An Approximation of Solutions to Heat Equations defined by Generalized Measure Theoretic Laplacians

We consider the heat equation defined by a generalized measure theoretic Laplacian on $[0,1]$. This equation describes heat diffusion in a bar such that the mass distribution of the bar is given by a non-atomic Borel probabiliy measure $\mu$, where we do not assume the existence of a strictly positive mass density. We show that weak measure convergence implies convergence of the corresponding generalized Laplacians in the strong resolvent sense. We prove that strong semigroup convergence with respect to the uniform norm follows, which implies uniform convergence of solutions to the corresponding heat equations. This provides, for example, an interpretation for the mathematical model of heat diffusion on a bar with gaps in that the solution to the corresponding heat equation behaves approximately like the heat flow on a bar with sufficiently small mass on these gaps.


Introduction
Let [a, b] ⊂ R be a finite interval, µ be a non-atomic Borel probability measure on [a, b] such that a, b ∈ supp(µ), L 2 ([a, b], µ) be the space of measurable functions f such that b a f 2 dµ < ∞ and L 2 ([a, b], µ) be the corresponding Hilbert space of equivalence classes with inner product f, g µ := b a f gdµ. We define The Krein-Feller operator with respect to µ is given as This definition involves the derivative with respect to µ. If a function f has a representation given by then d dµ f is called the µ-derivative of f . Consequently, in the above definition, (f ) µ is the µ-derivative of f . This operator has been widely studied, for example with an emphasis on addressing questions of the spectral asymptotics and further analytical properties [3, 4, 12-21, 23, 24, 34, 35, 37, 38], diffusion processes [29][30][31], wave equations [6] and higher-dimensional generalizations [22,36,39].
In order to connect these operators with diffusion equations from a physical point of view, we follow for example [27,Section 1.2] and consider a metallic rod of constant cross-sectional area oriented in the x-direction occupying a region from x = 0 to x = 1 such that all thermal quantities are constant across a section. We can thus consider the rod as one-dimensional. We investigate the conduction of heat energy on a segment from x = a to x = b. Let the temperature at the point x ∈ [a, b] and time t ∈ [0, ∞) denoted by u(t, x) and the total thermal energy in the considered segment at time t by e a,b (t). It is well-known that Hence, we can define the total heat energy even if µ has no density. The total heat energy changes only if heat energy flows through the boundaries x = a and x = b. We deduce for the rate of change of heat energy where φ(t, x) denotes the heat flux density, which gives the rate of thermal energy flowing through x at time t to the right. Assuming sufficient regularity, we can rewrite (1)  Since this is valid for all a, b ∈ [0, 1], a < b, it follows for t ∈ [0, ∞) and µ-almost all x ∈ [0, 1] Applying the definition of the Krein-Feller operator leads to the generalized heat equation with Dirichlet boundary conditions u(t, 0) = u(t, 1) = 0 for all t ≥ 0 if we assume that the temperature vanishes at the boundaries or with Neumann boundary conditions ∂u ∂x (t, 0) = ∂u ∂x (t, 1) = 0 if the boundaries are assumed to be perfectly insulated. This provides a physical motivation for a mass distribution having full support even if it possesses no Lebesgue density. However, it is still not clear how to interpret the equation if the support of the mass distribution is not the whole interval, in particular for singular measures, such as measures on the Cantor set.
The problem then is to describe heat flow on a rod with massless parts. Krein-Feller operators defined by measures on the classic Cantor set or, more general, Cantor-like sets with gaps have been extensively studied in recent years (see e.g. [2,[17][18][19][20]). In this paper, we give an interpretation of a solution to (2) in the case where µ is not of full support. We approximate the solution by a sequence of solutions to heat equations defined by µ n for n ∈ N such that µ n is of full support and converges weakly to µ for n → ∞.
To this end, let b ∈ {N, D} represent the boundary condition, where N denotes Neumann and D Dirichlet boundary conditions and we give our basic assumption. Assumption 1.1: Let (µ n ) n∈N be a sequence of non-atomic Borel probability measures on [0, 1] such that 0, 1 ∈ supp(µ n ) and µ n µ, n → ∞, where denotes weak measure convergence.
It is well-known that ∆ b µ is a non-positive self-adjoint operator (see, e.g., [14]) and thus the generator of a strongly continuous semigroup T b t t≥0 (see, e.g. [25,Lemma 1.3.2]). If u 0 ∈ L 2 ([0, 1], µ), then the unique solution to the initial value problem is given by u(t) = T b t u 0 , according to a generalized solution concept we introduce later. This motivates the investigation of strong semigroup convergence. However, for different measures, the corresponding semigroups are defined on different spaces. For the special case supp(µ) = supp(µ n ) = [0, 1] for all n ∈ N, the results in [7] can be applied to obtain strong semigroup convergence on the space of continuous functions on [0, 1]. To formulate a strong semigroup convergence result without that assumption, we restrict the semigroup T N t t≥0 associated to ∆ N µ on L 2 ([0, 1], µ) to the subspace of continuous functions, denoted by C([0, 1]) N µ , which is a Banach space with the uniform norm. The semigroup T D t t≥0 is restricted to the Banach space of continuous functions satisfying Dirichlet boundary conditions, denoted by C([0, 1]) D µ . We show that the restricted semigroup, which we denote by T b t t≥0 , is, again, a strongly continuous contraction semigroup and the infinitesimal generator is given bȳ µn , where we denote the embedding by π n . We will see that in this case, strong semigroup convergence is equivalent to strong resolvent convergence and strong resolvent convergence is what we will establish. More precisely, let f ∈ C([0, 1]) b µ , λ > 0 and n ∈ N. We The main tool for proving (4) is the generalization of the hyperbolic functions sinh and cosh, defined by generalizing the series We replace x k k! by generalized monomials defined by a measure µ. This extends the theory of measure theoretic functions, developed for trigonometric functions in [2]. Then, we show that the resolvent density of the operator ∆ b µ is a product of such generalized hyperbolic functions. This leads to the desired strong resolvent convergence by proving convergence of these generalized hyperbolic functions. We obtain After that, we will see that for f ∈ (C[0, 1]) b µ u(t) =T b t f : t ≥ 0 is the unique solution to the initial value problem in the sense that t → u(t) satisfies (5) for all t > 0 and is continuous with respect to (C[0, 1]) b µ for all t ≥ 0. Analogously, u n (t) =T b t,n f : t ≥ 0 is the unique solution to the initial value problem ∂u n ∂t We obtain a meaningful interpretation for the diffusion of heat in the case of a mass distribution with gaps in that the heat in a rod with mass distribution µ diffuses approximately like the heat on a rod with mass distribution µ n for sufficiently large n.
This paper is structured as follows. In the following section, we recall definitions related to Krein-Feller operators. In Section 3, we introduce the concept of generalized hyperbolic functions and the connection to resolvent operators. Section 4 is devoted to the restriction of the Krein-Feller operator semigroup to the spaces (C[0, 1]) b µ for b ∈ {N, D}. After these preparations, in Section 5 we develop the central convergence results, namely the convergence of the hyperbolic functions and the strong resolvent convergence in Section 5.1, the graph norm convergence of the considered operators in Section 5.2 and finally, the strong semigroup convergence and convergence of solutions to heat equations in Section 5.3. In Section 6 we show how to apply the results in three examples. Lastly, in Section 7, we discuss some open problems.

Generalized Hyperbolic Functions and the Resolvent Operator
Let b ∈ {N, D} and let µ be defined as before. In this section we develop a useful representation for the resolvent density of ∆ b µ . Let λ > 0. We consider the initial value problem on L 2 ([0, 1], µ). The problem (7) possesses a unique solution (see [14,Lemma 5.1]), which we denote by g λ 1,N . Further, under the initial conditions and respectively, the above eigenvalue problem also possesses a unique solution (see [14,Remark 5.2]), and we denote it by g λ 2,N , g λ 1,D and g λ 2,D , respectively. The resolvent density is then given as follows.
where the resolvent densities are given by It is well-known that if µ = λ 1 , the solutions to (7) and (9) are given by respectively. We generalize the notion of hyperbolic functions by solving (7) and (9) for an arbitrary measure µ according to the given conditions. To this end, we introduce generalized monomials as in [2].
It follows that in this case and analogously cosh z (x) = cosh(zx).
Proof. The assertion for g λ 1,D was proven in [19,Lemma 2.3]. The proof for g λ 1,N works analogously. We verify the assertion for g λ 2,N . Let x ∈ [0, 1]. Then, Due to estimate (11) we can use the dominated convergence theorem and obtain and in particular The latter equation can be written as ∆ µ f = λf. It remains to verify the initial conditions. Obviously, f (1) = cosh √ λ (0) = 1. Using (11) again, we have The proof for g λ 2,D follows using the same ideas.
This leads to the following representation for the resolvent density:

The Restricted Semigroup
Let b ∈ {N, D} and let µ be defined as before. It is well-known that ∆ b µ is the generator of a strongly continuous Markovian semigroup T b t t≥0 of contractions on L 2 ([0, 1], µ).
This is called the heat kernel of ∆ b µ .
The heat kernel is the integral kernel of the semigroup T b t t≥0 . That is, for t > 0 and f ∈ L 2 ([0, 1], µ), we can write In this section, we restrict these semigroups to appropriate spaces of equivalence classes of continuous functions. The space where f is the continuous representative of f that is linear on all intervals in [0, 1] \ supp(µ). To simplify the notation, we henceforth write 1], µ) and let t > 0. It holds and thus T b t u ∈ D ∆ b µ . Hence, the following inclusion holds: This motivates the definition of the restricted semigroup , which is When evaluating an element of (C[0, 1]) b µ pointwise, we always evaluate the representative that is linear on all intervals in [0, 1] \ supp(µ).
The goal of this section is to show that T b t t≥0 again defines a strongly continuous contraction semigroup. It is obvious that the semigroup property holds. Note that by the Markov property of ( Hence, (T b t ) t≥0 is a semigroup of contractions. It remains to prove the strong continuity. To this end, we need some preparations. We write Proof. We follow the proof of [28,Lemma 5.2.8]. Let f ∈ F. Then, by the Cauchy-Schwarz inequality for all x, y ∈ [0, 1] Now, let g ∈ F 0 . Then, by setting y = 0 in the previous calculation, we get and consequently, Further, for a given f ∈ F, let f 0 be the unique harmonic function that coincides with f on the boundary, that is f 0 ( Then, and thus Combining (14) and (15), Since the space of harmonic functions on [0, 1] is two-dimensional, there exists a constant c 2 > 0 such that for all f ∈ F, the corresponding harmonic function f 0 satisfies Combining the previous inequalities, Proof. We follow the proof of [ By the fact that F is dense in (C[0, 1]) N µ and that, for t ≥ 0, T N t is continuous on (C[0, 1]) N µ , we obtain the assertion for b = N . To verify the case b = D, we prove that F 0 is dense in (C[0, 1]) D µ . Let f ∈ (C[0, 1]) D µ . Then, by the density of F in (C[0, 1]) N µ , there exists a sequence (f n ) n∈N with f n ∈ F for each n ∈ N such that We define for n ∈ N which is an element of F 0 . Further, we have that since f satisfies Dirichlet boundary conditions. This along with (16) implies for n ∈ N The main result of this section now follows immediately.

Strong Resolvent Convergence
Let µ be defined as before and let F be the distribution function of µ. Further, let (µ n ) n∈N satisfy Assumption 1.1 and let F n be the distribution function of µ n for n ∈ N.
First, we give convergence results for the generalized hyperbolic functions introduced in Section 3 using results from [20]. Let p k , q k , k ∈ N be defined by µ and p k,n , q k,n , k ∈ N be defined by µ n for n ∈ N.  For z ∈ R let cosh z , sinh z be defined by µ and cosh z,n , sinh z,n be defined by µ n for n ∈ N. We obtain a result for the generalized hyperbolic functions, comparable to that for the trigonometric functions in [20].
Proof. Let x ∈ [0, 1] and n ∈ N. Then, Further, note that and With that, Finally, We turn to the main result of this section. For b ∈ {N, D} and λ > 0, let R b λ be defined by µ and R b λ,n be defined by µ n . We assume supp(µ) ⊆ supp(µ n ) for all n ∈ N. Then, the mapping Proof. We simplify the notation in this proof by omitting all embeddings. If we evaluate on supp(µ n ) \ supp(µ), we always evaluate the representative that is linear on each interval I ⊆ supp(µ n ) \ supp(µ). First, we consider the case b = N. Let λ > 0, n ∈ N, x, y ∈ [0, 1] with x ≤ y.
Using the triangle inequality, We have and similarly cosh √ λ,n (1) ≥ λ. Applying this along with Lemma 5.3, we get and thus with (12) For the second term on the right-hand side of inequality (18), we calculate Treating the third term analogously and using the above calculations in (18) yields Further, by (12) and (19), Due to weak measure convergence, and consequently, We get the same result for x ≥ y and obtain Now, let b = D. Again using the triangle inequality, for n ∈ N, x, y ∈ [0, 1], x ≤ y, We have and thus Arguing in the same way as before, we get Due to the weak measure convergence, this goes to zero as n tends to ∞. Deducing the same result for x ≥ y and combining the above inequalities,

Graph Norm Convergence
Let µ be defined as before and let λ > 0. Analogously to the restricted semigroup, we define the restricted resolvent operator bȳ Further, we define the operators∆ N µ and∆ D µ bȳ which are called the part of the operator ∆ N µ in C[0, 1]) N µ and the part of the operator ∆ D µ in C[0, 1]) D µ , respectively. The following Lemma shows how the restricted semigroup, the restricted resolvent and the part of the operator are connected. For that, let b ∈ {N, D}.
Lemma 5.5: (i) The infinitesimal generator of the strongly continuous contraction semigroup Corollary 4.5). We thus can apply [10, II.2.3 Proposition] to verify (i). We turn to part (ii). Let λ > 0 and let R b λ be the resolvent of∆ b µ . By part (i) and [10, 1.10 Theorem], this operator is well-defined and given by Further, by definition of T b t t≥0 andR b λ , We are now able to establish graph norm convergence. To this end, let (µ n ) n∈N satisfy Assumption 1.1 and we assume supp(µ) ⊆ supp(µ n ) for all n ∈ N.

Strong Semigroup Convergence
For b ∈ {N, D} let T b t t≥0 be defined by µ, T b t,n t≥0 be defined by µ n and analogously the restricted semigroups T b t t≥0 and T b t,n t≥0 be defined by µ and µ n , respectively. The main result of this paper is a direct consequence of the previous results.
Proof of Theorem 1.2. For n ∈ N, π n is a bounded linear transformation between Banach spaces. Further, T b t t≥0 and T b t,n t≥0 , n ∈ N are strongly continuous contraction semigroups on their respective spaces (see Corollary 4.5). Hence, due to [11,Theorem 6.1], the assertion is a direct consequence of Theorem 5.6.
Strong semigroup convergence can be interpreted as convergence of solutions to heat equations. The connection is given as follows (see [10,Proposition 6.2]). as n → ∞, assuming that π n f ∈ D ∆ b µn . However, the assumption f ∈ D ∆ b µ and π n f ∈ D ∆ b µn for all n ∈ N is very restrictive, as the following example illustrates.
on the Hilbert space L 2 ([0, 1], µ) is given by v(t) = T b t f (see [28,Theorem B.2.6]). We now show that a solution to (24), which we denote by u, is also a solution to (27). The continuity with respect to L 2 ([0, 1], µ) follows from Therefore, u is a solution to (27). This proves the uniqueness. We can follow the same arguments to verify thatT b t,n π n f is the unique solution to (25) for n ∈ N. Then, (26) is a direct consequence of Theorem 1.2.
In the previous example, µ could be chosen to be an absolutely continuous measure, for example , or to be a singular measure, as a self-similar measure on the Cantor set. Furthermore, it is not required that the approximating measures have full support.
Example 6.2: Let w 1 , w 2 ∈ (0, 1) such that w 1 + w 2 = 1 and let µ be the unique invariant Borel probabiliy measure on [0, 1] given by the IFS consisting of S 1 (x) = x 3 and S 2 (x) = 2 3 + x 3 , x ∈ [0, 1] and weights w 1 and w 2 , i.e. µ is a so-called Cantor measure. Following [20], for n ∈ N we define the approximating Cantor measures of level n by  where The approximating Cantor measures of levels n = 0, 1, 2 are illustrated in Figure 1. We denote the distribution function of µ by F and the distribution function of µ n by F n for n ∈ N. Then, F − F n ∞ → 0 (see [20,Proposition 4.2]) as well as supp(µ) ⊂ supp(µ n ) for n ∈ N and Theorem 5.10 can be applied. Hence, for with respect to the uniform norm as n tends to infinity.
Finally, we connect both applications. Example 6.3: Let ε > 0, n ∈ N and let µ, µ n , {u(t) : t ≥ 0} and {u n (t) : t ≥ 0} be defined as in Example 6.2. We define µ n,ε by µ n,ε := µ n + ελ 1 1 + ε , i.e. analogously to Example 6.1, and {u n,ε (t) : t ≥ 0} to be the solution to du n,ε dt (t) =∆ b µn,ε u n,ε (t), u n,ε (0) = π n,ε f, where π n,ε is an embedding as previously defined. Further, let t ∈ [0, ∞) and δ > 0. By Example 6.2, there exists n 0 ∈ N such that for all n ≥ n 0 it holds By Example 6.1, for each n ≥ n 0 there exists ε n > 0 such that for all ε < ε n it holds Hence, for all n ≥ n 0 , ε < ε n it holds Hence, the heat on a rod with mass distribution given by a Cantor measure diffuses approximately like the heat on a rod possessing a strictly positive mass density which is small off the Cantor set.

Directions for Further Research
Remark 7.1: Consider the heat equation (2) with initial value given by the Delta distribution δ y : g → g(y) for y ∈ supp(µ). Then, the heat kernel solves the equation in the distributional sense, where λ b k , k ≥ 1 are the ascending ordered eigenvalues and ϕ b k , k ≥ 1 the L 2 ([0, 1], µ)-normed eigenfunctions of ∆ b µ on L 2 ([0, 1], µ). The heat kernel is of particular importance in the context of the associated Markov process (see the remark below) and stochastic partial differential equations (see [8,9]). It is an open question whether weak measure convergence implies convergence of the corresponding heat kernels in an appropriate sense.
Remark 7.2: The operator ∆ b µ on L 2 ([0, 1], µ) is the infinitesimal generator of a Markov process, called a quasi-diffusion (see, e.g., [29][30][31][32]). Convergence of semigroups raises the question whether the associated Markov processes also converge weakly. If µ n µ, our results imply that for each f ∈ (C[0, 1]) b µ , t ∈ [0, ∞) and each starting point x ∈ [0, 1] where X b is associated to ∆ b µ and X b n is associated to ∆ b µn . This would need to be extended to a proof of convergence of all finite-dimensional distributions, and tightness would also be required, to establish that X b n → X b weakly in the Skorokhod space of càdlàg functions. on L 2 ([0, 1], µ). This hyperbolic equation describes the motion of a vibrating string with mass distribution µ such that, if it is deflected, a tension force drives it back towards its state of equilibrium. If µ were not of full support, the string would have massless parts. It is not clear how to interpret massless parts of a string. We suppose that the motion of such a string behaves approximately like the motion of a string with very little mass on these gaps, analogous to our results about the diffusion of heat. Assume that u(0) ∈ D ∆ b µ and, for reasons of simplicity, that the initial velocity vanishes. Then, there exists a unique solution on L 2 ([0, 1], µ) given by u(t) = C(t)u(0), t ≥ 0, where {C(t) : t ≥ 0} denotes the strongly continuous cosine family of ∆ b µ (see, e.g., [40]). We have already shown that µ n µ implies strong resolvent convergence of the corresponding operators restricted to continuous functions. It is well-known that this implies convergence of the corresponding cosine families {C n (t) : t ≥ 0}, which implies convergence of the solutions to the corresponding wave equation, provided that there exists M > 0 and w ≥ 0 such that for all n ≥ 1, t ≥ 0 C n (t) ≤ M e w|t| (see [26]). Proving that the restriction of C(t) to (C[0, 1]) b µ is the cosine family of∆ b µ (and analogously for µ n ) and verifying the above estimate would be a way to establish the desired convergence of solutions to the wave equation.