The Strong Maximum Principle for Schr\"{o}dinger operators on fractals

We prove a strong maximum principle for Schr\"odinger operators defined on a class of fractal sets and their blowups without boundary. Our primary interest is in weaker regularity conditions than have previously appeared in the literature; in particular we permit both the fractal Laplacian and the potential to be Radon measures on the fractal. As a consequence of our results, we establish a Harnack inequality for solutions of these operators.


Introduction
The goal of this note is to prove a strong maximum principle and related results for Schrödinger operators L = ∆ − ν where ∆ is a fractal Laplacian (to be defined below) and the potential ν is a non-negative measure on the fractal. When the fractal K is the standard Sierpinski gasket or its unbounded analogue, Strichartz established strong maximum principles for solutions of the nonlinear equation ∆u = F(x, u) where F : K × R → R is continuous and nonnegative in the sense that if u(x) ≥ 0 then F(x, u(x)) ≥ 0, see [1]. We consider algebraically simpler operators because our interest is in weakening the regularity conditions on both the Laplacian ∆ and the potential, both of which we will permit to be measures.
In Section 2 we recall some basic facts about analysis on fractals and fractal blowups, details of which may be found in [2,3]. Section 3 contains the proof of the maximum principle and some comments on a Hopf-type lemma, and Section 4 has the proof of a Harnack inequality.

Preliminaries
We consider a connected self-similar set K generated by an iterated function system {F 1 , . . . , F N } consisting of contractive maps on a complete metric space. To a finite word ω = ω 1 ω 2 · · · ωn ∈ {1, . . . , N} n we associate Fω = Fω 1 ∘ · · · ∘ Fω n and a cell Cω = Fω(K). We assume K is post-critically finite, whence there is a finite set V 0 so that for any words ω, and thereby obtain a countable dense subset V * = ∪n Vn of K. The topology of K is generated by cells in the sense that any x / ∈ V * has a neighborhood base consisting of interiors of cells, while any x ∈ Vn has a neighborhood base in which each set is a finite union of interiors of cells adjoined at x. We let µ denote the standard self-similar measure on K.
Following Kigami [2] we make the strong assumption that there is a resistance form E, also called the energy, with domain dom E ⊆ L 2 (K, µ) that is obtained from a regular self-similar harmonic structure. This means that there is an irreducible, non-negative, symmetric, quadratic form E 0 defined on the (finitedimensional) vector space of functions on V 0 , and factors 0 < r i < 1 for each i = 1, . . . , N, such that setting and E(u, u) = limn En(u, u), where the latter sequence is non-decreasing (see Sections 2.3 and 2.4 of [2]). Those functions on which En is constant for n ≥ m are called piecewise harmonic at scale m. It follows that the pair (E, dom E) has the following properties: 1) E is a non-negative symmetric quadratic form on the linear space dom E, 2) E vanishes exactly on the constants and dom E modulo constants is a Hilbert space under E, 3) any function on a finite subset of K has an extension in dom E, 4) for any p, q ∈ K the quantity is finite, and 5) if u ∈ dom E then so isū = max{0, min{u, 1}} and E(ū,ū) ≤ E(u, u). Moreover, R(p, q) is a metric on V * with completion homeomorphic to (and hence identified with) K, to which the continuous extension of R is called the resistance metric; any function u ∈ dom E satisfies It follows that any u ∈ dom E is continuous with respect to the resistance metric; and E is a Dirchlet form on L 2 (K, µ), see [4]. From the general theory of Dirichlet forms (see, for example, [5]) one then defines a nonpositive definite, self-adjoint (Dirichlet) Laplacian operator ∆ for which u ∈ dom ∆ and ∆u = f ∈ C(K) by requiring for all w ∈ dom E such that w| V0 = 0. If we only assume that f ∈ L p (K, µ), then we say that u ∈ dom L p ∆. We will primarily work with an extension of the above definition in which ∆u is a finite signed Radon measure (see [6,Definition 4.1] and [7, Definition 6.7]): we say that u ∈ dom M ∆ and for all w ∈ dom E such that w| V0 = 0. Of course, we can view any function f ∈ L 1 (K, µ) as the measure f dµ. We assume that the Laplacian is self-similar in the sense that for all ω ∈ {1, . . . , N} n and n ≥ 1, where µ i are the weights of the self-similar measure µ and µω = ∏︀ n j=1 µω j and rω is defined similarly (see the comments following Definition 4.2 of [6] why the definition of the localization of the measure valued Laplacian to a cell is not as straightforward as it may seem).
Our main results are also valid on bounded subsets of the fractal blowups considered by Strichartz [8]. An infinite blow-up without boundary points of the p.c.f. fractal K is defined using a sequence α ∈ {1, . . . , N} N such that α is not eventually constant. For n ≥ 1 set Then {Kn} is an increasing sequence of sets and the infinite blow-up is defined to be K∞ = ⋃︀ ∞ n=1 Kn. Both the energy E and the measure µ extend to K∞ in the obvious fashion, and we write E∞ and µ∞ for these extensions. The Laplacian ∆∞ is defined weakly as before.

Maximum principle
Let ν be a finite non-negative Radon measure. The Schrödinger operator L with potential ν is defined on dom M ∆ by Lu := ∆u − uν.
where the cpq > 0 are constants depending only on En and q ∼n p means that q and p are neighbors in Vn (this is established in a similar manner as [2, Lemma 3.5.1]). Using this inductively beginning at n = 1 and considering each point in Vn, we deduce that u is bounded on V * by max V0 u, whence the desired inequality follows by continuity of u. Now suppose u attains an interior maximum at x. We distinguish two cases according to whether x ∈ V * or x / ∈ V * . If x ∈ V * then it is in Vn for some n. Let q ∼n x and from (3.2) and u(x) ≥ u(q) deduce both that u(q) = u(x) and that ∫︀ hx ∆u = 0. Since hx > 0 on the interiors of the n-cells containing x, we conclude that ∆u has no mass on these cells and thus that u is harmonic on them. As u is harmonic and all its boundary values on these cells equal u(x) it is a constant function, and hence u ≡ u(x) on a neighborhood of x.
If x / ∈ V * we fix an n and the n-cell Cn containing x. Let h be harmonic on Cn with h = u on the boundary ∂Cn. We use another result of Kigami [2, Proposition 3.5.5 and Theorem 3.5.7] and [7, Theorem 6.8], namely that there is a non-negative Green kernel g that inverts −∆ on Cn with Dirichlet boundary data. Thus, for y ∈ Cn, which simply says u is subharmonic. However we then have where the first inequality is (3.3), the second is the maximum principle for harmonic functions, the equality is u = h on ∂Cn, and the final inequality is that u(x) is the maximum of u. Since equality must hold throughout we conclude u = u(x) on ∂Cn. However (3.3) must also be an equality, and since g(y, z) > 0 unless z ∈ ∂C we find that ∆u has no mass on the interior of C whence u is harmonic and therefore constant. Again we have found u ≡ u(x) on a neighborhood of x.
We conclude by noting {y : u(y) = u(x)} is closed by continuity of u, open by the preceding reasoning, and contains x, so by connectivity it is K.
The preceding argument extends readily to our class of Schrödinger operators. For u a function on K or K∞ let u + (x) = max{u(x), 0}.

Moreover, if u achieves a positive maximum at an interior point x ∈ E then u is constant on the connected component of E containing x.
Proof. Observe that the asserted inequality is trivial if u ≤ 0. Accordingly we may assume U := {x ∈ E : u(x) > 0} ≠ ∅. Then ∆u is a non-negative measure on U, so Proposition 1 is applicable to each cell contained in U. Moreover the proof of Proposition 1 implies there cannot be a strict maximum at p ∈ V * if there is a scale n such that all neighbors q ∼n p are in U. Thus the maximum of u cannot only occur at an interior point to U because every such point has a neighborhood in U that is a cell or finite union of cells at a single scale. Since u = 0 at any point p ∈ ∂U that is interior to E, the maximum must be achieved on ∂U ∩ ∂E, which implies the stated inequality. In the classical setting of a Euclidean space, one standard approach to obtaining the strong maximum principle from the weak maximum principle is to use the Hopf lemma. It is perhaps interesting to note that in the fractal setting we can prove a Hopf-type lemma at points in V * but have no corresponding results at points of K \ V * and therefore cannot use this approach to obtain a strong maximum principle.
To state our Hopf-type lemma we recall that the normal derivative [2, Definition 3.7.6] of a function at a point p ∈ V 0 may be written using the scale m piecewise harmonic function h (m) p which is 1 at p and zero on Vm \ {p} as ∂n u(p) = lim Lemma 3. Let ν be a non-negative Radon measure on K and suppose u ∈ dom M ∆ satisfies Lu = ∆u − uν is also non-negative measure. If there is p ∈ V 0 such that u(p) > u(x) for all x ∈ K \ {p} and also u(p) > 0 then ∂n u(p) > 0.
Proof. If n < m the difference k (n,m) = h (n) p − h (m) p is zero at p and the points q ∼n p, and is equal to h (n) p (q) > 0 at each q ∼m p; it is otherwise harmonic, so the minimum principle for harmonic functions implies it is nonnegative. Now u(p) > 0 and u is continuous so there is n such that u is positive on the support of k (n,m) . For this n and any m > n we see ∆u ≥ uν is a non-negative measure on the support of k (n,m) and therefore for some values cpq > 0 depending on En. The fact that u(p) > u(q) for all q concludes the proof.

Harnack inequality
In the classical setting the strong maximum principle for Lu ≥ 0 implies a Harnack inequality for solutions of Lu = 0, see [10,Section 8.8]. We show that this is the case in our setting.
Before stating the Harnack inequality, we note that the equation Lu = 0 for L as in (3.1) has solutions for sufficiently small measures ν on K, because with boundary data a harmonic function h, the operator defined using the continuous non-negative Green kernel g(x, y) by

y)u(y)ν(dy)
is contracting in the uniform norm provided ∫︀ g(x, y)ν(dy) < 1. Moreover, on the cell Fω(K) the Green kernel is rω g(F −1 ω (x), F −1 ω (y)) where rω = ∏︀ |ω| j=1 rω j by [2,Proposition 3.5.5]. It follows for any ν that we can take |ω| sufficiently large so that contractivity of the analogous operator on Fω(K) is valid, ensuring local solutions exist everywhere. This argument extends to the case when one has a finite union of cells.
Theorem 4 (Harnack inequality). On K, fix a non-negative finite Radon measure ν. Suppose u ∈ dom M (∆) satisfies both u ≥ 0 and Lu = 0, where L is as in (3.1) and the latter is an equality of measures. If E is a compact subset of a connected component of K \ V 0 then there is a constant A that depends only on E such that max E u ≤ A min E u.
The proof makes use of the following result, which may be of independent interest. If instead, u| V0 ≠ 0, then choose a harmonic function h so (u − h)| V0 = 0 and since |h| ≤ 1 on V 0 we have |h(x) − h(y)| 2 ≤ 2R(x, y), so (4.1) holds with M replaced by M + 2. However an estimate of the type (4.1) implies A is equicontinuous, and since the definition of A implies it is equibounded, an application of the Arzela-Ascoli theorem then yields that it is precompact in the uniform norm. Now suppose {un} ⊂ A and un → u uniformly. If w ∈ dom(E) and w| V0 = 0 we know w is continuous and therefore bounded so Hölder's inequality provides and therefore un is Cauchy in dom(E). However dom(E) is a Hilbert space under the norm E + ‖ · ‖∞, so we conclude that u ∈ dom(E) and E(u, w) = − ∫︀ wudν, whence ∆u = uν. This implies u ∈ A, so A is closed, and in light of our Arzela-Ascoli argument, compact in the uniform norm. Evidently this is a closed subset of the space of functions considered in Proposition 5 and is therefore compact in the uniform norm. Hence the quantity a = inf A min x∈E u(x) is achieved by someũ ∈ A. Either a = 1 orũ is non-constant and the hypothesisũ ≥ 0 implies a > 0 by the strong minimum principle. Now take u as in the statement of the theorem. It is continuous and the strong maximum principle is applicable, so it achieves its maximum on V 0 . Accordingly, u/(max V0 u) ∈ A and has minimum at least a on E. The result follows with A = 1/a. Remark 6. The above immediately implies the corresponding result in the setting of a bounded subset of K∞ because such a set is contained in a (sufficiently large) copy of K, and we can transfer the theorem directly to this setting using the self-similarity of the energy.