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<p< , and 1<q , then accord- ∞ p ≤∞ 1,q ing to the Aubin–Lions lemma the inclusionL ( 0,T X) W ( 0,T X ) L p( 0,T H) is compact.