) D(h(u)φ) = fh(u)φ. (3.4.26) for all h Cc 1 (R), φ W 1,p( ) () L (). To this end, we take h Cc 1 (R), φ W 1,p( ) () L () arbitrary and plug h l (u n )h(u)φ into (3.4.1) to obtain where I 1 n,l = I 2 n,l = I 3 n,l = I 4 n,l = I 1 n,l + I 2 n,l + I 3 n,l = I 4 n,l, (3.4.27) b n h l (u n )h(u)φ a(x, Du n ) D(h l (u n )h(u)φ) F (u n ) D(h l (u n )h(u)φ) f n h l (u n )h(u)φ. Step 1: Passing to the limit with n Applying the convergence results from Lemma we get lim n I1 n,l = bh l (u)h(u)φ, (3.4.28) lim n I2 n,l = fh l (u)h(u)φ. (3.4.29) Let us write where I 2,1 n,l = I 2,2 n,l = I 2 n,l = I 2,1 n,l + I2,2 n,l, h l (u n )a(x, Du n ) D(h(u)φ), h l(u n )a(x, Du n ) Du n h(u)φ. With similar arguments as in the proof of (3.2.25) it follows that lim n I2,1 n,l = h l (u)a(x, Du) D(h(u)φ). (3.4.3)

33 CHAPTER 3. THE ELLIPTIC CASE 32 By (3.4.18), we get the estimate lim I 2,2 n n,l h L () φ L () ( ) σc 4 l (p 1) + f { f >σ} (3.4.31) for all n N and all l 1, σ >. Next, we write I 3 n,l = I 3,1 n,l + I3,2 n,l, where lim n I3,1 n,l = lim n I3,2 n,l = h l (u)f (u) D(h(u)φ), (3.4.32) h l(u)f (u) Du h(u)φ (3.4.33) follows with the same arguments as in (3.2.26) - (3.2.3). Step 2: Passage to the limit with l. Combining (3.4.27) with (3.4.28) - (3.4.33) we get for all σ > and all l 1 I 1 l + I 2 l + I 3 l + I 4 l + I 5 l = I 6 l (3.4.34) where Il 1 = bh l (u)h(u)φ, Il 2 = h l (u)a(x, DT l+1 (u)) D(h(u)φ), ( Il 3 h L () φ L () σc 4 l (p 1) + { f >σ} ) f, for any σ > and I 4 l = I 5 l = I 5 l = h l(u)f (u) h(u)φdu, h l (u)f (u) D(h(u)φ), fh l (u)h(u)φ.

34 CHAPTER 3. THE ELLIPTIC CASE 33 Choosing m > such that supp h [ m, m], we can replace u by T m (u) in Il 1,..., I6 l, hence lim l I1 l = bh(u)φ, (3.4.35) lim l I2 l = a(x, Du) D(h(u)φ), (3.4.36) lim l I3 l h L () φ L () f, (3.4.37) { f >σ} lim l I4 l =, (3.4.38) lim l I5 l = F (u) D(h(u)φ), (3.4.39) lim l I6 l = f h(u)φ (3.4.4) for all σ >. Combining (3.4.34) with (3.4.35) - (3.4.4) we finally obtain that (3.4.1) holds for all h C 1 c (R) and all φ W 1,p( ) () L () Existence in one space dimension For N = 1, = (a, b) with a, b R, a < b the following improved existence result holds: Proposition For f L 1 (a, b) there exists at least one weak solution (u, b) to (E, f) in the sense of Defintion Proof: We fix f L 1 (a, b). By the continuous embedding of W 1,p( ) (a, b) into C([a, b]) we have L 1 (a, b) (W 1,p( ) (a, b)). Now it follows from [55], Theorem 2.7 that for any ε > there exists u ε W 1,p( ) (a, b) such that β ε (u ε ) (a(x, (u ε ) x ) + F (u ε )) x = f (3.4.41) holds in D (a, b). For right hand sides f m,n L (a, b) as defined in Section 3.2, all a priori estimates stated in Lemma hold uniformly in ε >. Moreover, the sequence (u ε m,n) is uniformly bounded in L (a, b) for ε > and m, n N. Therefore, using similar arguments as in the conclusion of the proof of Theorem 3.1.2, we find that (u, b) is a weak solution to (E, f).

35 CHAPTER 3. THE ELLIPTIC CASE Uniqueness of renormalized solutions In this section, we prove a uniqeness result for renormalized solutions to the problem (E, f) with f L 1 (). Theorem For f, f L 1 () let (u, b) be a renormalized solution of (S, f) and (ũ, b) be a renormalized solution of (S, f). Then, the following comparison principle holds: (b b) + (f f) sign + (u ũ) + (f f) sign + (b b). (3.5.1) {u=ũ} Proof: We choose π W 1,p( ) () L () such that π 1 almost everywhere in. For l > arbitrary, we use h l (u)h + δ (T l+1(u) T l+1 (ũ) + δπ) as a test function in the renormalized formulation for (u, b) and h l (ũ)h + δ (T l+1(u) T l+1 (ũ) + δπ) as a test function in the renormalized formulation for (ũ, b). Subtracting the resulting equalities, we obtain I 1 l,δ + I 2 l,δ + I 3 l,δ + I 4 l,δ + I 5 l,δ + I 6 l,δ + I 7 l,δ = I 8 l,δ, (3.5.2) where M := { < T l+1 (u) T l+1 (ũ) + δπ < δ} and Il,δ 1 = (bh l (u) bh l (ũ))h + δ (T l+1(u) T l+1 (ũ) + δπ), Il,δ 2 = (h l(u)a(x, Du) Du h l(ũ)a(x, Dũ) Dũ)H + δ (T l+1(u) T l+1 (ũ) + δπ), Il,δ 3 = (h l (u)a(x, Du) h l (ũ)a(x, Dũ)) D(T l+1 (u) T l+1 (ũ)), δ M Il,δ 4 = (h l (u)a(x, Du) h l (ũ)a(x, Dũ)) Dπ, Il,δ 5 = M (h l(u)f (u) Du h l(ũ)f (ũ) Dũ)H + δ (T l+1(u) T l+1 (ũ) + δπ), Il,δ 6 = (h l (u)f (u) h l (ũ)f (ũ)) D(T l+1 (u) T l+1 (ũ)), δ M Il,δ 7 = (h l (u)f (u) h l (ũ)f (ũ)) Dπ, M Il,δ 8 = (fh l (u) fh l (ũ))h + δ (T l+1(u) T l+1 (ũ) + δπ). 1. Step: Passage to the limit as δ Since H + δ (T l+1(u) T l+1 (ũ)+δπ) sign + (T l+1 (u) T l+1 (ũ))+χ {Tl+1 (u)=t l+1 (ũ)}π

36 CHAPTER 3. THE ELLIPTIC CASE 35 as δ almost everywhere in it follows that lim Il,δ 1 = (bh l (u) bh l (ũ))(sign + (T l+1 (u) T l+1 (ũ)) + χ {Tl+1 (u)=t l+1 δ (ũ)}π), (3.5.3) lim Il,δ 7 =, (3.5.4) δ lim Il,δ 8 = (fh l (u) fh l (ũ))(sign + (T l+1 (u) T l+1 (ũ)) + χ {Tl+1 (u)=t l+1 δ (ũ)}π). (3.5.5) Let us recall that, because of the definition of h l, we can replace u by T l+1 (u) and ũ by T l+1 (ũ) which belong to W 1,p( ) () in Il,δ 1,..., I8 l,δ and so DT l+1(u) = DT l+1 (ũ) almost everywhere in {T l+1 (u) = T l+1 (ũ)}. Therefore, lim Il,δ 2 = δ and Let us write where I 3,1 l,δ = 1 δ I 3,2 l,δ = 1 δ By (A3), I 3,2 l,δ I 3,1 l,δ (h l(u)a(x, Du) Du h l(ũ)a(x, Dũ) Dũ)(sign + (T l+1 (u) T l+1 (ũ)) M M (3.5.6) lim δ I 4 l,δ =. (3.5.7) I 3 l,δ = I 3,1 l,δ + I3,2 l,δ, (h l (u) h l (ũ))a(x, DT l+1 (u)) D(T l+1 (u) T l+1 (ũ)), h l (ũ)(a(x, DT l+1 (u)) a(x, DT l+1 (ũ))) D(T l+1 (u) T l+1 (ũ)). is nonnegative. As h l 1 for all l >, we have the estimate {<T l+1 (u) T l+1 (ũ)<δ} and from (3.5.8) it follows that Now, we write I 5 l,δ = a(x, DT l+1 (u)) D(T l+1 (u) T l+1 (ũ)) (3.5.8) lim sup Il,δ 3. (3.5.9) δ ( ) Tl+1 (u) div h l(r)f (r)dr H δ (T l+1 (u) T l+1 (ũ) + δπ) = I 5,1 l,δ + I5,2 l,δ, T l+1 (ũ)

37 CHAPTER 3. THE ELLIPTIC CASE 36 where I 5,1 l,δ = I 5,2 l,δ = 1 δ {<T l+1 (u) T l+1 (ũ)+δπ<δ} {<T l+1 (u) T l+1 (ũ)+δπ<δ} It is easy to calculate that and from I 5,2 l,δ max F (s) s [ l 1,l+1] it follows that Let us write where I 6,1 l,δ = 1 δ I 6,2 l,δ = 1 δ M M Tl+1 (ũ) T l+1 (u) Tl+1 (u) T l+1 (ũ) h l(r)f (r)dr Dπ, h l(r)f (r)dr D(T l+1 (u) T l+1 (ũ)). lim I 5,1 l,δ = (3.5.1) δ {<T l+1 (u) T l+1 (ũ)<δ} D(T l+1 (u) T l+1 (ũ)) (3.5.11) lim δ I 5 l,δ =. (3.5.12) I 6 l,δ = I 6,1 l,δ + I6,2 l,δ, h l (u)(f (T l+1 (u)) F (T l+1 (ũ))) D(T l+1 (u) T l+1 (ũ)), (h l (u) h l (ũ))f (T l+1 (ũ)) D(T l+1 (u) T l+1 (ũ)). Let L F > be the Lipschitz constant of F. Then we find I 6,1 l,δ L F D(T l+1 (u) T l+1 (ũ)), (3.5.13) {<T l+1 (u) T l+1 (ũ)<δ} I 6,2 l,δ max F (s) s [ l 1,l+1] {<T l+1 (u) T l+1 (ũ)<δ} hence, from (3.5.13) and (3.5.14) it follows that Combining (3.5.2) with (3.5.3) - (3.5.15) we obtain D(T l+1 (u) T l+1 (ũ)), (3.5.14) lim Il,δ 6 =. (3.5.15) δ I 1 l + I 2 l I 3 l, (3.5.16)