Uniqueness of diffusion on domains
[2mm] with rough boundaries
[2mm] Juha Lehrbäck and Derek W. Robinson
31st March 2015
Let be a domain in and a quadratic form on with domain where the are real symmetric -functions with for almost all . Further assume there are such that for where is the Euclidean distance to the boundary of .
We assume that is Ahlfors -regular and if , the Hausdorff dimension of , is larger or equal to we also assume a mild uniformity property for in the neighbourhood of one . Then we establish that is Markov unique, i.e. it has a unique Dirichlet form extension, if and only if . The result applies to forms on Lipschitz domains or on a wide class of domains with a self-similar fractal. In particular it applies to the interior or exterior of the von Koch snowflake curve in or the complement of a uniformly disconnected set in .
AMS Subject Classification: 47D07, 35J70, 35K65.
|1.||Department of Mathematics||2.||Centre for Mathematics|
|and Statistics||and its Applications|
|University of Jyvaskyla||Mathematical Sciences Institute|
|PO Box 35 (MaD)||Australian National University|
|FI-400014 University of Jyvaskyla||Canberra, ACT 0200|
The theory of diffusion processes has a distinct probabilistic character and is most naturally studied on -spaces. Consequently much of the analysis of such processes has relied on methods of stochastic differential equations or stochastic integration. Our aim, however, is to examine symmetric diffusion problems on domains of Euclidean space with the techniques of functional analysis and semigroup theory. In particular we focus on the characterization of uniqueness of the -theory on domains with rough or fragmented boundaries. First we formulate the problem of diffusion as a problem of finding extensions of a given elliptic operator which generate semigroups with the general characteristics suited to the description of diffusion.
Let be a domain in , i.e. a non-empty open connected subset, with boundary and a strongly continuous, positive, contraction semigroup on . If the positive normalized functions in are viewed as probability distributions then has the basic properties required for description of their evolution with time. For brevity we refer to as a diffusion semigroup. We define to be symmetric if
for all , all and all . It follows that extends by continuity from to a weakly continuous semigroup on which we also denote by . The extended semigroup is automatically equal to the adjoint semigroup . Then can be defined on for each by interpolation. In particular is a self-adjoint, positive, contraction semigroup on . If is the positive, self-adjoint generator of it then follows from the Beurling–Deny criteria (see, for example, [RS78]) that the corresponding quadratic form with is a Dirichlet form. Therefore the semigroup is submarkovian, i.e. if then for all , by the theory of Dirichlet forms [BH91] [FOT94].
Next define the operator on the domain by
where are real and the matrix of coefficients for all in the sense of matrix order. The corresponding diffusion problem consists of classifying all extensions of to which generate symmetric diffusion semigroups. One can establish the existence of at least one such extension by quadratic form techniques. Let be the positive, quadratic, form associated with on , i.e.
for all . Since is a symmetric operator on the form is closable and the closure, which we denote by , is automatically a Dirichlet form [BH91] [FOT94]. The corresponding positive, self-adjoint operator , the Friedrichs’ extension of , generates a positive, contraction semigroup on which extends to a similar semigroup on each of the -spaces. The extension to automatically satisfies the symmetry relation (1). Therefore generates a symmetric diffusion semigroup on . The extension corresponds to Dirichlet boundary conditions on . But the same argument establishes that each Dirichlet form extension of determines the generator of a symmetric diffusion semigroup on . Therefore there is a one-to-one correspondence between extensions of on which generate symmetric diffusion semigroups and Dirichlet form extensions of on . The classification of extensions of which generate symmetric diffusion semigroups on is now reduced to the more amenable and transparent problem of classifying the Dirichlet form extensions of on .
The Dirichlet form extensions of have a fundamental ordering property. The closure is the smallest Dirichlet form extension of but there is also a largest such extension . The maximal extension is defined on the domain
where is the carré du champ, by setting
for . Then is a Dirichlet form and the associated operator is the extension of corresponding to generalized Neumann boundary conditions. But if is a general Dirichlet form extension of then (see [FOT94], Section 3.3, [RS11a], Theorem 1.1, or [RS11b], Theorem 2.1). Thus in the sense of ordering of quadratic forms. Clearly all the Dirichlet form extensions are equal in the interior of and differ only by their behaviour at the boundary. If then a classification of the possible extensions can be extracted from the general analyis of Feller [Fel54]. But a classification of the extensions in terms of boundary conditions seems well beyond reach if . The multi-dimensional problem is complicated by the wide range of geometric possibilities for and the wide variety of possible boundary conditions. Nevertheless these observations give a direct approach to the characterization of uniqueness of a Dirichlet form extension and consequently the uniqueness of the solution to the diffusion problem.
First define the form to be Markov unique if the closure is the unique Dirichlet form extension. Thus is Markov unique if and only if . It was established in [RS11a] [RS11b] (see also Section 2) that this latter condition is equivalent to the boundary having capacity zero measured with respect to the form . This criterion is a property which depends on the degeneracy of the coefficients near the boundary together with the regularity and uniformity properties of . It does not, however, depend on any smoothness of the coefficients. Therefore in the subsequent analysis of the uniqueness problem we replace the assumption by the weaker assumption . Specifically we now assume that is defined by (3) with real coefficients and with for almost all . Then for each compact subset there is a such that for almost all . Hence is closable (see [MR92], Section II.2b) and one can again define as the closure. Moreover , defined as above, is again a Dirichlet form (see [OR12], Proposition 2.1). Therefore one can analyze the Markov uniqueness condition in this broader framework. This equality depends critically on the behaviour of the coefficients on the boundary and we next formulate an appropriate degeneracy condition.
Let denote the Euclidean distance from to and the open Euclidean ball with centre and radius . Further set for each non-empty subset . Then, denoting the boundary of by , we assume there is a and for each bounded non-empty subset there are such that
for almost all where is the (inner) -neighbourhood of .
Next we place some mild geometric restraints on the domain .
First we suppose that satisfies a property of Ahlfors -regularity. Specifically we assume that there is a regular Borel measure on and an such that for each subset , with and , one can choose so that
for all and . This is a locally uniform version of the Ahlfors regularity property used in the theory of metric spaces (see, for example, the monographs [DS97], [Sem01], [Hei01] or [MT10]). It implies that and the Hausdorff measure on are locally equivalent and , the Hausdorff dimension of . The terminology regular is somewhat misleading as an Ahlfors regular boundary can be quite ‘rough’, e.g. the boundary of the von Koch snowflake (see Section 5) is Ahlfors regular. Condition (5) does, however, imply that is regular in the sense that each of the subsets with has Hausdorff dimension .
Secondly, if we assume a local form of the uniformity property introduced by Martio and Sarvas [MS79]. If and set . Then is defined to be -uniform if there is a such that for all there is a rectifiable curve with , and length at most such that for all . Note that the curve is in but is not constrained to .
The local uniformity condition has two elements. First, if an arbitrary pair of points can be joined by a rectifiable curve in then must belong to a connected component of (but note that need not be connected). Secondly, it is necessary for the detailed properties of the curves that the boundary subset has the characteristics of the boundary of a uniform domain. For example, if then outward pointing parabolic cusps, inward pointing antennae or slits which separate locally are all forbidden.
The foregoing assumptions allow a rather simple characterization of Markov uniqueness in terms of the order of degeneracy of the coefficients of the form at the boundary and the Hausdorff dimension .
Let be a domain in with boundary . Assume satisfies the Ahlfors -regularity property with . Further, if assume there is a and an such that is -uniform. Finally assume the coefficients of the form satisfy the degeneracy condition for .
Then the form is Markov unique, i.e. , if and only if .
Theorem 1.1 is a straightforward illustration of our principal results. In the sequel (see Section 4) we describe situations with the index of regularity and the order of degeneracy taking different values on distinct components and faces of the boundary. This introduces a number of extra complications but the basic elements of the proofs are already contained in the proof of the simpler theorem.
Despite the relative simplicity of Theorem 1.1 it does cover a variety of interesting examples. First if is a Lipschitz domain then the regularity and uniformity assumptions of the theorem are valid with (see Section 5). Therefore is Markov unique if and only if . Secondly, the theorem also applies to a broad class of domains whose boundaries are self-similar fractals. In particular it is applicable if and is the interior, or exterior, of the von Koch snowflake. Therefore is Markov unique if and only if with , the Hausdorff dimension of the snowflake. Thirdly, the conclusions of the theorem are stable under the subtraction of Ahlfors -regular subsets of the interior of with (see Corollary 3.7). Finally let be a uniformly disconnected subset of (see, for example, [Hei01] Section 14.24). MacManus, [Mac99] page 275, observed that by the compactness argument of Väisälä, [Väi88] Theorem 3.6, the complement is a uniform domain. (We give an explicit proof of this result in Lemma 3.9.) Therefore Theorem 1.1 in combination with this observation immediately gives the following corollary.
Let where is a closed uniformly disconnected set satisfying the Ahlfors -regularity property with . Further assume the coefficients of the form satisfy the degeneracy condition for .
Then the form is Markov unique, i.e. , if and only if .
In particular if is the usual Cantor dust with granular ratio then is uniformly disconnected and . This is of interest as the Hausdorff dimension can take all values between and as varies from to . Note that by setting and one deduces that the Laplacian defined on is Markov unique if and only if .
In this section we gather some preliminary results which are needed in the proof Theorem 1.1. First we recall some earlier results which characterize Markov uniqueness by a zero capacity condition on the boundary . Secondly, we establish some implications of Ahlfors regularity and local uniformity of the boundary.
The Markov uniqueness criterion is by definition equivalent to the density, with respect to the -graph norm , of in . But this criterion is a boundary condition and in earlier papers [RS11a] [RS11b] [Rob13] it was established that it is equivalent to having zero capacity relative to the form . These earlier results were stated for forms with coefficients or but in fact no smoothness of the coefficients is necessary. The property of importance is the density in , equipped with the graph norm, of the subspace of bounded functions in with compact support in . This density property follows from the boundedness of the coefficients of . In fact the density holds for large classes of coefficients which grow at infinity but fails in general (see [Maz85], Section 2.7, or [OR12], Lemma 2.3, and [Rob13], Section 4).
The capacity of a subset of relative to the form is defined by
The definition of is analogous to the canonical definition of the capacity associated with a Dirichlet form [BH91] [FOT94] and if the two definitions coincide. The two capacities share many general characteristics. If is measurable then where denotes the Lebesgue measure of . Moreover, the map is monotonic, if is an increasing family of measurable sets then and if is a decreasing family of compact sets then .
The following proposition is a slight extension of Theorem 1.2 of [RS11a].
Let be a domain in with boundary and the quadratic form with -coefficients defined by .
The following conditions are equivalent:
Theorem 1.2 of [RS11a] gives a similar statement for . This smoothness property ensures that is the form of a symmetric operator. But the argument used to establish the statement does not depend on smoothness. It is a quadratic form argument which applies equally well for
The degeneracy conditions (4) and regularity conditions (5) have not been assumed in Proposition 2.1. The upper bounds of (4) and the lower bounds of (5) are, however, critical for the subsequent verification of the criterion (see Section 3). Another crucial factor is the growth in volume of inner neighbourhoods of subsets of . The simplest estimates on the growth are given by the following lemma.
Let be a bounded non-empty subset of the boundary of the domain and a regular Borel measure on .
If there exist and such that for all and all then there exists a such that
for all .
If there exist and such that and in addition for all and all then there exists a such that
for all .
Proof If and is a general bounded non-empty subset then the proposition follows from Lemma 2.1 in [Sal91]. The proof of the latter lemma is based on a standard packing/covering argument.
The lower bounds on given by the second statement of Lemma 2.2 depend on the bounds . These latter bounds are not generally valid but require some additional assumptions. One general result in this direction is the following.
Assume the boundary satisfies the Ahlfors -regularity condition with . If for some and some then there are such that
for all .
Proof The proof follows by establishing that for all and .
Set . If denotes the interior of the complement of then and . But it follows from the Ahlfors regularity that . Now assume . Then separates and , i.e. each rectifiable curve in starting at and ending at has an intermediate point in . Therefore has topological dimension . Consequently, the Hausdorff dimension of is greater or equal to (see [HW48], Section VII.4, or [Hei01], Section 8.13). But this is a contradiction so one must have . Hence . Now the lower bound (8) follows from Statement II of Proposition 2.2.
The lower bounds of Proposition 2.3 are independent of any uniformity property of . But if there is a and such that is -uniform then one has similar bounds for in the neighbourhood of the point of uniformity.
Assume the boundary satisfies the Ahlfors -regularity condition with . Further assume there is a and an such that is -uniform.
It follows that if then there exist such that
for all .
Proof It suffices to prove that there are and such that
The estimate (10) is, however, a consequence of the uniformity of by the following argument.
Fix and with . Therefore . Let be a curve joining and which satisfies the local uniformity properties. Then . Thus for each . But one also has . In particular if is the midpoint of then and . Consequently, . Then replacing by one deduces that for all with and . Then (10) follows immediately with .
The bounds on for the subsets of the boundary of are fundamental for the proof of Theorem 1.1. The estimates are also related to the existence of the Minkowski dimension of . There are a number of possible definitions of the Minkowski dimension but the appropriate definition in the current context would be
whenever the limit exists. It follows, however, from the Ahlfors -regularity of that the limit exists and . (For a fuller discussion of Ahlfors regularity property and the equality of various possible dimensions see [Leh08b], Lemma 2.1, and [LT13], Theorems 4.1 and 4.2).
Next we note that the Ahlfors -regularity property (5) implies local equivalence of Hausdorff measure and Hausdorff content. This is directly related to the observation that the regularity property implies local equivalence of the measure and the Hausdorff measure . In fact the lower bound in the Ahlfors property (5) implies that if with and then there is an such that for all Borel subsets . Conversely, the upper bound of (5) implies that there is a such that for all . Hence and are equivalent on and . (See, for example, [Hei01], Section 8.7.)
Now the Hausdorff measure of each Borel subset of is defined by
for all . But the Hausdorff content of is defined as , i.e. there is no restriction on the diameters of the sets in the cover. Moreover, in these definitions it suffices to consider covers of by balls with .
If satisfies the Ahlfors -regularity property and with and then there is a such that
for all Borel subsets .
Proof The lower bound follows directly from the definition of and . To establish the upper bound let be a covering of by balls with and . Then
and taking the infimum over the possible covers one deduces that .
Finally we derive an estimate which is relevant to the derivation of a local version of the weighted Hardy inequality. This will be of importance in the sequel.
Assume the boundary of the domain satisfies the Ahlfors -regularity condition . Fix and . Then there is an such that
for all .
Proof First if with then . Hence . Secondly, if then . Thirdly, it follows from the foregoing that the Hausdorff content and the Hausdorff measure are equivalent on . Therefore to prove (11) it suffices to prove that there is a such
for all . But since is locally equivalent to one has
for all by the regularity assumption.
3 Markov uniqueness
In this section we give the proof of Theorem 1.1. It is in two parts. First we prove that the degeneracy bounds imply Markov uniqueness. This part of the proof is based on an argument given in [RS11a]. Secondly, we use local versions of the weighted Hardy inequalities derived in [KL09] [Leh14] to prove that Markov uniqueness implies the degeneracy bounds.
The first part of the proof is based on the observation of Proposition 2.1 that Markov uniqueness of is equivalent to the property . But it follows from the general monotonicity properties of the capacity that if and only if for all bounded non-empty subsets of . The latter property is, however, a consequence of the arguments of [RS11a], Proposition 4.2.
Let be a bounded non-empty subset of . Assume there are and such that
for almost all . Further assume there is a Borel measure on and such that
for all and .
If where then .
Thirdly, define a sequence of functions by for all , for and if . Then set . It follows that , and if . Therefore to prove that it suffices to show that .
where we have used the degeneracy bounds (12). If then and . Since this conclusion holds for all small , and , one deduces that . Hence .
If, however, then
where we have used the assumptions and . It follows by combination of these estimates that if then
for all . Therefore and .
The assertion in Theorem 1.1 that the bound suffices to establish Markov uniqueness is now a corollary of Propositions 2.1 and 3.1. First note that for all so the bounds (12) formulated with follow from the similar bounds formulated with . Therefore the upper bound of the degeneracy condition (4) is sufficient to deduce from Proposition 3.1 that for all the subsets with . Then by the monotonicity properties of the capacity. Finally by Proposition 2.1.
Proposition 3.1 can be strengthened by comparison of the capacity as a non-additive measure on and the Hausdorff measure. The argument adapts well known results for the Laplacian on and the classical capacity (see, for example, [EG92], Section 4.7 or [MZ97], Section 2.1.7). This approach was used in [RS11b] Proposition 4.4. In particular if one obtains a bound . But the regularity assumption (13) implies that . Therefore if then and . Hence . If the argument is slightly more intricate. Then the regularity property implies that and this suffices to deduce that by adapting the reasoning of Section 4.7.2 in [EG92] or Theorem 2.52 in [MZ97].
Next we turn to the proof of the converse statement in Theorem 1.1, the assertion that Markov uniqueness of the form implies that . The proof is based on weighted Hardy inequalities which are local versions of the Hardy inequalities given by Theorem 1.4 of [KL09] and Theorem 1.2 of [Leh14]. In conformity with these references we state the following propositions for all although in the current context they are only of interest for the case . We begin with the case where the degeneracy parameter , i.e. the weight exponent, does not exceed . Again for and .
Let be a domain in with boundary . Fix and . Assume there exist and such that
for all .
Then for each and there exists such that the local weighted Hardy inequality
is valid for all .