ANALYSIS AND BOUNDARY VALUE PROBLEMS ON SINGULAR DOMAINS: AN APPROACH VIA BOUNDED

GEOMETRY

BERND AMMANN, NADINE GROSSE, AND VICTOR NISTOR

Abstract. We prove well-posedness and regularity results for elliptic boundary value problems on certain singular domains that are conformally equivalent to manifolds with boundary and bounded geometry. Our assumptions are satisfied by the domains with a smooth set of singular cuspidal points, and hence our results apply to the class of domains with isolated oscillating conical singularities. In particular, our results generalize the classical L^{2}-well-posedness result of Kondratiev for the Laplacian on domains with conical points. However, our domains and coefficients are too general to allow for singular function expansions of the solutions similar to the ones in Kondratiev’s theory. The proofs are based on conformal changes of metric, on the differential geometry of manifolds with boundary and bounded geometry, and on our earlier geometric and analytic results on such manifolds.

Version française abrégée. Nous prouvons des résultats de solvabilité et régularité pour des systèmes satisfaisants la condition de Legendre forte avec conditions au bord mixtes de type Dirichlet-Neumann sur certains domaines singuliers. Notre classe de domaines singuliers contient la classe des domaines avec des singularités coniques isolées. Nos résultats généralisent ainsi le théorème d’isomorphisme de Kondratiev [17]. Dans la suite, M sera une variété riemannienne lisse à bord de dimension m et E → M sera un fibré vectoriel hermitien équipé d’une connexion. Pour nos résultats, nous allons aussi supposer que M ait une géométrie bornée. Soit a une forme sesquilinéaire lisse sur T^{∗}M ⊗ E et P_{a}: H^{1}(M; E) → H^{1}(M; E)^{∗} défini par la formule ⟨P_{a}u, v⟩ := ∫_{M} a(∇u, ∇v)d vol_{g}, pour u, v ∈ H^{1}(M; E). Nos espaces de fonctions seront les espaces de Sobolev pondérés de type Kondratiev, voir Équation (1). Nous supposons donée une partition ∂M = ∂_{0}M ⊔ ∂_{1}M du bord en deux sous-ensembles disjoints et ouverts, ainsi que des conditions au bord B_{j} d’ordre j, B_{j} sur ∂_{j} M. Nos résultats sont alors:

(i) P satisfait la régularité dans les espaces pondérés fK_{(ρ)}^{ℓ,2} de l’équation (1) si, et seulement si les conditions au bord B = (B_{0}, B_{1}) satisfont la condition de régularité de Shapiro-Lopatinski uniforme. Ces conditions sont satisfaites pour les opérateurs satisfaisant les conditions de Legendre fortes avec des conditions au bord mixtes (Dirichlet/Neumann). On obtient en particulier des résultats de régularité pour l’opérateur de Laplace avec conditions au bord mixtes, Théorème 6.

B.A. and N.G. have been partially supported by SPP 2026 (Geometry at infinity), funded by the DFG. B.A. has also been partially supported by the DFG SFB 1085 (Higher Invariants). V.N.

has been partially supported by ANR-14-CE25-0012-01.

AMS Subject classification (2010): 58J32 (primary), 35J57, 58J32, 35B65.

(ii) Si, en plus des conditions de (i), P satisfait une inégalité de Hardy-Poincaré,

alors le problème au bord associé à P est également bien posé. En principe, la classe des domaines à laquelle nos résultats s’appliquent est assez large, mais pour des raisons d’espace et afin de réduire au minimum les détails techniques, nous considérons dans cette note principalement les exemples de domaines cuspidaux et wedge. L’ensemble des points singuliers de ces domaines est une sous-variété lisse compacte.

Introduction. We prove well-posedness and regularity results for systems of partial differential equations satisfying the strong Legendre condition with mixed Dirichlet-Neumann boundary conditions on certain singular domains. Our class of singular domains includes the class of domains with isolated conical singularities and thus they generalize the classical well-posedness result of Kondratiev [17]. Unlike Kondratiev’s theory, singular functions expansions are not possible in our setting.

Let us briefly state our main result. Here are first, our assumptions. Throughout this paper, (M, g) will be a smooth, m-dimensional Riemannian manifold with boundary and E → M will be a hermitian vector bundle with connection ∇ such that its curvature R^{E} and all its covariant derivatives ∇^{j} R^{E}, j ≥ 1, are bounded. For our results, we shall also assume that M has bounded geometry (a concept recalled below, see, however [4, 5, 7, 14] for the concepts not recalled in this paper). Let a be a smooth, sesquilinear form on T^{∗}M ⊗E and P_{a}: H^{1}(M; E) → H^{1}(M; E)^{∗} be defined by the formula ⟨P_{a}u, v⟩ := ∫_{M} a(∇u, ∇v)d vol_{g}, for u, v ∈ H^{1}(M; E). Recall that if a is uniformly positive definite, then a is said to satisfy the strong Legendre condition. If P = P_{a} + Q, where Q is of order ≤ 1, we shall say that P satisfies the strong Legendre condition if, and only if, a does. This implies that P is strongly elliptic. For scalar operators, the condition that P satisfies the strong Legendre condition is actually equivalent to P being uniformly strongly elliptic. A smooth function f: M → (0, ∞) will be called an admissible weight if f^{−1} df has bounded covariant derivatives of all orders. Let f and ρ be admissible weights on M. If g_{0} := ρ^{2}g and ∇_{0} is the Levi-Civita connection associated to g_{0}, we can describe our function spaces as the following Kondratiev-type weighted Sobolev spaces

fK_{(ρ)}^{ℓ,p}(M, g_{0}; E) := {ψ | ρ^{j} ∇_{0}^{j}(f^{−1} ψ) ∈ L^{p}(M, g_{0}; T^{∗}M^{⊗j} ⊗ E), (∀) j ≤ ℓ }.  (1)

In our applications and in some of our results, the weight ρ is bounded. For simplicity, we shall assume this throughout the paper. We also assume that we have a partition ∂M = ∂_{0}M ⊔ ∂_{1}M of the boundary in two disjoint, open subsets. We assume that we are given boundary conditions B_{j} of order j, B_{j} on ∂_{j} M, satisfying the boundedness and smoothness conditions stated before Theorem 6. Our results are then as follows (for (M, g) with bounded geometry):

(i) P satisfies regularity in the weighted spaces fK_{(ρ)}^{ℓ,2} of Equation (1) if, and only if B = (B_{0}, B_{1}) satisfies the uniform Shapiro-Lopatinski regularity conditions. These conditions are satisfied for operators satisfying the strong Legendre conditions with mixed (Dirichlet/Neumann) boundary conditions. In particular, the Laplace operator satisfies regularity for mixed boundary conditions, Theorem 6.

(ii) If, in addition to the conditions of (i), P satisfies a Hardy-Poincaré inequality, then P also satisfies a well-posedness result. We provide several examples of how to obtain the Hardy-Poincaré inequality.

In principle, the class of domains to which our results apply is pretty large, but for reasons of space and in order to keep the technicalities to a minimum, we mostly consider the examples of canonical cuspidal and wedge domains introduced by H. Amann [1], whose definition is recalled below. The set of singular points of such a domain is smooth and compact (without corners). It is even a finite set for cuspidal domains. Some very general and nice results were obtained in [12, 18] for certain domains with isolated point cusp singularities. Our methods are quite different, relying heavily on differential geometry, and thus allowing us to treat a large class of domains. Moreover, our coefficients are less regular than the ones in the references, but we lose the Fredholm properties and the singular function expansions obtained in [12, 18]. Algebras of pseudodifferential operators on manifolds with cuspidal points were considered in [25]. The index problem on such manifolds was considered in [19]. We thank Herbert Amann for useful comments.

Manifolds with boundary and bounded geometry. In this paper, (M, g) will always be a smooth, m-dimensional Riemannian manifold with boundary and E → M will be an vector bundle with metric and metric preserving connection. A smooth function f: M → (0, ∞) will be called a g-admissible weight if f^{−1} df has bounded covariant derivatives of all orders. We shall say that E has totally bounded curvature if its curvature and all of its covariant derivatives are bounded. We endow T M with the Levi-Civita connection ∇ associated to g. Recall that M is said to have bounded geometry if its injectivity radius r_{inj} (M) > 0 is positive and if T M has totally bounded curvature. We assume from now on that E is complex and it has totally bounded curvature.

Let us consider a codimension one submanifold H ⊂ M (i.e. hypersurface). Assume that H carries a globally defined unit normal vector field ν and let exp^{⊥} (x, t) := exp_{x}^{M} (tν_{x}) be the exponential in the direction of the chosen unit normal vector. By II^{H} we denote the second fundamental form of H. The following two definitions are from [5].

Definition 1. Let (M̂, ĝ) be a Riemannian manifold with bounded geometry. We say that H ⊂ M̂ is a bounded geometry hypersurface in M if it is a closed subset of M, if ∥∇^{k}II^{H} ∥_{L^{∞}} < ∞ for all k ≥ 0, and if here is r_{∂} > 0 such that exp^{⊥} : H × (−r_{∂}, r_{∂}) → M̂ is a diffeomorphism onto its image.

Definition 2. A Riemannian manifold (M, g) with boundary is said to have bounded geometry if there is a Riemannian manifold M̂ with bounded geometry containing M as an open subset such that ∂M is a bounded geometry hypersurface in M̂.

Remark 3. In [1], Amann has introduced the class of “singular manifolds.” A singular manifold (M, g_{0}, ρ) is a Riemannian manifold with boundary (M, g_{0}) together with a singularity function ρ satisfying suitable properties. In particular, (M, ρ^{−2} g_{0}) is assumed to be a manifold with boundary and bounded geometry. Conversely, if (M, g) has bounded geometry and ρ is a g-admissible weight, then (M, g_{0} := ρ^{2}g) is a singular manifold with singularity function ρ. In the boundaryless case, this was first noticed in [13] (see also [2].) The singularity function ρ is seen to be a gadmissible weight. For manifolds with boundary, this result follows from [5] or [15]. The results of [1] apply therefore to the setting of manifolds with boundary and bounded geometry endowed with an admissible weight. A triple (M, g, ρ) consisting of a manifold with boundary and bounded geometry and a bounded g-admissible weight ρ will be called an Amann triple.

Conformal changes of metric. If h_{1}, h_{2} : X → (0, ∞), we shall write h_{1} ∼ h_{2} if h_{1}/h_{2} and h_{2}/h_{1} are both bounded. Let g_{0} be a second Riemannian metric on M, whose Levi-Civita connection is denoted ∇_{0}. Let ρ, f: M → (0, ∞) be measurable functions and p ∈ [1, ∞]. Recall then from Equation (1) the definition of the spaces fK_{(ρ)}^{ℓ,p}(M, g_{0}; E), which reduce to the usual Sobolev spaces if ρ, f ∼ 1. More precisely, if g := ρ^{−2} g_{0} and if ρ is a g-admissible weight and f is continuous, then the weighted and classical spaces are related by

fK_{(ρ)}^{ℓ,p}(M, g_{0}; E) = fρ^{−m/p} W^{ℓ,p}(M, g; E), 1 ≤ p ≤ ∞.  (2)

(see [1, 3, 6] and Remark 3). We drop the superscript p for p = 2: K_{(ρ)}^{ℓ}(M, g_{0}; E) :=

K_{(ρ)}^{ℓ,2}(M, g_{0}; E) and so on. We assume from now on that g = ρ^{−2} g_{0}.

Example 4. A typical example is when M ⊂ ℝ^{m} is the closed unit ball, g_{0} is the euclidean metric, and ρ = r^{λ}, where r is the distance to the origin. Then (M, g := ρ^{−2} g_{0}) has bounded geometry if, and only if, λ ≥ 1. Moreover,

f := \{ \begin{matrix}e^{−( \frac{r}{ϵ} )^{−ϵ}} , & if λ = 1 + ϵ > 1, \\ r = ρ, & if λ = 1. \end{matrix}

is a g-admissible weight. This example is adapted to a domain with conical points (for instance, a polygonal domain) with set of vertices V by taking f(x) = ρ(x) := ∏ _{P ∈V} |x − P | and λ = 1. The extra weight f becomes then unnecessary (for λ = 1) and the weighted Sobolev spaces K_{(ρ)}^{ℓ}(M) := {u | ρ^{|α|} ∂^{α} u ∈ L^{2}(M), (∀) |α| ≤ ℓ } are the spaces introduced by Kondratiev [17].

We shall assume from now that (M, g, ρ) is an Amann triple (see Remark 3) and that f: M → (0, ∞) be a second g-admissible weight. In particular, ρ is a bounded g-admissible weight. We have seen in Equation (2) how the Sobolev spaces change with respect to conformal changes of metric. Recall that g_{0} = ρ^{2}g. For differential operators, a simple calculation based on L^{∞} (M; E ⊗ T M^{⊗p} ⊗ T M^{∗⊗q} , g) = ρ^{p−q} L^{∞} (M; E ⊗ T M^{⊗p} ⊗ T M^{∗⊗q} , g_{0}) and the fact that ρ is bounded gives:

Lemma 5. Let P be an order k differential operator on M and P_{1} := ρ^{k}P. We have that P satisfies the strong Legendre condition with respect to the metric g_{0} if, and only if, P_{1} satisfies the strong Legendre condition with respect to the metric g = ρ^{−2} g_{0}. If P has coefficients in W^{∞,∞} (g_{0}), then P_{1} has coefficients in W^{∞,∞} (g).

A similar result is valid for the boundary differential operators appearing as boundary conditions.

Regularity and well-posedness. Let P be a second order differential operator. We assume from now on that we are given a partition of the boundary ∂M = ∂_{0}M ⊔ ∂_{1}M into two disjoint, open subsets, as in [5], and order i differential boundary conditions B_{i} on ∂_{i}M. See, for example, [7, 20] for general results on boundary value problems on smooth domains, [11, 18, 23, 22] for the case of non-smooth domains, and [4, 14] for more general boundary conditions involving projections. We assume that ρ^{2}P, ρB_{1}, and B_{0} have coefficients in W^{∞,∞} (g). The typical assumption is that P, B_{1}, and B_{0} have coefficients in W^{∞,∞} (g_{0}), which means that they “stabilize” towards the singular points, as in [18], and this is a necessary condition for the existence of singular function expansions. In view of Lemma 5, our assumptions are thus weaker, but singular functions expansions are no longer available in general in our setting. Our more general setting may be needed in applications to nonlinear PDEs and uncertainty quantification. Also, recall from [14] the uniform Shapiro-Lopatinski regularity conditions and that they are invariant with respect to conformal changes of metric. Combining this property with Equation (2) and with Lemma 5, we get:

Theorem 6. Let P be a g_{0}-uniformly elliptic second order differential operator acting on sections of E → M and B = (B_{0}, B_{1}) be a boundary differential operator. We assume that P and B satisfy the g_{0}-uniform Shapiro-Lopatinski regularity conditions. Then, for any ℓ ∈ ℕ, there exists C > 0 such that, for all u ∈ fK_{(ρ)}^{1}(M, g_{0}; E)

∥u∥_{fK_{(ρ)}^{ℓ+1} (M,g_{0} ;E)} ≤ C( ∥P u∥_{fρ^{−2} K_{(ρ)}^{ℓ−1} (M,g_{0} ;E)} + ∥u∥_{fK_{(ρ)}^{1} (M,g_{0} ;E)}

+ ∥B_{0}u∥_{fρ^{− \frac{1}{2}} K_{(ρ)}^{ℓ+ \frac{1}{2}} (∂_{0} M,g_{0} ;E)} + ∥B_{1}u∥_{fρ^{− \frac{3}{2}} K_{(ρ)}^{ℓ− \frac{1}{2}} (∂_{1} M,g_{0} ;E)} ).

In particular, we can take P to be a uniformly strongly elliptic scalar operator (such as the Laplacian P = Δ_{g_{0}} ), B_{0}u = u|_{∂_{D} M} (the restriction) and B_{1}u = ∂_{ν}^{a}u.

Let (M, g_{0}) be a Riemannian manifold with boundary. We now turn to the wellposedness on (M, g_{0}). Let h: M → (0, ∞), and A ⊂ ∂M be a measurable subset. We shall say that (M, A, E, g_{0}, h) satisfies the Hardy-Poincaré inequality if there exists a constant C > 0 such that, for any u ∈ H_{loc}^{1}(M, g_{0}; E), u = 0 in L^{2}(A), we have ∫_{M} |du|_{g_{0}}^{2} dvol_{g_{0}} ≥ C ∫_{M} h^{−2} u^{2} dvol_{g_{0}} . The Hardy-Poincaré inequality implies coercive estimates, and hence well-posedness also for the associated parabolic and hyperbolic equations, as in [20]. The Hardy-Poincaré inequality is related to the Poincaré inequality, and hence to the concept of “finite width.” If A ⊂ ∂M, recall that (M, A) is said to have finite width if dist(x, A) is uniformly bounded on M [5] (the distance between two disjoint connected components of M is +∞). Typically in our results, the set A will be an open and closed subset of ∂M.

Example 7. Again, a typical application is when g_{0} is the euclidean metric on ℝ^{m}, r is the distance to the origin, and λ > 0, as in Example 4. However, in this case, we let ρ = r^{λ} only for r < 1/2, but set ρ = r for r > 1 and M ⊂ ℝ^{m} is a closed, infinite cone with base a smooth domain of the unit sphere and with vertex at the origin. Then again, (M, g) has bounded geometry if, and only if, λ ≥ 1. Also, (M, ∂M, g) has finite width if, and only if λ = 1. Finally, (M, ∂M, g_{0}, ρ) satisfies the Hardy-Poincaré inequality (for ρ) if, and only if, λ ≤ 1.

Recall that we have assumed (M, g) to be a Riemannian manifold with boundary and bounded geometry, ρ, f: M → (0, ∞) to be g-admissible weights, and g_{0} := ρ^{2}g. We define P_{a} by (P_{a}u, v)_{g_{0}} := ∫_{M} a(∇u, ∇v)d vol_{g_{0}} , with a sesquilinear form a satisfying the strong Legendre condition with respect to g_{0}. Let ∂_{ν}^{a} be the conormal derivative associated to P, see [14]. Combining Theorem 6 with the Lax-Milgram Lemma and with the fact that the Dirichlet and Neumann boundary conditions satify the uniform Shapiro-Lopatinski regularity conditions [5, 14], we obtain:

Theorem 8. We assume that (M, ∂_{0}M, E, g_{0}, ρ) satisfies the Hardy-Poincaré inequality. Let P = P_{a} satisfy the strong Legendre condition with all ∇^{j} a bounded.

Then there exists η_{a,f} > 0 such that, for |s| < η_{a,f} and ℓ ≥ 1, we have an isomorphism

P_{a}: ρf^{s}K_{(ρ)}^{ℓ+1} (M, g_{0}; E) ∩ {u|_{∂_{0} M} = 0, ∂_{ν}^{a}u|_{∂_{1} M} = 0} → ρ^{−1} f^{s}K_{(ρ)}^{ℓ−1} (M, g_{0}; E).

In particular, we can take P = Δ_{g_{0}} , the Laplacian associated to g_{0}. For ℓ = 0 the result remains true, once one reformulates it in a variational (i.e. weak) sense.

Examples. We include some basic examples.

Two dimensional domains. We consider a (disjoint) partition of the boundary ∂M = ∂_{0}M ⊔ ∂_{1}M as above (so ∂_{0}M and ∂_{1}M are open and closed). Recall that P_{a} is a second order differential operators on E → M with coefficients in W^{∞,∞} (g) and satisfying the strong Legendre condition with respect to g_{0}. For dimension two domains M, the Poincaré inequality for (M, A, g) is equivalent to the Poincaré inequality for (M, A, g_{0}) (same proof as the conformal invariance of the Laplacian in two dimensions). The Poincaré inequality of [5] then gives:

Theorem 9. Assume that (M, ∂_{0}M, g) has finite width and m := dim(M) = 2. Let P = P_{a} satisfy the strong Legendre condition with all ∇^{j} a bounded. Then there exists η_{a,f} > 0 such that, for |s| < η_{a,f} and ℓ ∈ ℤ_{+} , we have an isomorphism

P_{a}: ρf^{s}K_{(ρ)}^{ℓ+1} (M, g_{0}; E) ∩ {u|_{∂_{0} M} = 0, ∂_{ν}^{a}u|_{∂_{1} M} = 0} → ρ^{−1} f^{s}K_{(ρ)}^{ℓ−1} (M, g_{0}; E).

In particular, we can take P_{a} = Δ_{g_{0}} .

Canonical cuspidal and wedge domains. We continue with some concrete examples. The simplest examples in higher dimensions are those of “model cuspidal and wedge domains.” We follow the presentation in [1]. Let 1 < α < ∞ and B ⊂ ℝ^{m−1} a compact submanifold, possibly with boundary, and

K_{α}^{m}(B) := {(r, r^{α} y) ∈ ℝ^{m} | 0 < r < 1, y ∈ B},  (3)

which will be called a model canonical cusp of order α. For α = 1, we take B a subset of the unit sphere. A domain with canonical cuspidal singularities is a bounded domain Ω ⊂ M̂ in a Riemannian manifold (M̂, g_{0}) such that, around each singular point P of the boundary, it is locally diffeomorphic to K_{α_{P}}^{m} (B_{P}) via a diffeomorphism defined in a neighborhood of the ambient manifold. Let V be the set of singular points of the boundary, then V is finite and we let M := \overline{Ω} ∖ V. If α_{P} = 1 for all P ∈ V, we obtain a domain with conical points. If we replace K_{α}^{m}(B) with K_{α}^{m−k} (B) × [0, 1]^{k}, k ≥ 0, we obtain domains with canonical wedge singularities, in which case, of course, the set V of singular points of ∂Ω will no longer be finite.

Let us fix λ_{P} ≥ 1 for each singular point P ∈ V. The weight functions ρ and f are then chosen, around each P ∈ V as in Example 4 for λ = λ_{P}. Let g := ρ^{−2} g_{0}, as before. If λ_{P} ≥ α_{P} for all P, then (M, g) has bounded geometry (proved in [2] if λ_{P} = α_{P} for all P ∈ V) and consequently, we have regularity in the weighted spaces for the mixed Dirichlet-Neumann problem for operators satisfying the strong Legendre condition. If λ_{P} ≤ α_{P} and ∂_{0}M intersects each V_{P}, then (M, ∂_{0}M, g) satisfies the Hardy-Poincaré inequality. This follows from the usual Poincaré inequality on each {r} × r^{α} B by also rescaling in r. We work in generalized spherical coordinates (r, y) ∈ (0, ∞) × S^{m−1} , so dx = r^{m−1} drdy. This gives r^{−2α} ∫_{r^{α} B} |u(r, y)|^{2}dy ≤ C ∫_{r^{α} B} |∇_{y}u(r, y)|^{2}dy the Hardy-Poincaré inequality ∫_{K_{α}^{m} (B)} r^{−2α} |u(r, y)|^{2}dx ≤ C ∫_{K_{α}^{m} (B)} |∇u(r, y)|^{2}dx for u = 0 on ∂_{1}B. Let us fix λ_{P} = α_{P}. By considering the Amann triple (M, g := ρ^{−2} g, ρ), we obtain that our domain with canonical wedge singularities satisfies the conclusion (isomorphism) of Theorem 9. For canonical cuspidal domains and constant coefficient operators, this theorem was first proved in [18]. See also [9, 10, 12, 16, 21].

Other examples. Certain simple examples are not “canonical.”

Example 10. Let Ω = {(x, y) ∈ ℝ^{2} | x, y > 0, (y − 1)^{2} + x^{2} > 1} ∩ [−2, 2] × [0, 2] (see the picture). The corners {A_{i}}_{i=1}^{4} of Ω are conical points and are treated as above.

Close to O we have two cuspidal open sets (U and its mirror image), similar to the ones treated in the previous subsection, but not canonical. We have λ = 2 for these open sets. (We double O, in a certain sense.) Let r be the distance to O. Close to O, we then choose ρ ∼ r^{2}, and f ∼ e^{−r^{−1}} . More precisely, ρ(x) = r^{2} ∏ _{i=1}^{4} |x − A_{i}| and f = e^{−r^{−1}} ∏ _{i=1}^{4} |x − A_{i}|. Note that we could have also chosen ρ ∼ x^{2} near O.

\begin{matrix}Ω \\ O \\ U \\ x \\ y \end{matrix}

Let g := ρ^{−2} g_{E}, where g_{E} is the standard (flat) Euclidean metric. We then have that (M, g), M := \overline{Ω} ∖ \{O, A_{i}, i = 1, 4\}, is a manifold with boundary and bounded geometry. Assume that ∂_{0}M touches each singular point (where O is considered as a double point as above). We can then prove that (M, ∂_{0}M, g_{E}) satisfies the Hardy-Poincaré inequality as in the previous example, and hence Theorem 9 applies. See also [16, 21].

Example 11. Let f_{0}, f_{1} : ℝ → (0, 2π) satisfy ∥f_{i}^{(k)}∥_{∞} < ∞, k ≥ 0, and f_{1} −f_{0} ≥ ϵ >

0. Let Ω_{f_{0} ,f_{1}} := {(r cosθ, r sinθ) | f_{0}(logr) < θ < f_{1}(logr)} and Ω be a bounded domain, smooth away from a finite number of points, at which it coincides, up to a diffeomorphism, with a neighborhood of 0 in a set of the form Ω_{f_{0} ,f_{1}} . Then Ω is a domain with oscillating conical singularities, similar to the ones studied by [24]. Theorem 9 holds for this domain with ρ = f = r = (x^{2} + y^{2})^{1/2} . In general, a domain with oscillating conical points is not a domain with conical points.

Finally, in [8], it was proved that the assumptions and the conclusions of Theorem 8 are fulfilled by any polyhedral domain Ω ⊂ ℝ^{m} (defined as a suitable stratified space) with the Euclidean metric g_{0} for a suitable g-admissible weight ρ ∼ the distance to the singular points of the boundary (g = ρ^{−2} g_{0}).

References

[1] Amann, H. Function spaces on singular manifolds. Math. Nachr. 286, 5-6 (2013), 436–475.

[2] Amann, H. Uniformly regular and singular Riemannian manifolds. In Elliptic and parabolic equations, vol. 119 of Springer Proc. Math. Stat. Springer, Cham, 2015, pp. 1–43.

[3] Amar, E. Sobolev embeddings with weights in complete riemannian manifolds. Preprint (arxiv:1803.07811.

[4] Ammann, B., Große, N., and Nistor, V. The strong Legendre condition and the wellposedness of mixed Robin problems on manifolds with bounded geometry. Accepted in: Revue Roumaine de Mathématiques Pures et Appliquées (2018). Special issue “Spectral theory and applications to mathematical physics.” arXiv:1810.06926 .

[5] Ammann, B., Große, N., and Nistor, V. Well-posedness of the Laplacian on manifolds with boundary and bounded geometry. Math. Nachr. (2019)

    https://doi.org/10.1002/mana.201700408 (Online first).

[6] Ammann, B., Ionescu, A. D., and Nistor, V. Sobolev spaces on Lie manifolds and regularity for polyhedral domains. Doc. Math. 11 (2006), 161–206 (electronic).

[7] Brezis, H. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.

[8] Băcuţă, C., Mazzucato, A., Nistor, V., and Zikatanov, L. Interface and mixed boundary value problems on n-dimensional polyhedral domains. Doc. Math. 15 (2010), 687–745.

[9] Cardone, G., Nazarov, S. A., and Sokolowski, J. Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary. Asymptot. Anal. 62, 1-2 (2009), 41–88.

[10] Costabel, M., Dauge, M., and Nicaise, S. Analytic regularity for linear elliptic systems in polygons and polyhedra. Math. Models Methods Appl. Sci. 22, 8 (2012), 1250015, 63.

[11] Dauge, M. Elliptic boundary value problems on corner domains, vol. 1341 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions.

[12] Dauge, M. Strongly elliptic problems near cuspidal points and edges. In Partial differential equations and functional analysis, vol. 22 of Progr. Nonlinear Differential Equations Appl. Birkhäuser Boston, Boston, MA, 1996, pp. 93–110.

[13] Disconzi, M., Shao, Y., and Simonett, G. Some remarks on uniformly regular Riemannian manifolds. Math. Nachr. 289, 2-3 (2016), 232–242.

[14] Große, N., and Nistor, V. Uniform Shapiro-Lopatinski conditions and boundary value problems on manifolds with bounded geometry. Potential Analysis (2019).

https://doi.org/10.1007/s11118-019-09774-y (Online First).

[15] Große, N., and Schneider, C. Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces. Math. Nachr. 286, 16 (2013), 1586–1613.

[16] Kamotski, I. V., and Maz’ya, V. G. On the third boundary value problem in domains in cusps. J. Math. Sci. (N.Y.) 173, 5 (2011), 609–631. Problems in mathematical analysis. No. 54.

[17] Kondrat′ev, V. A. Boundary value problems for elliptic equations in domains with conical or angular points. Transl. Moscow Math. Soc. 16 (1967), 227–313.

[18] Kozlov, V. A., Maz′ ya, V. G., and Rossmann, J. Elliptic boundary value problems in domains with point singularities, vol. 52 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997.

[19] Lesch, M., and Peyerimhoff, N. On index formulas for manifolds with metric horns. Comm. Partial Differential Equations 23, 3-4 (1998), 649–684.

[20] Lions, J.-L., and Magenes, E. Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.

[21] Munnier, A., and Ramdani, K. Asymptotic analysis of a Neumann problem in a domain with cusp. Application to the collision problem of rigid bodies in a perfect fluid. SIAM J. Math. Anal. 47, 6 (2015), 4360–4403.

[22] Nazarov, S., and Popoff, N. Self-adjoint and skew-symmetric extensions of the Laplacian with singular Robin boundary condition. C. R. Math. Acad. Sci. Paris 356, 9 (2018), 927–932.

[23] Nazarov, S. A., and Plamenevsky, B. A. Elliptic problems in domains with piecewise smooth boundaries, vol. 13 of de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin, 1994.

[24] Rabinovich, V., Schulze, B.-W., and Tarkhanov, N. C^{∗}-algebras of singular integral operators in domains with oscillating conical singularities. Manuscripta Math. 108, 1 (2002), 69–90.

[25] Schulze, B.-W., Sternin, B., and Shatalov, V. An operator algebra on manifolds with cusp-type singularities. Ann. Global Anal. Geom. 16, 2 (1998), 101–140.

B. Ammann, Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Email address: bernd.ammann@mathematik.uni-regensburg.de

N. Große, Mathematisches Institut, Universität Freiburg, 79104 Freiburg, Germany

Email address: nadine.grosse@math.uni-freiburg.de

V. Nistor, Université de Lorraine, CNRS, IECL, F-57000 Metz, France and Inst. Math. Romanian Acad. PO BOX 1-764, 014700 Bucharest Romania

Email address: victor.nistor@univ-lorraine.fr