Heat equation on the hypergraph containing vertices with given data

This paper is concerned with the Cauchy problem of a multivalued ordinary differential equation governed by the hypergraph Laplacian, which describes the diffusion of ``heat'' or ``particles'' on the vertices of hypergraph. We consider the case where the heat on several vertices are manipulated internally by the observer, namely, are fixed by some given functions. This situation can be reduced to a nonlinear evolution equation associated with a time-dependent subdifferential operator, whose solvability has been investigated in numerous previous researches. In this paper, however, we give an alternative proof of the solvability in order to avoid some complicated calculations arising from the chain rule for the time-dependent subdifferential. As for results which cannot be assured by the known abstract theory, we also discuss the continuous dependence of solution on the given data and the time-global behavior of solution.


Introduction
Let V = {v 1 , . . ., v n , v n+1 , . . ., v n+m } be a finite set, E ⊂ 2 V be a family of subsets consisting of two or more elements of V , and w : E → (0, ∞).The triplet G = (V, E, w), which is called hypergraph, can be interpreted as a model of a network structure in which vertices v 1 , . . ., v n+m ∈ V are connected by each hyperedge e ∈ E (see Fig. 1).In order to investigate the structure of network represented by a hypergraph, Yoshida [16] introduced a nonlinear set-valued operator on R V (the family of mappings x : V → R) as follows.Define f e (x) := max u,v∈e (x(u) − x(v)) with respect to each e ∈ E and (1) ϕ G,p (x) := 1 p e∈E w(e)(f e (x)) p p ∈ [1, ∞).
These are convex functionals with domains D(f e ) = D(ϕ G,p ) = R V and then subdifferentiable at every x ∈ R V .By the chain rule (see, e.g., [2]) and the maximum rule (see, e.g., [13,Proposition 2.54]), the subdifferential of ϕ G,p coincides with where B e ⊂ R V , called base polytope for e ∈ E, is defined by This operator L G,p = ∂ϕ G,p : R V → 2 R V is called hypergraph (p-)Laplacian, where 1 ≤ p < ∞ (see also [7]).When p = 2 and every e ∈ E consists of two elements of V , then L G,2 becomes a linear operator on R V , i.e., a square matrix of order #V = (n + m) and represents the movement of "particles" on a vertex to another adjacent vertex through an edge e ∈ E Figure 2: We consider the case where the "heat" of some vertices (colored in the above) is given as x n+i (t) = a i (t).
in the random walk model on the graph.By analogy with this, we can regard the ODE x ′ (t) + L G,p (x(t)) ∋ 0 with respect to x : [0, T ] → R V as a diffusion model of "heat" or "particles" on each vertex v i , which is described by x(t)(v i ) = x i (t).In fact, the timeglobal behavior of solution to this ODE (studied in [7]) quite resembles those to the PDE ∂ t u − ∇ • (|∇u| p−2 ∇u) = 0.In information science, this ODE x ′ (t) + L G,p (x(t)) ∋ 0 is used to investigate the eigenvalues of L G,p , prove the Cheeger like inequality, and analyze the cluster structure of network represented by hypergraph (see, e.g., [3,6,12,15]).
Here let the heat on two vertices v i , v j be fixed as x(v i ) = α and x(v j ) = β (α > β).Under this situation, the final state x ∞ = lim t→∞ x(t) of the diffusion process x ′ (t) + L G,p (x(t)) ∋ 0 might describe the thermal gradient in each path and then indicate the optimal path of heat flux from vertex v i to v j .As a generalization of this problem, we deal with the case where the heat at vertices v n+1 , . . ., v n+m are determined by some given functions a j : [0, T ] → R (j = 1, . . ., m), i.e., the heat equation governed by L G,p under the condition (5) x n+j (t) = x(t)(v n+j ) = a j (t) j = 1, . . ., m (see Fig. 2).This can be interpreted as a situation where the observer internally manipulates the heat of the network.As another example, the graph Laplacian (2) and its energy functional (1) on the usual graph can be found in the definition of Laplacian on the fractal (see, e.g., [10,11]).In this case, the assumption a j ≡ 0 corresponds to a homogeneous Dirichlet boundary condition on the fractal.We here state the setting of our problem more precisely.Without loss of generality, we assume the hypergraph G = (V, E, w) is connected, i.e., for every u, v ∈ V there exist some u 1 , . . .u N −1 ∈ V and e 1 , e 2 , . . .e N ∈ E such that u j−1 , u j ∈ e j holds for any j = 1, 2, . . ., N, where u 0 = u and u N = v (if G is disconnected, we only have to divide G into connected components and consider the problem on each component, see [7]).
By letting x i := x(v i ), we can identify R V (the family of mappings from V onto R) with the Euclidean space R n+m ((4) coincides with a unit vector of the canonical basis).In this sense, R V is a Hilbert space endowed with the standard inner product x • y := v∈V x(v)y(v) and the norm x := v∈V x(v) 2 .For x : [0, T ] → R V , we shall write x i (t) := x(t)(v i ) for the same reason.
Let K a (t) ⊂ R V be a constraint set which corresponds to the condition (5) such that the former n components x 1 , . . ., x n are chosen freely and the latter m components x n+1 , . . ., x n+m are fixed by given data a 1 (t), . . ., a m (t).Namely, define Note that K a (t) is a closed convex subset in R V and the projection onto K a (t) can be defined by (7) Proj Ka(t) x := (x 1 , . . ., x n , a 1 (t), . . ., a m (t)), where x = (x 1 , . . ., x n+m ).
Moreover, let We can define the subdifferential of I Ka(t) on K a (t).Here when ψ : R V → (−∞, +∞] is a proper (ψ ≡ +∞) lower semi-continuous and convex function, its subgradient at x ∈ R V is defined by Then the diffusion process on the hypergraph with the condition (5) and the initial state x 0 ∈ R V can be represented by the following Cauchy problem of an evolution equation with constraint condition: x(0) = x 0 , where h : [0, T ] → R V is the given external force.Since ϕ G,p (x) < ∞ holds for every x ∈ R V (i.e., D(ϕ G,p ) = R V ), we have ∂(ϕ G,p + I Ka(t) ) = ∂ϕ G,p + ∂I Ka(t) .Therefore (P) a,h,x 0 can be reduced to a problem of evolution equation governed by a subdifferential operator of time-dependent functional ϕ t := ϕ G,p + I Ka(t) , which has been investigated in numerous papers, e.g., [4,5,8,9,14].In this paper, however, we aim to introduce a simpler method for the constraint problem (P) a,h,x 0 by focusing on features of ∂ϕ G,p and ∂I Ka(t) stated in the next section.We prove the existence of solution to (P) a,h,x 0 in §3.As for a result which cannot be assured by the known abstract theory, we discuss the continuous dependence of solution on the given data a(t) in §4.Finally, we consider the time-global behavior of solution in §5 by using the facts given in previous parts.

Preliminary
In this section, we check and review some basic facts for later use.First, the subgradient of ( 8) can be written specifically as follows: . By the definition of subdifferntial ( 9), 0 ≥ ξ • (z − x) holds for every z ∈ K a (t).When x, z ∈ K a (t), we have This implies that ξ 1 , . . ., ξ n should be equal to zero and ξ n+1 , . . ., ξ n+m can be chosen arbitrarily.
We next check the following Poincaré type inequality by repeating the argument in [7].Here and henceforth, ϕ G,p in (1) will be abbreviated to ϕ. Theorem 2.2.Let the hypergraph G = (V, E, w) be connected and x ∈ K a (t).Then there exists some constant C depending only on p and G such that Proof.Let the index j satisfy |a j (t)| = min 1≤i≤m |a i (t)|.Since G is assumed to be connected, for every fixed v k (k = 1, . . ., n) there exist some u 1 , . . ., u N −1 ∈ V and e 1 , e 2 , . . ., e N ∈ E s.t.u l−1 , u l ∈ e l (l = 1, 2, . . ., N) holds, where u 0 = v k and u N = v n+j .By x(v n+j ) = a j (t) and Hölder's inequality, we have where p ′ := p/(p − 1) is the Hölder conjugate exponent.Then we obtain (10) with the positive constant In addition, we can easily show that x by definition and b ≤ 2 for any b ∈ B e since 1 u ∈ R V can be identified with a unit vector of R n+m .Hence as for the boundedness of ϕ and ∂ϕ, hold for every x ∈ R V and η ∈ ∂ϕ(x), where #E is the number of hyperedges.We also recall a variant of Gronwall's inequality as follows (see [1, Lemme A.5]).Let L q (0, T ; R V ) (q ∈ [1, ∞)) and L ∞ (0, T ; R V ) denote the family of g : [0, T ] → R V which satisfies T 0 g(t) q dt < ∞ and sup 0≤t≤T g(t) < ∞, respectively.Moreover, g ∈ W 1,q (0, T ; R V ) implies that g and its derivative g ′ belong to L q (0, T ; R V ).
This result can be assured by known results for abstract theory of nonlinear evolution equations with time-dependent subdifferential.Indeed, we can apply [9, Theorem 1.1.2]since ϕ t := ϕ + I Ka(t) satisfies the condition (H) 2 in [9, §1.5] if a ∈ W 1,2 (0, T ; R V ).In this section, however, we shall give an alternative proof which only relies on the basic properties of subdifferential in order to avoid some complicated calculations arising from the chain rule of d dt ϕ t (x(t)).
Proof.Remark that the Moreau-Yosida regularization of I Ka(t) and its subdifferential (the Yosida approximation of ∂I Ka(t) ) coincide with We first replace ∂I Ka(t) in (P) a,h,x 0 with (∂I Ka(t) ) λ and consider the following approximation problem: By the Lipschitz continuity of 1 λ (x−Proj Ka(t) x), the abstract theory (see e.g, [1, Théorème 3.6]) yields the existence of a unique solution x λ ∈ W 1,2 (0, T ; R V ) to (P) λ for every We establish a priori estimates independent of λ.Since by x λ − a and using (9), we have 1 2 Hence if x λ (0) = x 0 = (x 01 , . . ., x 0n , a 1 (0), . . ., a n (0)) ∈ K a (0), we obtain From this and ( 12), there exists some constant where η λ is the section of (P) λ , i.e., fulfills for a.e.t ∈ [0, T ].We next test (P) λ by (x λ (t) − a(t)) ′ .Since ϕ does not depend on the time, the standard chain rule (η and ( 14), we can derive By x λ,n+j (0) = a j (0), we get By ( 14) and ( 15), we can apply the Ascoli-Arzela theorem and extract a subsequence of {x λ } λ>0 which converges uniformly in [0, T ] (we omit relabeling since the original sequence also converges as will be seen later).Let x ∈ C([0, T ]; R V ) be its limit.Furthermore, ( 14) and ( 15) also lead to and the demiclosedness of maximal monotone operators implies that the limit η satisfies η(t) ∈ ∂ϕ(x(t)) for a.e.t ∈ (0, T ).Since (15) yields we have x n+j (t) = a j (t), i.e., x(t) ∈ K a (t) for every t ∈ [0, T ].By the equation, By taking its limit as λ → 0, we obtain Thus the limit x fulfills all requirements of the solution to (P) a,h,x 0 .The uniqueness of solution can be guaranteed by the monotonicity of ∂ϕ and ∂I Ka(t) .Therefore the limit x is determined independently of the choice of subsequences and then the original sequence {x λ } also converges to x. Remark 3.2.We can find several abstract results for the case where the movement of the constraint set is not smooth (see, e.g., [4, §4]).However, it seems to be difficult to mitigate our assumption a ∈ W 1,2 (0, T ; R V ) since K a (t) does not possess any interior point and the smoothness of movement of K a (t) must exactly coincide with the regularity of the solution.

Continuous Dependence of Solutions on Given data
We next consider the dependence of solutions with respect to the given data.
Remark 4.3.Theorem 4.1 implies that the solution to (P) a,h,x 0 is 1/2-Hölder continuous with respect to the given data a(t).This seems to be optimal and it is difficult to derive better estimate in general.Indeed, let , and η k (t) ∈ ∂ϕ(x k (t)) be the section of (P) a k ,h k ,x k 0 .Moreover, we assume the initial data for x 1 satisfies with some i 1 , i 2 ∈ {1, . . ., n} and for x 2 fulfills with some j 1 , j 2 ∈ {1, . . ., m}.By the continuity of solutions and a k , there exists some T 0 > 0 such that the solutions still satisfy Hence we can obtain which appears in the estimate above.It is not obvious whether we can deduce the term x 1 (t) − x 2 (t) α (α > 1) from this.
We next deal with the case where a depends on t and h(t), a(t) → 0 as t → ∞.
Proof.Henceforth, let c 0 , C 0 denote general constants independent of T .Note that the assumptions lead to a, h ∈ L ∞ (0, ∞; R V ) and a ∞ = h ∞ = 0.

Figure 1 :
Figure 1: When #e = 2 for every e ∈ E, each edge e ∈ E can be regarded as a segment connecting two vertices (left figure, called usual graph).Hypergraph is a generalization of usual graph which represents the connection and grouping of multiple members (right figure).