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 [13, Chap. 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 quasilinear PDI in Theorem 2.6 using the aforementioned BPL for linear elliptic PDI, refining an analogous statement in [13, Theorem 2.7.1]. We provide a proof using the approach outlined in [13, Sect. 2.7] where it is noteworthy that a full proof is omitted. To conclude the section, we give a simple counterexample to [13, Theorem 2.7.1] and provide a further example to highlight the importance of specific conditions in Theorem 2.6 that are not present in [13, Theorem 2.7.1].

Notation and definitions

For a set (Xsubset mathbb{R}^{n}), we denote (partial X=bar{X} setminus operatorname{int}(X)), to be the boundary of X. In addition, throughout this note, (Omega subset mathbb{R}^{n}) denotes an open connected bounded set (a bounded domain), and we denote the set (B_{R}(x_{0})subset mathbb{R}^{n}) to be an open n-dimensional ball of radius R (with respect to the Euclidean distance) centered at (x_{0}in mathbb{R}^{n}). We also denote the origin in (mathbb{R}^{n}) by O. Furthermore, we denote (R(X)) to be the set of real-valued functions with domain X, (C(X)subset R(X)) to be the set of all continuous functions in (R(X)), and (C^{i}(X)subset C(X)) to be the set of i-times continuously differentiable functions in (C(X)) for each (iin mathbb{N}). Additionally, for (uin C^{2}(Omega )) and (mathcal{S}subset Omega ), we consider the linear elliptic operator (L:C^{2}(Omega )to R(Omega setminus mathcal{S} )) given by

$$begin{aligned} L[u]:= sum_{i,j=1}^{n} a_{ij}u_{x_{i}x_{j}} + sum_{i=1}^{n} b_{i}u_{x_{i}} + cu quadtext{in } Omega setminus mathcal{S}, end{aligned}$$

(2.1)

with (a_{ij},b_{i},c:Omega setminus mathcal{S} to mathbb{R}) prescribed functions for (i,j=1,dots,n), and such that there exists a nonnegative function (Lambda:Omega setminus mathcal{S}to mathbb{R}) for which,

$$begin{aligned} vert y vert ^{2} leq sum _{i,j=1}^{n} a_{ij}(x)y_{i}y_{j} leq Lambda (x) vert y vert ^{2}quad forall xin Omega setminus mathcal{S}, yin mathbb{R}^{n} . end{aligned}$$

(2.2)

We refer to the set (mathcal{S}) where the linear elliptic operator is not defined for u, as the singular set. Additionally, note that by rescaling the coefficients in the operator in (2.1) by ϵ, the left-hand side of (2.2) can be expressed as (epsilon |y|^{2}), i.e., with an equivalent frequently used ellipticity condition. Moreover, for (uin C^{2}(Omega )) 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 (mathcal{S}subset Omega ). We refer to (mathcal{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 (mathcal{S}). We note that in [1], alternatively, two-sided ‘hour-glass’ conditions are employed for regularity conditions on singular sets that complement the following definition.

Definition 2.1

Let (Omega subset mathbb{R}^{n}) be a domain and (Ssubset Omega ). We say that (mathcal{S}) satisfies an outward ball property if, given any nonempty relatively closed set (mathcal{T}subset Omega ) that is a strict subset of Ω, there exists (R>0) and (x_{0}in Omega setminus (mathcal{T}cup mathcal{S})) such that

$$begin{aligned} B_{R}(x_{0})subset Omega setminus ( mathcal{T}cup mathcal{S}) quadtext{and} quadpartial B_{R}(x_{0}) cap mathcal{T} neq emptyset. end{aligned}$$

(2.3)

To illustrate some geometric aspects of sets that satisfy an outward ball property, consider the following:

  1. (i)

    If (mathcal{S}) consists solely of a finite number of points in Ω then (mathcal{S}) satisfies the outward ball property. This follows by considering (d_{H’}:mathcal{P}(mathbb{R}^{n})times mathcal{P}(mathbb{R}^{n}) to [0,infty )) with (mathcal{P}(X)) denoting the power set of X, and

    $$begin{aligned} d_{H’}(X,Y) = sup_{xin X} Bigl( inf _{yin Y} vert x-y vert Bigr) quadforall X,Yin mathcal{P} bigl(mathbb{R}^{n}bigr), end{aligned}$$

    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 (mathcal{T}) be as in Definition 2.1. Then, since (mathcal{T}) is nonempty and (mathcal{T}neqOmega ), it follows that (partial mathcal{T}cap Omega neqemptyset ). If (d_{H’}(partial mathcal{T} cap Omega, mathcal{S})=0), it follows that (mathcal{T} subseteq mathcal{S}), and we can choose a point (x_{0}in Omega setminus ( mathcal{T}cup mathcal{S})) sufficiently close to (mathcal{T}) such that there exists a ball (B_{R}(x_{0})) that satisfies (2.3). Alternatively, if (d_{H’}(partial mathcal{T} cap Omega, mathcal{S})>0), then since (Omega setminus (mathcal{T}cup mathcal{S})) is a nonempty open set, we can chose (x_{0}in Omega setminus (mathcal{T}cup mathcal{S})) so that (d_{H’}({ x_{0}},mathcal{T})<frac {1}{2} d_{H’}({ x_{0}}, mathcal{S}cup partial Omega )). Thus, there exists a ball (B_{R}(x_{0})) that satisfies (2.3).

  2. (ii)

    If (Omega =(-1,1)^{2}subset mathbb{R}^{2}) and

    $$begin{aligned} mathcal{S}={}&Bigl{ (x_{1},x_{2})in Omega: (x_{1},x_{2}) = bigl(phi _{1}(t), phi _{2}(t)bigr) forall tin (0,1)text{ with } \ &{} phi: (0,1)to Omega text{ twice continuously differentiable and injective on } \ &{} (0,1), phi ‘>0 text{ on } (0,1), lim_{tto 0} phi (t) neq lim_{tto 1} phi (t) text{ and } \ &{} lim_{tto 0}phi (t), lim_{tto 1}phi (t) neq phi (s) forall sin (0,1) Bigr} , end{aligned}$$

    then (mathcal{S}) satisfies the outward ball property. To see this, let (mathcal{T}) be as in Definition 2.1. If (d_{H’}(partial mathcal{T} cap Omega, mathcal{S})>0), then a ball that satisfies (2.3) is guaranteed to exist, following the justification in (i). Alternatively, if (d_{H’}(partial mathcal{T} cap Omega, mathcal{S})=0), then it follows that (partial mathcal{T} cap Omega subseteq overline{mathcal{S}}). Suppose there exists (s_{0} in partial mathcal{T}cap mathcal{S}). Then, since (mathcal{S}) is given by a sufficiently smooth curve, for (s_{0}), there exists a ball (B_{R}(s_{0})subset Omega ) such that

    $$begin{aligned} mathcal{S} cap bar{B}_{R}(s_{0}) & = bigl{ bigl(phi _{1}(t),phi _{2}(t)bigr):t_{1} leq t leq t_{2} bigr} \ & =: mathcal{S}_{R} cup bigl{ bigl(phi _{1}(t_{1}), phi _{2}(t_{1})bigr), bigl( phi _{1}(t_{2}), phi _{2}(t_{2})bigr) bigr} . end{aligned}$$

    Thus, (partial mathcal{T}cap B_{R} (s_{0}) subset mathcal{S}_{R}) and hence, via the Jordan Curve Theorem, (B_{R}(s_{0})) can be decomposed into the disjoint sets (mathcal{S}_{R}), (B_{R}^{1}(s_{0})) and (B_{R}^{2}(s_{0})) with (B_{R}^{1}(s_{0})) the connected open set with boundary (mathcal{S}_{R}) and the arc on (partial B_{R}(s_{0})) connecting ((phi _{1}(t_{1}),phi _{2}(t_{1}))) to ((phi _{1}(t_{2}),phi _{2}(t_{2}))) in a clockwise direction ((B_{R}^{2}(s_{0})) is defined similarly to (B_{R}^{1}(s_{0})) with clockwise replaced by anticlockwise). Thus, (mathcal{T} cap B_{R}(s_{0})) is either: (mathcal{S}_{R}cap mathcal{T}), (mathcal{S}_{R}cup B_{R}^{1}(s_{0})) or (mathcal{S}_{R} cup B_{R}^{2}(s_{0})), and in each case, since (mathcal{S}_{R}) is defined by a (C^{2}) curve, there exists a ball (B_{R_{1}}(x)subset B_{R}(s_{0}) setminus (mathcal{T}cup mathcal{S})) that satisfies (2.3). If instead (d_{H’}(partial mathcal{T} cap Omega, mathcal{S})=0) and (partial mathcal{T}cap mathcal{S} = emptyset ) 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 (Omega subset mathbb{R}^{n}) also satisfies the outward ball property.

  3. (iii)

    If (Omega = (-1,1)^{2}) and

    $$begin{aligned} mathcal{S}’ = bigl{ (x_{1},x_{2})in Omega: x_{1}=0 text{ or }x_{2}=0 bigr} , end{aligned}$$

    then (mathcal{S}’) does not satisfy the outward ball property. This follows by considering (mathcal{T}={ (0,0) }) and observing that every ball (B_{R}(x)subset Omega ) such that (partial B_{R}(x)cap mathcal{T} neq emptyset ), also satisfies (B_{R}(x)cap mathcal{S}’ neqemptyset ). However, if instead (Omega = (-1,1)^{2}) and

    $$begin{aligned} mathcal{S} = bigl{ (x_{1},x_{2})in [0,1)times (-1,1): x_{1}=0 text{ or }x_{2}=0bigr} end{aligned}$$

    then (mathcal{S}) satisfies the outward ball property.

  4. (iv)

    If (mathcal{S}’) is locally dense on (B_{R}(x)subset Omega ) then (mathcal{S}’) does not satisfy the outward ball property. This can be observed by choosing (mathcal{T}) to contain any point in (mathcal{S}’cap B_{R}(x)). Consequently, sets that satisfy Definition 2.1 are necessarily measure zero sets with respect to the Lebesgue measure.

  5. (v)

    If (mathcal{S}) satisfies Definition 2.1, then (mathcal{S}) is 1-porous at each (sin mathcal{S}) with respect to [20, Definition 2.1]. This follows by considering (mathcal{T}={ s}). However, not all subsets of Ω that are 1-porous at every point necessarily satisfy Definition 2.1. For example, consider (Omega = (-1,1)^{2}) with

    $$begin{aligned} mathcal{S}’ = biggl{ (x_{1},x_{2})in Omega: x_{1} = frac {1}{2n} text{ for } nin mathbb{N} biggr} . end{aligned}$$

    Since (mathcal{S}’) consists of a countable set of isolated lines, it follows immediately that (mathcal{S}’) is 1-porous at each (sin mathcal{S}’). However, by considering (mathcal{T}= (-1,0]times (-1,1)subset Omega ), it follows that (mathcal{S}’) does not satisfy the outward ball property.

Later in this section, for (uin C^{2}(Omega )), we consider the quasilinear operator (Q:C^{2}(Omega )to R(Omega )) given by,

$$begin{aligned} Q[u]:= sum_{i,j=1}^{n} A_{ij}(x,u,Du)u_{x_{i}x_{j}} + B(x,u,Du) quadtext{in } Omega, end{aligned}$$

(2.4)

with (A_{ij},B:Omega times mathbb{R}times mathbb{R}^{n} to mathbb{R}) prescribed functions. Specifically, we refer to Q as elliptic with respect to a specific (uin C^{2}(Omega )) if (2.2) holds for (a_{ij}(x) = A_{ij}(x,u(x),Du(x))) for all (xin Omega ).

CSMP and BPL for linear elliptic PDI

Before we establish a CSMP and BPL for classical solutions to (L[u]geq 0) with L given by (2.1), we give the following lemma that guarantees the existence of a suitable comparison function.

Lemma 2.2

Let (R,m>0) be constants and set (k=2n ( frac {2}{R} + 1 ) + 3). Additionally, suppose that there exists a constant (epsilon >0) and a continuous nonincreasing function (Lambda:(0,epsilon ]to (0,infty )) such that (Lambda in L^{1}((0,epsilon ))),

$$begin{aligned} epsilon in biggl( 0, min biggl{ 1, frac {R}{2} biggr} biggr)quad textit{and}quad int _{0}^{epsilon}Lambda (s) ,ds < frac{1}{k}, end{aligned}$$

(2.5)

and moreover, for (Omega = B_{R}(O)setminus bar{B}_{R-epsilon}(O)) that

$$begin{aligned} & vert y vert ^{2} leq sum _{i,j=1}^{n} a_{ij}(x)y_{i}y_{j} leq Lambda bigl(R- vert x vert bigr) vert y vert ^{2}quad forall xin Omega, yin mathbb{R}^{n}, end{aligned}$$

(2.6)

$$begin{aligned} &biglvert b_{i}(x) bigrvert leq Lambda bigl(R- vert x vert bigr)quad forall xin Omega, end{aligned}$$

(2.7)

$$begin{aligned} &{ -}c(x) leq frac{Lambda (R- vert x vert )}{(R- vert x vert )} quadforall xin Omega. end{aligned}$$

(2.8)

Then, if L is a linear elliptic operator with coefficients that satisfy (2.6)(2.8), there exists (v:bar{Omega} to [0,m]), such that:

  1. (i)

    (v=0) on (partial B_{R}(O)), (v=m) on (partial B_{R-epsilon}(O)), and (v>0) on Ω.

  2. (ii)

    (vin C^{1}(bar{Omega}) cap C^{2}(Omega )).

  3. (iii)

    (L[v]>0) on Ω.

  4. (iv)

    (partial _{nu}v < 0) on (partial B_{R}(O)), where (partial _{nu}v) denotes the outward (to Ω) directional derivative of v normal to Ω.

Proof

Define (f:[0,epsilon ]to [0,infty )) to be

$$begin{aligned} f(r) = r + k int _{0}^{r} int _{0}^{s} Lambda (t) ,dt ,ds quadforall rin [0, epsilon ]. end{aligned}$$

(2.9)

It follows immediately that

$$begin{aligned} fin C^{1}bigl([0,epsilon ]bigr)cap C^{2}bigl((0,epsilon ]bigr), end{aligned}$$

(2.10)

with

$$begin{aligned} & f'(r) = 1+ k int _{0}^{r} Lambda (t) ,dt quadforall rin [0, epsilon ], end{aligned}$$

(2.11)

$$begin{aligned} & f”(r) = kLambda (r)quad forall rin (0, epsilon ]. end{aligned}$$

(2.12)

Now, we define (tilde{v}:bar{Omega}to mathbb{R}) to be

$$begin{aligned} tilde{v} (x) = fbigl(R- vert x vert bigr) quadforall xin bar{Omega}. end{aligned}$$

(2.13)

It follows from (2.9)–(2.13) that

$$begin{aligned} &tilde{v}in C^{1}(bar{Omega} )cap C^{2}( Omega ), end{aligned}$$

(2.14)

$$begin{aligned} &tilde{v} >0 quadtext{on } Omega, end{aligned}$$

(2.15)

and

$$begin{aligned} L[tilde{v}](x) = {}& frac{ ( f”(R- vert x vert ) vert x vert + f'(R- vert x vert ) )}{ vert x vert ^{3}}sum _{i,j=1}^{n} a_{ij}(x)x_{i}x_{j} \ &{} – frac{f'(R- vert x vert )}{ vert x vert } sum_{i=1}^{n} bigl( a_{ii}(x) + b_{i}(x)x_{i} bigr) + fbigl(R- vert x vert bigr)c(x) end{aligned}$$

(2.16)

for all (xin Omega ). It now follows from substituting (2.9), (2.11), and (2.12) into (2.16), and using (2.5)–(2.8), that

$$begin{aligned} L[tilde{v}](x) ={}& frac{1}{ vert x vert ^{3}} biggl( kLambda bigl(R- vert x vert bigr) vert x vert + biggl(1+k int _{0}^{R- vert x vert } Lambda (t) ,dt biggr) biggr) sum _{i,j=1}^{n} a_{ij}(x) x_{i}x_{j} \ & {}- frac{1}{ vert x vert } biggl(1+k int _{0}^{R- vert x vert } Lambda (t) ,dt biggr) sum _{i=1}^{n} bigl(a_{ii}(x) + b_{i}(x)x_{i}bigr) \ & {}+ c(x) biggl( bigl(R- vert x vert bigr) + k int _{0}^{R- vert x vert } int _{0}^{s} Lambda (t) ,dt ,ds biggr) \ geq{}& Lambda bigl(R- vert x vert bigr) biggl( k – 2n biggl(frac{2}{R} + 1 biggr) – 2 biggr) \ ={}& Lambda bigl(R- vert x vert bigr) \ >{}&0 end{aligned}$$

(2.17)

for all (xin Omega ). Now, define (v:bar{Omega}to mathbb{R}) to be

$$begin{aligned} v(x) = frac{tilde{v}(x)m}{f(epsilon )} quadforall xin bar{Omega}. end{aligned}$$

(2.18)

Then, via (2.18), (2.13), (2.15), and (2.9), v satisfies (i). Also, via (2.14) v satisfies (ii). Additionally, from (2.17) and (2.18), v satisfies (iii). Moreover, via (2.18), (2.13), (2.11), and (2.9), it follows that

$$begin{aligned} partial _{nu}v (x)|_{ vert x vert =R} = frac{-m}{f(epsilon )} < 0, end{aligned}$$

and hence v satisfies (iv), as required. □

We now establish a CSMP for linear elliptic PDI that 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 [4, 10, 13]) by considering (mathcal{S}=emptyset ) with Λ a sufficiently large constant.

Theorem 2.3

((CSMP))

Let (Omega subset mathbb{R}^{n}) and (mathcal{S}subset Omega ) satisfy the outward ball property. Suppose that (uin C^{2}(Omega )) satisfies the linear elliptic PDI (L[u]geq 0) on (Omega setminus mathcal{S}). In addition, suppose that for each (B_{R}(x_{0})subset (Omega setminus mathcal{S})) for which (partial B_{R}(x_{0}) cap partial Omega = emptyset ), there exists a function (Lambda: ( 0,frac{R}{2} ] to (0,infty )) that is continuous nonincreasing and such that (Lambda in L^{1} ( ( 0,frac {R}{2} ) )), and such that the coefficients of L satisfy

$$begin{aligned} & vert y vert ^{2} leq sum _{i,j=1}^{n} a_{ij}(x)y_{i}y_{j} leq Lambda bigl(dbigl({ x },partial B_{R}(x_{0}) bigr)bigr) vert y vert ^{2} quadforall xin B_{R}(x_{0}), yin mathbb{R}^{n}, end{aligned}$$

(2.19)

$$begin{aligned} & biglvert b_{i}(x) bigrvert leq Lambda bigl(d bigl({ x },partial B_{R}(x_{0})bigr)bigr) quadforall xin B_{R}(x_{0}), end{aligned}$$

(2.20)

$$begin{aligned} &{-}c(x) leq frac{Lambda (d({ x},partial B_{R}(x_{0}))) }{d({ x},partial B_{R}(x_{0}))} quadforall xin B_{R}(x_{0}). end{aligned}$$

(2.21)

Additionally, let

$$begin{aligned} M_{u}=sup_{xin Omega}u(x) end{aligned}$$

(2.22)

and suppose that either (M_{u}=0), or (M_{u} >0) with c nonpositive. Then, (M_{u} > u(x)) for all (xin Omega ) or u is constant on Ω.

Proof

Suppose that u is not constant on Ω, and

$$begin{aligned} mathcal{T} = bigl{ xin Omega: u(x) = M_{u} bigr} end{aligned}$$

(2.23)

is not empty. Since (mathcal{T}) is a relatively closed strict subset of Ω and (mathcal{S}) satisfies the outward ball property, it follows that there exists a sufficiently small (B_{R}(x_{0})) such that (B_{R}(x_{0})subset Omega setminus (mathcal{T}cup mathcal{S})), (partial B_{R}(x_{0}) cap mathcal{T} = { y_{0} }) and (partial B_{R}(x_{0}) cap partial Omega = emptyset ). Moreover, it follows from (2.19)–(2.21) and the hypotheses on L and Λ, that Lemma 2.2 can be applied to a linear elliptic operator defined in (Omega _{0} = B_{R}(O) setminus bar{B}_{R-epsilon}(O)), for sufficiently small (epsilon in (0,R)), with coefficients given by

$$begin{aligned} tilde{a}_{ij}(x)=a_{ij}(x+x_{0}),qquad tilde{b}_{i}(x) = b_{i}(x+x_{0}),qquad tilde{c}(x) = frac{-Lambda (R- vert x vert )}{(R- vert x vert )},quad forall x in Omega _{0} end{aligned}$$

(2.24)

with

$$begin{aligned} m=M_{u}- Bigl( sup_{partial B_{R-epsilon}(x_{0})} u Bigr) >0, end{aligned}$$

to guarantee the existence of (v:bar{Omega}_{0}to [0,m]) that satisfies the conclusions of Lemma 2.2. Now, define (w:overline{Omega}_{0}to mathbb{R}) to be

$$begin{aligned} w(x) = u(x+x_{0}) + v(x) – M_{u} quadforall x in overline{Omega}_{0} . end{aligned}$$

(2.25)

It follows that (win C^{2}(Omega _{0})cap C^{1}(overline{Omega}_{0})) and (sup_{partial Omega _{0}}w = w(y_{0}-x_{0})=0). Additionally, it follows that (w leq 0) on (Omega _{0}), for suppose that the converse holds, i.e., that there exists (x^{*}in Omega _{0}) such that (sup_{xin Omega _{0}}w(x) = w(x^{*})>0). From the hypotheses and Lemma 2.2(iii), we have (L[u](x^{*}+x_{0})geq 0) and (tilde{L}[v](x^{*}) >0), respectively. Thus, via (2.24) and (2.25) we have,

$$begin{aligned} & sum_{i,j=1}^{n} tilde{a}_{ij}bigl(x^{*}bigr)w_{x_{i}x_{j}} bigl(x^{*}bigr) + sum_{i=1}^{n} tilde{b}_{i}bigl(x^{*}bigr) w_{x_{i}} bigl(x^{*}bigr) \ &quad > -cbigl(x^{*}+x_{0}bigr)u bigl(x^{*}+x_{0}bigr)-tilde{c}bigl(x^{*} bigr)vbigl(x^{*}bigr) \ &quad geq frac{Lambda (R- vert x^{*} vert )}{(R- vert x^{*} vert )}bigl(min bigl{ ubigl(x^{*}+x_{0} bigr), 0bigr} + vbigl(x^{*}bigr)bigr) \ &quad > 0 end{aligned}$$

(2.26)

with the last two inequalities following from (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 semidefinite. Consequently, via the Schur Product Theorem, the left-hand side of (2.26) is nonpositive, which gives a contradiction, and hence, (w leq 0) on (Omega _{0}). Therefore, (partial _{nu}w(y_{0}-x_{0}) geq 0), and hence (partial _{nu}u(y_{0}) geq -partial _{nu}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 (partial _{nu}u(y_{0}) >0). Therefore, either u is constant on Ω, or (u< M_{u}) on Ω, as required. □

Remark 2.4

Note that in Theorem 2.3, the conditions on the coefficients of L apply on balls that satisfy (partial B_{R}(x_{0})cap bar{mathcal{S}}neq emptyset ) but not on balls that satisfy (B_{R}(x_{0})cap bar{mathcal{S}}neq emptyset ). Thus, although the coefficients of L can, under constraints (2.19)–(2.21), blow-up as (xto bar{mathcal{S}}), they cannot blow-up (except c negatively) as (xto x_{0}) for (x_{0}in Omega setminus bar{mathcal{S}}). Moreover, observe that the coefficients of L can blow-up as (xto partial Omega ) with conditions (2.19)–(2.21) not required to hold on (B_{R}(x_{0})) such that (partial B_{R}(x_{0})cap partial Omega neq emptyset ). However, for a BPL to hold for a linear elliptic operator L on Ω, conditions (2.19)–(2.21) are required to hold on balls (B_{R}(x_{0})) such that (partial B_{R}(x_{0})cap partial Omega neq emptyset ). This is the principal difference in hypothesis between BPL and CSMP for linear elliptic PDI.

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 2.3hold, with the restriction that ‘for which (partial B_{R}(x_{0}) cap partial Omega = emptyset )’ is omitted.Footnote 1In addition, suppose that (uin C^{1}(bar{Omega})) and (sup_{xin Omega} u(x)=u(x_{b})) for some (x_{b}in partial Omega ) such that there exists (B_{R_{b}}(x_{b}’)subset Omega setminus mathcal{S}) that satisfies (x_{b} in partial B_{R_{b}}(x_{b}’)). If u is not constant on Ω, then (partial _{nu}u(x_{b}) >0).

Proof

Since u satisfies the conditions of Theorem 2.3 and is not constant, it follows that (u(x) < u(x_{b})) for all (xin B_{R_{b}}(x_{b}’)). 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 (partial _{nu}u(x_{b})>0), as required. □

Comparison-type BPL for elliptic classical solutions to quasilinear PDI

In this subsection we establish a comparison-type BPL for classical elliptic solutions to quasilinear PDI using the approach described in [13, Chap. 2]. Specifically, via an application of Theorem 2.5, a BPL for classical elliptic solutions to quasilinear PDI can be established. Although the proof is standard, we provide it to inform the discussion that follows.

Theorem 2.6

((BPL))

Suppose that (u,v:bar{Omega}to mathbb{R}) satisfy (u,vin C^{2}(Omega )cap C^{1}(bar{Omega })) and the quasilinear PDI (Q[u]geq 0) and (Q[v]leq 0) on Ω. Furthermore, suppose that Q is elliptic with respect to u, with (v_{x_{i}x_{j}}) bounded on Ω (or instead suppose that Q is elliptic with respect to v, with (u_{x_{i}x_{j}}) bounded on Ω) for (i,j=1,ldots,n). Suppose that (u< v) in Ω and (u=v) at (x_{b}in partial Omega ) for which there exists (B_{R_{b}}(x_{b}’)subset Omega ) with (x_{b} in partial B_{R_{b}}(x_{b}’)). Suppose that there exists a continuous nonincreasing function (Lambda: ( 0, frac{R_{b}}{2} ] to (0,infty )) such that (Lambda in L^{1} ( ( 0, frac{R_{b}}{2} ) )),

$$begin{aligned} & biglvert A_{ij}(x,z_{1},eta _{1}) – A_{ij}(x,z_{2},eta _{2}) bigrvert \ & quadleq Lambda bigl(dbigl({ x},partial B_{R_{b}} bigl(x_{b}’bigr) bigr)bigr) Biggl( frac{ vert z_{1}-z_{2} vert }{d({ x},partial B_{R_{b}}(x_{b}’))} + sum_{l=1}^{n} vert eta _{1l}-eta _{2l} vert Biggr) end{aligned}$$

(2.27)

for all ((x,z_{1},eta _{1}),(x,z_{2},eta _{2})in B_{R_{b}}(x_{b}’) times [-M_{z},M_{z}] times [-M_{eta}, M_{eta}]^{n}), and

$$begin{aligned} & B(x,z_{1}, eta _{1} ) – B(x,z_{2},eta _{2} ) \ & quadgeq – Lambda bigl(dbigl({ x},partial B_{R_{b}} bigl(x_{b}’bigr) bigr)bigr) Biggl( frac{(z_{1} – z_{2})}{d({ x}, partial B_{R_{b}}(x_{b}’) )} + sum_{l=1}^{n} vert eta _{1l}-eta _{2l} vert Biggr) end{aligned}$$

(2.28)

for all ((x,z_{1},eta _{1}),(x,z_{2},eta _{2})in B_{R_{b}}(x_{b}’) times [-M_{z},M_{z}] times [-M_{eta}, M_{eta}]^{n}) with (z_{1} geq z_{2}), with

$$begin{aligned} M_{z}=sup_{xin B_{R_{b}}(x_{b}’)} bigl{ biglvert u(x) bigrvert , biglvert v(x) bigrvert bigr} quadtextit{and}quad M_{eta}= sup_{substack{xin B_{R_{b}}(x_{b}’) \ i=1,dots,n}}bigl{ biglvert u_{x_{i}}(x) bigrvert , biglvert v_{x_{i}}(x) bigrvert bigr} . end{aligned}$$

Then, (partial _{nu}u(x_{b}) > partial _{nu}v(x_{b})).

Proof

Let (w=u-v) on (overline{B_{R_{b}}(x_{b}’)}). Then, on (B_{R_{b}}(x_{b}’)),

$$begin{aligned} 0 leq{}& sum_{i,j=1}^{n} bigl( A_{ij} (cdot,u,Du) u_{x_{i}x_{j}} – A_{ij} (cdot,v,Dv) v_{x_{i}x_{j}} bigr) + B(cdot,u,Du) – B( cdot,v,Dv) \ = {}&sum_{i,j=1}^{n} bigl(A_{ij}(cdot,u,Du) (u_{x_{i}x_{j}} – v_{x_{i}x_{j}})+ bigl(A_{ij}(cdot,u,Du) -A_{ij}(cdot,u,Dv)bigr) v_{x_{i}x_{j}} \ &{} + bigl( A_{ij}(cdot,u,Dv) -A_{ij}(cdot,v,Dv) bigr) v_{x_{i}x_{j}} bigr) \ & {}+ bigl( B(cdot,u,Du) – B(cdot,u,Dv) bigr) + bigl( B( cdot,u, Dv) – B(cdot, v, Dv) bigr) \ leq {}&sum_{i,j=1}^{n} A_{ij}(cdot, u, Du) w_{x_{i}x_{j}} \ & {}+ Lambda bigl(dbigl(cdot, partial B_{R_{b}} bigl(x_{b}’bigr) bigr)bigr) sum _{i=1}^{n} Biggl( operatorname{sgn} (w_{x_{i}}) + sum_{k,l=1}^{n} v_{x_{k}x_{l}} operatorname{sgn} (v_{x_{k}x_{l}}w_{x_{i}}) Biggr)w_{x_{i}} \ &{} + Biggl( frac{Lambda (d(cdot, partial B_{R_{b}}(x_{b}’) ))}{d(cdot, partial B_{R_{b}}(x_{b}’) )} sum_{k,l=1}^{n} v_{x_{k}x_{l}}operatorname{sgn} (v_{x_{k}x_{l}}w) \ & {} + frac{ ( B(cdot,u, Dv) – B(cdot, v, Dv) )}{w} Biggr) w \ =:{}& sum_{i,j=1}^{n} tilde{a}w_{x_{i}x_{j}} + sum_{i=1}^{n} tilde{b}_{i}w_{x_{i}} + tilde{c}w, end{aligned}$$

where, for (i,j=1,dots,n), (tilde{a}_{ij},tilde{b}_{i},tilde{c}: Omega to mathbb{R}) are given by,

$$begin{aligned} &tilde{a}_{ij}=A_{ij}(cdot,u,Du) , end{aligned}$$

(2.29)

$$begin{aligned} &tilde{b}_{i}= Lambda bigl(dbigl(cdot,partial B_{R_{b}} bigl(x_{b}’bigr) bigr)bigr) Biggl( operatorname{sgn}(w_{x_{i}}) + sum_{k,l=1}^{n} v_{x_{k}x_{l}}operatorname{sgn}(v_{x_{k}x_{l}}w_{x_{i}}) Biggr), end{aligned}$$

(2.30)

$$begin{aligned} &tilde{c} = frac{Lambda (d(cdot,partial B_{R_{b}}(x_{b}’) ))}{d(cdot,partial B_{R_{b}}(x_{b}’) )} sum_{k,l=1}^{n} v_{x_{k}x_{l}} operatorname{sgn}(v_{x_{k}x_{l}}w) + frac{B(cdot,u,Dv)-B(cdot,v,Dv)}{(u-v)} end{aligned}$$

(2.31)

$$begin{aligned} &phantom{tilde{c} }geq – frac{Lambda (d(cdot,partial B_{R_{b}}(x_{b}’) ))}{d(cdot,partial B_{R_{b}}(x_{b}’) )}Bigl(n^{2} sup_{substack{k,l=1,ldots,n \ xin B_{R_{b}}(x_{b}’)}} biglvert v_{x_{k}x_{l}}(x) bigrvert +1Bigr), end{aligned}$$

(2.32)

on (B_{R_{b}}(x_{b}’)). Thus, it follows that 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 (partial _{nu}w >0) at (x_{b}) and hence,

$$begin{aligned} partial _{nu}u(x_{b}) > partial _{nu}v (x_{b}), end{aligned}$$

as required. □

Remark 2.7

Note that conditions (2.27) and (2.28) ensure that: (A_{ij}) are locally Lipschitz continuous in z and η on Ω; B is locally lower Lipschitz in z and Lipschitz continuous in η; and the associated Lipschitz and lower Lipschitz constants for (A_{ij}) and B can tend to ∞ as (xto partial Omega ) but are constrained by the integrability condition on Λ. Additionally, observe that the conditions in Theorem 2.6 can be readily altered to accommodate (d(x,partial Omega )) instead of (d(x,partial B_{R_{b}}(x_{b}’))).

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 (Omega subset mathbb{R}^{n}) and for (x_{b}in partial Omega ) there exists (B_{R_{b}}(x_{b}’)subset Omega ) with (x_{b} in partial B_{R_{b}}(x_{b}’)). Consider (u:bar{Omega}to mathbb{R}) given by,

$$begin{aligned} u(x) = 0 quadforall xin bar{Omega}, end{aligned}$$

(2.33)

and (v:bar{Omega}to mathbb{R}) such that:

$$begin{aligned} & vin C^{infty}(bar{Omega}), end{aligned}$$

(2.34)

$$begin{aligned} & v>0 quadtext{in } Omega, end{aligned}$$

(2.35)

$$begin{aligned} & v(x_{b})=0 quadtext{and}quad partial _{nu}v(x_{b})=0. end{aligned}$$

(2.36)

Note that (2.34) implies that there exists (Mgeq 0) such that for (i,j=1,ldots, n),

$$begin{aligned} vert v vert , vert v_{x_{i}} vert , vert v_{x_{i}x_{j}} vert leq M quadtext{for all }xin bar{Omega}. end{aligned}$$

(2.37)

Now, for the quasilinear PDI in (2.4), set (A_{ij}:Omega times mathbb{R}times mathbb{R}^{n}to mathbb{R}) to be

$$begin{aligned} A_{ij}(x,z,eta ) = delta _{ij} quadforall (x,z,eta )in Omega times mathbb{R}times mathbb{R}^{n}, end{aligned}$$

(2.38)

for all (i,j=1,dots,n) and (B:Omega times mathbb{R}times mathbb{R}^{n}to mathbb{R}) to be

$$begin{aligned} B(x,z,eta ) = -frac{z}{v(x)}sum _{i=1}^{n}v_{x_{i}x_{i}}(x) quadforall (x,z,eta )in Omega times mathbb{R}times mathbb{R}^{n}. end{aligned}$$

(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, (Lambda = 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

$$begin{aligned} B(x,z_{1},eta _{1} )-B(x, z_{2}, eta _{2} ) = – frac{z_{1}-z_{2}}{v(x)}sum_{i=1}^{n}v_{x_{i}x_{i}}(x) geq -M_{K}(z_{1}- z_{2}), end{aligned}$$

(2.40)

for all ((x,z_{1},eta _{1} ),(x, z_{2},eta _{2})in K) such that (z_{1} geq z_{2} ), with (K= Omega ‘ times [-M,M]times [-M,M]^{n}) for any (Omega ‘) that is a compact subset of Ω, where via (2.35) and (2.37),

$$begin{aligned} M_{K}=frac{Mn}{inf_{xin Omega ‘}{v(x)}} > 0. end{aligned}$$

(2.41)

Therefore, it follows from (2.40) and (2.41) that (B(x,z,eta )) satisfies the conditions of Theorem 2.6 with the exception of the lower Lipschitz condition, which instead holds only locally on (Omega times mathbb{R}times mathbb{R}^{n}). It follows from (2.33)–(2.36), that (Q[u]geq 0) and (Q[v]leq 0) on Ω, for Q defined by (2.38) and (2.39). Moreover, via (2.33), (2.35), and (2.36) (u< v) in Ω and (u(x_{b})=v(x_{b})) for (x_{b}in partial Omega ). 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),

$$begin{aligned} partial _{nu}u(x_{b}) = partial _{nu}v(x_{b}), end{aligned}$$

which violates the conclusion of Theorem 2.6.

Remark 2.9

We note that u, v, (A_{ij}) and B in Example 2.8 satisfy all of the conditions of [13, Theorem 2.7.1], but violate the conclusion. This occurs since an unconstrained local lower Lipschitz constant is supposed on B with respect to z in [13, Theorem 2.7.1], which is an error. 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 [10, p.174, Theorem 7], as illustrated in [9]. 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.

Remark 2.10

If (partial _{nu nu} v(x_{b}) >0) in Example 2.8, by considering (K=B_{R}(x_{b}’)times [-M,M]times [-M,M]^{n}) with (0< R< R_{b}), it follows from (2.34) that as (Rto R_{b}),

$$begin{aligned} M_{K} geq frac{2Mn}{v_{nu nu}(x_{b}) ( R_{b}-R )^{2} + O( (R-R_{b})^{3})} geq frac{Mn}{v_{nu nu}(x_{b})d(partial B_{R}(x_{b}’), partial B_{R_{b}}(x_{b}’))^{2}} . end{aligned}$$

(2.42)

Thus, we observe that B in (2.40) satisfies the conditions of Theorem 2.6 with the exception of (Lambda in L^{1} ( ( 0, frac{R_{b}}{2} ) )) in (2.28). This follows from letting (Rto R_{b}) in (2.42), which implies that Λ necessarily satisfies

$$begin{aligned} Lambda (d) geq frac{Mn}{v_{nu nu}(x_{b})d} quadtext{as } dto 0^{+}. end{aligned}$$

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 [13, Theorem 2.7.1].

Example 2.11

Let (Omega = (0,1)subset mathbb{R}) and (u,v:bar{Omega}to mathbb{R}) be given by

$$begin{aligned} u(x) = x^{1+alpha }, v(x)=2x^{1+alpha} quadforall xin bar{Omega}, end{aligned}$$

(2.43)

with constant (alpha in (0,1)). It follows that (u,vin C^{1}(bar{Omega} )cap C^{2}(Omega )), (v>u) in Ω, and for (x_{b}=0in partial Omega ), we have (u(x_{b})=v(x_{b})=0). Now, consider the quasilinear operator Q with (A:Omega times mathbb{R}times mathbb{R} to mathbb{R}) given by

$$begin{aligned} A(x,z,eta ) = 1 + frac{2}{x^{1+alpha}} biggl( frac{3}{2}x^{1+ alpha} – z biggr)quad forall (x,z,eta )in Omega times mathbb{R}times mathbb{R} end{aligned}$$

(2.44)

with (B=0) on (Omega times mathbb{R}times mathbb{R} to mathbb{R}). Since

$$begin{aligned} Q[u]= A(cdot,u,Du)u_{xx} = 2u_{xx}geq 0,qquad Q[v]= A(cdot,v,Dv)v_{xx} leq 0, end{aligned}$$

(2.45)

on Ω, it follows that Q is elliptic with respect to u, and that Q satisfies the conditions (2.27) and (2.28) in Theorem 2.6 with (Lambda:(0,1]to (0,infty )) given by

$$begin{aligned} Lambda (d) = frac{2}{d^{alpha }}quad forall d in biggl( 0, frac {1}{2} biggr]. end{aligned}$$

Since Λ is continuous nonincreasing and (Lambda in L^{1} ( ( 0,frac {1}{2} ) )), it follows from (2.43)–(2.45) that u and v satisfy all of the conditions of Theorem 2.6 with the exception of (v_{xx}) being bounded on Ω. However, via (2.43),

$$begin{aligned} u_{nu}(x_{b})=v_{nu}(x_{b})=0, end{aligned}$$

which violates the conclusion of Theorem 2.6.

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