Sum of Lower Semicontinuous Function and Continuous Function is Lsc

Stationary Partial Differential Equations

Thomas Bartsch , ... Michel Willem , in Handbook of Differential Equations: Stationary Partial Differential Equations, 2005

Proof

By Theorem 1.1 and lower semicontinuity, it is easy to verify that

(1.3) $\begin{array}{lll}\mu =\hfill & \underset{||v|{|}_{L}p=1}{\underset{v\in H_0^1\left(\Omega \right)}{\text{inf}}}\hfill & {\displaystyle {\int }_{\Omega }|\nabla v{|}^2+a\left(x\right)v^2\text{d}x}\hfill \\ \hfill \end{array}$

is achieved by some $\overline{v}$ . After replacing $\overline{v}$ by | $\overline{v}$ |, we may assume that $\overline{v}⩾0$ . It follows from the Lagrange multiplier rule that

$-\Delta \overline{v}+a\left(x\right)\overline{v}=\mu {\overline{v}}^{p-1}.$

A solution of (1.1) is then given by $\overline{u}=u^{1/(p-2)}\overline{v}$ . Indeed, $\overline{u}>0$ on Ω by the strong maximum principle. □

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Nonlinearity and Functional Analysis

In Pure and Applied Mathematics, 1977

6.5A A sharpening of the steepest descent method

For any general study of critical points, it is necessary to supplement the notions of weak lower semicontinuity and coerciveness by more refined analytic and topological considerations. To this end, we first reconsider the method of steepest descent, introduced in Section 3.2. Suppose F(x) is a smooth C ² functional defined on a real Hilbert space H, and bounded from below on H. Then in Section 3.2 it was shown that the solutions x(t, x ₀) of the initial value problem

(6.5.1) $dx/dt=-F\prime \left(x\right),x\left(0,x_0\right)=x_0,$

,exist for all t ⩾ 0, with lim x(t, x ₀) as t → ∞ a critical point of F(x) provided that the critical points of F(x) are isolated and that F(x) satisfies the following compactness condition (mentioned earlier in (6.1.1′)).

(6.5.2) Condition (C) Any sequence {x_n } in H with |F(x_n )| bounded and ||F′(x_n )|| → 0 has a convergent subsequence.

Under hypothesis (6.2.2), the discussion of Section 3.2 shows that F(x) clearly attains its infimum on H. The following result shows the utility of (6.5.2) for the study of other types of critical points.

(6.5.3) Theorem Suppose that a C ² functional F(x) defined on H is bounded from below and satisfies (6.5.2), and has only isolated critical points. If F(x) possesses two isolated relative minima y ₁, y ₂, then the functional F(x) must possess a third critical point y ₃, distinct from y ₁ and y ₂, which is not an isolated relative minimum.

Proof: Suppose that F(x) does not have a third critical point. Then we shall show that H can be represented as the union of two open, disjoint subsets U ₁ and U ₂; which obviously contradicts the connectedness of H. To construct the sets U_i , suppose that x(t, x ₀) is the solution of (6.5.1). By (6.5.2), x(t, x ₀) exists for all t ⩾ 0 and lim x(t, x ₀) as t → ∞ is y_i (i = 1, 2). Let U_i = {x ₀ | lim x(t, x ₀) = y_i as t → ∞} (i = 1, 2). Clearly H = U ₁ ∪ U ₂, while U ₁ and U ₂ are disjoint. To show that the sets U_i are open in H, we first note that each y_i , being a strict relative minimum, has a neighborhood W_i such that any solution x(t, x ₀) which enters W_i remains in W_i and, in fact, converges to y_i as t → ∞. Indeed, for x ₀ sufficiently near y_i , since F(x(t, x ₀)) is a decreasing function of t, x(t, x ₀) → y_i . Thus, by virtue of the continuity of x(t, x ₀) with respect to the initial condition x ₀, if z ₀ ∈ U_i , then for ɛ > 0 sufficiently small with $||z_0-\tilde{z}||<\epsilon$ , there is a T such that both x(T, z ₀) and $x\left(T,{\overline{z}}_0\right)$ lie in W_i . Consequently $\tilde{z}\in U_i$ if z ₀ is. Therefore each U_i is open, and we have obtained the desired contradiction.

It is also immediate that the third critical point y ₃ shown to exist by the above argument cannot be another relative minimum, for if it were and if F(x) had no other critical points, then the argument just given would again lead to a contradiction.

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Stationary Partial Differential Equations

F. Brock , in Handbook of Differential Equations: Stationary Partial Differential Equations, 2007

Proof

(1) Assume U ∈ $C_{0+}^{0,1}$ (B_R ) for some R > 0, and

$B*\left(u\right):=\left\{\upsilon \in C_{0+}^{0,1}\left(B_R\right):\upsilon \sim u,{\omega }_{\upsilon }⩽{\omega }_u,J(\upsilon )⩽J\left(u\right)\right\},$

where

$J(\upsilon ):={\int }_{{ℝ}^N}G\left(\upsilon ,|\nabla \upsilon |\right)\text{d}x.$

Furthermore, let δ = inf{‖v − u ^★‖₂: v ∈ B ^★(u )}. In view of the weak lower semicontinuity of the functional J and the nonexpansivity of the Schwarz symmetrization, B ^★(u) is weakly closed in $C_{0+}^{0,1}$ (B_R ). Hence there exists some U ∈ B ^★(u) with δ = ‖U − u ^★‖₂. Since J(U_H ) = J(U) ∀H ∈ H ^★, we may then argue as in the proof of Theorem 4.1 to obtain that δ = 0, and thus U = u ^★. This shows (4.8).

(2) Let u ∈ $W_{+}^{1,p}$ (ℝ ^N ) for some p ∈ (1, ∞). We choose a sequence {u _n} ⊂ $C_{0+}^{0,1}$ (ℝ ^N ) which converges to u in W ^1,p (ℝ ^N ). By (4.8) with G(v, z) = z^p we then have that ‖∇u_n ‖ _p ⩾ ‖∇(u_n )^★‖ _p, n = 1, 2,…. Hence we find a subsequence { $(u_n^{\prime }{)}^{★}$ } and a function v ∈ W ^1,p (ℝ ^N ) such that $(u_n^{\prime }{)}^{★}$ ⇀ν weakly in W ^1,p (ℝ ^N ). But by the nonexpansivity of the Schwarz symmetrization in L^p we have that (u_n )^★ → u ^★ in L^p (ℝ ^N ), so that v = u ^★. Finally, the weak lower semicontinuity of the norm gives ‖∇u ^★‖ _p ⩽ lim inf ‖∇ $(u_n^{\prime }{)}^{★}$ ‖ _p ⩽ lim ‖∇u_n ‖ _p = ‖∇u‖ _p .

(3) Let u ∈ W ^1,∞ (ℝ ^N ) ∩ S ₊. By Rademacher's Theorem, there is a version u ∈ C ^0,1 (ℝ ^N ) ∩ L ^∞ (ℝ ^N ) ∩ S ₊. By Theorem 4.1 this implies ω _{u ^★} ⩽ ω _u and u ^★ ∈ C ^0,1(ℝ ^N ). Since also ‖∇ _u ‖_∞ = lim _t ↘ω _u (t)/t, (4.9) follows for p = ∞.

(4) Let u ∈ $W_{+}^{1,1}$ (ℝ ^N ). We choose a sequence {u_n } ⊂ $C_{0+}^{0,1}$ (ℝ ^N ) with u_n → u in W ^1,1(Ω). Then have that for every Young function G,

${\int }_{{ℝ}^N}G\left(|\nabla u_n|\right)\text{d}x⩾{\int }_{{ℝ}^N}G\left(|\nabla (u_n{)}^{★}|\right)\text{d}x,n=1,2,\dots$

By a result of [5] this implies that

$L^N\left\{|\nabla u_n|>t\right\}⩾L^N\left\{|\nabla (u_n{)}^{★}|>t\right\},n=1,2,\dots$

Due to a well-known weak compactness criterion in L ¹ this implies that there is a function v ∈ L ¹ (Ω) and a subsequence {(u_n^′ )^★} such that |∇(u_n^′ )^★)| ⇀ ν weakly in L ¹ (ℝ ^N ). Since (u_n )^★ → u in L ¹ (ℝ ^N ) this implies that |∇u ^★| = v and u ^★ ∈W ^1,1(ℝ ^N ). Finally, the inequality (4.9) for p = 1 follows from the weak lower semicontinuity of the norm.

The inequalities (4.10) and (4.11) in Theorem 4.4 below are well-known (see [72,101,18] and [37]), while experts seem to be familiar with (4.10), although we could not find a reference.

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Image Processing: Mathematics

G. Aubert , P. Kornprobst , in Encyclopedia of Mathematical Physics, 2006

Examples of Functional Spaces

In this section, we revisit some possible choices of functional spaces summarized in Table 1 .

Table 1. Examples of functional spaces and their norm (see model [8]).

Model	E and |u| E	ϕ(t)	G and |u| G	ψ(t)
(a)	$H^1(\text{Ω}),|u{|}_E={\left({\int }_{\text{Ω}}|\nabla u{|}^2\text{dx}\right)}^{1/2}$	t 2	L 2(Ω) with its usual norm	t 2
(b)	$\text{<mi mathvariant="italic" is="true">BV</mi>}(\text{Ω}),|u{|}_E={\int }_{\text{Ω}}|\text{<mi mathvariant="italic" is="true">Du</mi>}|$	t	L 2(Ω) with its usual norm	t 2
(c)	$\text{<mi mathvariant="italic" is="true">BV</mi>}(\text{Ω}),|u{|}_E={\int }_{\text{Ω}}|\text{<mi mathvariant="italic" is="true">Du</mi>}|$	t	$\{b\in L^2(\text{Ω});b=\text{div}\xi ,|\xi {|}_{L^{\infty }{(\text{Ω})}^2}\le 1,\xi \cdot N{|}_{\partial \text{Ω}}|=0\}$	t

The first case (a) was inspired by the classical Tikhonov regularization. The functional space $H^1(\text{Ω})(\text{Ω}\subset {\mathbb{R}}^2)$ is the space of functions in L ²(Ω) such that the distributional gradient Du is in L ²(Ω). Unfortunately, functions in H ¹(Ω) do not admit discontinuities across curves and this is a major problem with respect to image analysis since images are made of smooth patches separated by sharp variations.

Considering the problem reported in (a), Rudin et al. (1992) proposed to work on BV(Ω), the space of bounded variations (BV) Ambrosio et al. (2000) defined by

[9] $\begin{array}{l}\text{BV}(\text{Ω})=\left\{u\in L^1(\text{Ω});{\int }_{\text{Ω}}|Du|<\infty \right\}\\ \text{with}{\int }_{\text{Ω}}|Du|=sup\left\{{\int }_{\text{Ω}}u\text{div}\phi \text{d}x;\phi =({\phi }_1,{\phi }_2,\dots ,{\phi }_N)\in C_0^1{(\text{Ω})}^N,{|\phi |}_{L\infty (\text{Ω})}\le 1\right\}\end{array}$

It is equivalent to define BV(Ω) as the space of L ¹(Ω) functions whose distributional gradient Du is a bounded measure and [9] is its total variation. The space BV(Ω) has some interesting properties:

1.

lower semicontinuity of the total variation ∫ _Ω|Du| with respect to the L₁(Ω) topology,

2.

if u∈BV(Ω), we can define, for ${\mathcal{H}}^1$ almost everywhere x∈S _u , the complement of Lebesgue points (i.e., the jump set of u), a normal n _u (x) and two approximate "right" and "left" limits u ⁺(x) and u ⁻(x), and

3.

Du can be decomposed as a sum of a regular measure, a jump measure, and a Cantor measure:

$Du=\nabla u\text{d}x+(u^{+}-u^{-})n_u{\mathcal{H}}_{/S_u}^1+C_u$

where ∇u is the approximate gradient and ${\mathcal{H}}^1$ the one-dimensional Hausdorff measure.

This ability to describe functions with discontinuities across a hypersurface S _u makes BV(Ω) very convenient to describe images with edges. In this context, the image restoration problem is well posed and suitable numerical tools can be proposed (Chambolle and Lions 1997).

One criticism of the model (b) in Table 1 pointed out by Meyer (2001) is that if f is a characteristic function and if f is sufficiently small with respect to a suitable norm, then the model (Rudin et al. 1992) gives u=0 and η=f contrary to what one should expect (u=f and η=0). In fact, the main reason of this phenomenon is that the L ²-norm for the η component is not the right one since very oscillating functions can have large L ²-norm (e.g., f_n (x)=cos(nx)). To better describe such oscillating functions, Meyer (2001) introduced the space of functions which can be expressed as a divergence of L ^∞-fields. This work was developed in R^N and this framework was adapted to bounded 2D domains by Aubert and Aujol (2005) (see (c) in Table 1 ). An example of image decomposition is shown in Figure 3 .

Figure 3. Example of image decomposition (see Aubert and Aujol (2005)).

In this section, we have shown how the choice of the functional spaces is closely related to the definition of a variational formulation. The functionals are written in a continuous setting and they can usually be minimized by solving the discretized Euler equations iteratively, until convergence. These PDEs and the differential operators are constrained by the energy definition but it is also possible to work directly on the equations, forgetting the formal link with the energy. Such an approach has also been much developed in the computer vision community and it is illustrated in the next section.

We refer the reader to Aubert and Kornprobst (2002) for a general review of variational approaches and PDEs as applied to image analysis.

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Handbook of Dynamical Systems

Alexander Mielke , in Handbook of Dynamical Systems, 2002

Theorem 5.2

Under the above assumptions for each ɛ > 0 there exists an ℓ > 0 such that for all domains Ω ⊂ ℝ ^d with {x: |x| < ℓ} and all bc ∈ {Dir, Neu, Per} we have ${dist}_{L_w^p}\left(A_{bc}^{\Omega },A^{{ℝ}^d}\right)\le \varepsilon$ .

The remaining open problem is under what conditions we also have lower semicontinuity of the attractor, i.e., ${dist}_{L_w^p}\left({\mathcal{A}}^{{ℝ}^d},{\mathcal{A}}_{bc}^{\Omega }\right)\le \varepsilon$ for suitably large domains Ω. In the next subsection we give a simple example showing that lower semicontinuity is in general false. In our example the limit of ${\mathcal{A}}_{bc}^{\left(-l,l\right)}$ for ℓ → ∞ exists for each bc ∈ {Dir, Neu, Per}, but it depends on bc and is strictly contained in A ^ℝ.

The problem of convergence of $A_{{bc}_n}^{{\Omega }_n}to{A}^{{ℝ}^d}$ is a problem of interchanging the limit t → ∞ in the definition of the attractors with the limit Ω _n → ℝ ^d . The interchange of such limits would only be possible, if the attraction rates for the attractors ${\mathcal{A}}_{{bc}_n}^{{\Omega }_n}$ are uniform in n; however, this is in general not the case. In [65] a generalized limit attractor was introduced which allows for double limits where t _n → ∞ simultaneously with the parameter ɛ _n → 0. In our situation this leads to the following definition. Consider a sequence (bc _n , Ω _n ) _n∈ℕ of boundary conditions and domains which approach ℝ ^d in the sense that {x: |x| < ℓ _n } ⊂ Ω _n with ℓ _n → ∞. Now set

$\begin{array}{l}A*=\{u\in L_{ul}^p\left({ℝ}^d\right):\exists t_{n}with{t}_n\to \infty \exists u_n\in L_{ul}^p\left({\Omega }_n\right)\\ with\underset{n\in \mathbb{N}}{sup}{\Vert u_n\Vert }_{p,ul}<\infty suchthat\\ {\Vert S_{{bc}_n}^{{\Omega }_n}\left(t_n,u_n\right)-u\Vert }_{L_w^p\left({ℝ}^d\right)}\to 0forn\to \infty \}.\end{array}$

The motivation to consider A* rather than w-lime $A_{{bc}_n}^{{\Omega }_n}$ is that for practical purposes (e.g., in experiments or numerics) we always have to consider finite time and bounded domains. Thus, one should neither prescribe the limit t → ∞ before n → ∞ nor the opposite, see [65] for a further discussion.

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Reliable Methods for Computer Simulation

In Studies in Mathematics and Its Applications, 2004

5.4.1 Lower semicontinuous functionals

Recall that the functional J:V→R is called continuous at v ₀if for any sequence ${\left\{v_k\right\}}_{k=1}^{\infty }$ converging to v ₀ in V, we have

(5.4.3) $\underset{k\to \infty }{\lim }J\left({\upsilon }_k\right)=J\left({\upsilon }_0\right).$

Also, J is said to be continuous at v ₀if for any α and β such that α <J(v ₀) <β there exists a number ɛ > 0 such that

$\alpha <J(\upsilon )<\beta ,\forall \upsilon \in B\left({\upsilon }_0,\epsilon \right).$

Let ${\left\{{\xi }_k\right\}}_{k=1}^{\infty }$ be a sequence of real numbers. By $\underset{k\to \infty }{\underset{\_}{\lim }}{\xi }_k$ we denote the lower limit of this sequence, which may be finite or infinite.

Definition 5.4.1

We say that J:V→ $\overline{\text{R}}$ is lower semicontinuous at v ₀∈V if

(5.4.4) $\underset{k\to \infty }{\underset{\_}{\lim }}J\left({\upsilon }_k\right)\ge J\left({\upsilon }_0\right)$

for any sequence ${\left\{v_k\right\}}_{k=1}^{\infty }$ converging to v ₀ in V.

Another definition of lower semicontinuous functionals is as follows.

Definition 5.4.2

A functional J is said to be lower semicontinuous at v ₀∈V if the inequality J(v ₀) > αimplies that this inequality also holds in a ball centered at v ₀,i.e.,

$J(\upsilon )>\alpha ,\forall \upsilon \in B\left({\upsilon }_0,\epsilon \right),$

where ɛ is a positive number.

The relationship between these definitions is quite transparent. Indeed, assume that J is lower semicontinuous in the sense of Definition 5.4.2 and v _k →v ₀ in V. Take α such that J(v ₀) > α and choose a respective B(v ₀, ɛ). Then,v _k ∈ B(v ₀, ɛ), provided that k is greater than a certain positive integer N(ɛ). Consequently, J(v _k ) > α and lim J(v _k ) ≥J(v ₀). If V is a space with a countable base, then both definitions are equivalent. In general topological spaces, Definition 5.4.2 should be used.

Definition 5.4.3

A functional that is lower semicontinuous at any point is called lower semicontinuous or an l.s.c. functional.

Definition 5.4.4

A functional G is called upper semicontinuous if G=-J, where J is a lower semicontinuous functional.

Note that a functional is continuous if and only if it is simultaneously lower and upper semicontinuous.

Proposition 5.4.1

The sets

$V_{\alpha }:=\left\{\upsilon \in V|J(\upsilon )\le \alpha ,\alpha \in ℝ\right\}$

are closed if and only if J is an l.s.c. functional.

Proof

Let J be a lower semicontinuous functional. Then the set

$V\backslash V_{\alpha }=\left\{\upsilon \in V|J(\upsilon )>\alpha \right\}$

is open. Indeed, this fact follows from Definition 5.4.2, which says that any point v ₀belongs to V\V _αwith a certain neighborhood. Therefore,V _α is a closed set. If V _α is closed for any α, then V\V _αis open and, consequently, any v ₀∈V\V _αhas a neighborhood B(v ₀, ɛ) ⊂V\V _α. This means that J(v) > α for all v∈ B(v ₀, ɛ), i.e., J is a lower semicontinuous functional.

The theorem below shows the meaning of lower semicontinuity in proving existence theorems for variational problem.

Proposition 5.4.2

Assume that J:V→ $\overline{\text{R}}$ is an l.s.c. functional, the set V _α is compact for someα > inf P,and

$\underset{\upsilon \in V}{\inf }J(\upsilon )=\inf p>-\infty .$

Then there exists an element u∈V such that J(u) = inf P.

Proof

Let ${\left\{v_k\right\}}_{k=1}^{\infty }$ be a minimizing sequence. Then,v _k ∈V _α for k>N(α). Since V _α is compact, we can extract a subsequence {v _ks } such that v _ks →u∈V _α. Then

$\inf P\le J\left(u\right)\le \underset{k\to \infty }{\underset{\_}{\lim }}J\left({\upsilon }_{k_S}\right)=\inf P,$

which shows that J(u) = inf P.

However, this theorem is difficult to use, because in the majority of variational models that are of practical importance, the sets V _α are not compact. This suggests to impose stronger conditions on J with less demanding conditions for V _α . Along this way, the lower semicontinuity condition is replaced by the weak lower semicontinuity condition.

Definition 5.4.5

A functional J:V→ $\overline{\text{R}}$ is said to be weakly lower semicontinuous at v ₀∈V if

$\underset{k\to \infty }{\underset{\_}{\lim }}J\left({\upsilon }_k\right)\ge J\left({\upsilon }_0\right)$

for any sequence ${\left\{v_k\right\}}_{k=1}^{\infty }$ that weakly converges to v ₀.

The concept of weak lower semicontinuity plays an important role in the calculus of variations. First, we need a suitable criterion for verifying whether or not a functional possesses such a property. In general, this is not an easy task. However, the weak lower semicontinuity of convex functionals is found to follow from the lower semicontinuity, which makes the verification of it rather simple. Below we prove this fact for Hilbert spaces.

Proposition 5.4.3 (Banach, Saks, and Mazur)

Let{v _k }be a sequence of elements in the Hilbert space H, which weakly converges to v ₀∈H. Then one can find a subsequence{v _ki }such that the sequence $w_m=\frac{1}{m}\sum _{i=1}^m{\upsilon }_{ki}$ strongly converges to v ₀ in H.

Proof

Without loss of generality, we may consider only the case v ₀= 0. Let us construct a subsequence {v _ki } by the following procedure. Set v _{k 1} =v ₁. Since (v _{k 1} ,v) → 0 as k→ ∞, we find v _{k 2} such that |(v _{k 1} ,v _{k 2} )| < 1. Assume that v _{k 1} , …,v _{k i} are determined. Find v _{k i+1} such that

$\left|\left({\upsilon }_{k_1},{\upsilon }_{k_{i+1}}\right)\right|<\frac{1}{i},\left|\left({\upsilon }_{k_2},{\upsilon }_{k_{i+1}}\right)\right|<\frac{1}{i},\mathrm{\dots },\left|\left({\upsilon }_{k_i},{\upsilon }_{k_{i+1}}\right)\right|<\frac{1}{i}.$

The sequence {v _k } is bounded and, therefore, ||v _{k i} || ≤c< +∞. Note that the amount of products (v _{k i} ,v _{k i} ), where i<j, is 2(j- 1) (the multiplier 2arises due to the symmetry) and the value of such a product is no greater than 1/(j- 1). In view of this fact, we have

$\left|\right|w_m|{|}^2=\frac{1}{m^2}\left(\sum _{i=1}^m{\upsilon }_{k_i},\sum _{j=1}^m{\upsilon }_{k_j}\right)\le \frac{1}{m^2}\left(m{c}^2+2\left(m-1\right)\right).$

The right-hand side of this estimate tends to zero as m→ +∞. Hence, the required statement is proved.

Remark 5.4.1

Proposition 5.4.3 implies the following result: any convex closed set K is weakly closed. Indeed, if {v _k } ∈ K and v _k weakly converges to v ₀∈H(i.e.,v _k ⇀v ₀), then one can find a sequence of "averaged" elements {w _k } that converges to v ₀. Since K is closed, we see that v ₀∈K and, therefore, K is weakly closed.

Proposition 5.4.4

Let J:H→ $\overline{\text{R}}$ be a lower semicontinuous functional. If J is convex, then it is weakly lower semicontinuous.

Proof

Our aim is to show that

$\underset{k\to \infty }{\underset{\_}{\lim }}J\left({\upsilon }_k\right)\ge J\left({\upsilon }_0\right)$

for any sequence {v _k } such that v _k ⇀v ₀. By the definition of the lower limit, the sequence {v _k } contains a subsequence { $v_{k_s}$ } such that

$\underset{k\to \infty }{\underset{\_}{\lim }}J\left({\upsilon }_k\right)=\underset{s\to \infty }{\lim }J\left({\upsilon }_{k_s}\right).$

Obviously,v _{k s} weakly converges to v ₀. Thus, we may apply Proposition 5.4.3, which shows that {v _{k s} } contains a subsequence {v _i } such that $w_m=1/m\sum _{i=1}^m{\upsilon }_i$ strongly converges to v ₀. By the convexity of J, we obtain

$\frac{1}{m}\sum _{i=1}^{m}J\left({\upsilon }_i\right)\ge J\left(\frac{1}{m}\sum _{i=1}^m{\upsilon }_i\right)=J\left(w_m\right).$

Since J is lower semicontinuous,

$\underset{m\to \infty }{\lim }J\left(w_m\right)\ge J\left({\upsilon }_0\right).$

we see that

$\underset{k\to \infty }{\underset{\_}{\lim }}J\left({\upsilon }_k\right)=\underset{m\to \infty }{\lim }\frac{1}{m}\sum _{i=1}^{m}J\left({\upsilon }_i\right)\ge J\left({\upsilon }_0\right).$

Remark 5.4.2

Propositions 5.4.3 and 5.4.4 admit of wide generalizations. They remain valid for reflexive Banach spaces. After the replacement of weak convergence by the so-called *-weak convergence, they are also valid for topological vector spaces (see, e.g., [190]).

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Convex Analysis and Duality Methods

G. Bouchitté , in Encyclopedia of Mathematical Physics, 2006

Continuity and Lower-Semicontinuity

A first consequence of the convexity is the continuity on the topological interior of the domain. We refer for instance to Borwein and Lewis (2000) for a proof of

Theorem 1

Let $f:X\to \mathbb{R}\cup \{+\infty \}$ be convex and proper. Assume that ${\text{sup}}_{U}f<+\infty$ , where U is a suitable open subset of X. Then f is continuous and locally Lipschitzian on all int(dom f  ).

As an immediate corollary, a convex function on a normed space is continuous provided it is majorized by a locally bounded function. In the finite-dimensional case, it is easily deduced that a finite-valued convex function $f:{\mathbb{R}}^d\to \mathbb{R}$ is locally Lipschitz. Furthermore, by Aleksandrov's theorem, f is almost everywhere twice differentiable and the non-negative Hessian matrix ∇² f coincides with the absolutely continuous part of the distributional Hessian matrix D ² f (it is a Radon measure taking values in the non-negative symmetric matrices).

However, in infinite-dimensional spaces, for ensuring compactness properties (as, e.g., in condition (ii) of Theorem 4 below), we need to use weak topologies and the situation is not so simple. A major idea consists in substituting the continuity property with lower-semicontinuity.

Definition 2

A function $f:X\to \mathbb{R}\cup \{+\infty \}$ is τ-l.s.c. at $x_0\in X$ if for all $\alpha \in \mathbb{R}$ , there exists $U\in V_{x_0}$ such that f > α on U. In particular, f will be l.s.c. on all X provided $f^{-1}((r,+\infty ))$ is open for every $r\in \mathbb{R}$ .

Remark 3

(i)

The following sequential notion can be also used: f is τ-sequentially l.s.c. at x ₀ if

$\forall (x_n)\subset X{x}_n{\displaystyle \stackrel{\tau }{\to }}{x}_0\Rightarrow \underset{n\to +\infty }{\text{lim}\text{inf}}f(x_n)\ge f(x_0)$

It turns out that this notion (weaker in general) is equivalent to the previous one provided x ₀ admits a countable basis of neighborhoods.

(ii)

A well-known consequence of Hahn–Banach theorem is that, for convex functions, the lower-semicontinuity property with respect to the normed topology of X is equivalent to the weak (or weak sequential) lower-semicontinuity.

Theorem 4

(Existence).Let $f:X\to \mathbb{R}\cup \{+\infty \}$ be proper, such that

(i)

f is τ-l.s.c.,

(ii)

$\forall r\in \mathbb{R},f^{-1}((-\infty ,r])$ is τ-relatively compact.

Then there is ${\displaystyle \bar{x}}\in X$ such that $f({\displaystyle \bar{x}})=\text{inf}f$ and argmin $f:=\{x\in X|f(x)=\text{inf}f\}$ is τ-compact.

In practice, the choice of the topology τ is ruled by the condition (ii) above. For example, if X is a reflexive infinite-dimensional Banach space and if f is coercive (i.e., ${\text{lim}}_{\Vert x\Vert \to \infty }f(x)=+\infty$ ), we may take for τ the weak topology (but never the normed topology). This restriction implies in practice that the first condition in Theorem 4 may fail. In this case, it is often useful to substitute f with its lower-semicontinuous (l.s.c.) envelope.

Definition 5

Given a topology τ, the relaxed function ${\displaystyle \bar{f}}(={{\displaystyle \bar{f}}}^{\tau })$ is defined as

$\begin{array}{l}{\displaystyle \bar{f}}(x)=\text{sup}\{g(x)|g:X\to \mathbb{R}\cup \{+\infty \},\\ g\text{is}\tau -l.s.c.,g\le f\}\end{array}$

It is easy to check that f is τ-l.s.c. at x ₀ if and only if ${\displaystyle \bar{f}}(x_0)=f(x_0)$ . Futhermore,

${\displaystyle \bar{f}}(x)=\underset{U\in {\nu }_u}{\text{sup}}\underset{U}{\text{inf}}f,\text{epi}{\displaystyle \bar{f}}={\text{cl}}_{(X\times \mathbb{R})}(\text{epi}f)$

We can now state the relaxed version of Theorem 1.4.

Theorem 6

(Relaxation). Let $f:X\to \mathbb{R}\cup \{+\infty \}$ , then: $\text{inf}f=\text{inf}{\displaystyle \bar{f}}$ . Assume further that, for all real r, $f^{-1}((-\infty ,r])$ is $\mathcal{T}$ -relatively compact; then f attains its minimum and $\text{argmin}f=\text{arg}\text{min}{\displaystyle \bar{f}}\cap$ $\{x\in X|f(x)={\displaystyle \bar{f}}(x)\}$ .

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Gradient Flows of Probability Measures

Luigi Ambrosio , Giuseppe Savaré , in Handbook of Differential Equations: Evolutionary Equations, 2007

3.2 Examples ofconvex functionals in $\mathcal{P}$ ₂(ℝ^d)

In this section we introduce the main classes of geodesically convex functionals.

Example 3.4

(Potential energy). Let V : ℝ^d → (−∞, +∞] be a proper, lower semicontinuous function whose negative part has a quadratic growth, i.e.,

(3.6) $V\left(x\right)\ge -A-B{\left|x\right|}^2\mathrm{\forall }x\in {ℝ}^d\text{for some}A,B\in {ℝ}^{+}\mathrm{.}$

In $\mathcal{P}$ ₂ (ℝ ^d ) we define

(3.7) $\mathrm{V}\left(\mu \right):={\int }_{{ℝ}^d}V\left(x\right)\text{d}\mu \left(x\right)\mathrm{.}$

Evaluating $\mathcal{V}$ on Dirac's masses we check that $\mathcal{V}$ is proper; since V⁻ has at most quadratic growth Lemma 1.2 gives that $\mathcal{V}$ is lower semicontinuous in $\mathcal{P}$ ₂(ℝ ^d ). If V is bounded from below we have even lower semicontinuity w.r.t. narrow convergence.

The following simple proposition shows that $\mathcal{V}$ is convex along all interpolating curves induced by admissible plans; choosing optimal plans one obtains in particular that $\mathcal{V}$ is convex along geodesics.

Proposition 3.5

(Convexity of $\mathcal{V}$ ). If V is λ-convex then for every μ ¹ ,μ ² ∈ D( $\mathcal{V}$ ) and μ e Γ(μ ¹ , μ ²) we have

(3.8) $\begin{array}{c}\mathcal{V}\left({\mu }_t^{1\to 2}\right)\le (1-t)\mathcal{V}\left({\mu }^1\right)+t\mathcal{V}\left({\mu }^2\right)\\ -\frac{1}{2}t(1-t){\int }_{{ℝ}^d\times {ℝ}^d}{|x_1-x_2|}^2\text{d}\mu (x_1,x_2)\mathrm{.}\end{array}$

In particular $\mathcal{V}$ is λ-convex along geodesics.

Proof. Since V is bounded from below either by a continuous affine functional (if $\lambda \ge 0$ ) or by a quadratic function (if λ < 0) its negative part satisfies (3.6); therefore the definition (3.7) makes sense.

Integrating (3.3) along any admissible transport plan μ ∈ Γ(μ ¹, μ ²) with μ ¹, μ ² ∈ D( $\mathcal{V}$ ) we obtain (3.8), since

$\begin{array}{c}\begin{array}{c}\mathcal{V}\left({\mu }_t^{1\to 2}\right)\\ ={\int }_{{ℝ}^d\times {ℝ}^d}V\left((1-t)x_1+t{x}_2\right)\text{d}\mathbf{\mu }(x_1,x_2)\end{array}\\ \begin{array}{c}\le {\int }_{{ℝ}^d\times {ℝ}^d}\left((1-t)V\left(x_1\right)+tV\left(x_2\right)-\frac{\lambda }{2}t(1-t){|x_1-x_2|}^2\right)\text{d}\mathbf{\mu }(x_1,x_2)\\ =(1-t)\mathcal{V}\left({\mu }^1\right)+t\mathcal{V}\left({\mu }^2\right)-\frac{\lambda }{2}t(1-t){\int }_{{ℝ}^d\times {ℝ}^d}{|x_1-x_2|}^2\text{d}\mathbf{\mu }(x_1,x_2)\mathrm{.}\end{array}\end{array}$

Since $\mathcal{V}$ (δ_x ) = V(x), it is easy to check that the conditions on V are also necessary for the validity of the previous proposition.

Example 3.6

(Interaction energy). Let us fix an integer k > 1 and let us consider a lower semicontinuous function W:ℝ ^kd → (−∞, +∞], whose negative part satisfies the usual quadratic growth condition. Denoting by μ^×k the measure μ × μ × … × μ on ℝ ^kd , we set

(3.9) ${\mathrm{W}}_k\left(\mu \right):={\int }_{{ℝ}^d}W(x_1,x_2,\cdots ,x_k)\text{d}{\mu }^{\times k}(x_1,x_2,\cdots ,x_k)\mathrm{.}$

If

(3.10) $\exists x\in {ℝ}^d:W(x,x,\cdots ,x)<+\infty ,$

then W_k is proper; its lower semicontinuity follows from the fact that

(3.11) ${\mu }_n\to \mu \text{in}{\mathrm{P}}_2\left({ℝ}^d\right)\Rightarrow {\mu }_n^{\times k}\to {\mu }^{\times k}\text{in}{\mathrm{P}}_2\left({ℝ}^{kd}\right)\mathrm{.}$

Here the typical example is k = 2 and $W(x_1,x_2):=\tilde{W}(x_1-x_2)$ for some $\tilde{W}:{ℝ}^d\to (-\infty ,+\infty ]$ with $\tilde{W}\left(0\right)<+\infty$ .

Proposition 3.7

(Convexity of W). If W is convex then the functional W_k is convex along the interpolating curve ${\mu }_t^{1\to 2}$ induced by any μ e Γ(μ ¹, μ ²), in $\mathcal{P}$ ₂(ℝ ^d ).

Proof. Observe that W_k is the restriction to the subset

${\mathrm{P}}_2^{\times }\left({ℝ}^{kd}\right):=\{{\mu }^{\times k}\mu \in {\mathrm{P}}_2\left({ℝ}^d\right)\}$

of the potential energy functional $\mathcal{W}$ on $\mathcal{P}$ ₂(ℝ ^kd ) given by

$\mathrm{W}\left(\mu \right):={\int }_{{ℝ}^{kd}}W(x_1,\dots ,x_k)\text{d}\mu (x_1,\dots ,x_k)\mathrm{.}$

We consider the linear permutation of coordinates P : (ℝ^2d )^k → (ℝ ^kd )² defined by

$\mathcal{P}\left((x_1,y_1),(x_2,y_2),\dots ,(x_k,y_k)\right):=((x_1,\dots x_k),\left(y_{1,\dots }{y}_k\right))\mathrm{.}$

If μ ∈ Γ(μ ₁, μ ₂) then it is easy to check that ${\mathcal{P}}_{\#}{\mu }^{\times k}\in \mathrm{\Gamma }({\mu }_1^{\times k},{\mu }_2^{\times k})\subset \mathcal{P}\left({\left({ℝ}^{kd}\right)}^2\right)$ and

${\left({\pi }_t^{1\to 2}\right)}_{\#}{\mathcal{P}}_{\#}\left({\mu }^{\times k}\right)={\mathcal{P}}_{\#}{\left({\left({\pi }_t^{1\to 2}\right)}_{\#}\mu \right)}^{\times k}\mathrm{.}$

Therefore all the convexity properties of W_k follow from the corresponding ones of $\mathcal{W}$ .

Example 3.8

(Internal energy). Let F : [0, +∞) → (−∞, +∞] be a proper, lower semicontinuous convex function such that

(3.12) $F\left(0\right)=0\underset{s\downarrow 0}{\text{lim inf}}\frac{F\left(s\right)}{s^{\alpha }}>-\infty \text{for some}\alpha >\frac{d}{d+2}\mathrm{.}$

We consider the functional ℱ: $\mathcal{P}$ ₂(ℝ ^d ) → (−∞, +∞] defined by

(3.13) $\mathrm{F}\left(\mu \right):=\{\underset{+\infty }{{\int }_{{ℝ}^d}}F\left(u\left(x\right)\right)\text{d}{\mathrm{L}}^d\left(x\right)\underset{\text{otherwise.}}{\text{if}\mu =u}\cdot {\mathrm{L}}^d\in {\mathrm{P}}_2^{\text{a}}\left({ℝ}^d\right),$

Remark 3.9

(The meaning of condition (3.12)). Condition (3.12) simply guarantees that the negative part of F (μ) is integrable in ℝ ^d . For, let us observe that there exist nonnegative constants c ₁, c ₂ such that the negative part of F satisfies

$F^{-}\left(s\right)\le c_{1}s+c_2{s}^{\alpha }\forall s\in [0,+\infty ),$

and it is not restrictive to suppose $\alpha \le 1$ . Since μ = uℒ^d ∈ $\mathcal{P}$ ₂(ℝ ^d ) and 2α/(1 − α) < d we have

$\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}{\int }_{{ℝ}^d}{u}^{\alpha }\left(x\right)\text{d}{\mathcal{L}}^d\left(x\right)\end{array}\\ =\begin{array}{c}{\int }_{{ℝ}^d}{u}^{\alpha }\left(x\right){(1+\left|x\right|)}^{2\alpha }\end{array}{(1+\left|x\right|)}^{-2\alpha }\text{d}{\mathcal{L}}^d\left(x\right)\end{array}\\ \le {\left({\int }_{{ℝ}^d}u\left(x\right){(1+\left|x\right|)}^2\text{d}{\mathcal{L}}^d\left(x\right)\right)}^{\alpha }{\left({\int }_{{ℝ}^d}{(1+\left|x\right|)}^{-2\alpha /(1-\alpha )}\text{d}{\mathcal{L}}^d\left(x\right)\right)}^{1-a}\end{array}\\ <+8\end{array}$

and therefore F ⁻ (u) ∈ L ¹(ℝ ^d ).

Remark 3.10

(Lower semicontinuity of ℱ). General results on integral functionals (see, for instance, [8]) show that ℱ is narrowly lower semicontinuous if F is nonnegative and has a superlinear growth at infinity. Indeed, under this assumption sequences μ_n = u_nℒ^d on which ℱ is bounded have the property that (u_n ) is sequentially weakly relatively compact in L ¹(ℝ ^d ), and the convexity of ℱ together with the lower semicontinuity of F ensure the sequential lower semicontinuity with respect to the weak L ¹ topology.

In the next proposition we prove the geodesic convexity of the internal energy functional (3.13) by using the change of variable formula (1.24). This was first shown by McCann [66] with a different argument.

Proposition 3.11

(Convexity of ℱ). If F has a superlinear growth at infinity and

(3.14) $th\epsilon ma\mathcal{P}s\mapsto s^{d}F\left(s^{-d}\right)isconv\epsilon xandnonincr\epsilon asingin(0,+\infty ),$

then the functional ℱ is convex along geodesics in $\mathcal{P}$ ₂(ℝ ^d ).

Proof. We consider two measures μⁱ = uⁱℒ^d ∈ D(ℱ), i = 1, 2, and the optimal transport map r such that r#μ ¹ = μ ². Setting r _t := (1 − t)i + t r, by the characterization of constant speed geodesics we know that r _t is the optimal transport map between μ ¹ and μ _t := r _t# μ ¹ for any t ∈ [0, 1], and ${\mu }_t{L}^d\in {\mathcal{P}}_2^{\text{a}}\left({ℝ}^d\right)$ , with

$u_t\left({\mathbf{r}}_t\left(x\right)\right)=\frac{u^1\left(x\right)}{\text{det}\Delta {\mathbf{r}}_t\left(x\right)}\text{for}{\mu }^1\text{-a.e.}x\in {ℝ}^d\mathrm{.}$

By (1.24) it follows that

$\mathrm{F}\left({\mu }_t\right)={\int }_{{ℝ}^d}F\left(u_t\left(y\right)\right)\text{d}y\text{=}{\int }_{{ℝ}^d}F\left(\frac{u^1\left(x\right)}{\text{det}\Delta {\mathbf{r}}_t\left(x\right)}\right)\text{det}\Delta {\mathbf{r}}_t\left(x\right)\text{d}x\mathrm{.}$

Since for a diagonalizable map D with nonnegative eigenvalues

(3.15) $t\mapsto \det {\left(\right(1-t)I+tD)}^{1/d}\text{\hspace{0.17em}is concave}\text{in}[0,1],$

the integrand above may be seen as the composition of the convex and nonincreasing map s ↦ s^d F (u ¹(x)/s^d ) and of the concave map in (3.15), so that the resulting map is convex in [0, 1] for μ ¹-a.e. x ∈ ℝ ^d . Thus we have

$F\left(\frac{u^1\left(x\right)}{\text{det}\Delta {\mathbf{r}}_t\left(x\right)}\right)\text{det}\Delta {\mathbf{r}}_t\left(x\right)\le (1-t)F\left(u^1\left(x\right)\right)+tF\left(u^2\left(x\right)\right)$

and the thesis follows by integrating this inequality in ℝ^d .

In order to express (3.14) in a different way, we introduce the function

(3.16) $\underset{\text{which satisfies}-L_F\left({\text{e}}^{-z}\right){\text{e}}^z=\frac{\text{d}}{\text{d}z}F\left({\text{e}}^{-z}\right){\text{e}}^z;}{L_F\left(z\right):=z{F}^{\prime }\left(z\right)-F\left(z\right)}$

denoting by $\hat{F}$ the modified function F(e^−z )e ^z we have the simple relation

(3.17) $\begin{array}{c}\begin{array}{cc}{\hat{L}}_F\left(z\right)=-\frac{d}{dz}\hat{F}\left(z\right),& {\hat{L}}_F^2\left(z\right)=-\frac{d}{dz}\end{array}{\hat{L}}_F\left(z\right)=\frac{{\text{d}}^2}{{\text{d}z}^2}\hat{F}\left(z\right),\\ \text{where}{L}_F^2\left(z\right):=L_{L_F}\left(z\right)=z{L^{\prime }}_F\left(z\right)-L_F\left(z\right)\mathrm{.}\end{array}$

The nonincreasing part of condition (3.14) is equivalent to say that

(3.18) $L_F\left(z\right)\ge 0\forall z\in (0,+\infty ),$

and it is in fact implied by the convexity of F. A simple computation in the case F ∈ C ² (0, +∞) shows

$\frac{{\text{d}}^2}{\text{d}{s}^2}F\left(s^{-d}\right)s^d=\frac{{\text{d}}^2}{\text{d}{s}^2}\hat{F}(d\cdot \text{log}s)={\hat{L}}_F^2(d\cdot \text{log}s)\frac{d^2}{s^2}+{\hat{L}}_F(d\cdot \text{log}s)\frac{d}{s^2},$

and therefore

(3.19) $\left(3.14\right)\text{is equivalent to}{L}_F^2\left(z\right)\ge -\frac{1}{d}{L}_F\left(z\right)\forall z\in (0,+\infty ),$

i.e.,

(3.20) $\underset{\text{the map}z\mapsto z^{1/d-1}{L}_F\left(z\right)\text{is nonincreasing.}}{z{L^{\prime }}_F\left(z\right)\ge (1-\frac{1}{d})L_F\left(z\right),}$

Observe that the bigger is the dimension d, the stronger are the above conditions, which always imply the convexity of F.

Remark 3.12

(A "dimension free" condition). The weakest condition on F yielding the geodesic convexity of ℱ in any dimension is therefore

(3.21) $L_F^2\left(z\right)=z{L^{\prime }}_F\left(z\right)-L_F\left(z\right)\ge 0\forall z\in (0,+\infty )\mathrm{.}$

Taking into account (3.17), this is also equivalent to ask that

(3.22) $\text{the map}s\mapsto F\left({\text{e}}^{-s}\right){\text{e}}^{s}isconv\epsilon xandnonincr\epsilon asingin(0,+\infty )\mathrm{.}$

Among the functionals F satisfying (3.14) we quote

(3.23) $\text{the entropy functional:}F\left(s\right)=s\text{log}s,$

(3.24) $\text{the power functional:}F\left(s\right)=\frac{1}{m-1}{s}^m\text{for}m\ge 1-\frac{1}{d}\mathrm{.}$

Observe that the entropy functional and the power functional with m > 1 have a superlinear growth. In order to deal with the power functional with $m\ge 1$ , due to the failure of the lower semicontinuity property one has to introduce a suitable relaxation ℱ* of it, defined by [24,55]

(3.25) $\underset{\text{with}\mu =u\cdot {\mathrm{L}}^d+{\mu }_s,{\mu }_s\perp {\mathrm{L}}^d\mathrm{.}}{\mathrm{F}*\left(\mu \right):=\frac{1}{m-1}{\int }_{{ℝ}^d}{u}^m\left(x\right)\text{d}{\mathrm{L}}^d\left(x\right)}$

In this case the functional takes only account of the density of the absolutely continuous part of μ w.r.t. ℒ^d and the domain of ℱ* is the whole $\mathcal{P}$ ₂(ℝ ^d ). The functional ℱ* retains the convexity properties of ℱ, see [9].

Example 3.13

(The opposite Wasserstein distance). Let us fix a base measure μ ¹ ∈ $\mathcal{P}$ ₂(ℝ ^d ) and let us consider the functional

(3.26) $\varphi \left(\mu \right):=-\frac{1}{2}{w}_2^2({\mu }^1,\mu )\mathrm{.}$

Proposition 3.14

For each couple μ ², μ ³ ∈ $\mathcal{P}$ ₂(ℝ ^d ) and each transfer plan μ ²³ ∈ Γ(μ ², μ ³) we have

(3.27) $\begin{array}{c}\begin{array}{c}{W}_2^2({\mu }^1,{\mu }_t^{2\to 3})\\ \ge (1-t)W_2^2({\mu }^1,{\mu }^2)+t{W}_2^2({\mu }^1,{\mu }^3)\end{array}\\ -t(1-t){\int }_{{ℝ}^d\times {ℝ}^d}{|x_2-x_3|}^2\text{d}{\mu }^{23}(x_2,x_3)\forall t\in [0,1]\mathrm{.}\end{array}$

In particular the map $\varphi :\mu \mapsto -\frac{1}{2}{W}_2^2({\mu }^1,\mu )$ is (−1)-convex along geodesics.

Proof. For μ ²³ ∈ Γ (μ ², μ ³), we can find (see Proposition 7.3.1 of [9]) μ ∈ $\mathcal{P}$ (ℝ ^d × ℝ ^d × ℝ ^d ) whose projection on the second and third variable is μ ²³ and such that

(3.28) ${({\pi }^1,(1-t){\pi }^2+t{\pi }^3)}_{\#}\mathbf{\mu }\in {\mathrm{\Gamma }}_{\text{o}}({\mu }^1,{\mu }_t^{2\to 3}),$

with ${\mu }_t^{2\to 3}:={\left((1-t){\sigma }^2+t{\pi }^3\right)}_{\#}{\mathbf{\mu }}^{23}$ . Therefore

$\begin{array}{c}\begin{array}{c}\begin{array}{c}{W}_2^2({\mu }^1,{\mu }_t^{2\to 3})\\ ={\int }_{{ℝ}^{3d}}{\left|\right(1-t)x_2+t{x}_3-x_1|}^2\text{d}\text{d}\mu (x_1,x_2,x_3)\end{array}\\ ={\int }_{{ℝ}^{3d}}((1-t){|x_2-x_1|}^2+t{|x_3-x_1|}^2-t(1-t){|x_2-x_3|}^2)\text{d}\mu (x_1,x_2,x_3)\end{array}\\ \begin{array}{c}\ge (1-t)W_2^2({\mu }^1,{\mu }^2)+t{W}_2^2({\mu }^1,{\mu }^3)\\ -t(1-t){\int }_{{ℝ}^{2d}}{|x_2-x_3|}^2\text{d}{\mathbf{\mu }}^{23}(x_2,x_3)\mathrm{.}\end{array}\end{array}$

In particular, choosing optimal plans in (3.27), we obtain the semiconcavity inequality of the Wasserstein distance from a fixed measure μ ³ along the constant speed geodesics ${\mu }_t^{1\to 2}$ connecting μ ¹ to μ ²:

(3.29) $\begin{array}{c}{W}_2^2({\mu }_t^{1\to 2},{\mu }^3)\\ \ge (1-t)W_2^2({\mu }^1,{\mu }^3)+t{W}_2^2({\mu }^2,{\mu }^3)-t(1-t)W_2^2({\mu }^1,{\mu }^2)\mathrm{.}\end{array}$

According to Aleksandrov's metric notion of curvature (see [5,58]), this inequality can be interpreted by saying that the Wasserstein space is a positively curved metric space (in short, a PC-space). This was already pointed out by a formal computation in [74], showing also that generically the inequality is strict. An example where strict inequality occurs can be obtained as follows: let d = 2 and

$\underset{\begin{array}{c}{\mu }^3:=\frac{1}{2}({\delta }_{(0,0)}+{\delta }_{(0,-4)})\mathrm{.}\end{array}}{\begin{array}{c}{\mu }^1:=\frac{1}{2}({\delta }_{(1,1)}+{\delta }_{(5,3)}),\end{array}}\begin{array}{c}{\mu }^2:=\frac{1}{2}({\delta }_{(-1,1)}+{\delta }_{(-5,3)}),\end{array}$

then, it is immediate to check that $W_2^2({\mu }^1,{\mu }^2)=40,W_2^2({\mu }^1,{\mu }^3)=30$ and $W_2^2({\mu }^2,{\mu }^3)=30$ . On the other hand, the unique constant speed geodesic joining μ ¹ to μ ² is given by

${\mu }^1:=\frac{1}{2}({\delta }_{(1-6t,1+2t)}+{\mu }^1:=\frac{1}{2}{\delta }_{(5-6t,3-2t)})$

and a simple computation gives

$24=w_2^2\left({\mu }_{1/2,}{\mu }^3\right)>\frac{30}{2}+\frac{30}{2}-\frac{40}{4}\mathrm{.}$

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Basic Concepts and Results

Juan Ferrera , in An Introduction to Nonsmooth Analysis, 2014

1.2 Semicontinuity

Definition 1.8

We start by recalling a well-known definition. We say that a real function $f:X\to \mathbf{R}$ is continuous at $x\in X$ provided that for every $\epsilon >0$ there is a $\delta >0$ such that $\mid f(x)-f(y)\mid <\epsilon$ whenever $\mid x-y\mid <\delta$ . If this property holds for every $x\in X$ , we say that $f$ is continuous on $X$ .

It is easy to see that a function $f$ is continuous at $x$ if and only if for every sequence $\{x_n\}$ converging to $x$ , $\{f(x_n)\}$ converges to $f(x)$ .

Continuous functions have many good properties; they attain maxima and minima over compacts for instance. Many of these properties hold under weaker continuity conditions. The aim of this section is to introduce such conditions.

Definition 1.9

We say that a real function $f:X\to \mathbf{R}$ is lower semicontinuous ( $\mathit{lsc}$ ), respectively upper semicontinuous ( $\mathit{usc}$ ), at $x_0$ , provided that $f(x_0)\le \mathrm{lim\; inf}f(x_n)$ , respectively $f(x_0)\ge \mathrm{lim\; sup}f(x_n)$ , for every sequence $\{x_n\}\subset X$ satisfying ${\lim }_n{x}_n=x_0$ . If the property holds for every point $x_0\in X$ we say that $f$ is $\mathit{lsc}\text{,}\mathit{usc}$ respectively, on $X$ .

It is not difficult to see that lower semicontinuity at $x_0$ is equivalent to

$f(x_0)\le \underset{x\to x_0}{\mathrm{lim\; inf}}f(x)$

while upper semicontinuity is characterized by

$f(x_0)\ge \underset{x\to x_0}{\mathrm{lim\; sup}}f(x)$

The following proposition gives us a $\epsilon$ - $\delta$ characterization of lower semicontinuity.

Proposition 1.10

$f$ is $\mathit{lsc}$ at $x_0$ if and only if for every $\epsilon >0$ there is a $\delta >0$ such that $f(x_0)-f(x)<\epsilon$ whenever $\mid x-x_0\mid <\delta$ . A similar result holds for $\mathit{usc}$ functions

Proof

Assume that $f$ is $\mathit{lsc}$ at $x_0$ . Let $\alpha ={\mathrm{lim\; inf}}_{x\to x_0}f(x)$ then for every $\epsilon >0$ there is a $\delta >0$ such that $\mid \alpha -\inf \{f(x):x\in B(x_0\text{,}\delta )\}\mid <\epsilon$ . For every $x\in B(x_0\text{,}\delta )$ we have

$f(x_0)-f(x)\le \alpha -f(x)\le \alpha -\inf \{f(x):x\in B(x_0\text{,}\delta )\}<\epsilon$

Conversely, let $\epsilon >0$ , choose a positive ${\delta }_0$ such that $f(x_0)-f(x)<\frac{\epsilon }{2}$ for every $x\in B(x_0\text{,}{\delta }_0)$ . We take a point $\tilde{x}\in B(x_0\text{,}{\delta }_0)$ such that

$f(\tilde{x})<\inf \{f(x):x\in B(x_0\text{,}{\delta }_0)\}+\frac{\epsilon }{2}$

we have

$f(x_0)-\inf \{f(x):x\in B(x_0\text{,}\delta )\}<f(x_0)-f(\tilde{x})+\frac{\epsilon }{2}<\epsilon$

for every positive $\delta \le {\delta }_0$ . Let us observe also that the left-hand side of the inequality is positive, hence $\mid f(x_0)-\inf \{f(x):x\in B(x_0\text{,}\delta )\}\mid <\epsilon$ for every positive $\delta \le {\delta }_0$ . We have proved that

$f(x_0)=\underset{x\to x_0}{\mathrm{lim\; inf}}f(x)\square$

The following question arises: have we proved more than lower semicontinuity? Of course not, since we have defined lower limits in such a way that the inequality ${\mathrm{lim\; inf}}_{x\to x_0}f(x)\le f(x_0)$ always holds. The following result is a mere exercise.

Proposition 1.11

$f$ is continuous if and only if it is both $\mathit{lsc}$ and $\mathit{usc}$ .

The following proposition summarizes some elementary properties of semicontinuous functions. We suggest the reader work through the details of the proof.

Proposition 1.12

Let $f\text{,}g:X\to \mathbf{R}\text{,}\lambda >0$ . We have

(i)

$f$ is $\mathit{lsc}$ if and only if $-f$ is $\mathit{usc}$

(ii)

If $f\text{,}g$ are $\mathit{lsc}$ , respectively $\mathit{usc}$ , then $f+g$ is $\mathit{lsc}$ , respectively $\mathit{usc}$ .

(iii)

If $f$ is $\mathit{lsc}$ , respectively $\mathit{usc}$ , then $\lambda f$ is also $\mathit{lsc}$ , respectively $\mathit{usc}$ .

One of the goals of this book is to establish the involved ideas in a geometrical form as well as an analytical one. For this we introduce two sets associated to a given function, which will allow us to present the analytical properties that we have just defined as geometric properties.

Definition 1.13

For a given function $f:X\to \mathbf{R}$ , we define its epigraph, respectively hypograph,as the following set: $\mathit{epif}=\{(x\text{,}r)\in X\times \mathbf{R}:f(x)\le r\}$ , respectively $\mathit{hypof}=\{(x\text{,}r)\in X\times \mathbf{R}:f(x)\ge r\}$ .

Despite its simplicity the next result remarks the relation between analytic and geometrical properties.

Proposition 1.14

Let $f:X\to \mathbf{R}$ be a function, we have that $f$ is $\mathit{lsc}$ if and only if $\mathit{epif}$ is a closed set, similarly $f$ is $\mathit{usc}$ if and only if $\mathit{hypof}$ is closed (see Fig. 1.1 ).

Proof

Assume that $f$ is $\mathit{lsc}$ , let $\{(x_n\text{,}{r}_n)\}\subset \mathit{epif}$ be a sequence converging to $(x\text{,}r)$ , $f(x_n)\le r_n$ implies that $\mathrm{lim\; inf}f(x_n)\le r$ , hence $f(x)\le r$ since $f$ is $\mathit{lsc}$ , and consequently $(x\text{,}r)\in \mathit{epif}$ .

Conversely we assume that $\mathit{epif}$ is closed. Fix a point $x_0$ , given a sequence $\{x_n\}$ converging to $x_0$ , such that ${\lim }_{n}f(x_n)={\mathrm{lim\; inf}}_{x\to x_0}f(x)$ , we have that $(x_n\text{,}f(x_n))\in \mathit{epif}$ and consequently $(x\text{,}{\mathrm{lim\; inf}}_{x\to x_0}f(x))$ belongs to $\mathit{epif}$ too since it is closed. This implies that $f(x)\le {\mathrm{lim\; inf}}_{x\to x_0}f(x)$ , in other words: $f$ is $\mathit{lsc}$ at $x_0$ . The proof for $\mathit{usc}$ is similar. $\square$

Figure 1.1. Graphs of semicontinuous functions.

As a consequence of the preceding results, a continuous function satisfies that its graph, which can be represented as $\mathit{epif}\cap \mathit{hypof}$ , is closed. The converse is not true, consider for instance the function $f:\mathbf{R}\to \mathbf{R}$ defined as $f(x)=\frac{1}{x}$ if $x>0\text{,}f(x)=0$ otherwise. Its graph and epigraph are closed, but $\mathit{hypof}$ does not have this property.

An easy and important consequence of the preceding proposition is the stability of semicontinuous functions under some operations. For instance:

Corollary 1.15

Let ${\{f_i\}}_{i\in I}$ be a family of $\mathit{lsc}$ , respectively $\mathit{usc}$ functions. Then ${\sup }_{i\in I}{f}_i$ is $\mathit{lsc}$ , respectively ${\inf }_{i\in I}{f}_i$ is $\mathit{usc}$ .

Proof

The result is a consequence of the fact that

$\mathit{epi}({\displaystyle \underset{i\in I}{\sup }}{f}_i)={\displaystyle \bigcap _{i\in I}}{\mathit{epif}}_i\text{and}\mathit{hypo}({\displaystyle \underset{i\in I}{\inf }}{f}_i)={\displaystyle \bigcap _{i\in I}}{\mathit{hypof}}_i\square$

These results are no longer true for continuous functions.The $\sup$ of the following family of continuous functions is not continuous.

Example

$f_n:\mathbf{R}\to \mathbf{R}$ defined by $0$ if $x\le 0\text{,}1$ if $x\ge \frac{1}{n}$ , and $\mathit{nx}$ if $x\in \left[0\text{,}\frac{1}{n}\right]$ (see Fig. 1.2).

Figure 1.2. Continuity is not preserved by sup.

A well-known result establishes that every continuous function attains its minimum in any compact set. The next result extends this property.

Proposition 1.16

Let $K$ be a compact subset of $X$ . Every $\mathit{lsc}$ function $f:X\to \mathbf{R}$ attains its minimum with respect to $K$ .

Proof

Let $r_0=\inf \{f(x):x\in K\}$ . If $r_0=-\infty$ , there would be a sequence $\{x_n\}\subset K$ such that $\lim f(x_n)=-\infty$ . We may assume without loss of generality that $\{x_n\}$ converges to $x_0\in K$ by compactness. Lower semicontinuity of $f$ would imply that $f(x_0)\le -\infty$ , which is not possible. Hence $r_0\in \mathbf{R}$ , and we take a sequence $\{x_n\}\subset K$ , that we may consider convergent to a $x_0\in K$ again, such that $\lim f(x_n)=r_0$ . Finally, $f(x_0)\le \lim f(x_n)=r_0$ implies $f(x_0)=r_0$ , hence $x_0$ is a minimum. $\square$

Sometimes we will consider functions defined on a subset instead of on the whole space $X$ . The set where the function is defined is called domain of $f$ , and we will denote it by $\mathit{domf}$ . In this case, continuity concepts are defined as above, imposing that all the arguments lie in $\mathit{domf}$ .

We will sometimes allow our functions to reach the value $+\infty$ , in other words: $f:X\to (-\infty \text{,}+\infty ]$ . Many definitions make sense in this new context, since we can calculate limits and do algebraic operations (except for $-(+\infty )$ !). Accepting this new value, we may extend functions defined in a subset of $X$ to the whole space, assuming that the value of the function is $+\infty$ out of its domain. Some properties of the function are stable under this kind of extension, the minimum value for instance. The reason for choosing the value $+\infty$ instead of $-\infty$ is precisely because we are going to deal with minima, and $\mathit{lsc}$ functions. Replacing function $f$ by $-f$ , minima are maxima, lower semicontinuity transforms into upper semicontinuity, and we extend the function to $-\infty$ . Both derivations are trivially equivalent. However we will not admit the function $f(x)\equiv +\infty$ . Even though it is probably clear how to define $\mathit{lsc}$ functions for functions taking values in $(-\infty \text{,}+\infty ]$ , we still write it.

Definition 1.17

We say that a function $f:X\to (-\infty \text{,}+\infty ]$ is $\mathit{lsc}$ at a point $x_0\in X$ if $f(x_0)\le {\mathrm{lim\; inf}}_{x\to x_0}f(x)$ .

Let us observe that lower limits are defined exactly as in Definition 1.5 if the range of the function is $(-\infty \text{,}+\infty ]$ . If $x_0\notin \mathit{domf}$ , then $\mathit{lsc}$ implies that ${\mathrm{lim\; inf}}_{x\to x_0}f(x)$ , and consequently ${\lim }_{x\to x_0}f(x)$ are $+\infty$ . Upper semicontinuity is defined for functions $f:X\to [-\infty \text{,}+\infty )$ in a similar way.

An extremely useful function in this setting is the $\mathit{indicator}$ function ${\delta }_A$ of a set $A$ , defined by

${\delta }_A(x)=0\text{if}x\in A\text{,}{\delta }_A(x)=+\infty \text{if}x\notin A$

It is left as an exercise for the reader to see that ${\delta }_A$ is $\mathit{lsc}$ if and only if $A$ is closed.

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Handbook of Dynamical Systems

G. Raugel , in Handbook of Dynamical Systems, 2002

Definition 3.5

Let ${\lambda }_0\in \overline{L}$ , where L is a subset of Λ. The ω-limit set ${\tilde{\omega }}_{\text{L}}\left(A.,{\lambda }_0\right)$ of the family of sets $A_{\lambda },\lambda \in N_{\Lambda }({\lambda }_0,\eta )\cap L,\eta >0$ , is defined by

(3.11) ${\tilde{\omega }}_L(A.,{\lambda }_0)=\bigcap _{0<\delta <\eta }C{l}_X\bigcup _{\lambda \in N_{\Lambda }({\lambda }_0.\delta )\cap L}{A}_{\lambda }.$

Several properties of the set ${\tilde{\omega }}_L\left(A.,{\lambda }_0\right)$ are given in [111]. If ${\lambda }_0\in L$ and, for each $\lambda \in N_{\Lambda }\left({\lambda }_0,\eta \right)\cap L,S_{\lambda }$ has a compact global attractor, then the upper semicontinuity (respec tively the lower semicontinuity) of the attractors at λ = λ ₀ implies that ${\tilde{\omega }}_L\left(A.,{\lambda }_0\right)\subset A_{{\lambda }_0}$ (respectively $A_{{\lambda }_0}\subset {\tilde{\omega }}_L\left(A.,{\lambda }_0\right))$ . If ${\text{Cl}}_X\left({\cup }_{\lambda \in L\cap N_{\Lambda }({\lambda }_0,\eta )}{A}_{\lambda }\right)$ is compact, it follows from the inclusion ${\tilde{\omega }}_L\left(A.,{\lambda }_0\right)\subset A_{{\lambda }_0}$ that the attractors are upper semicontinuous at λ₀. We remark that the inclusion $A_{{\lambda }_0}\subset {\tilde{\omega }}_L\left(A.,{\lambda }_0\right)$ does not imply lower semicontinuity of the attractors A _λ. Indeed, consider the ODE $\dot{x}=-x\left({\left(-1\right)}^n{\lambda }_n+{\left(x-1\right)}^2\right)$ with ${\lambda }_n=1/n$ , that is $L=\{1,1/2,\dots ,1/n,\dots \}$ . There is no continuity of the attractors at Λ = 0; however, ${\tilde{\omega }}_L\left(A.,0\right)=A_0=\left[\mathrm{0,1}\right]$ .

We notice that ${\tilde{\omega }}_L\left(A.,{\lambda }_0\right)$ does not involve directly the semigroup $S_{{\lambda }_0}$ . In particular, $S_{{\lambda }_0}$ could be conservative.

The following question then arises: how much information can we obtain about a conservative system by considering the limit of dissipative systems, when the dissipation goes to zero? We cannot hope to obtain too many specific properties of the dynamics of the limit system in this way, but one should be able to obtain some information about the manner in which the orbits of the dissipative systems wander over the level sets of the energy of the limit system.

Consider the ODE $\dot{u}=v,\dot{v}=f(u)-\beta v$ , where $\beta ⩾0$ is a constant, $f\in {\mathcal{C}}^2\left(ℝ,ℝ\right)$ has only simple zeros and f (u) is dissipative (i.e., lim $lim{sup}_{\left|u\right|\to +\infty }\frac{f\left(u\right)}{u}⩽\alpha <0)$ . The energy functional is $\Phi \left(u,v\right)=(1/2)v^2-{\int }_0^{u}f(s)\text{d}s$ . For $\beta >0$ , the ODE is a gradient system and has a global attractor $A_{\beta }$ . Let $\left\{s_j,j=1,2,\dots ,M\right\}$ be the set of the saddle equilibrium points of the system. If $\Phi \left(s_j\right)\ne \Phi \left(s_k\right)$ , for $j\ne k,j,k=1,2,\dots ,M$ , then, for any interval $(0,{\beta }_0]$ ,

${\tilde{\omega }}_L(A.,0)=\left\{(u,\upsilon )\in {ℝ}^2|\Phi (u,\upsilon )⩽c_M\right\},$

, where $c_M=max\left\{\Phi \left(s_j\right)j=1,2,\dots ,M\right\}$ (for details, see [111]).

The limit ${\tilde{\omega }}_L\left(\mathcal{A}.,{\lambda }_0\right)$ only uses information about the attractors. As a consequence, the transient behaviour of the semigroups S _Λ for initial data not on the attractors is completely ignored. To gain some information about this transient behaviour, one can consider the following concept of ω-limit set:

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