Energy Spaces, Dirichlet Forms and Capacities in a Nonlinear Setting

In this article we study lower semicontinuous, convex functionals on real Hilbert spaces. In the first part of the article we construct a Banach space that serves as the energy space for such functionals. In the second part we study nonlinear Dirichlet forms, as defined by Cipriani and Grillo, and show, as it is well known in the bilinear case, that the energy space of such forms is a lattice. We define a capacity and introduce the notion quasicontinuity associated with these forms and prove several results, which are well known in the bilinear case.


Introduction
The theory of gradients and subgradients of convex functions on a Hilbert space, as presented in [Roc81,Lio69] or [Bre73], can be seen as a nonlinear counterpart to the theory of symmetric bilinear forms.
In the setting of Lions, one is given a Gelfand triple V ֒→ H ֒→ V ′ and a convex, differentiable function E : V → R. Then, the gradient is a nonlinear operator from V to V ′ . In this situation, the Banach space V is sometimes called the energy space.
In the setting of Brezis, where E : H → [0, ∞] is merely a convex, lower semicontinuous function on the Hilbert space H, such an energy space is not explicitly given. One goal of this article is to show that nevertheless a Banach space serving as an energy space can be constructed naturally, and therefore to partly unify the approaches of Lions and Brezis. In [CM18] the authors equipped the effective domain of the functional with a metric but here we are interested in a linear space.
In the case of quadratic forms, this energy space is the domain of the bilinear form equipped with the usual Hilbert space structure, and in the general case the energy space is a Banach space. For example the energy space of the p-Laplace operator on an open domain X ⊂ R n with Neumann boundary conditions is the Sobolev space W 1 p,2 (X) = {u ∈ L 2 (X, m) | ∇u ∈ L p (X, m)}.
The second goal of this article is to study these energy spaces, when the functional is a nonlinear Dirichlet form. These forms were introduced by Cipriani and Grillo in [CG03] as convex, lower semicontinuous functionals on an L 2 space which generate order preserving and L ∞ contractive semigroups of nonlinear operators. Furthermore, they showed, using earlier results form Barthelemy [Bar96] as well as Bénilan and Picard [BC91], that this definition is equivalent to the intrinsic Definition 4.1 which we use in this paper.
In the theory of bilinear Dirichlet forms the Dirichlet spaces and the capacity are important building blocks of the vast theory of these forms and the semigroups as well as the Markov processes they generate [FOT11,BH91,MR92]. We show that the Dirichlet space of a nonlinear Dirichlet form is a lattice and, under some assumptions, the lattice operation are continuous. We define a capacity and quasicontinuous functions and show that many of the results from the bilinear world still can be transferred to our setting.
In a forthcoming paper we want to use the results presented here to investigate boundary conditions and perturbations of Dirichlet forms.
We point out that in the article [Bir04,BV05] a capacity is defined in the nonlinear setting, too. The assumption (H0) in both articles assumes that the Dirichlet space exists and that the functional is p-homogeneous. It turns out that both assumptions are not necessary to define a capacity and we give an explicit way to construct the Dirichlet space. In addition, non homogeneous examples like the energy of the ∞-Laplacian are also covered in our approach. Other examples include the p or p(x)-Laplacian on subsets of R d or arbitrary Riemannian manifolds, fractional versions of these operators and sums thereof.

Energy Spaces of Symmetric Functionals
In the following H is a real Hilbert space and E : H → [0, ∞] always denotes a convex and lower semicontinuous functional. We define the effective domain Remark 2.1. For a convex functional E : H → [0, ∞], the condition E(−x) = E(x) already implies that 0 is a global minimizer of E. In the study of lower semicontinuous functionals in the context of partial differential equations and calculus of variations we are mostly interested in the minimizers and not the minimum itself. So without loss of generality we can assume that E(0) = 0.
We now construct a Banach space associated with the convex, lower semicontinuous functional E. This construction in based on the ideas of modular spaces used in the construction of Musielak-Orlicz space and variable exponent Lebesgue space. For reference of this procedure see [Mus83] or [DHHR11]. Let and · D : H → [0, ∞] be defined as We define the energy space of E by Proof. Observe that for all x ∈ H we have Since . H is a norm, this yields x D = 0 if and only if x = 0.
To show homogeneity, let µ > 0. Then, Homogeneity for µ < 0 follows, since For the triangle inequality, let x, y ∈ D and a > x D , b > y D . Then taking the infimum over all possible a and b yields Thus · D is a norm. Since we already showed that we know that the embedding ι : D → H is continuous.
It is easy to see, that E 1 (x) ≤ 1 if and only if x D ≤ 1. Thus, Since E 1 is lower semicontinous, the set is also closed. Therefore · D is lower semicontinuous on H.
Let (x n ) n be a Cauchy sequence in D. Then (x n ) n is a Cauchy sequence in H. Since H is complete, there is an x ∈ H with x n → H x and, since the norm · D is lower semicontinuous on L 2 (X, m) and Cauchy sequences are bounded, Hence x ∈ D. Let ǫ > 0. We choose N ∈ N such that for every n, m ≥ N Thus, the lower semicontinuity of the norm yields for every n ≥ N . Therefore, x n → D x. Thus, D is complete.
Let α > 0, Similar arguments as in the first part of the previous proof show, that · D,α are norms for every α > 0 and | · | D is a seminorm.
Proof. At first, we can use exactly the same proof as for · D to show that D together with · D,α is a Banach space. Let 0 < α 1 < α 2 . Then for every x ∈ D, since x D,α2 takes the infimum over a larger set. The open mapping theorem implies that comparable complete norms are equivalent. Thus, there is a C > 0 such that for every x ∈ D.
For the second part, let x ∈ D. Obviously Hence On the other hand, let λ 1 , λ 2 such that x H ≤ λ 1 and |x| D ≤ λ 2 . Then Hence, Thus, Taking the infimum over all possible λ 1 and λ 2 yields Since x D,2 and x D are equivalent, there is a C > 0 such that Hence all the norms are equivalent.
Theorem 2.4. Let E be a symmetric, convex and lower semicontinuous functional. Then D is a dual space.
Proof. Kaijser proved in [Kai77], that a Banach space Y is a dual space if there is a set of continuous linear functionals E on Y that separates the points of Y and the closed unit ball of Y is compact in the weak topology generated by E.
Let E = H ′ = H. By the Hahn-Banach theorem, E separates the points of H. Hence, it separates the points of D. On the other hand the closed unit ball B 1 of D is closed and bounded in H. Therefore, it is compact in the weak* topology by Banach-Alaoglu. But weak and weak* topology coincide on reflexive spaces. Hence, B 1 is compact in the weak topology induced by E. Thus D is a dual space.
Note that Kaijser also showed that the predual is then given by the weak* closure of span E in D ′ .
Proof. For both theorems we omit the proof, see [DHHR11, Lemma 2.1.9, 2.1.14 and Corollary 2.1.15] instead. Note that the convexity and lower semicontinuity of E and E 1 imply that E 1 is left continuous and therefore a modular in the notation of this reference.

Energy Spaces of non Symmetric Functionals
Definition 3.1. Let E : H → [0, ∞] be a lower semicontinuous, convex functional on a real Hilbert space H, such that E(0) = 0. We define the symmetric Moreover, any f ∈ dom(sym E) can be written as Proof. We need to show that dom(sym E) ⊂ span dom(E). For this purpose, let Since sym E is the lower semicontinuous closure of After choosing a subsequence, we may assume that λ n → λ ∈ [0, 1]. Now, let us assume λ = 0. Since r n is positive, Hence, (s n ) n∈⋉ is bounded since λ = 0. Again, after choosing a subsequence if necessary, there is a s ∈ R such that s n → s. By the definition of the epigraph, we know Hence, (u n ) n∈N is bounded in H. Thus there is a subsequence, again denoted by (u n ) n∈N , such that u n ⇀ u. Since E is lower semicontinuous with respect to the weak topology on H, u ∈ dom E. On the other hand, If λ = 0 then 1 − λ = 0 and switching r and s as well as u and v in the previous argument yields the conclusion.
be a lower semicontinuous, convex functional on a Hilbert H, such that E(0) = 0. We call the space D associated with sym E the energy space of E.
Remark 3.4. If E is already symmetric, then sym E = E 1 and both definitions of D coincide.
Remark 3.5. In this article we always start with a Hilbert space H. For the results up to this point this is in fact not necessary. Similar results hold in the case of reflexive Banach spaces like for example L p for 1 < p < ∞. It might be more natural in some applications, for example if E itself is p-homogeneous, to define E 1 (·) = · p L p + E(·), which leads again to a p-homogeneous functional.
Remark 3.6. If 0 is not a global minimizer of E, but if nevertheless E possesses a global minimizer, then we can shift the functional E and still define an energy space. This shifted functional is still a Dirichlet form if E is a Dirichlet form. But one looses the property dom E ⊂ D if one defines D by using the shifted functional.
Remark 3.7. We also assume that E is convex. But in fact we only need that E 1 is convex, or more general that is convex for some α > 0. Then E α generates a energy space and the norms are equivalent for any α such that E α is convex. To prove this, let α < β such that E α and E β are convex. Then E α resp. E β generate an energy space D α resp. D β . By Theorem 2.3, the norms · Dα and · D β are equivalent to · H + | · | Dα and · H + | · | D β . Obviously |x| Dα ≤ |x| D β by definition. On the other hand, Hence, the norms on D α and D β are equivalent. Thus the energy space does not depend on the choice of α and we can define an energy space for any functional E such that is convex. Such functionals E are called ω-semiconvex, where ω = 1 α . Taking the previous remarks into account we can define an energy space for any lower semicontinuous, semiconvex functional E on any reflexive Banach space.

Dirichlet Forms and Dirichlet Spaces
Let X be a countably generated Borel space and m a σ-finite Borel measure such that supp(m) = X. The following definition was first introduced by [CG03]. and for every u, v ∈ L 2 (X, m), α > 0.
If E(0) = 0 we call the energy space D of E the Dirichlet space.
Remark 4.2. It is well known, that a convex, lower semicontinuous functional with dense domain generates a semigroup of nonlinear contractions. In the theory of bilinear forms, Dirichlet forms are exactly those forms, which generate sub-Markovian semigroups, that is order-preserving and L ∞ -contractive semigroups. The two conditions in the previous definition hold if and only if the semigroup generated by E is sub-Markovian. Definition 4.4. We call p ∈ W 1,∞ (R) a normal contraction if 0 ≤ p ′ ≤ 1 and p(0) = 0.
Remark 4.5. Note that any such normal contraction p induces an operator We do not distinguish between p and T p and denote both objects by p.
for every u, v ∈ L 2 (X, m) and every normal contraction p. Theorem 4.7. Let E be a Dirichlet form on L 2 (X, m) such that E(0) = 0. Then the space D is a Riesz subspace of L 2 (X, m). That is, for u, v ∈ D we have Proof. Let g ∈ D. Since there is a λ ≥ 0 and a function f ∈ dom(sym E) such that g = λf . We showed in Theorem 3.2, that there are u, v ∈ dom(E) such that f = u − v. Hence, Since u ∨ v, v ∈ dom(E), g ∨ 0 ∈ span dom(E) = D. The same argument works for g ∧ 0. This implies the claim, since x ∧ y = (x − y) ∧ 0 + y.
Theorem 4.8. Let E be a symmetric Dirichlet form on L 2 (X, m). Then Proof. Let us assume the contrary. Then, there are u, v ∈ D such that Note that, either u or v is non zero. Therefore, we can choose a λ ∈ 1 u∧v D , 1 u D + v D . This implies λ u ∧ v D > 1 and λu D , λv D < 1. Hence, by Theorem 2.6, Since E 1 is a Dirichlet form, the previous inequality is a contradiction.
One could hope that the following even stronger inequality holds But this is false, as the next example illustrates.
The functional E is lower semicontinuous, by Fatou's Lemma. Additionally, it is convex, since χ [0,1] is convex and it is easy to verify that E satisfies the projection inequalities of Definition 4.1. Hence, it is a symmetric Dirichlet form. The norm · D is induced by the functional By definition, the Dirichlet space D is given by all elements u ∈ L 2 (X, m) such that there is a λ > 0 satisfying which is equivalent to almost everywhere, or in other words |u(x)| ≤ λ for almost all x ∈ X Thus, u ∈ D if and only if u ∈ L ∞ (X, m). Analogously the norm is given by Since ([0, 1], m) is a probability space, the norm · D is equal to · ∞ . Now let Thus, Therefore, the inequality does not hold for the energy norm of a Dirichlet form in general.
Theorem 4.10. Let E be a quasilinear, symmetric Dirichlet form and v ∈ D.
Then the lattice operations are continuous.
Proof. At first, let v ∈ D such that v ≥ 0, let (u n ) n∈N be a sequence in D converging to 0 and λ > 0. Then u n ∧ v → 0 and u n ∨ v → v in L 2 (X, m) for n → ∞. Hence, the lower semicontinuity of E implies where we used lower semicontinuity and the first inequality from Definition 4.1.
Since E 1 is quasilinear, D = dom E 1 , and since E 1 is convex and lower semicontinuous on D, it is continuous on D. Therefore, since E 1 (0) = 0 the previous inequality yields We already know, that Thus, lim n→∞ E 1 λ(u n ∧ v) = 0 and Theorem 2.5 implies u n ∧ v → 0 in D. In We now use the first part again, which yields Finally, let (u n ) n be an arbitrary sequence in D converging to u. Then u n − u converges to 0 and Hence, the infimum is separately continuous. Now, let (u n ) n∈N and (v n ) n∈N in D converging to u and v. Then Hence, The claim for the supremum follows by using the identity Lemma 4.11. If E is a symmetric Dirichlet form, then Proof. The map p : R → R given by p(x) = (−n)∨x∧n is a normal contraction. Hence, Theorem 4.6 implies Setting v = 0 implies since E 1 (0) = 0 and E 1 ≥ 0.
For the second part, let us assume that E is quasilinear. By Property (2) in the definition of Dirichlet forms, we have for every u, v ∈ L 2 (X, m), α > 0. Plugging in v = 0 and replaying α by n yields Let λ > 0. Since lim n→∞ −n ∨ u ∧ n = u in L 2 (X, m), the lower semicontinuity of E 1 and the previous inequality imply Since E 1 is lower semicontinuous, Since E 1 is quasilinear, E 1 (λu) is finite and Theorem 2.5 implies 1 2 u − (−n) ∨ u ∧ n → 0 for n → ∞ in D. Since u was arbitrary, this yields (−n) ∨ u ∧ n → u in D for every u ∈ D and n → ∞.  Let us show that E is lower semicontinuous. Let (u n ) n∈N be a sequence in L 2 (X, m) converging to u such that E(u n ) = 0 for every n ∈ N. Then, u ′ n L ∞ ≤ 1 and there exists v ∈ L ∞ (X), v ∞ ≤ 1 such that, up to a subsequence, u ′ n → v in the weak* topology of L ∞ (X). For every φ ∈ C ∞ c (X) Passing to the limit yields Therefore u ′ = v and E(u) = 0.
It is easy to see that E is a Dirichlet form. The Dirichlet space D coincides with W 1,∞ (X). Let f (x) = x and g n (x) = 1 n . Then g n → 0 in D but f ∧ g n D ≥ 1 for every n ∈ N. Hence, f ∧ g n does not converge to 0 in D.

Capacity
One tool in the study of classical Dirichlet forms is the capacity. For a reference in the classical case, see for example [FOT11]. From now on let us assume that X is a topological measure space and supp m = X.
In this section E denotes a symmetric Dirichlet form and (D, . D ) the associated Dirichlet space.
Definition 5.1. Let A ⊂ X. We define the set Proof. For the first part, let us consider two arbitrary functions f A ∈ L A , f B ∈ L B . Then Since f A , f B are arbitrary, this implies the first claim. The second part follows directly from the definition and properties of the infimum.
Proof. Let A n , A ⊂ X as above.
Let us assume ∞ n=1 Cap D (A n ) < ∞. Otherwise there is nothing to show. Let ǫ > 0. By the properties of the infimum, we can choose functions u n ∈ L 2 (X, m) and open sets U n for every n ∈ N such that u n ≥ 1 on U n and A n ⊂ U n u n D ≤ Cap D (A n ) + ǫ2 −n .
Without loss of generality we assume u n ≤ 1, otherwise take u n ∧ 1. Set By Theorem 4.8, Since · L 2 (X,m) ≤ · D , the sequence g n is bounded in L 2 . Thus, there exists a weakly convergent subsequence, again denoted by g n , such that g n ⇀ g for some g ∈ L 2 (X, m). For every k ∈ N we have g n ⇀ g in L 2 ( k j=1 U j , m), but g i = 1 in L 2 ( k j=1 U j , m) for every i > k. Hence, g = 1 on U k for every k ∈ N. Thus, g = 1 on U and, by the weak lower semicontinuity of · D on L 2 (X, m), we have Cap D (A n ) + ǫ.
Since ǫ was arbitrary, this concludes the proof.
Lemma 5.4. Let K n ⊂ X be compact subsets such that K n ↓ K. Then

Proof. By the monotonicity of Cap
For the converse inequality, let us choose an ǫ > 0, an open neighborhood U of K and a function f ∈ L U such that We know For every i ∈ N, K i ⊂ K 1 and U c ∩K 1 ⊂ K 1 . Since K 1 is compact, we can apply the finite intersection property. Hence, there are finitely many K i1 , . . . , K in such that Thus, U is an open neighborhood of K in and, by the definition of the capacity, Since ǫ is arbitrary, the claim follows.
Definition 5.5. We call a set A ⊂ X polar, if Cap D (A) = 0, and some property holds quasi everywhere (q.e.), if it holds up to a polar set.
Example 5.6. Let X ⊂ R n a domain, m the Lebesgue measure and p ≥ 1. Then, is a convex, lower semicontinuous Dirichlet form. The Dirichlet space D = W 1 p,2 (X) = {u ∈ L 2 (X, m) | ∇u ∈ L p } and the capacity of E is the relative p-capacity (or the usual p-capacity if X = R n ). Let α ≥ 2. The perturbation is also a convex, lower semicontinuous Dirichlet form.
Let A ⊂ X be a Cap DE -polar set. Then there is a sequence (u n ) n∈N in D such that u n = 1 on a neighbourhood of A and u n → 0 in D. Hence, u n → 0 in L 2 (X, m). By the Hölder inequality u n → 0 in L α (X). Therefore A is Cap DE α -polar. The converse implication, namely that every Cap DE α -polar set is Cap D E -polar, is clear. Hence, the capacities possess the same polar sets. Note carefully that the corresponding Dirichlet spaces need not be the same.

Quasicontinuity
In this section E denotes a symmetric Dirichlet form. Additionally we call f ∈ L 2 (X, m) quasicontinuous, if there is a representative which is quasicontinuous. Whenever this is the case, we denote this representative again by f .
is open in X by the definition of the induced topology. Thus, Since ǫ was arbitrary we have Cap D (N ) = 0. The other direction follows directly from Lemma 6.2.
Corollary 6.4. Let f 1 , f 2 : X → R be two quasicontinuous representatives of some f ∈ L 2 (X, m). Then f 1 = f 2 quasi everywhere on X. Proof. The proof works exactly like the one in the bilinear case [FOT11].
Theorem 6.7. Let f ∈ D be a quasicontinuous function and λ > 0. Then Proof. Let λ > 0 and ǫ > 0. Since f is quasicontinuous, there is an open set U ǫ with Cap D (U ǫ ) ≤ ǫ and f is continuous on U c ǫ . There is a function g ǫ , such that g ǫ ≥ 1 on U ǫ and g ǫ D ≤ 2ǫ. Additionally, the set O = {|f | > λ} ∪ U ǫ is open. Note that (λ −1 f ) ∨ g ǫ ≥ 1 on O. Hence, Since ǫ is arbitrary, this implies the claim.
Theorem 6.8. Let (f n ) n be a sequence of quasicontinuous functions in D and f ∈ D with f n → f in D. Then f is quasicontinuous and there exists a subsequence which converges pointwise quasi everywhere and quasi uniformly, that is for every ǫ > 0 there is a open set U ⊂ X such that Cap D (U ) ≤ ǫ and f n converges uniformly to f on U c .
Proof. Let us choose a subsequence (f n ) n such that Then, since |f n − f n+1 | is quasicontinuous, Theorem 6.7 yields Hence, for every m ∈ N we have In addition, let ǫ > 0. Since f n is quasicontinuous, we can choose an open set U n such that Cap D (U n ) ≤ ǫ2 −n and f n | U c n is continuous. Hence, Therefore, the sequence (f n | U c ) n∈N of continuous functions converges uniformly on A m = n≥m {|f n − f | ≤ 2 −n }. Thus, f | U c ∩Am is continuous, since it is the uniform limit of continuous functions, and This implies, that f is quasicontinuous. Next, we show that a subsequence of f n converges pointwise quasi everywhere. For this purpose, let us choose a different subsequence such that Let m ∈ N be arbitrary and x ∈ n≥m {|f n − f | ≤ 2 −n }. Then f n (x) → f (x). Corollary 6.9. Let f ∈ D ∩ C(X) D . Then f is quasicontinuous on X.
Proof. Every continuous function is quasicontinuous. Hence, the previous theorem implies the claim.
Remark 6.10. For E = · 2 this theorem is a version of Lusin's Theorem and Egorov's theorem. Furthermore, Theorem 6.7 is a version of the Markov or weak L 1 inequality.