Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation

We study whether the solutions of a parabolic equation with diffusion given by the fractional Laplacian and a dominating gradient term satisfy Dirichlet boundary data in the classical sense or in the generalized sense of viscosity solutions. The Dirichlet problem is well posed globally in time when boundary data is assumed to be satisfied in the latter sense. Thus, our main results are \emph{a)} the existence of solutions which satisfy the boundary data in the classical sense for a small time, for all H\"older-continuous initial data, with H\"older exponent above a critical a value, and \emph{b)} the nonexistence of solutions satisfying the boundary data in the classical sense for all time. In this case, the phenomenon of loss of boundary conditions occurs in finite time, depending on a largeness condition on the initial data.


Introduction
The present work is a contribution to the study of the qualitative properties of a nonlinear parabolic equation involving nonlocal diffusion. Specifically, we study the occurrence of loss of boundary conditions (LOBC, for short; see Sec. 2 for a precise definition) for the following problem: u t + (−∆) s u = |Du| p in Ω × (0, T ), (1.1) u = 0 in R N \Ω × (0, T ), (1.2) u(x, 0) = u 0 (x) in Ω. (1.3) Here, Ω ⊂ R N is a bounded domain with C 2 boundary, T > 0, and (−∆) s denotes the well-known fractional Laplacian operator, defined as (1.4) (−∆) s u(x, t) = C N,s P.V.
where C N,s is a normalization constant. See [13] for details. In addition to s ∈ (0, 1), we impose the following restrictions on s and p, (1.5) s + 1 < p < s 1 − s .
As with (1.5), (1.6) might not be optimal, and is explained in the context of our Theorem 1.1 (see below). Equation (1.1) can be seen as a generalization of the so-called viscous Hamilton-Jacobi equation, (1.8) u t − ∆u = |Du| p in Ω × (0, T ).
For p = 2, this corresponds to the deterministic Kardar-Parisi-Zhang equation, proposed by these authors in [21] as a model for the profile of a growing interface. Due to its mathematical relevance as a simple model for an equation with nonlinear dependence on the gradient of the solution, (1.8) has been studied from numerous points of view and with different qualitative results in mind: existence and uniqueness of classical solutions ( [16]); existence and nonexistence of global, classical solutions and gradient blow-up and related phenomena ( [31], [32], [34], [23]; see also [26] and the references therein for a broader context); global existence of viscosity solutions, assuming boundary conditions in the viscosity sense ( [4]); and, closest to our work, regarding LOBC, [24]. Some of the previous results have been extended to more general equations, still in the second-order setting: e.g., to degenerate equations in [1], [2]; and, by the authors, to fully nonlinear, uniformly parabolic equations in [25], which the present work closely parallels.
Under the structural assumptions of nondegeneracy of the diffusion and coercivity of the first order term, which are easily shown to be satisfied by (1.1) (see Remark 2.2), and the compatibility condition (1.7), together with the notion of boundary conditions in the viscosity sense, the existence of a unique solution of (1.1)-(1.2)-(1.3) defined globally in time, u ∈ C(Ω × [0, ∞)) ∩ L ∞ (Ω × (0, T )) for all T > 0, is shown in [7], Theorem 7.1. This result is proved by means of a comparison result ( [7], Theorem 3.2), and a subsequent application of Perron's method. See Sec. 2 for a precise definition of the notion of solution employed and further remarks on the application of the results of [7] to our problem.
We remark that there exist certain results concerning the regularity of solutions for problems related to (1.1)-(1.2)-(1.3) (see, e.g., [5], [6]). However, even if they were adapted to our setting, they do not provide the regularity needed for the proof of Theorem 1.2. For this reason we resort to a regularization procedure. See the remarks after the statement of Theorem 1.2.
1.1. Main results. The first of our results concerns local existence, i.e., the existence of solutions which, for small time, satisfiy (1.2) in the classical sense. Due to the results of [7], it suffices to show that the globally defined viscosity solution of (1.1) satisfies (1.2) in the classical sense (see Sec. 2). This is accomplished by a barrier argument, i.e., the construction of a supersolution of (1.1)-(1.2)-(1.3) in a neighborhood of ∂Ω. It is here that the restriction (1.5) comes into play. Consider s ∈ (0, 1) fixed. The upper bound p < s 1−s implies for the critical exponent in (1.6) that β * < s, while the construction of the barrier ultimately relies on computing for β ∈ (0, 2s); more precisely, on the fact that F (β) < 0 for each β ∈ (0, s) (see [11], [27] for details). We note that neither the corresponding local existence result for (1.8) in [16] nor its extension to the fully nonlinear case in [25] require an upper bound for p (or, more generally, for the rate of growth of the gradient nonlinearity). In this sense our result might not be optimal. It is stable, however, in the sense that the restriction disappears as we approach the second-order case, since s 1−s → ∞ as s → 1 − .
The statement of our second result, concerning LOBC, involves the principal eigenfunction of the fractional Laplacian. We denote by (λ 1 , ϕ 1 ) the solution pair for where the solution is normalized so that ϕ then the viscosity solution of (1.1)-(1.2)-(1.3) has LOBC at some finite time prior to T .
The proof of Theorem 1.2 uses a key argument from Theorem 2.1 in [31], sometimes referred to as the "principal eigenfunction method." In adapting this argument to the current setting, the main difficulty is the lack of regularity of solutions. More precisely, we would need solutions to satisfy the equation either in the weak sense, or in some pointwise sense; in the latter case, all terms in the equation must be summable. As mentioned earlier, the existing theory for our problem does not provide such regularity. We note that even in the case of (1.8) (for which LOBC is obtained in [24], among other results), where viscosity solutions are shown to be smooth, some approximation procedure is necessary to "integrate" the equation.
We remedy this problem by using regularization by inf-sup-convolutions, a method introduced in [22]. Afterwards, we require that various estimates related to (1.9) remain uniform with respect to the regularization parameters. In particular, we obtain the stability of solutions to (1.9) with respect to the varying domain (see Subsec. 4.2). For this part we also rely on fundamental estimates for the Dirichlet problem for the fractional Laplacian from [27].
The part of (1.5) that is relevant to Theorem 1.2 is s + 1 < p. This assumption appears only in the crucial Lemma 4.16, which states that a certain negative power (given by p and s) of the principal eigenfunction on an approximate domain is summable. The restriction (1.5) is related to our method of proof. However, a lower bound for p in terms of s is necessary for LOBC to occurr: it is known that solutions of (1.1) satisfying (1.2) in the classical sense for all T > 0 exist for s ∈ [0, 1] and p ≤ 2s (see [3], Theorem 4, for the nonlocal case and [4] for the local case, i.e., s = 1). A natural question which we leave open is whether there is classical solvability or if LOBC occurs when s ∈ [0, 1) and 2s < p ≤ s + 1.
For simplicity, we have restricted our analysis to the case of homogeneous boundary conditions, as in (1.2). Local existence for more general boundary conditions can be obtained in the same way as in Theorem 1.1, following the construction of [11]. Theorem 1.2 applies to the case of general boundary conditions with practically no modification (see Remark 5.1).
The methods of Theorem 1.2 apply to more general nonlinear operators as well, provided they satisfy the nonlocal equivalent of having divergence form (see [30], Sec. 3.6). This restriction is due to the essential use of "integration by parts" in the so-called principal eigenfunction method. For instance, the result can be extended to an equation with diffusion given by the so-called p-fractional Laplacian, defined for s ∈ (0, 1), p > 1 and x ∈ R N as In this case the technical results of Sec. 4 can be reproduced following [20] and [12].

1.2.
Organization of the article. In Sec. 2 we recall the notion of solution from [7], which is used throughout our work, and provide some remarks on the relevant results contained in that work. Sec. 3 is devoted to the proof of Theorem 1.1. In Sec. 4 we provide the technical results required for the proof of Theorem 1.2, which is proved in Sec. 5.
1.3. Notation. We write d : R N , d = d(x) for the distance to the boundary of the set Ω, extended by zero to the whole of R N , i.e., For δ > 0, we write Ω δ = {x ∈ Ω : d(x) < δ}, where d = d(x) is defined as above. Similarly, Ω δ = {x ∈ Ω : d(x) > δ}. However, to avoid confusion, we abstain from using both notations in the same section: e.g., in Sec. 3 we use the notation Ω δ , and from Sec. 4 onwards we use only Ω δ . The closure and boundary operation on sets is performed "after" specifying a subset in terms of the distance: e.g., Ω δ = {x ∈ Ω : d(x) ≤ δ}.
In Sec. 4 and Appendix A we write, for η > 0, d η = d η (x) for the distance to the boundary of Ω η (extended by zero outside this set, as in (1.12)).
Nonnegative constants whose precise value does not affect the argument are denoted collectively by C, and the value of C may change from line to line. When convenient, dependence of C on certain parameters is indicated in parentheses, e.g., as in (1.11). Dependency on Ω, N, s, and p is sometimes omitted for simplicity. Constants we wish to keep track of are numbered accordingly (c 1 , c 2 , . . . , C 0 , C 1 , etc.)
Remark 2.2. To be precise, the equation to which the results of [7] apply is which differs from (1.1) only in the sign of the nonlinearity (here Ω, T , s, and p are as before). This does not in any way complicate the analysis, since a simple sign change allows us to go from one equation to the other; i.e., if u is a subsolution (resp. supersolution) of (2.4), thenũ = −u is a supersolution (resp. subsolution) of (1.1).
Remark 2.3. Definition 2.1 also interprets the initial condition (1.3) in the viscosity sense, given that Ω × {t = 0} (the "bottom" of the domain) is part of the parabolic boundary of Ω × (0, T ). However, as noted in [7], Lemma 4.1, there is no LOBC on this set. That is, if u and v are respectively a bounded, upper-semicontinuous subsolution and a bounded, lower-semicontinuous supersolution of (1.
Similarly, as a consequence of [7], Proposition 4.3, and Remark 2.2 above, there is no LOBC for supersolutions of (1. Remark 2.4. An important consequence of the comparison result of [7] is that solutions of (1.1)-(1.2)-(1.3) are uniformly bounded for all 0 ≤ t ≤ T . Indeed, from the assumptions on the initial data, we have that v ≡ 0 and are respectively sub-and supersolutions to (

Local existence
From the discussion in the Introduction and from Remark 2.3, Theorem 1.1 follows if we can construct a suitable supersolution satisfying (1.2) in the classical sense. To this end we follow the corresponding construction in [11], which addresses a similar (stationary) problem. For convenience, we state the key estimates obtained therein.
Lemma 3.1. Let Ω ⊂ R N be a bounded, C 2 domain and s ∈ (0, 1). Then, there exists a δ > 0 such that, for each 0 < α < s there exists c 1 > 0 such that The constant c 1 depends on N, s and α, and is such that Proof. This is a special case of Lemma 3.1 in [11].
Proof. This is a special case of Lemma 3.3 in [11].
Since v y − u 0 is strictly positive over the compact set Ω\Ω δ , there exists ǫ > 0 such that Recall that the continuous viscosity solution u satisfies (1.3) in the classical sense (see Remark 2.3). This implies that u(·, t) → u 0 as t → 0 + uniformly over Ω. Therefore, there exists T * > 0 such that Applying Lemmas 3.1 and 3.2, and using that d(x) ≤ |x − y|, we have for all sufficiently small δ > 0. On the other hand, using that α < 1 and again that Combining these estimates, we obtain We now take µ > 0 large enough, so that µc 1 − λc 2 > µc1 2 , then take δ > 0 small enough, so that Thus, which gives that v y satisfies (3.1). By standard arguments, the function is a viscosity supersolution of (3.1). It also satisfies (3.3) and (3.4), since these are satisfied by v y for all y ∈ ∂Ω. Furthermore, v is continuous across ∂Ω and, by (3.6), Therefore, applying the comparison principle of [7] over the domain Ω δ × (0, Remark 3.3. Local existence can be proven for initial data with "critical" regularity, i.e., u 0 ∈ C β * (Ω), β * = p−2s p−1 , in exactly the same way, provided [u 0 ] β * ,Ω is sufficiently small. More precisely, we define Proceeding as above, we set µ > 0 so that µc 1 − c 2 > µc1 2 , and require that is satisfied, instead of (3.8). Note that the estimate corresponding to (3.7) is satisfied automatically.

Technical results
4.1. Regularization. In this section we use regularization by inf-sup-convolution, introduced in [22], to obtain a supersolution of (1.1) which has the regularity needed for the proof of Theorem 1.2. This function approximates the viscosity solution u of (1.1) uniformly over Ω × [0, T ] for any T > 0 as the regularization parameters tend to zero.
We can also define v ǫ,κ and v ǫ similarly. Note the use of just one superscript when regularization if performed only in the space variable.
We collect a series of well-known facts regarding these operations which will be used shortly.
(i) Both operations preserve both pointwise upper and lower bounds, i.e., where inf and sup are taken over Ω × (0, T ).
In other words, the sup and inf in the definition of the convolutions are achieved, provided we are at sufficient distance from the boundary. (iii) Both u ǫ,κ and u ǫ,κ are Lipschitz continuous in Similarly, they are Lipschitz continuous in t with constant K √ κ . (iv) u ǫ,κ , u ǫ,κ → u uniformly as ǫ, κ → 0, and similarly for u ǫ .
(vi) With the notation above, The easier proofs follow more or less directly from the definitions (see e.g., [14]), while (vii) and (viii) may be found in [9]; (ix) is Proposition 4.5 in [9]. Property (v) uses the well-known theorems of Rademacher and Alexandrov on the differentiability of Lipschitz and convex functions, respectively; see [15] and the Appendix of [10].
For a given v ∈ C(Ω × [0, T ]), we will obtain a function which is Lipschitz continuous with respect to t and C 1,1 with respect to x, following [9].
First, denote byṽ the lower "0-extension" of v, as defined in (2.3). That is, v = v g with g ≡ 0 (this is only to avoid the notation "v 0 "). For v ∈ C(Ω × [0, T ]) with v| ∂Ω ≡ 0, this leads toṽ ∈ BU C(R N × [0, T ]). We remark that this is precisely what a solution of (1.1)-(1.2)-(1.3) with no LOBC satisfies. We then iterate the convolution operators defined above: where ǫ, δ, κ > 0. The expression furthest to the right follows from (vii) Proposition 4.2, (vii). As a first step we recall that inf-convolution (resp., sup-convolution) preserves supersolutions (resp. subsolutions) in the viscosity sense, albeit in a proper subset of the original domain.  Proof. This is a time-dependent version of Proposition 5.5 in [8], in the particular case where the equation in its entirety is translation invariant, i.e., when there is no "(x, t)-dependence". In this case, the regularized function satisfies exactly the same inequality as the original supersolution. . A simple computation then shows that |ŷ −x| ≤ ǫ * , |ŝ−t| ≤ κ * , as defined in Lemma 4.3. Therefore, to ensure that (ŷ,ŝ) ∈ Ω × (0, T ), so that we can test the equation at this point, we require that d(x) > ǫ * and κ * < t < T − κ * . We also remark that, although slightly different from the one given in Definition 2.1, the notion of solution given in [8] is essentially the same when concerning the behavior of either sub-and supersolutions at interior points (in particular, for the purposes of Lemma 4.3).
We now state a key proposition concerning the eigenvalues of a regularized function.
Proposition 4.5. Let v ∈ BU C(R N ), δ > 0 and suppose that w = (v δ ) δ is differentiable everywhere. If for somex, w(x) < v(x) and w is twice differentiable atx, then D 2 w(x) has − 1 δ as an eigenvalue. Proof. This is Proposition 4.4 in [9], save for the order in which the inf-and supconvolutions are performed. The proof is entirely analogous.

4.2.
The principal eigenvalue problem for the Dirichlet fractional Laplacian. In this section we provide some results regarding the principal eigenvalue problem for the fractional Laplacian on domains approximating Ω, i.e., The existence of a solution pair (λ η 1 , ϕ η 1 ) of (4.5) where λ η 1 > 0, and ϕ η 1 is nonnegative in Ω η and unique up to a multiplicative constant is proved in [29], Proposition 9. The solution obtained in this work is in the weak, or variational, sense. In particular, ϕ η 1 ∈ H s (Ω η ). Furthermore, in [28], Proposition 4, it is proved that ϕ η 1 ∈ L ∞ (Ω η ). We set (4.6) ϕ η 1 ∞ = 1 for all η > 0. From here, it is possible to apply the results of [27] to obtain that ϕ η 1 ∈ C s (R N ). Once the "right-hand side" of (4.5) is continuous, the notions of weak and viscosity solution coincide (see [27], Remark 2.11). Moreover, "bootstrapping" the results contained in [27] (see also [8]), the solution can be shown to be regular enough in the interior of Ω η for (4.5) to hold in a classical, pointwise sense.
Additionally, since ∂Ω η is smooth (see Remark 4.8), it can be shown that as a consequence of Hopf's lemma and the strong maximum principle ( [18]) that Towards the proof of our main result, our aim is to provide estimates that remain uniform with respect to the varying domain (i.e, independent of η). To this end, we recall a few basic facts concerning the geometry of the domains Ω η , η > 0. First, since Ω is C 2 by assumption, there exists an η 0 > 0 such that the distance function d| Ω\Ω 2η 0 is C 2 ; in particular Ω η is C 2 for all η ∈ (0, 2η 0 ).
Remark 4.10. In particular, a C 2 domain satisfies an exterior uniform sphere condition. Furthermore, at any given point of the boundary, the radius of the exterior tangent sphere is bounded by below by the smallest of the radii of curvature, which are equal to the inverses of the principal curvatures. Proposition 4.9 allows us to extend the uniform exterior sphere condition to domains close to Ω in a uniform way. More precisely, there exists a positive constant ρ 0 , depending only on Ω and η 0 , as given by Remark 4.8, such that for all η ∈ (0, η 0 ), and for all y ∈ ∂Ω η , there exists y 1 ∈ R N \Ω η such that B ρ0 (y 1 ) ∩ Ω η = {y}.

Stability of eigenfunctions.
Theorem 4.11. Let η 0 be as in Remark 4.8. Then, there exists C depending only on Ω, N , and s, such that, for all η ∈ (0, η 0 ), the positive solution of (4.5), normalized as above, satisfies Proof. Theorem 4.11 follows readily from the corresponding estimate for the Dirichlet problem. Indeed, we apply Proposition 1.1 from [27] to the solution of (4.5), recalling the normalization (4.6), and obtain for some positive C 2 depending on N , s, and Ω η . For the last inequality we used the fact that Ω η0 ⊂ Ω η for η < η 0 , and therefore λ η 1 ≤ λ η0 1 , by (4.7). It remains only to verify that, once η 0 is fixed, the constant C 2 = C 2 (N, s, Ω η ) in (4.9) (i.e., the constant in Proposition 1.1 from [27]) can be taken uniformly for η ∈ (0, η 0 ). We perform this analysis in Appendix A.
Remark 4.13. The following is a consequence of Lemma 4.12 that will be useful later on: given K ⊂⊂ Ω and η ′ > 0 small enough, there exists a positive constant c, depending on K and η ′ , such that, for all η ∈ (0, η ′ ), ϕ η 1 (x) > c for all x ∈ K.

A uniform Hopf 's lemma.
Lemma 4.14. Let η 0 be as in Remark 4.8. Then, there exists C 3 , depending only on N, s, and Ω, such that, for all η ∈ (0, η 0 ), For a fixed domain, this is contained in Lemma 3.9 in [27]. For completeness, we go into the details of the proof of this result to show that the right-hand side of (4.10) is uniformly bounded. To this end, we also use elements from the proof of the corresponding result for the (more general) case of the fractional p-Laplacian in [20] (Theorem 3.6). Additionally, we employ Proposition 4.9.
This, together with (4.15), (4.16) and (4.17), implies that . . , κ η N −1 ). Combining the above estimates with (4.12) and Proposition 4.9, we conclude that f η ∞ is uniformly bounded by a constant that depends only on N, s, and Ω; more specifically, on N, s, and the the principal curvatures of ∂Ω.
The following computation is adapted from [12].  N, s, and Ω, such that for all η ∈ (0, η 0 ), the solution of (4.5) satisfies Proof. Write K 0 = Ω η0 for short and define, for where χ K0 denotes the characteristic function of K 0 . Note that v ∈ U SC(R N ) and v ∈ C(R N \K 0 ). From Lemma 4.14, for where C 3 is the constant from Lemma 4.14 and C(K 0 ) > 0. Hence, for sufficiently large A, (−∆) s v(x) ≤ 0.
Using the change of variables Ψ η ∈ C 2 (R N , R N ) introduced in Lemma 4.14, together with a covering argument and the estimates provided therein, we have By assumption (1.5), − s p−1 > −1, hence this last integral is finite. On the other hand, using that Ω η0 ⊂⊂ Ω, together with Lemma 4.12 and Remark 4.13, we have ϕ η 1 (x) > c 4 > 0 for all x ∈ Ω η0 , where c 4 depends only on Ω, N and s (through η 0 ). Hence,

Nonexistence of global solutions and LOBC
As in our previous work [25], the proof of Theorem 1.2 uses key ideas from that of Theorem 2.1 in [31]. For completeness, we reproduce some of the elements of [25]. We remark that some care is required in choosing certain parameters appearing in our argument in the correct order, a difficulty which is not present in [31]. Specifically, we first choose u 0 large in an appropriate sense, then take η (which depends on u 0 and the regularization parameters of Sec. 4) sufficiently small.
Proof of Theorem 1.2. Consider the differential inequality where C, M 0 > 0. We can integrate (5.1) explicitly to obtain . Since 1 − p < 0, this implies y(t) → +∞. Alternatively, for a fixed t 1 > t 0 , blow-up occurs for t < t 1 provided we have So fix T > 0 and assume that the viscosity solution u of (1.1) satisfies (1.2) in the classical sense. We will later specify the largeness condition on u 0 in terms of M 0 above, but may consider it set from now on, since it depends only on constants already available. The constant C in (5.1) is also specified later, depending only on the appropriate quantities. In particular, it is independent of both η and u 0 .
Recall the approximate equation obtained in Proposition 4.6, where now w is obtained by regularization of the viscosity solution u of (1.1). Define z(t) = Ω η w(x, t)ϕ η 1 (x) dx, where ϕ η 1 is the unique positive solution of (4.5) normalized so that ϕ η 1 ∞ = 1. From Proposition 4.2 (iv) and Remark 2.4, we obtain, for sufficiently small η, that w ∞ ≤ u ∞ + 1 ≤ u 0 ∞ + 1. Thus z(t) is uniformly bounded for 0 ≤ t ≤ T . In what remains of the proof, we show that z satisfies (5.1) by using the assumption that the solution u satisfies (1.2) in the classical sense, and hence blows-up, a contradiction.
Since w(·, t) ∈ C 1,1 (R N ) for all t ∈ [t 0 , t 1 ], (−∆) s w(·, t) is classically defined a.e. in Ω, precisely at the points where w has a second order expansion. Moreover, at such points we have that the integral with respect to y converges as ω → 0 and, by the standard computation, Hence by the dominated convergence theorem, We note that the integrand is no longer singular, and write dV for the measure on R 2N . Applying first Fubini's theorem, and then by symmetry (i.e., interchanging x and y), we have Starting from the last integral, we repeat the above computation "in reverse" to pass the operator onto ϕ η 1 and use the associated eigenvalue problem (4.5). Since now w ≡ 0 in R N \Ω, this gives Using that w, ϕ η 1 ≥ 0, and again that ϕ η Therefore, that p > 1, hence z(t) p is the dominating term in the right-hand side of (5.7). Therefore, taking and η sufficiently small ensures that bothż(t) ≥ C5 2 z(t) p for t > t 0 and that z(t 0 ) ≥ M 0 , which together are equivalent to (5.1). This gives the desired contradiction.
Remark 5.1. Given the indirect nature of the preceding proof, we would like to highlight the role played by the main assumptions leading to LOBC: the fact that the gradient term is "dominating" in the equation, i.e., p > s + 1, from (1.5), is used only in Lemma 4.16. On the other hand, the assumption that leads to contradiction, that (1.2) is satisfied in the classical sense, is used only in (5.6) and in applying Poincaré's inequality.
The case of more general boundary conditions can be treated in exactly the same way as above. Assuming u = g in R N \Ω × (0, T ) is satisfied in the classical sense, with g ∈ C b (R N \Ω × (0, T )), we obtaiṅ z(t) ≥ −λ η 1 z(t) + Cz(t) p − C 6 , instead of (5.7), where C 6 depends on g L ∞ (∂Ω×(0,T )) . From here on, the proof continues as above.
Remark 5.2. Assuming higher regularity for the initial data, e.g., u 0 ∈ C 2 (Ω) it is possible to obtain estimates for u t (see, e.g., [33], Proposition 4.1, for an example of this method in the local setting). This allows the application of regularity results available for stationary problems (e.g, those of [7]) to our problem, essentially by treating u t as a bounded "right-hand side". Global Hölder estimates can then be obtained for the solution of (1. This situation is analogous to that of gradient blow-up for (1.8).
Appendix A. Uniform C s regularity for the approximate domains In this Appendix we state a version of results from [27] which concern the regularity of solutions to the Dirichlet problem for the fractional Laplacian. We revisit the corresponding proofs to show that the estimates are uniform with respect to varying domains such as those appearing in (4.5). In this way we conclude the analysis postponed in the proof of Theorem 4.11.
Proposition A.1. Let u be a solution of (A.1). Then u ∈ C s (R N ) and where C is a constant depending only on Ω and s. In particular, the constant C can be taken uniformly for η ∈ (0, η 0 ). This is Proposition 1.1 from [27], save for the dependency of C on the parameter η, which we require to be uniform. To this end, we outline the manner in which this result was obtained. We begin stating the key Lemmas leading up to it.
Lemma A.2. Assume that w ∈ C ∞ (R N ) is a solution of (−∆) s w = h in B 2 . Then, for every β ∈ (0, 2s), where the constant C depends only on N, s and β.
Remark A.4. Lemmas A.2 and A.3 have no dependence on the domains Ω nor Ω η . Therefore, they apply directly to our setting.
Let Ω η and g be as above, and let u be the solution of A.1. Then where C depends only on Ω and s. In particular, C can be taken uniformly in η ∈ (0, η 0 ).
Lemma A.5 relies on the following result: Lemma A.6. Let Ω be a bounded domain and let g ∈ L ∞ (Ω η ). Let u be the solution of (A.1). Then where C is a constant depending only on N and s.
Proof of Lemma A.5: This is Lemma 2.7 in [27]. For points near ∂Ω η , the estimate is obtained by scaling the supersolution from Lemma A.3 to the annular region B 2ρ0 \B ρ0 , where B ρ0 is an exterior tangent ball to ∂Ω, and applying comparison. Owing to Remark 4.10, the scaling can be done uniformly with respect to η ∈ (0, η 0 ). For the remaining points in Ω η , Lemma A.6 is employed.
Proof. This is a special case of Lemma 2.9 in [27]. Although more intricate than that of the previous results, the proof of this result uses only a scaling of the interior estimate of Lemma A.2 to the ball B R (x 0 ), the use of the upper barrier for u L ∞ (Ω) obtained in A.5, and a covering argument. As such, the constant C in Lemma A.7 now depends on the measure of Ω as well. This quantity, however, varies continuously for Ω η with η ∈ (0, η 0 ).
Proof of Proposition A.1: It remains only to extend the estimate from Lemma A.7 up to the boundary. For this we provide an argument from [20]. Through a covering argument, Lemma A.7 extends to an interior bound on any compact subset of Ω η .