2.2 Existence of Minimizers 29

Remarks 2.8 (i) Regularity of minimizers, e.g.,u H 2(Ω), can be proved if the boundary ofΩ isC 2-regular andW is strongly convex∈ , i.e.,D 2W(A) B,B c B 2 m d [ ]≥ | | for allA,B R × andc>0. (ii) If the∈ minimization problem involves a constraint, such asG(u(x)) 0 for = almost everyx Ω with a continuously differentiable functionG R m R, then one can formally∈ consider a saddle point problem to derive the: Euler–Lagrange→ equations, e.g., forW(A) A 2/2, the problem =| |   1 inf sup u 2 dx λG(u)dx. u H 1(Ω) λ L q (Ω) 2 |∇ | + ∈ 0 ∈ Ω Ω

The optimality conditions are withg DG given by =    u vdx λg(u) vdx 0, μG(u)dx 0 ∇ ·∇ + · = = Ω Ω Ω

1 q for allv H 0 (Ω R ) and allμ L (Ω). The unknown variableλ is the Lagrange multiplier∈ associated; to the constraint∈ G(u(x)) 0 for almost everyx Ω. = ∈ (iii) On the part∂Ω Γ D where no Dirichlet boundary conditions u Γ u D are \ | D = imposed, the homogeneous Neumann boundary conditions DW0( u) n 0 are satisfied. Inhomogeneous Neumann conditions can be specified through∇ · = a function q m g L (ΓN R ) and a corresponding contribution to the energy functional, e.g., ∈ ;  

I(u) W0( u(x))dx guds. = ∇ − Ω ΓN

1,p 1,p (iv) The Euler–Lagrange equations define an operatorL W (Ω) W (Ω) and we look foru W 1,p (Ω) withL(u) b for a given: right-hand→ sideb 1,p ∈ = ∈ W (Ω) . Under certain monotonicity conditions onL, the existence of solutions for this equation can be established with the help of discretizations andfixed-point theorems. This is of importance when the partial differential equation is not related to a minimization problem.

2.3 Gradient Flows

The direct method in the calculus of variations provides existence results for global minimizers of functionals but its proof is nonconstructive. In practice, the most robust methods tofind stationary points are steepest descent methods. These can often be regarded as discretizations of time-dependent problems. To understand the stability and convergence properties of descent methods, it is important and insightful to analyze the corresponding continuous problems. Infinite-dimensional situations we 30 2 Analytical Background may think of a functionV R n R and the ordinary differential equation : →

y V(y),y(0) y 0. = −∇ = IfV C 2(Rn), then the Picard–Lindelöf theorem guarantees the existence of a ∈ unique local solutiony ( T, T) R n. Taking the inner product of the differential equation withy and using: − the chain→ rule to verify V(y(t)) y (t) (V y) we ∇ · = ◦ have after integration over(0,T )

T 2 y dt V(y(T )) V(y(0)). | | + = 0

This is called an energy law and shows that the functiont V(y(t)) is decreasing. Since the evolution becomes stationary if V(y(t)) 0, this → allows us tofind critical points ofV with small energy. It is the aim∇ of this= section to justify gradientflows for functionals on infinite-dimensional spaces. For more details on this subject, we refer the reader to the textbooks [2, 6–8].

2.3.1 Differentiation in Banach Spaces

We consider a Banach spaceX and a functionalI X R. : → Definition 2.3 (a) We say thatI is Gâteaux-differentiable atv 0 X if for allh X the limit ∈ ∈ I(v 0 sh) I(v 0) δI(v 0,h) lim + − = s 0 s → exists and the mappingDI(v 0) X R,h δI(v 0,h) is linear and bounded. : → → (b) We say thatI is Fréchet-differentiable atv 0 X if there exist a bounded linear ∈ operatorA X R and a functionϕ R R with lim s 0 ϕ(s)/s 0 such that : → : → → =

I(v 0 h) I(v 0) Ah ϕ( h X ). + − = +

In this case we defineDI(v 0) A. = Remark 2.9 IfI is Gâteaux-differentiable at every point in a neighborhood ofv 0 and DI is continuous atv 0, thenI is Fréchet-differentiable atv 0. The gradient of a functional is the Riesz representative of the Fréchet derivative with respect to a given scalar product. Definition 2.4 LetH be a Hilbert space such thatX is continuously embedded in H. IfI is Fréchet-differentiable atv 0 X withDI(v 0) H , then the H-gradient ∈ ∈ H I(v 0) H is defined by ∇ ∈ 2.3 Gradient Flows 31

( H I(v 0),v) H DI(v 0) v ∇ = [ ] for allv H. ∈  Example 2.2 ForX H 1(Ω) andI(u) (1/2) u 2 dx, we have = 0 = Ω |∇ |   1 1 I(u sv) I(u) u 2 2s u v s 2 v 2 dx u 2 dx + − = 2 |∇ | + ∇ ·∇ + |∇ | − 2 |∇ | Ω  Ω s2 s u vdx v 2 dx = ∇ ·∇ + 2 |∇ | Ω Ω  andI is Fréchet differentiable withDI(u) v Ω u vdx. ForH X 1 [ ] = ∇ ·∇ = = H (Ω) with scalar product(v,w) H 1 v wdx, we thus have H 1 I(u) u. 0 0 = Ω ∇ ·∇ ∇ 0 = Ifu H 2(Ω) H 1(Ω), then Green’s formula shows that ∈ ∩ 0   DI(u) v u vdx ( u)vdx, [ ]= ∇ ·∇ = − Ω Ω so thatDI(u) is a bounded linear functional onL 2(Ω). ForH L 2(Ω) with scalar = product(v,w) Ω vwdx, we therefore have L2 I(u) u. = ∇ =− Remark 2.10 The Euler–Lagrange equations forI(u) Ω W(x,u, u)dx in the 2 = ∇ strong form corresponds to a vanishingL -gradient ofI , i.e., 2 I(u) 0. ∇ L =

2.3.2 Bochner–Sobolev Spaces

For evolutionary partial differential equations we will consider functionsu : 0,T X for a time interval 0,T R. We assume that the Banach space [ ]→ [ ]⊂ X is separable and say thatu 0,T X is weakly measurable if, for allϕ X , the functiont ϕ,u(t) is Lebesgue:[ ]→ measurable. In this case the Bochner∈ integral →  T u(t)dt 0      T  T is well defined with 0 u(t)dt X 0 u(t) X dt. The duality pairing between X andX will be denoted by , . ≤ · · Definition 2.5 For 1 p , the Bochner space L p( 0,T X) consists of all ≤ ≤∞ [ ]; weakly measurable functionsu 0,T X with u L p( 0,T X) < , where :[ ]→ [ ]; ∞ 32 2 Analytical Background ⎧ ⎨ esssupt 0,T u(t) X ifp ,   ∈[ ]  =∞ u L p( 0,T X) p 1/p [ ]; = ⎩ u(t) dt if 1 p< . 0,T ≤ ∞ [ ] Remark 2.11 The spaceL p( 0,T X) is a Banach space when equipped with the [ ]; norm L p( 0,T X) . · [ ]; Definition 2.6 Foru L 1( 0,T X) we say thatw L 1( 0,T X) is the gener- alized derivative ofu∈, denoted[ by];u w if for allφ∈ C (([ 0,T];)), we have = ∈ c∞ T T

φ (t)u(t)dt φ(t)w(t)dt. =− 0 0

Remark 2.12 SinceX is separable one can show that the generalized derivativeu coincides with the weak derivative∂ t u defined by

T T

u,∂ t φ dt ∂t u,φ dt   =−   0 0 for allφ C 1( 0,T X ) withφ(0) φ(T) 0. ∈ [ ]; = = Definition 2.7 The Sobolev–Bochner space W 1,p ( 0,T X) consists of all func- tionsu L p( 0,T X) withu L p( 0,T X) and[ is equipped]; with the norm ∈ [ ]; ∈ [ ]; ⎧ ⎪ ⎨esssupt 0,T ( u(t) X u (t) X ) ifp ,  T ∈[ ] +  =∞ u W 1,p ( 0,T X) ⎪ p p 1/p [ ]; = ⎩ u(t) X u (t) X dt if 1 p< . 0 + ≤ ∞

We writeH 1( 0,T X) forW 1,2( 0,T X). [ ]; [ ]; Remarks 2.13 (i) We have thatW 1,p ( 0,T X) is a Banach space for 1 p . IfX is a Hilbert space andp 2, thenW[ ];1,2( 0,T X) is a Hilbert space≤ ≤∞ denoted byH 1( 0,T X). = [ ]; (ii) For[ 1 p]; andu W 1,p ( 0,T X), we have thatu C( 0,T X) with ≤ ≤∞ ∈ [ ]; ∈ [ ]; maxt 0,T u(t) c u W 1,p ( 0,T X) . ∈[ ] ≤ [ ]; Definition 2.8 IfH is a separable Hilbert space that is identified with its dualH and such that the inclusionX H is dense and continuous, then(X,H,X ) is called a Gelfand or an evolution triple⊂ .

Remark 2.14 For a Gelfand triple(X,H,X ) the duality pairing ϕ,v forϕ X andv X is regarded as a continuous extension of the scalar product on H, i.e.,∈ if ϕ X∈ H , then ∈ ∩ ϕ,v (ϕ,v) H .  = 2.3 Gradient Flows 33

Below, we always consider an evolution triple(X,H,X ). The Sobolev–Bochner spaces then have the following important properties.

Remarks 2.15 (i) Ifu L p( 0,T X) withu L p ( 0,T X ), thenu ∈ [ ]; ∈ [ ]; ∈ C( 0,T H) with max t 0,T u(t) H c( u L p( 0,T X) u p ) and L ( 0,T X ) the[ integration-by-parts]; ∈ formula[ ] ≤ [ ]; + [ ];

t2

(u(t2),v(t 2))H (u(t 1),v(t 1))H u (t),v(t) v (t),u(t) dt − =  +  t1

p p holds for allv L ( 0,T X) withv L ( 0,T X) andt 1,t 2 0,T . In particular, we∈ have [ ]; ∈ [ ]; ∈[ ] 1 d 2 u(t) u (t),u(t) 2 dt H =  for almost everyt 0,T . (ii) IfX is compactly∈[ embedded] inH,1

2.3.3 Existence Theory for Gradient Flows

We consider a Fréchet-differentiable functionalI X R withDI X X and we want to derive conditions that guarantee existence: → of solutions: → for theH-gradient flow ofI formally defined by

∂t u H I(u),u(0) u 0. = −∇ =

We always let(X,H,X ) be an evolution triple and assume that an abstract Poincaré inequality holds, i.e., that for a seminorm X onX we have |·|

u X c P ( u X u H ). ≤ | | + for allu X. ∈ 34 2 Analytical Background

p Definition 2.9 Givenu 0 H, we say thatu L ( 0,T X) is a solution of the ∈ p ∈ [ ]; H-gradientflow for I ifu L ( 0,T X ) and for almost everyt 0,T and everyv X, we have that ∈ [ ]; ∈[ ] ∈

u (t),v DI(u) v 0  + [ ]= andu(0) u 0. =  Example 2.3 ForI(u) (1/2) u 2 dx defined onH 1(Ω), theL 2(Ω)-gradient = Ω |∇ | 0 flow is the linear heat equation∂ t u u 0. − = We follow the Rothe method to construct solutions. This method consists of three steps: First, we consider an implicit time discretization that replaces the time deriv- ative by difference quotients and establishes the existence of approximations. In the second step, a priori bounds that allow us to extract weakly convergent subsequences of the approximations as the time-step size tends to zero are proved. Finally, we pass to the limit and try to show that weak limits are solutions of the gradientflow.

Definition 2.10 The functionalI X R is called semicoercive if there exist : → s>0,c 1 > 0, andc 2 R such that ∈ s 2 I(v) c 1 v c 2 v ≥ | |X − H for allv X. ∈ Proposition 2.1 (Implicit Euler scheme) Assume that I is semicoercive and weakly lower semicontinuous. Then for everyτ>0 with4τc 2 <1 and k 1,2,...,K, = K T/τ , the functionals I k X R, =  : →

k 1 k 1 2 u I (u) u u − I(u), → = 2τ − H +

0 with u u 0 have minimizers that satisfy = k k (dt u ,v) H DI(u ) v 0 + [ ]= k k k 1 for all v X with the backward difference quotientd t u (u u )/τ. ∈ = − − Proof SinceI k is coercive, bounded from below, and weakly lower semicontinuous, the direct method in the calculus of variations implies the existence of a minimum. SinceI andv v 2 are Fréchet-differentiable, the minimizers satisfy the asserted → H equations. � Remarks 2.16 (i) More generally, one can consider a pseudomonotone operatorA k k k : X X and look for a solutionu of the equation(d t u ,v) H A(u ) v 0 for all→v X. + [ ]= (ii) We∈ have thatu k is uniquely defined ifI k is strictly convex. This is often satisfied forτ sufficiently small, i.e., ifI is semiconvex. 2.3 Gradient Flows 35

For the proof of the a priori bounds two important ingredients are required. The first is based on the binomial formula 2(a b)a (a b) 2 (a 2 b 2) and shows that − = − + −   1 k k 1 k 1 k k 1 2 1 k 2 k 1 2 (u u − ,u )H u u − u u − τ − = 2τ − H + 2τ H − H fork 1,2,...,K . Equivalently, we have =

k k τ k 2 1 k 2 (dt u ,u )H dt u dt u = 2 H + 2 H

2 which is a discrete version of the identity 2 u ,u (d/dt) u H . The second ingredient is the following discrete Gronwall lemma. =

Lemma 2.2 (Discrete Gronwall lemma) Let(y ) 0,1,...,L be a sequence of non- = negative real numbers such that for nonnegative real numbers a0,b 0,b 1,...,b L 1 and 0,1,..., L, we have − =  1 − k y a 0 bk y . ≤ + k 0 =    1 Then we have max 0,1,...,L y a 0 exp k−0 bk . = ≤ = Proof The proof follows from an inductive argument. � We also have to assume a coerciveness property for the mappingDI X X . : → Definition 2.11 We say thatDI X X is semicoercive and bounded if there : → existp (1, ),c >0,c R, andc > 0 such that ∈ ∞ 1 2 ∈ 3 p 2 DI(v) v c v c v [ ]≥ 1| |X − 2 H and p 1 DI(v) c (1 v − ) X ≤ 3 + X for allv X. ∈ Proposition 2.2 (A priori bounds) Suppose that DI X X is semicoercive and bounded. If4τc 1, then we have : → 2 ≤ K K K k p k p k p max u H τ u τ dt u τ DI(u ) C 0 0,1,...,K + X + X + X ≤ = k 1 k 1 k 1 = = = with a constant C0 >0 that depends on p, T , u 0, cP , c1 , c2 , and c3 . 36 2 Analytical Background

k k Proof Since(d t u ,v) H DI(u ) v 0 for allv X, we obtain by choosing v u k that + [ ] = ∈ = τ k 2 1 k 2 k p k 2 k k k k dt u dt u c u c u (d t u ,u )H DI(u ) u 0. 2 H + 2 H + 1| |X − 2 H ≤ + [ ] = Multiplication byτ and summation overk 1,2,..., for 1 K lead to = ≤ ≤    1 2 τ k 2 k p 1 0 2 k 2 u τ dt u c τ u u c τ u , 2 H + 2 H + 1 | |X ≤ 2 H + 2 H k 1 k 1 k 1 = = =  k 2 2 0 2 where we used the telescope effectτ k 1 dt u H u H u H . Since the = = − second term on the left-hand side is nonnegative and since 4c2 τ 1 so that we can absorbc τ u 2 on the left-hand side, wefind that ≤ 2 H   1 1 2 k p 1 0 2 − k 2 u c τ u u c τ u . 4 H + 1 | |X ≤ 2 H + 2 H k 1 k 1 = = For 0,1,...,K , we set =  1 2 k p y u c τ u , = 4 H + 1 | |X k 1 = 0 2 a0 (1/2) u andb 4c τ so that = H = 2  1 − k y a 0 by ≤ + k 1 = and we are in the situation to apply the discrete Gronwall lemma. This shows that

K  K1   1 2 k p 1 0 2 − 1 0 2 max u H c 1 τ u u H exp 4c2 τ u H exp 4c2 T) 0,...,K 4 + | |X ≤ 2 ≤ 2 = 1 k 1 = = and proves the bound for thefirst term on the left-hand side. The second bound follows with the abstract Poincaré inequality u X c P ( u X u H ) and ≤ | | +   k p p k k p p p 1 k p p k p u c ( u X u H ) c 2 − u c u . X ≤ P | | + ≤ P | |X + P H For the third and fourth term on the left-hand side of the bound, we note that with the boundedness ofDI and(p 1)p p, it follows that − = 2.3 Gradient Flows 37

L K K k p p k p 1 p p p 1 k p τ DI(u ) τ(c 3 ) (1 u − ) τ(c 3 ) 2 − (1 u ), X ≤ + X ≤ + X k 1 k 1 k 1 = = = and the right-hand side is bounded according to the previous bounds. This proves the fourth estimate and the third bound follows immediately since d uk t X k k k = DI(u ) due to the identity(d t u ,v) DI(u ) v for allv X. X =− [ ] ∈ � To identify the limits of the approximations we define interpolants of the approx- k imations(u )k 0,...,K . = k Definition 2.12 Given a time-step sizeτ> 0 and a sequence(u )k 0,...,K H for = ⊂ K T/τ , we sett k kτ fork 0,1,...,K and define the piecewise constant =  = = and piecewise affine interpolantsu τ −,u τ+, τu 0,T H fort (t k 1,t k) by :[ ]→ ∈ −

k 1 k t t k 1 k tk t k 1 u−(t) u − ,u +(t) u , u τ (t) − − u − u − . τ = τ = = τ + τ The construction of the interpolants is illustrated in Fig. 2.14. 1, k Remarks 2.17 (i) We have uτ W ∞( 0,T H) with u τ d t u on(t k 1,t k) for k 1,2,...,K . Moreover,u ∈,u L [ ( 0];,T H) and, e.g.,= − = τ+ τ− ∈ ∞ [ ]; K p k p u+ p τ u τ L ( 0,T X) ≤ X [ ]; k 1 = with equality ifKτ T. = (ii) We haveu τ+(t) u τ−(t) τ u τ (t) and uτ (t) u τ−(t) (t t k 1) τ u(t) − = = + − − = uτ+(t) (t k t) uτ (t) for almost everyt (t k 1,t k). − − ∈ − (iii) If u 1 c for allτ> 0, then it follows thatu u 0 and τ L ( 0,T X ) τ+ τ− [ ]; 1 ≤ − → τu u τ± 0 inL ( 0,T X ) asτ 0. In particular, all interpolants have the same− limit→ if these exists.[ ]; →

Lemma 2.3 (Discrete evolution equation) With the interpolants of the approxima- k tions(u )k 0,...,K , we have =

( u(t),v) H DI(u +(t)) v 0 τ + τ [ ]=

� + uτ uτ uτ−

t t t

Fig. 2.14 Continuous interpolant uτ (left) and piecewise constant interpolantsu τ+ (middle) andu τ− (right) 38 2 Analytical Background for all v X and almost every t 0,T . Moreover, we have for allτ>0 ∈ ∈[ ]

u u p u DI(u ) C . τ+ L∞( 0,T H) τ+ L ( 0,T X) τ W 1,p ( 0,T X ) τ+ L p ( 0,T X ) 0 [ ]; + [ ]; + [ ]; + [ ]; ≤ k k Proof The identity follows directly from(d t u ,v) H DI(u ) v 0 fork + [ ] = = 1,2,...,K and allv X with the definitions of the interpolants u τ andu . With ∈ τ+ the triangle inequality and t t k τ fort (t k 1,t k), we observe that | − |≤ ∈ − u c T 1/p u τ u . τ L p ( 0,T X ) P + L∞( 0,T H) τ L p ( 0,T X ) [ ]; ≤ [ ]; + [ ]; The a priori bounds of Proposition 2.2 together with, e.g.,

K T k p p τ u u+ dt, X ≥ τ X k 1 = 0 where we usedKτ T , imply the a priori bounds. ≥ � The bounds for the interpolants allow us to select accumulation points.

Proposition 2.3 (Selection of a limit) Assume that X is compactly embedded in H. Then there exist u L p( 0,T X) W 1,p ( 0,T X ) andξ L p ( 0,T X ) ∈ [ ]; ∩ [ ]; ∈ [ ]; such that for a sequence(τ n)n N of positive numbers withτ n 0 as n , we have ∈ → →∞

τun ,u τn ∗ u in L∞( 0,T H), p [ ]; τu ,u τ  u in L ( 0,T X), n n [ ]; 1,p τu  u in W ( 0,T X ), n [ ]; p DI(u + )ξ in L ( 0,T X ). τn [ ];

We have u C( 0,T H) with u(0) u 0 and ∈ [ ]; =

u (t),v ξ(t),v 0  + = for almost every t 0,T and all v X. In particular, ifξ DI(u), then u is a solution of the H-gradient∈[ ] flow for∈ I . =

Proof For a sequence(τ n)n N of positive numbers withτ n 0 asn , the a priori bounds yield the existence∈ of weak limits for an appropriate→ →∞ subsequence which is not relabeled. Due to the bound for τ u, the weak limits coincide. Multiplying the discrete evolution equation of Lemma 2.3 byφ C( 0,T ) and integrating the resulting identity over 0,T wefind that ∈ [ ] [ ] 2.3 Gradient Flows 39

T

u,φv DI(u + ) φv dt 0  τn + τn [ ] = 0 for everyv X. Sinceφv L p( 0,T X) we can pass to the limitn in the equation and∈ obtain ∈ [ ]; →∞ T

u ,φv ξ,φv dt 0.  +  = 0

Since this holds for everyφ C( 0,T ) we deduce the asserted equation. The map- ∈ [ ] p 1,p pingv v(0) defines a bounded linear operatorL ( 0,T X) W ( 0,T X ) → [ ]; ∩ [ ]; → H which is weakly continuous. Since uτn (0) u 0 for alln N, we deduce that = ∈ u(0) u 0. By continuous embeddings we also haveu C( 0,T H) which implies = ∈ [ ]; the continuous attainment of the initial data. �

Remark 2.18 The assumed identityξ DI(u) inL p ( 0,T X ), i.e., the con- = [ ]; vergenceDI(u τ+n )DI(u) can in general only be established under additional conditions onDI and requires special techniques from nonlinear functional analy- sis, e.g., based on concepts of pseudomonotonicity.

1 2 Example 2.4 ForF C (R) with 0 F(s) c F (1 s ) andf(s) F (s) such that f(s) c (1∈ s ), we consider≤ ≤ +| | = | |≤ F +| |   1 I(u) u 2 dx F(u)dx. = 2 |∇ | + Ω Ω

1 2 Then, forX H 0 (Ω) andH L (Ω), the conditions of the previous propositions are satisfied= withp 2 and = =   DI(u) v u vdx f(u)vdx. [ ]= ∇ ·∇ + Ω Ω

 2( , 1(Ω))  2( , 2(Ω d )) Wehaveu τ+n u L 0 T H 0 so that u τ+n u inL 0 T L R and thus ∈ [ ]; ∇ ∇ [ ]; ; T  T 

u + wdxdt u wdxdt ∇ τn ·∇ → ∇ ·∇ 0 Ω 0 Ω for allw L 2( 0,T H 1(Ω)). The compactness of the embedding ∈ [ ]; 0 L2( 0,T H 1(Ω)) W 1,2( 0,T H 1(Ω) ) L 2( 0,T L 2(Ω)) L 2( 0,T Ω) [ ]; 0 ∩ [ ]; 0 → [ ]; = [ ]× in combination with the generalized dominated convergence theorem shows that 40 2 Analytical Background

T  T 

f(u + )wdxdt f(u)wdxdt. τn → 0 Ω 0 Ω

Altogether this proves that

T T  T 

DI(u + ) w dt u wdxdt f(u)wdxdt, τn [ ] → ∇ ·∇ + 0 0 Ω 0 Ω i.e.,ξ DI(u). = Remarks 2.19 (i) For the semilinear heat equation∂ t u u f(u) of Example 2.4, one can establish the existence of a solution under more= general− conditions onf. Moreover, one can prove stronger a priori bounds and the energy law

T 2 I(u(T )) u (t) 2 dt I(u 0) + L (Ω) ≤ 0

1 for almost everyT 0,T providedu 0 H 0 (Ω). The key ingredient is the convexity ofI in the highest-order∈[ ] term. ∈ (ii) An alternative method to establish the existence of solutions for gradientflows is the Galerkin method which is based on a discretization in space. This leads to a sequence of ordinary differential equations onfinite-dimensional spaces and with appropriate a priori bounds, one can then show under appropriate conditions that the approximate solutions converge to a solution as the dimension tends to infinity.

2.3.4 Subdifferential Flows

The estimates for discretized gradientflows can be significantly improved if the functionalI is convex, since then we would have

k k k 1 k k 1 I(u ) DI(u ) u − u I(u − ). + [ − ]≤ k k k In particular, choosingv d t u in the identity(d t u ,v) H DI(u ) v 0 gives = + [ ]= k 2 k k 1 τ d t u I(u ) I(u − ) H + ≤  L L k 2 0 and a summation overk yields the a priori boundI(u ) τ k 1 dt u H I(u ). With these observations it is possible to establish a theory+ for convex= functionals ≤ that are not differentiable. We always consider a Hilbert spaceH that is identified with its dual. 2.3 Gradient Flows 41

Fig. 2.15 Subdifferential of the functionx x at x 0; the arrows →| | x = | | s1 s1,s 2,s 3,s 4 indicate subgradients at 0 which are s2 the slopes of supporting hyperplanes at 0 s3 s4

Definition 2.13 We say that a functionalI H R belongs to the class : → ∪ {+∞} Γ(H) if it is convex, lower semicontinuous, i.e.,I(u) lim inf n I(u n) whenever ≤ →∞ un u inH asn , and proper, i.e., there existsu H withI(u) R. → →∞ ∈ ∈ We assume thatI Γ(H) below. ∈ Definition 2.14 The subdifferential∂I H 2 H ofI associates to everyu H the set : → ∈

∂I(u) v H I(w) I(u) (v,w u) H for allw H . ={ ∈ : ≥ + − ∈ } The elements in∂I(u) are called subgradients ofI atu.

Example 2.5 ForF(x) x ,x R, we have∂F(0) 1,1 , cf. Fig. 2.15. =| | ∈ =[− ]

Remarks 2.20 (i) The subdifferential∂I(u) consists of all slopes of affine functions that are belowI and that intersect the graph ofI atu. (ii) For allu 1,u 2 H andv 1 ∂I(u 1),v 2 ∂I(u 2) we have the monotonicity estimate ∈ ∈ ∈ (v1 v 2,u 1 u 2)H 0. − − ≥ (iii) We have 0 ∂I(u) if and only ifu H is a global minimum forI. (iv) We have∂I(∈u) s fors H if and∈ only ifI is Gâteaux-differentiable atu. (v) ForI,J Γ(H) we={ have} ∂(∈I J) ∂I ∂J, and if there exists a point at whichI and∈J arefinite andI or+J is⊂ continuous,+ we have equality.

Theorem 2.7 (Resolvent operator) Let I Γ(H). For every w H andλ>0 there exists a unique u H with ∈ ∈ ∈ u λ∂I(u) w. +  1 This defines the resolvent operatoru R λ(w) (Id λ∂I) (w). = = + − Proof For a short proof we make the simplifying but nonrestrictive assumption that

I(v) c 1 c 2 v H . ≥− − 42 2 Analytical Background

Forλ> 0 andw H we consider the minimization problem defined through the functional ∈

1 2 1 2 1 1 2 Iλ,w(u) u w I(u) u (u,w) H w I(u). = 2λ − H + = 2λ H − λ + 2λ H +

The identity 2(a2 b 2) (a b) 2 (a b) 2 and the convexity ofI show that for + = + + − u1,u 2 H we have ∈   1 1 u1 u 2 Iλ,w(u1) Iλ,w(u2) I λ,w + 2 + 2 − 2     1 2 1 1 u1 u 2 1  2 u1 u 2 I(u 1) I(u 2) I + u1 u 2 , = 8λ − H + 2 + 2 − 2 ≥ 8λ − H i.e.,I λ,w is strictly convex. Thus, ifI λ,w has a minimizer, then it is unique. Moreover, u H minimizesI λ,w if and only if 0 ∂I λ,w(u) (1/λ)(u w) ∂I(u). It remains ∈ ∈ = − + to show that there exists a minimizer. SinceI λ,w is convex and lower semicontinuous it follows thatI is weakly lower semicontinuous. We also have thatI λ,w is coercive since two applications of Young’s inequality lead to

1 2 1 1 2 Iλ,w(v) v (v,w) H w c 1 c 2 v H ≥ 2λ H − λ + 2λ H − − 1 2 4 2 2 v w c 1 4λc . ≥ 4λ H − λ H − − 2 This estimate also proves the boundedness from below. The direct method in the calculus of variations thus implies the existence of a minimizer.�

Definition 2.15 The Yosida regularization Aλ H H is forw H defined by : → ∈ Aλ(w) (1/λ)(w R λ(w)). = − Remark 2.21 The resolvent operator satisfies limλ 0 Rλ(w) w. We have thatA λ is Lipschitz continuous with Lipschitz constant 2/λ→ and approximates= ∂I in the sense thatA λ(w) ∂I(R λw). ∈ The theorem about the resolvent operator implies that for a time-step sizeτ>0 0 k and an initialu H, there exists a unique sequence(u )k 0,...,L H with ∈ = ⊂ k k dt u ∂I(u ) ∈ − k k 1 since this is equivalent tou R τ (u − ). We expect that asτ 0 the approxima- tions converge to a solution of= the subdifferentialflow →

u ∂I(u),u(0) u 0. ∈ − = Related a priori bounds that permit a corresponding passage to a limit will be dis- cussed in Chap.4. 2.3 Gradient Flows 43

Theorem 2.8 (Subdifferentialflow, [ 2]) For every u0 H such that∂I(u 0) and every T>0, there exists a unique function u C(∈0,T H) with u = ∅ ∈ [ ]; ∈ L ( 0,T H) such that u(0) u 0,∂I(u(t)) for every t 0,T , and ∞ [ ]; = =∅ ∈[ ]

u (t) ∂I(u(t)) ∈− for almost every t 0,T . ∈[ ] Proof The existence of a solution is established by considering for everyλ> 0 the problem ∂t uλ A λ(uλ),u λ(0) u 0 =− = and studying the limitλ 0. Uniqueness of solutions follows from the convexity → ofI , i.e., ifu 1 andu 2 are solutions then the monotonicity property ofI shows that

(u (t) u (t),u 1(t) u 2(t)) H 0 − 1 − 2 − ≥ for almost everyt 0,T and this implies that ∈[ ]

1 d 2 (u1 u 2)(t) (u (t) u (t),u 1(t) u 2(t)) H 0. 2 dt − H = 1 − 2 − ≤

Sinceu 1(0) u 2(0) we deduce thatu 1(t) u 2(t) for everyt 0,T . = = ∈[ ] � Remarks 2.22 (i) Negative subgradients are in general no descent directions. For 0 the subdifferentialflow one can however show thatu (t) ∂ I(u(t)) for almost everyt 0,T , where∂ 0 I(v) is the subgradients ∂I(v) with=− minimal norm, i.e., 0 ∈[ ] ∈ ∂ I(v) H minr ∂I(v) r H . = ∈ (ii) If∂I(u 0) , then there exists a unique solutionu C( 0,T H) such that t1/2u L 2( =0,∅T H). ∈ [ ]; ∈ [ ];

References

1. Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces. MPS/SIAM Series on Optimization, vol. 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006) 2. Brézis, H.R.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland Publishing Co., Amsterdam (1973). North-Holland Math- ematics Studies, No. 5. Notas de Matemática (50) 3. Ciarlet, P.G.: Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity. Studies in Mathe- matics and its Applications, vol. 20. North-Holland Publishing Co., Amsterdam (1988) 4. Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences, vol. 78, 2nd edn. Springer, New York (2008) 5. Eck, C., Garcke, H., Knabner, P.: Mathematische Modellierung, 2nd edn. Springer Lehrbuch. Springer, Berlin (2011) 6. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence (2010) 44 2 Analytical Background

7. Roubíˇcek,T.: Nonlinear Partial Differential Equations with Applications. International Series of Numerical Mathematics, vol. 153, 2nd edn. Birkhäuser/Springer Basel AG, Basel (2013) 8. Ružiˇcka,M.: Nichtlineare Funktional Analysis. Springer, Berlin-Heidelberg-New York (2004) 9. Salsa, S., Vegni, F.M.G., Zaretti, A., Zunino, P.: A Primer on PDEs. Unitext, vol. 65, italian edn. Springer, Milan (2013) 10. Struwe, M.: Variational Methods, 4th edn. Springer, Berlin (2008) 11. Temam, R., Miranville, A.: Mathematical Modeling in Continuum Mechanics, 2nd edn. Cam- bridge University Press, Cambridge (2005) 102 4 Concepts for Discretized Problems and with u v , we deduce that ∇ = ∇  a(u,v) sup c H u . v H 1(Ω) v ≥ ∇  ∈ 0 ∇  The second condition of the proposition is a direct consequence of the requirement thatω 2 is not an eigenvalue of . − Remark 4.5 Proposition 4.2 is important for the analysis of saddle-point prob- lems; the seminal paper [7] provides conditions that imply the assumptions of the proposition.

4.2.3 Abstract Subdifferential Flow

The subdifferentialflow of a convex and lower semicontinuous functionalI H : → R arises as an evolutionary model in applications, and can be used as a basis for∪ numerical{+∞} schemes to minimizeI . The corresponding differential equation seeks u 0,T H, such thatu(0) u 0 and :[ ]→ =

∂t u ∂I(u), ∈− i.e.,u(0) u 0 and = ( ∂t u,v u) H I(u) I(v) − − + ≤ for almost everyt 0,T and everyv H. An implicit discretization of this nonlinear evolution∈[ equation] is equivalent∈ to a sequence of minimization problems k k k 1 involving a quadratic term. Werecall thatd t u (u u − )/τ denotes the backward difference quotient. = −

Theorem 4.7 (Semidiscrete scheme [15, 17]) Assume that I 0 and for u 0 H k ≥ ∈ let(u )k 1,...,K H be minimizers for = ⊂

k 1 k 1 2 I (w) w u − I(w) τ = 2τ  − H + for k 1,2,..., K . For L 1,2,..., K , we have = = �L L k 2 0 I(u ) τ dt u I(u ). +  H ≤ k 1 = With the computable quantities

k 2 k k 1 Ek τ d t u I(u ) I(u − ) =−  H − + 4.2 Approximation of Equilibrium Points 103

k and the affine interpolant�uτ 0,T H of the sequence(u )k 0,...,K we have the a posteriori error estimate:[ ]→ =

�L � 2 0 2 max u u H u0 u H τ Ek. t 0,T  −  ≤  −  + ∈[ ] k 1 = We have the a priori error estimate

k 2 0 2 0 max u(tk) u H u0 u H τI(u ), k 0,...,K  −  ≤  −  + = and under the condition∂I(u 0) , the improved variant =∅ k 2 0 2 2 o 0 2 max u(tk) u H u0 u H τ ∂ I(u ) H , k 0,...,K  −  ≤  −  +   = where∂ o I(u 0) H denotes the element of minimal norm in∂I(u 0). ∈ Proof The direct method in the calculus of variations yields that fork 1,2,...,K, k k k = k there exists a unique minimizeru H forI , and we haved t u ∂I(u ), i.e., ∈ τ ∈ − k k k ( dt u ,v u )H I(u ) I(v) − − + ≤ for allv H; the choice ofv u k 1 implies that ∈ = − k 2 k k 1 Ek τ d t u I(u ) I(u − ) 0 − =  H + − ≤ k with 0 E k τd t I(u ). A summation overk 1,2,...,L yields the asserted ≤ ≤ − = k stability estimate. If�uτ is the piecewise affine interpolant of(u )k 0,...,K associated = k to the time stepst k kτ,k 0,1,...,K , andu τ+ is such thatu τ+ (tk 1 ,tk ) u for = = | − = k 1,2,... andt k kτ, then we have = =

( ∂t�τu,v u +)H I(u +) I(v) − − τ + τ ≤ for almost everyt 0,T and allv H. In introducing ∈[ ] ∈

Cτ (t) ( ∂ t�τu,u + �uτ )H I(u +) I(�uτ ) = − τ − − τ + we have ( ∂t�τu,v �uτ )H I(�uτ ) I(v) C τ (t). − − + ≤ +

The choice ofv u in this inequality andv �u τ in the continuous evolution equation yield = =

d 1 2 u �u ( ∂ t u �uτ ,�τu u) H C τ (t). dt 2 − H = − [ − ] − ≤ 104 4 Concepts for Discretized Problems

� � Noting uτ u τ+ (t t k)∂t τufort (t k 1,t k) and using the convexity ofI , i.e., − = − ∈ −

tk t k 1 t t k 1 k I(�τu) − I(u − ) − − I(u ), ≤ τ + τ

k we verify fort (t k 1,t k) usingu τ+ u that ∈ − =

2 tk t k 1 t t k 1 k tk t Cτ (t) (t t k) ∂t�τu I(u +) − I(u − ) − − I(u ) − Ek. ≤ −  H − τ + τ + τ = τ

k WithE k τd t I(u ) andI 0 we deduce that ≤ − ≥ �t L �L �L � � 2 k L 0 0 Cτ (t)dt τ Ek τ dt I(u ) τ I(u ) I(u ) τI(u ), ≤ ≤ − =− − ≤ k 1 k 1 0 = =

which implies the a posteriori and thefirst a priori error estimate. Assume that 0 1 0 0 1 o 0 ∂I(u ) and defineu − H so thatd t u (u u − )/τ ∂ I(u ), i.e., the discrete =∅ evolution equation∈ also holds fork =0, − =− = 0 0 0 ( dt u ,v u )H I(u ) I(v) − − + ≤ k k 1 for allv H. Choosingv u in the equation ford t u − ,k 1,2,...,K , we observe∈ that = = k 1 k k 1 k 1 k ( dt u − ,u u − )H I(u − ) I(u ), − − + ≤ k k k 1 i.e., τd t I(u ) τ(d t u ,d t u )H , and it follows that − ≤ − k k k k k k 1 k Ek τ(d t u ,d t u )H τd t I(u ) τ(d t u ,d t u )H τ(d t u − ,d t u )H =− − ≤− + 3 2 2 k k 2 dt k 2 τ 2 k 2 2 dt k 2 τ (d u ,d t u )H τ dt u d u τ dt u . =− t =− 2  H − 2  t H ≤ − 2  H

This implies that

�t L �L 2 2 τ 0 2 τ o 0 2 Cτ (t)dt τ Ek dt u ∂ I(u ) , ≤ ≤ 2  H = 2  H k 1 0 =

which proves the improved a priori error estimate. � Remarks 4.6 (i) The condition∂I(u 0) is restrictive in many applications. =∅ (ii) Subdifferentialflows∂ t u ∂I(u), i.e., Lu 0 for Lu ∂ t u v with ∈−  = + v ∂I(u), and with a convex functionalI H R define monotone ∈ : → ∪ {+∞} 4.2 Approximation of Equilibrium Points 105

problems in the sense that � � � Lu1 Lu 2,u 1 u 2 (∂ t (u1 u 2) (v 1 v 2),u 1 u 2 − − H = − + − − H � � 1 d 2 ∂t (u1 u 2),u 1 u 2 u1 u 2 ≥ − − H = 2 dt  − H

foru 1,u 2 andv 1,v 2 withv i ∂I(u i ),i 1,2. ∈ = (iii) IfI H R is strongly monotone in the sense that(u 1 u 2, : → ∪ {+∞} 2 − v1 v 2)H α u 1 u 2 H wheneverv ∂I(u ), 1, 2, and if there exists a solution− u ≥H of the− stationary inclusion ∈v 0 ∂I( =u), then we haveu(t) u ast .∈ A proof follows from the estimate= ∈ → →∞

1 d 2 2 u u (v v,u u) H α u u , 2 dt  −  H =− − − ≤ −  −  H

wherev ∂ t u ∂I(u), and an application of Gronwall’s lemma. =− ∈

4.2.4 Weak Continuity Methods

Let(u h)h>0 X be a bounded sequence in the reflexive, separable Banach spaceX such that there⊂ exists a weak limitu X of a subsequence that is not relabeled, i.e., ∈ we haveu h u ash 0. For an operatorF X X , we define the sequence → : →  (ξh)h>0 X throughξ h F(u h), and if the sequence is bounded inX , then ⊂  =  there existsξ X , such that for a further subsequence(ξ h)h>0 which again is not ∈  relabeled, we haveξ h ∗ ξ. The important question is now whether we have weak continuity in the sense that F(u) ξ. = Notice that weak continuity is a strictly stronger notion of continuity than strong con- tinuity. For partial differential equations, this property is called weak precompactness of the solution set of the homogeneous equation, i.e., if(u j ) j is a sequence with ∈N F(u j ) 0 for allj N andu j u asj then we may deduce thatF(u) 0. Such implications= ∈ may also be regarded→∞ as properties of weak stability since= they imply that ifF(u j ) r j with r j ε j andε j 0 asj , then we have =  X ≤ → →∞ F(u) 0 for every accumulation point of the sequence(u j ) j . = ∈N Theorem 4.8 (Discrete compactness) Forevery h>0 let u h X h solve Fh(uh) 0. ∈ = Assume that Fh(uh) X with F(u h) c for all h>0 and F is weakly contin- ∈   X ≤ uous on X, i.e., F(u j ) v F(u) v for all v X whenever u j  u in X. Suppose [ ]→ [ ] ∈ that for every bounded sequence(w h)h>0 X with wh X h for all h>0, we have ⊂ ∈

F(w h) F h(wh) X 0  −  h → 106 4 Concepts for Discretized Problems

as h 0 and(X h)h>0 is dense in X with respect to strong convergence. If → (uh)h>0 X is bounded, then there exists a subsequence(u ) > and u X ⊂ h h 0 ∈ such that uh  u in X and F(u) 0. = Proof After extraction of a subsequence, we may assume thatu h u inX ash 0 → for someu X. Fixingv X and usingF h(uh) vh 0 for everyv h X h, we have ∈ ∈ [ ] = ∈ F(u h) v F(u h) v v h F(u h) vh F h(uh) vh . [ ]= [ − ] + [ ]− [ ]

For a sequence(v h)h>0 X withv h X h for everyh> 0 andv h v inX, we find that ⊂ ∈ → F(u h) v v h F(u h) v v h X 0 | [ − ]|≤ X  −  →

ash 0. The sequences(u h)h>0 and(v h)h>0 are bounded inX and thus →

F(u h) vh F h(uh) vh F(u h) F h(uh) X vh X 0 | [ ]− [ ]|≤ −  h   → ash 0. Together with the weak continuity ofF wefind that →

F(u) v lim F(u h) v 0. [ ]= h 0 [ ]= → Sincev X was arbitrary this proves the theorem. ∈ � The crucial part in the theorem is the weak continuity of the operatorF. We include an example of an operator related to a constrained nonlinear partial differ- ential equation that fulfills this requirement.

1 3 Example 4.16(Harmonic maps) Let(u j ) j H (Ω R ) be a bounded sequence ∈N ⊂ ; such that u j (x) 1 for allj N and almost everyx Ω. Assume that for every | | = 1 3∈ 3 ∈ j N and allv H (Ω R ) L ∞(Ω R ), we have ∈ ∈ ; ∩ ; � � 2 F(u j ) v u j vdx u j u j vdx 0. [ ]= ∇ ·∇ − |∇ | · = Ω Ω

The choice ofv u j w shows that we have = × � � F(u j ) w u j (u j w)dx 0 [ ]= ∇ ·∇ × = Ω

1 3 3 for allw H (Ω R ) L ∞(Ω R ). Using∂ ku j ∂ k(u j w) ∂ ku j (u j ∂ kw) fork 1,∈2,...,d, we; find∩ that ; · × = · × = �d � � F(u j ) w ∂ku j (u j ∂ kw)dx 0. [ ]= · × = k 1 = Ω 4.2 Approximation of Equilibrium Points 107

1 3 2 3 Ifu j u inH D(Ω R ), thenu j u inL (Ω R ) and thus, for everyfixed 3 ; → ; w C ∞(Ω R ), we can pass to the limit andfind that ∈ ; F�(u) w 0. [ ]=

Since up to a subsequence we haveu j (x) u(x) for almost everyx Ω, we verify that u(x) 1 for almost everyx Ω.→ A density result shows that∈ this holds | 1| = 3 3 ∈ for allw H (Ω R ) L ∞(Ω R ). Reversing the above argument by choosing w u v∈ and employing; ∩ the identity; a (b c) (b a)c (c a)b shows that = × 1 3 × × = 3· − · F(u) v 0 for allv H (Ω R ) L ∞(Ω R ). [ ]= ∈ ; ∩ ; A general concept for weak continuity is based on the notion of pseudomonotonicity.

Example 4.17(Pseudomonotone operators) The operatorF X X  is a pseudo- monotone operator if it is bounded, i.e., F(u) c(1 : u→ s ) for somes 0,   X ≤ +  X ≥ and wheneveru j u inX, we have the implication that

lim sup F(u j ) u j u 0 F(u) u v lim inf F(u j ) u j v . j [ − ]≤ =⇒ [ − ]≤ j [ − ] →∞ →∞

For such an operator we have that ifF(u h) vh (v h) for allv h X h with a [ ] = ∈ strongly dense family of subspaces(X h)h>0 andu h u ash 0, thenF(u) . → = To verify this, letv X and(v h)h>0 withv h X h such thatv h u and note that ∈ ∈ →

lim sup F(u h) uh u lim sup F(u h) uh v h F(u h) vh u h 0 [ − ]= h 0 [ − ] + [ − ] → → lim sup (uh v h) F(u h) vh u 0. = h 0 − + [ − ]= →

Pseudomonotonicity yields for everyv h>0 Xh that h ∈ ∪

F(u) u v h lim inf F(u h) uh v h lim (uh v h ) (u v h ). [ −  ]≤ h 0 [ −  ] = h 0 −  = −  → →

With the density of(X h)h>0 inX, we conclude thatF(u) u v (u v) for all v X and withv u w, wefind thatF(u) w (w) for all[w−X].≤ − ∈ = ± [ ]= ∈ Remarks 4.7 (i) Radially continuous bounded operators are pseudomonotone. Here, radial continuity means thatt F(u tv) v is continuous fort R and allu,v X. These operators allow → us to apply+ [ Minty’s] trick to deduce∈ from the inequality∈ (u v) F(v) u v 0 for allv X thatF(u) . To prove this implication, note− − that with[ −v ]≥u εw, wefind∈ that (w) =F(u εw) w 0 and by radial continuity forε =0,+ it follows that (w) F−(u) w+ 0[ and]≤ hence F(u) . → − [ ]≤ = (ii) Pseudomonotone operators are often of the formF F 1 F 2 with a monotone = + operatorF 1 and a weakly continuous operatorF 2, e.g., a lower-order term described byF 2. 108 4 Concepts for Discretized Problems

Example 4.18(Quasilinear diffusion) The concept of pseudomonotonicity applies to the quasilinear elliptic equation � � p 2 div u − u g(u) f inΩ,u ∂Ω 0 on ∂Ω, − |∇ | ∇ + = | = r 1 withg C(R) such that g(s) c(1 s − ) and 1

4.3 Solution of Discrete Problems

We discuss in this section the practical solution of discretized minimization problems of the form �

MinimizeI h(uh) W( u h) g(u h)dx amongu h A h. = ∇ + ∈ Ω

In particular, we investigate four model situations with smooth and nonsmooth inte- grands and smooth and nonsmooth constraints included inA . The iterative algo- rithms are based on an approximate solution of the discrete Euler–Lagrange equa- tions. More general results can be found in the textbooks [4, 12].

4.3.1 Smooth, Unconstrained Minimization

Suppose that 1 m Ah uh S (Th) u h Γ u D,h = { ∈ : | D = } 1 m d 1 m andI h is defined as above with functionsW C (R × ) andg C (R ). The ∈ ∈ caseΓ D is not generally excluded in the following. A necessary condition for a =∅ 1 m minimizeru h A h is that for allv h S (Th) , we have ∈ ∈ D �

Fh(uh) vh DW( u h) v h Dg(u h) v h dx 0. [ ] = ∇ ·∇ + · = Ω

Steepest descent methods successively lower the energy by minimizing in descent directions defined through an appropriate gradient. 1 m Algorithm 4.1(Descent method) Let( , ) H be a scalar product onS D(Th) and 0 · · j j 1 μ (0,1/2). Given u h A h, compute the sequence(u h) j 0,1,... via uh+ = j∈ j j 1 ∈ m = u α j d with d S (Th) such that h + h h ∈ D j j (d ,v h)H F h(u ) vh h =− h [ ]