A note on boundary point principles for partial differential inequalities of elliptic type

In this note we consider boundary point principles for partial differential inequalities of elliptic type. Firstly, we highlight the difference between conditions required to establish classical strong maximum principles and classical boundary point lemmas for second order linear elliptic partial differential inequalities. We highlight this difference by introducing a singular set in the domain where the coefficients of the partial differential inequality need not be defined, and in a neighborhood of which, can blow-up. Secondly, as a consequence, we establish a comparison-type boundary point lemma for classical elliptic solutions to quasi-linear partial differential inequalities. Thirdly, we consider tangency principles, for $C^1$ elliptic weak solutions to quasi-linear divergence structure partial differential inequalities. We highlight the necessity of certain hypotheses in the aforementioned results via simple examples.


Introduction
In this note we consider boundary point principles (BPP) for solutions to elliptic partial differential inequalities (PDI). Specifically, we first give a relaxation of Hopf's [3] classical strong maximum principle (CSMP) for classical solutions to linear elliptic PDI, which here, allows the coefficients in the PDI to be unbounded in a neighborhood of a sufficiently regular subset of the spatial domain. The boundary point lemma (BPL) for linear elliptic PDI is obtained a consequence of this CSMP and complements available results (see [4] and [11]). Although coefficients in the PDI in the BPL are not necessarily bounded, they are constrained by growth conditions detailed in Section 2. As a secondary consideration, we illustrate how to extend BPL for classical solutions to linear elliptic PDI, to comparison-type BPL for elliptic classical solutions of quasi-linear PDI, and highlight the importance of specific conditions on this extension via examples. Consequently, we demonstrate that the BPL, as stated in [11,Theorem 2.7.1] is erroneous. Thirdly, we give an extension of a tangency principle for C 1 elliptic weak solutions to divergence structure quasi-linear PDI in domains with boundaries that satisfy an interior cone condition, which appeared in [16]. We also highlight that the tangency principle in [11,Theorem 2.7.2], an extension of that in [16], is erroneous. Corrections to the aforementioned erroneous theorem statements are provided.
We now give a brief account of the historical development of results in this note. The CSMP and BPL for classical solutions to linear elliptic PDI were established by Hopf in [3] for linear elliptic PDI with bounded (uniformly elliptic) coefficients. Although Hopf considered generalizations of the CSMP and BPL to elliptic solutions of nonlinear PDI in [3], more general statements of these results were established by McNabb in [5]. Extensions to the CSMP and BPL for classical solutions to linear elliptic PDI with coefficients that can blow-up or degenerate have been considered by numerous authors, as summarised in [2], [4] and [10]. Additionally, due to the the development of a theory for weak solutions to boundary value problems for divergence structure quasi-linear elliptic PDE, tangency principles for C 1 elliptic weak solutions to quasilinear PDI were established by Serrin in [16], and extended in [11]. We note that the proof of Serrin relies on an iteration method developed by Moser [6] and a Harnack inequality for quasilinear divergence structure elliptic PDI established by Trudinger [17]. More recently tangency principles have been established for C 1 elliptic weak solutions to quasi-linear PDI which have conclusions more similar to that of Hopf type BPL (see [13], [14] and [15]). A broader historical overview of the development of this theory can be found in [9, p.156-158 and p.193-194], [11, p.46], [7], [1] and [12].
The remainder of the note is presented as follows. In Section 2, we prove the CSMP and BPL for classical solutions to linear elliptic PDI, and consequently, we establish a comparison-type BPL for elliptic classical solutions to quasi-linear PDI. Furthermore, we provide examples which highlight the need for specific conditions given in the statement of the BPL as given here, one of which, is a counter-examples to [11,Theorem 2.7.1]. In Section 3, we establish a comparisontype tangency principle for C 1 elliptic weak solutions to quasi-linear divergence structure PDI in domains which satisfy an interior cone condition at boundary points. The necessity of several conditions in the BPL statement are highlighted, and furthermore, we demonstrate that [11,Theorems 2.7.2 and 2.7.3], are erroneous. In Section 4, we discuss how results in this note can be generalised and placed in a wider context.

Classical Theory
In this section, we establish a CSMP in Theorem 2.3 and BPL in Theorem 2.5 for classical solutions to linear elliptic PDI. The CSMP is noteworthy in that it allows coefficients in the PDI, under constraint, to blow-up in the interior of the domain in the neighborhood of a singular set. After defining the regularity of the singular set and constructing a suitable auxiliary function, the proofs of these results largely follow the description of related proofs available in [11,Chapter 2]. This allows us to highlight a distinction between the conditions required to establish a CSMP and BPL for classical solutions to linear elliptic PDI. Consequently, we also establish a comparison type BPL for classical elliptic solutions to quasi-linear PDI in Theorem 2.6 using the aforementioned BPL for linear elliptic PDI, refining an analogous statement in [11,Theorem 2.7.1]. We provide a proof using the approach outlined in [11,Section 2.7] where it is noteworthy that a full proof is omitted. To conclude the section, we give a simple counter-example to [11,Theorem 2.7.1] and provide a further example to highlight the importance of specific conditions in Theorem 2.6 which are not present in [11, Theorem 2.7.1].

Notation and Definitions
For a set X ⊂ R n , we denote ∂X =X ∖int(X), to be the boundary of X. In addition, throughout this note, Ω ⊂ R n denotes an open connected bounded set (a bounded domain), and we denote the set B R (x 0 ) ⊂ R n to be an open n-dimensional ball of radius R (with respect to the Euclidean distance) centred at x 0 ∈ R n . Furthermore, we denote R(X) to be the set of real-valued functions with domain X, C(X) ⊂ R(X) to be the set of all continuous functions in R(X) and C i (X) ⊂ C(X) to be the set of i-times continuously differentiable functions in C(X) for each i ∈ N. Additionally, for u ∈ C 2 (Ω) and S ⊂ Ω, we consider the linear elliptic operator L ∶ C 2 (Ω) → R(Ω ∖ S) given by with a ij , b i , c ∶ Ω ∖ S → R prescribed functions for i, j = 1, . . . , n, and such that there exists a non-negative function Λ ∶ Ω ∖ S → R for which, We refer to the set S where the linear elliptic operator is not defined for u, as the singular set. Additionally, note that by re-scaling the coefficients in the operator in (2.1) by ǫ, the left hand side of (2.2) can be expressed as ǫ y 2 i.e. with an equivalent frequently used ellipticity condition. Moreover, for u ∈ C 2 (Ω) we denote Du and D 2 u to be the gradient of u and the Hessian of u on Ω, respectively. To establish the CSMP in this note, we give the following definition, which will be used to define the structure of the singular set S ⊂ Ω. We refer to S as the singular set since the coefficients a ij , b i or c of L are allowed, with constraint, to blow up in neighborhoods of S. We note that in [1], alternatively, two-sided 'hour glass' conditions are employed for regularity conditions on singular sets which complement the following definition.
Definition 2.1. Let Ω ⊂ R n be a domain and S ⊂ Ω. We say that S satisfies an outward ball property if, given any nonempty relatively closed set T ⊂ Ω that is a strict subset of Ω, there exists R > 0 and x 0 ∈ Ω ∖ (T ∪ S) such that To illustrate some geometric aspects of sets that satisfy an outward ball property, consider the following: (i) If S consists solely of a finite number of points in Ω then S satisfies the outward ball property. This follows by considering d H ′ ∶ P(R n ) × P(R n ) → [0, ∞) with P(X) denoting the power set of X, and i.e. one component of the Euclidean Hausdorff distance between X and Y . Note that if X = 1, then d H ′ is the Euclidean Hausdorff distance between the two sets X and Y , denoted here by d(X, Y ). Now, let T be as in Definition 2.1. Then since T is nonempty and T = Ω, it follows that ∂T ∩ Ω = ∅. If d H ′ (∂T ∩ Ω, S) = 0, it follows that T ⊆ S, and we can choose a point x 0 ∈ Ω ∖ (T ∪ S) sufficiently close to T such that there exists a ball then S satisfies the outward ball property. To see this, let T be as in Definition 2.1. If d H ′ (∂T ∩ Ω, S) > 0, then a ball that satisfies (2.3) is guaranteed to exist, following the justification in (i). Alternatively, if d H ′ (∂T ∩ Ω, S) = 0, then it follows that ∂T ∩ Ω ⊆ S. Suppose ∂T ∩ S ⊃ {s 0 }. Then since S is given by a sufficiently smooth curve, for s 0 , there exists a ball B R (s 0 ) ⊂ Ω such that Thus, ∂T ∩B R (s 0 ) ⊂ S R and hence, via the Jordan Curve Theorem, B R (s 0 ) can be decomposed into the disjoint sets S R , B 1 R (s 0 ) and B 2 R (s 0 ) with B 1 R (s 0 ) the connected open set with boundary S R and the arc on ∂B R (s 0 ) connecting (φ 1 (t 1 ), φ 2 (t 1 )) to (φ 1 (t 2 ), φ 2 (t 2 )) in a clockwise direction (B 2 R (s 0 ) is defined similarly to B 1 R (s 0 ) with clockwise replaced by anti-clockwise). Thus, T ∩ B R (s 0 ) is either: , and in each case, since S R is defined by a C 2 curve, there exists a ball B R 1 (x) ⊂ B R (s 0 ) ∖ (T ∪ S) that satisfies (2.3). If instead d H ′ (∂T ∩ Ω, S) = 0 and ∂T ∩ S = ∅ then a similar argument to that in (i) can be used to demonstrate that a ball that satisfies (2.3) exists. It follows analogously from the Jordan-Brouwer Separation Theorem that any set of finitely many disjoint compact (n − 1)−dimensional sufficiently smooth C 2 manifolds in a domain Ω ⊂ R n also satisfies the outward ball property.
then S ′ does not satisfy the outward ball property. This follows by considering T = {(0, 0)} and observing that every ball then S satisfies the outward ball property.
Since S ′ consists of a countable set of isolated lines, it follows immediately that S ′ is 1porous at each s ∈ S ′ . However by considering T = (−1, 0] × (−1, 1) ⊂ Ω, it follows that S ′ does not satisfy the outward ball property. It is noteworthy that the review articles [18] and [19] do not indicate that a link has been established between porous sets and singular sets for elliptic PDI.

CSMP and BPL for linear elliptic PDI
Before, we establish a CSMP and BPL for classical solutions to L[u] ≥ 0 with L given by (2.1), we give the following lemma which guarantees the existence of a suitable comparison function.
We now establish a CSMP for linear elliptic PDI which allows coefficients of L to blow-up in neighborhoods of interior points of Ω. We note that one can recover a standard CSMP for linear elliptic PDI with bounded coefficients of appropriate sign (see for instance [3], [9] or [11]) by considering S = ∅ with λ a sufficiently large constant.

Theorem 2.3 (CSMP).
Let Ω ⊂ R n and S ⊂ Ω satisfy the outward ball property. Suppose that u ∈ C 2 (Ω) satisfies the linear elliptic PDI L[u] ≥ 0 on Ω ∖ S. In addition, suppose that for each which is continuous non-increasing and such that Λ ∈ L 1 0, R 2 , and such that the coefficients of L satisfy and suppose that either Proof. Suppose that u is not constant on Ω, and is not empty. Since T is a relatively closed strict subset of Ω and S satisfies the outward ball property, it follows that there exists a sufficiently small , for sufficiently small ǫ ∈ (0, R), with coefficients given bỹ Additionally, it follows that w ≤ 0 on Ω 0 , for suppose that the converse holds i.e. that there exists x * ∈ Ω 0 such that > 0 via (2.21), (2.25) and the hypotheses. However, since there is a local maxima of w at x * , then Dw(x * ) = 0, and D 2 w(x * ) is negative semi-definite. Consequently, via the Schur Product Theorem, the left hand side of (2.26) is non-positive, which gives a contradiction, and hence, w ≤ 0 on Ω 0 . Therefore, ∂ ν w(y 0 − x 0 ) ≥ 0, and hence ∂ ν u(y 0 ) ≥ −∂ ν v(y 0 − x 0 ) > 0. However, since there is a local maxima of u at y 0 , it follows from the regularity of u that Du(y 0 ) = 0, which contradicts ∂ ν u(y 0 ) > 0. Therefore, either u is constant on Ω, or u < M u on Ω, as required. A straightforward application of Theorem 2.3 gives an associated BPL for classical solutions to linear elliptic PDI. Theorem 2.5 (BPL). Suppose that the hypotheses of Theorem 2.3 hold, with the restriction that 'for which ∂B R (x 0 )∩∂Ω = ∅' is omitted. 1 In addition, suppose that u ∈ C 1 (Ω) and sup x∈Ω u( Proof. Since u satisfies the conditions of Theorem 2.3 and is not constant, it follows that u( A function analogous to w in (2.25) can now be constructed, from which, we can conclude (as in the proof of Theorem 2.3) that ∂ ν u(x b ) > 0, as required.

Comparison-type BPL for elliptic classical solutions to quasi-linear PDI
In this subsection we establish a comparison type BPL for classical elliptic solutions to quasilinear PDI using the approach described in [11,Chapter 2]. Specifically, via an application of Theorem 2.5, a BPL for classical elliptic solutions to quasi-linear PDI can be established. Although the proof is standard, we provide it to inform the discussion that follows.
where, for i, j = 1, . . . , n,ã ij ,b i ,c ∶ Ω → R are given by, . Thus, it follows thatL is a linear elliptic operator on B R b (x ′ b ), that satisfies the conditions of Theorem 2.5, provided that we consider Λ in Theorem 2.5 as that in (2.29)-(2.32) after multiplication by a sufficiently large constant. An application of Theorem 2.5 yields ∂ ν w > 0 at x b and hence, as required. We now demonstrate that if the bound on the lower Lipschitz constant for B in Theorem 2.6 is relaxed to a mere local lower Lipschitz condition, then the conclusion of Theorem 2.6 does not necessarily hold.
Example 2.8. Suppose that Ω ⊂ R n and for for all i, j = 1, . . . , n and B ∶ Ω × R × R n → R to be (2.39) Since the coefficients of A ij define a Laplacian, it can be seen that A ij satisfies the conditions of Theorem 2.6 (with, for example, Λ = 1), and also, that Q is elliptic with respect to u with v x i x j bounded on Ω. Moreover, observe that B is independent of η, and In conclusion, although u, v, A ij and B satisfy all of the conditions of Theorem 2.6 (with the exception of the lower Lipschitz condition on B, or alternatively (2.28)), via (2.36), which violates the conclusion of Theorem 2.6. It is noteworthy that essentially the same error can be found in the statement of a BPL for classical solutions to linear parabolic PDI given in [9,p.174,Theorem 7], as illustrated in [8].
We also highlight that in both of these instances, a direct proof of the associated BPL is not given, but instead, only the main ideas of the proofs are described. (2.42) Thus, we observe that B in (2.40) satisfies the conditions Theorem 2.6 with the exception of Λ ∈ L 1 0, R b 2 in (2.28). This follow from letting R → R b in (2.42) which implies that λ necessarily satisfies Now, we highlight the necessity of the bound on v x i x j (or u x i x j ) in Theorem 2.6. Note that this condition is not present in [11, Theorem 2.7.1].

Weak Theory
In this section, we establish a comparison-type tangency principle, for C 1 weak elliptic solutions to divergence structure PDI which is a correction of that stated in [

Notation and Definitions
The quasi-linear divergence structure PDI we consider are given by: div(A(⋅, u, Du)) + B(⋅, u, Du) ≥ 0 on Ω, with A ∶ Ω × R × R n → R n and B ∶ Ω × R × R n → R. Specifically, we consider C 1 weak solutions to (3.1) (and analogously (3.2)) that satisfy: u ∈ C 1 (Ω), A(⋅, u, Du), B(⋅, u, Du) ∈ L 1 loc (Ω) and for any test function ψ ∈ C 1 (Ω) such that ψ ≥ 0 on Ω and ψ has compact support in Ω. Moreover, we say that u (and analogously v) is an elliptic solution to (3.1) if a ij (x) = (A i ) η j (x, u(x), Du(x)) satisfies the left inequality in (2.2) for all x ∈ Ω. Furthermore, in this section we consider Ω with boundary ∂Ω that satisfies an interior cone condition i.e. at each point x b ∈ ∂Ω there exists a cone of finite height in Ω with apex x b . We denote the interior of such a cone by Ω b .

A comparison-type tangency principle for weak elliptic solutions to quasilinear divergence structure PDI
Theorem 3.1 ( Tangency Principle). Let x b ∈ ∂Ω satisfy the interior cone condition, and u, v ∶ Ω → R be such that: u, v ∈ C 1 (Ω); u, v satisfy (3.1) and (3.2) respectively; A ∶ Ω × R × R n → R n is continuous and continuously differentiable with respect to z and η; A z is uniformly bounded and A η is uniformly continuous on . Then the zero of v − u at x b is of finite order.
Proof. For a contradiction, assume that w = v − u has a zero of infinite order at x b ∈ ∂Ω. Via regularity on w, it follows that Dw(x b ) = 0. Moreover, for each ǫ ∈ (0, min {M z , M η 2}), there exists a cone of finite height in Ω with apex x b , without loss of generality denoted by Ω b , such and = div(Ã(x, w, Dw)) +B(x, w, Dw) ≤ 0 (3.7) on Ω b . The functionÃ ∶ Ω b × R × R n → R n arises from repeated application of the mean value theorem in (3.7), e.g.
Since Q is elliptic with respect to u, it follows from (3.5)-(3.8) that Additionally, via the regularity hypotheses on A and B it follows that there exist constants a η , b η , b z ≥ 0 such that Now, since x b is the apex of the cone Ω b ⊂ Ω, it follows that there exists a sequence of balls {B r k (y k )} k∈N 0 : that have boundaries that tangentially intersect ∂Ω b ; such that B r k 3 (y k ) ⊂ B 2r k+1 3 (y k+1 ) for all k ∈ N 0 ; for which y k → x b as k → ∞; r k+1 < r k for k ∈ N 0 ; and by denoting θ to be the half-angular opening of the cone, we can set for all k ∈ N 0 . It follows immediately that with ω n denoting the volume of a Euclidean unit ball in R n . By combining (3.13) and (3.15), we have min − a2u 2 and B(x, u, p) ≥ −b1 p − b2u for constants a2, b1, b2 ≥ 0 and a5 > 0. Now, via our initial assumption, w has a zero of infinite order at x b and hence via (3.14), for each m ∈ N there exists a positive constant c independent of k such that  .13) and (3.16), the constant c that arises, as in (3.18), is not necessarily independent of k, which is the source of the error in the proof.
Example 3.3. Suppose that Ω ⊂ R n and for x b ∈ ∂Ω satisfies an interior cone condition. Consider u ∶Ω → R and v ∶Ω → R as given in Example 2.8 such that additionally, v has a zero of infinite order at For the quasi-linear partial differential inequalities in (3.1) and (3. which violates the conclusion of Theorem 3.1. We also note here that the conditions on A η and A z in Theorem 3.1 cannot be relaxed to those in [ To conclude the section, we note that in [11,Theorem 2.7.3], a strong maximum principle and tangency principle is stated with the regularity conditions on u and v in [11, Theorem 2.7.2] relaxed to u, v ∈ C(Ω) but so that u and v also possess strong derivatives in L 2 loc (Ω). To compensate for these relaxed regularity conditions on u and v, stricter regularity conditions are imposed on A and B which we now demonstrate, are insufficient to establish the conclusion. This establishes that all three theorems in [11, Section 2.7] are erroneous. Observe that u, v ∈ C ∞ (Ω) and that the zero of v − u on ∂B 1 (0) is of infinite order. Additionally, note that A is locally bounded on R n and B is locally bounded and locally lower Lipschitz on Ω × R. Furthermore, for i = 1, . . . , n, we have For such ǫ > 0, it follows that B(x, z), as given by (2.39), is non-increasing in z on Ω. Therefore, although Ω, u, v, A and B satisfy the conditions of [11,Theorem 2.7.3], the conclusion that the zero of v − u on ∂B 1 (0) is of finite order is violated.

Conclusion
In Theorem 2.3, the outward ball condition on S in Definition 2.1 can be generalised to an outward C 1,Dini condition, provided that the conditions on the coefficients of L are appropriately constrained. This can be achieved with more restrictive conditions in the statement of Theorem 2.3, by replacing the function in Lemma 2.2 with a suitable alternative (for instance, the regularized distance functions constructed in [4, Sections 1 and 2]). In relation to Theorem 2.6, a fully nonlinear version can be established without substantial additional technicality (see, for example [3] or [11]). Moreover, the condition bounding v x i x j can be relaxed provided that the right hand side of (2.29)-(2.32) can be expressed (for instance, by further constraining the growth of Λ(d) as d → 0) so that Theorem 2.5 can be applied.
With regard to Theorem 3.1, we note that allowable blow-up in A and B as x → x b can be accommodated by using the more general integrability conditions on coefficients in Theorem [17,Theorem 1.2] i.e. by using Theorem [17,Theorem 5.1]. Also, complementary results are contained in [13], [14] and [15] where BPL for quasi-linear elliptic PDI are established under more regular domain and PDI constraints, but which guarantee the existence of non-zero (first) outward directional derivatives. It is also pertinent to note that in [15] the author highlights two further distinct incorrect statements of BPP from those highlighted here and in [8].