Strict Positivity for the Principal Eigenfunction of Elliptic Operators with Various Boundary Conditions

Wolfgang Arendt; A. F. M. ter Elst; Jochen Glück

doi:10.1515/ans-2020-2091

Article Open Access

Strict Positivity for the Principal Eigenfunction of Elliptic Operators with Various Boundary Conditions

Wolfgang Arendt , A. F. M. ter Elst and Jochen Glück

Published/Copyright: May 12, 2020

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From the journal Advanced Nonlinear Studies Volume 20 Issue 3

Abstract

We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain Ω⊂ℝd and show that the first eigenfunction v satisfies v⁢(x)≥δ>0 for all x∈Ω¯, even if the boundary ∂⁡Ω is only Lipschitz continuous. Under such weak regularity assumptions the Hopf–Oleĭnik boundary lemma is not available; instead we use a new approach based on an abstract positivity improving condition for semigroups that map Lp⁢(Ω) into C⁢(Ω¯). The same tool also yields corresponding results for Dirichlet or mixed boundary conditions. Finally, we show that our results can be used to derive strong minimum and maximum principles for parabolic and elliptic equations.

Keywords: Maximum Principle; Irreducible Semigroup; Elliptic Operator

MSC 2010: 35P15; 35B50; 35K08

1 Introduction

A frequent situation occurring in the study of elliptic but also parabolic boundary value problems with real coefficients on a bounded domain Ω⊂ℝd is the following. The solutions satisfy a weak maximum principle and there exists a principal eigenvalue with a principal eigenfunction u0 satisfying u0⁢(x)>0 a.e. on Ω. By elliptic regularity one also shows that u0∈C⁢(Ω¯). But what is not known is whether u0⁢(x)≥δ>0 for all x∈Ω¯. We shall show this under very weak regularity assumptions. Such a result has applications for the construction of super- and subsolutions (see Daners–López-Gómez [14]), but also for the asymptotic behaviour of parabolic problems.

Let us describe a concrete situation. Let Ω⊂ℝd be a bounded Lipschitz domain and β∈L∞⁢(∂⁡Ω). Given u0∈C⁢(Ω¯) there exists a unique u∈C⁢([0,∞)×Ω¯)∩C∞⁢((0,∞)×Ω) such that

∂ ∂ ⁡ t ⁢ u = Δ ⁢ u ,

u ⁢ ( 0 , x ) = u 0 ⁢ ( x ) for all ⁢ x ∈ Ω ¯ ,

( ∂ ν ⁡ u ) ⁢ ( t , z ) + β ⁢ ( z ) ⁢ u ⁢ ( t , z ) = 0 for all ⁢ z ∈ ∂ ⁡ Ω ⁢ and ⁢ t > 0 .

We shall show in Theorem 4.5 that if u0⁢(x)≥0 and u0≠0, then u⁢(t,x)>0 for all x∈Ω¯ and t>0. This implies in particular that the principal eigenfunction v∈C⁢(Ω¯) of the Robin Laplacian is strictly positive; that is, there exists a δ>0 such that v⁢(x)≥δ for all x∈Ω¯. If ∂⁡Ω and the eigenfunction are smooth enough, this property is known and can then be deduced from Hopf’s maximum principle [11, 21] (on the interior) and the Hopf–Oleĭnik boundary lemma (see for instance [2]). We shall prove the result for Lipschitz domains and arbitrary elliptic operators in divergence form with bounded real measurable coefficients, without any assumptions on the smoothness of the eigenfunction. This new result is important for applications to non-linear problems (see for example [14]).

Our arguments are best placed in a more abstract situation. Let S be a C0-semigroup on L2⁢(Ω) which is positive and holomorphic. Then S is irreducible (see below for the definition) if and only if S is positivity improving in the sense that if u≥0 and u≠0, then for each t>0 one has (St⁢u)⁢(x)>0 for almost every x∈Ω. Irreducibility on L2⁢(Ω) is very easy to prove by the use of the Beurling–Deny–Ouhabaz criterion [25, Theorem 2.10] and implies for the principal eigenfunction v that v⁢(x)>0 almost everywhere. In contrast to this, irreducibility in C⁢(Ω¯) is much stronger: it implies that v⁢(x)≥δ>0 for all x∈Ω¯ and some δ>0. Our main argument in Section 3 shows that irreducibility in L2⁢(Ω) already implies irreducibility in C⁢(Ω¯) if St⁢L2⁢(Ω)⊂C⁢(Ω¯) for all t>0 and if (St|C⁢(Ω¯))t>0 is a C0-semigroup on C⁢(Ω¯) (see Theorem 3.1 and Corollary 3.2).

In Section 4 we will apply this result not only to elliptic problems with Robin boundary conditions, but also to mixed boundary conditions, where we impose Neumann boundary conditions on a relatively open subset N of ∂⁡Ω and where we prove that (St⁢u)⁢(x)>0 for all t>0 and x∈Ω∩N whenever u≥0 and u≠0.

In Section 5 we will also prove a strong minimum principle for the heat equation. Given a continuous function ψ on the parabolic boundary ∂*⁡ΩT of the cylinder ΩT=(0,T)×Ω, there is a unique solution u∈C⁢(ΩT¯) of the heat equation ut=Δ⁢u which coincides with ψ on ∂*⁡ΩT. We shall show that if ψ≥0 and u⁢(t0,x0)=0 for some (t0,x0)∈ΩT, then u⁢(t,x)=0 for all (t,x)∈ΩT such that t<t0. Again, this also remains true if the Laplacian Δ is replaced with an elliptic operator (see Theorem 5.2). As a nice consequence, we also obtain a new proof of the strong parabolic maximum principle for elliptic operators in divergence form with bounded real measurable coefficients.

The paper is organised as follows. After a general introduction to irreducibility in Section 2, we establish our main abstract result in Section 3. Principal eigenvectors for elliptic problems with diverse boundary conditions are considered in Section 4 and the strong minimum principle is established in Section 5. For the parabolic operator we have two notions of solutions: mild and weak. For the mild solution we do not need any regularity on the coefficients of the operator, but in order to define the weak solutions we need some differentiability. Under these differentiability conditions we show that weak solutions and mild solutions are equivalent. For the latter equivalence we need a regularity result, for which we provide an elementary proof in the appendix.

2 Preliminaries: Irreducibility

In this section we recall the notions of positivity and irreducibility as well as some results which are used later. General references for this topic are [7] and [10].

Throughout the section, let E be a Banach lattice over 𝕂, where the field 𝕂 is either ℝ or ℂ. We are especially interested in the following cases.

Example 2.1.

Let Ω⊂ℝd be an open non-empty set. The following spaces are examples of Banach lattices:

E = L p ⁢ ( Ω ) , where p∈[1,∞).
E = C ⁢ ( Ω ¯ ) , if Ω is bounded.
E = C 0 ⁢ ( Ω ) , the closure in L∞⁢(Ω) of the space Cc⁢(Ω) of all continuous functions with compact support.

Let E+={u∈E:u≥0} be the positive cone of E. An ideal of E is a subspace J of E such that

if u∈J, then |u|∈J and
if u∈J, v∈E and 0≤v≤u, then v∈J.

The closure of an ideal is again an ideal. The closed ideals can be characterised in the case of Example 2.1. Let Ω⊂ℝd be an open set. If p∈[1,∞) and E=Lp⁢(Ω), then J⊂E is a closed ideal if and only if there exists a measurable subset B of Ω such that J={u∈E:u|B=0⁢ almost everywhere} (see [28, Section III.1, Example 1]). If Ω is bounded, and E=C⁢(Ω¯), then J⊂E is a closed ideal if and only if there exists a closed set B⊂Ω¯ such that J={u∈C⁢(Ω¯):u|B=0} (see [28, Section III.1, Example 2]). Finally, if E=C0⁢(Ω), then J⊂E is a closed ideal if and only if there exists a closed set B⊂Ω such that J={u∈C0⁢(Ω):u|B=0} (see [10, Proposition 10.14]).

Note that u≥0 in Lp⁢(Ω) means that u⁢(x)∈[0,∞) for almost every x∈Ω, whilst u≥0 in C⁢(Ω¯) means that u⁢(x)∈[0,∞) for allx∈Ω. We write u>0 if u≥0 and u≠0. Note that u≠0 in Lp⁢(Ω) means that {x∈Ω:u⁢(x)≠0} is not a null set.

If u≥0, then we denote by

E u = { v ∈ E : there exists an n ∈ ℕ such that ⁢ | v | ≤ n ⁢ u }

the principal ideal generated by u. It is easy to verify that this is indeed an ideal. We write u≫0 if Eu¯=E. In the literature of Banach lattices, such an element u is called a quasi-interior point. As a remark, quasi-interior points can be characterized by an approximation condition. Schaefer [28, Theorem II.6.3] proved that a vector u∈E+ is a quasi-interior point if and only if limn→∞⁡v∧n⁢u=v for every v∈E+.

If E=Lp⁢(Ω), then u≫0 if and only if u⁢(x)>0 for almost every x∈Ω. If Ω is bounded and E=C⁢(Ω¯), then u≫0 if and only if u⁢(x)>0 for all x∈Ω¯. So by compactness, u≫0 if and only if there exists a δ>0 such that u⁢(x)≥δ for all x∈Ω¯, which is the case if and only if u is an interior point of the positive cone E+. If E=C0⁢(Ω), then u≫0 if and only if u⁢(x)>0 for all x∈Ω. Note that the interior of E+ is empty if E=C0⁢(Ω) or E=Lp⁢(Ω).

Let also F be a Banach lattice. A linear map R:E→F is called positive if R⁢E+⊂F+. Positivity implies that R is continuous by [28, Theorem II.5.3]. We write R≥0 to express that R is positive. The set of all positive linear functionals on E is denoted by E+′. If E=Lp⁢(Ω), then E+′=Lp′⁢(Ω)+, where p′∈(1,∞] is the dual exponent. If Ω is bounded and E=C⁢(Ω¯), then E+′ is isomorphic to all finite (positive) Borel measures on Ω¯. If E=C0⁢(Ω), then E+′ is isomorphic to all (positive) finite Borel measures on Ω. For a proof of the last two statements, see [19, Theorem 14.1].

An operator R:E→F is called positivity improving if R⁢u≫0 in F for all u∈E with u>0. Positivity improving operators will be of central interest in this paper.

By a semigroup on E we mean a map S:(0,∞)→ℒ⁢(E) such that St+s=St⁢Ss for all s,t∈(0,∞), where we write St=S⁢(t). We say that S is a C0-semigroup if in addition limt↓0⁡St⁢u=u for all u∈E. A semigroup S is called positive if St≥0 for all t>0 and S is called positivity improving if St is positivity improving for all t>0. A semigroup S is called irreducible if it does not leave invariant any non-trivial closed ideal; that is, if J⊂E is a closed ideal and St⁢J⊂J for all t>0, then J=E or J={0}.

Irreducibility is independent of p for compatible semigroups.

Lemma 2.2.

Let Ω⊂Rd be open and p1,p2∈[1,∞). Let S(1) and S(2) be semigroups on Lp1⁢(Ω) and Lp2⁢(Ω). Suppose that S(1) and S(2) are compatible, that is, we have St(1)⁢u=St(2)⁢u almost everywhere for all t>0 and u∈Lp1⁢(Ω)∩Lp2⁢(Ω). Then S(1) is irreducible if and only if S(2) is irreducible.

Proof.

Suppose that S(1) is irreducible. Let B⊂Ω be measurable and set J2={u∈Lp2⁢(Ω):u|B=0}. Furthermore, suppose that St(2)⁢J2⊂J2 for all t>0. Let J1={u∈Lp1⁢(Ω):u|B=0}. Let t>0 and u∈J1. For all n∈ℕ let Vn={x∈Ω:∥x∥≤n⁢ and ⁢|u⁢(x)|≤n}. Then u⁢𝟙Vn∈J2∩Lp1⁢(Ω). So St(1)⁢(u⁢𝟙Vn)=St(2)⁢(u⁢𝟙Vn)∈J2. Therefore (St(1)⁢(u⁢𝟙Vn))|B=0 and St(1)⁢(u⁢𝟙Vn)∈J1. Then St(1)⁢u=limn→∞⁡St(1)⁢(u⁢𝟙Vn)∈J1 since J1 is closed. So St(1)⁢J1⊂J1. Because S(1) is irreducible, one concludes that |B|=0 or |Ω∖B|=0 and S(2) is irreducible. ∎

In general, a positive and irreducible C0-semigroup does not need to be positivity improving. An counterexample is the rotation semigroup on L2⁢(𝕋), where 𝕋 is the unit circle in ℂ. The situation changes, however, if the semigroup is also holomorphic.

Theorem 2.3.

Let S be a positive irreducible holomorphic C0-semigroup on E. Then S is positivity improving.

Proof.

See Majewski–Robinson [23, Theorem 3]. ∎

In the following proposition we collect a number of known spectral theoretic properties of positive semigroups.

Proposition 2.4.

Let S be a positive irreducible C0-semigroup in E and suppose that its generator -A has compact resolvent. Then one has the following:

σ ⁢ ( A ) ≠ ∅ .
The number λ 1 := inf ⁡ { Re ⁡ λ : λ ∈ σ ⁢ ( A ) } is an eigenvalue of A (and consequently, the infimum is actually a minimum).
There exists a u ∈ D ⁢ ( A ) such that A ⁢ u = λ 1 ⁢ u and u ≫ 0 .
The algebraic multiplicity of the eigenvalue λ 1 is one.

Proof.

a We may assume that dim⁡E≥2. Then by a result of de Pagter [26, Theorem 3] every compact, positive and irreducible operator on E has non-zero spectral radius. If we apply this to the resolvent of A, the assertion follows.

b See [10, Corollary 12.9].

c It follows from the Krein–Rutman theorem, see for example [10, Theorem 12.15], that there exists a u∈D⁢(A) with A⁢u=λ1⁢u and u>0. Then the statement follows from [10, Proposition 14.12 (a)].

d This follows from [7, Proposition C-III.3.5]. ∎

Note that since A has compact resolvent, λ1 is an isolated point of the spectrum. Therefore Proposition 2.4 d means that the spectral projection for λ1 has rank one.

If S is a positive irreducible C0-semigroup whose generator -A has compact resolvent, then we call min⁡{Re⁡λ:λ∈σ⁢(A)} the principal eigenvalue of A. It follows from Proposition 2.4 that the principal eigenvalue has a unique eigenvector u such that u≥0 and ∥u∥=1. We call u the principal eigenvector of A. One has u≫0.

3 Irreducibility on C⁢(Ω¯) and C0⁢(Ω)

In this section we consider a positive irreducible holomorphic C0-semigroup on Lp⁢(Ω) which maps Lp⁢(Ω) into C⁢(Ω¯) or C0⁢(Ω). Under a mild additional condition we shall prove that the semigroup obtained by restriction to C⁢(Ω¯) or C0⁢(Ω) is again irreducible.

In Subsection 3.1 we prove a not too difficult but very powerful abstract result that is the basis of everything that follows. In Subsections 3.2 and 3.3 we consider the special cases C⁢(Ω¯) and C0⁢(Ω), respectively. We close the section with a brief remark on the long-term behaviour of positive semigroups in Subsection 3.4.

3.1 An Abstract Positivity Improvement Result

We start with a general theorem about positivity in a single point. It is the main ingredient for our proofs of strict positivity in Section 4. Let Ω⊂ℝd be an open non-empty set and X a set such that Ω⊂X⊂Ω¯. If p∈[1,∞) and u∈Lp⁢(Ω), then we say that u∈C⁢(X) if there exists a (necessarily unique) u~∈C⁢(X) such that u~|Ω=u almost everywhere on Ω. Note that ∂⁡Ω might have positive Lebesgue measure. In the sequel we will identify u and u~. For example, in the next theorem we identify St⁢u and (St⁢u)∼.

Theorem 3.1.

Let Ω⊂Rd be an open non-empty set and p∈[1,∞). Let S be a positive irreducible holomorphic C0-semigroup on Lp⁢(Ω). Next let X be a set such that Ω⊂X⊂Ω¯. Finally, let x∈X. Suppose

S t ⁢ L p ⁢ ( Ω ) ⊂ C ⁢ ( X ) for all t > 0 and
there are t > 0 and w ∈ L p ⁢ ( Ω ) such that ( S t ⁢ w ) ⁢ ( x ) ≠ 0 .

Then (St⁢u)⁢(x)>0 for all t>0 and u∈Lp⁢(Ω) with u≥0 and u≠0.

In what follows, typical choices for X are X=Ω or X=Ω¯. We also have, however, an application in Theorem 4.10 for elliptic operators with mixed boundary conditions, where X is chosen strictly between Ω and Ω¯. Let us also remark that, while Theorem 3.1 works pointwise, we are in fact most interested in the case where condition II, and then also the conclusion of the theorem, is valid for all x∈X instead of merely a single point.

Proof of Theorem 3.1.

The map u↦(St⁢u)⁢(x) from Lp⁢(Ω) into ℂ is positive, hence by [28, Theorem II.5.3] it is continuous. By assumption II there exist s>0 and w∈Lp⁢(Ω) such that (Ss⁢w)⁢(x)≠0. Without loss of generality we may assume that w≥0. Therefore (Ss⁢w)⁢(x)>0.

Let t∈(0,∞) and u∈Lp⁢(Ω) with u>0. There are t1,t2∈(0,∞) such that t=t1+t2 and t1<s. According to Theorem 2.3 we have St2⁢u≫0 in Lp⁢(Ω), so it follows from the Lebesgue dominated convergence theorem that

lim n → ∞ ⁡ S s - t 1 ⁢ w ∧ n ⁢ S t 2 ⁢ u = S s - t 1 ⁢ w

with respect to the norm in Lp⁢(Ω). By the continuity that we established in the beginning, it follows that

lim n → ∞ ⁡ ( S t 1 ⁢ ( S s - t 1 ⁢ w ∧ n ⁢ S t 2 ⁢ u ) ) ⁢ ( x ) = ( S s ⁢ w ) ⁢ ( x ) > 0 .

Consequently, there exists an n∈ℕ such that (St1⁢(Ss-t1⁢w∧n⁢St2⁢u))⁢(x)>0. But then

0 < ( S t 1 ⁢ ( S s - t 1 ⁢ w ∧ n ⁢ S t 2 ⁢ u ) ) ⁢ ( x ) ≤ ( S t 1 ⁢ ( n ⁢ S t 2 ⁢ u ) ) ⁢ ( x ) = n ⁢ ( S t 1 + t 2 ⁢ u ) ⁢ ( x ) = n ⁢ ( S t ⁢ u ) ⁢ ( x )

and the theorem follows. ∎

For the convenience of the reader, as well as for the sake of later reference, we explicitly state a few consequences of Theorem 3.1 in the following subsections.

3.2 Irreducibility on C⁢(Ω¯)

As a special case of Theorem 3.1 one obtains the following result for X=Ω¯ if Ω is bounded.

Corollary 3.2.

Let Ω⊂Rd be a bounded open set and p∈[1,∞). Let S be a positive irreducible holomorphic C0-semigroup on Lp⁢(Ω). Suppose

S t ⁢ L p ⁢ ( Ω ) ⊂ C ⁢ ( Ω ¯ ) for all t > 0 and
for all x ∈ Ω ¯ there are t > 0 and w ∈ L p ⁢ ( Ω ) such that ( S t ⁢ w ) ⁢ ( x ) ≠ 0 .

Then the following holds:

For all t > 0 the operator S t : L p ⁢ ( Ω ) → C ⁢ ( Ω ¯ ) is positivity improving. This means ( S t ⁢ u ) ⁢ ( x ) > 0 for all x ∈ Ω ¯ , t>0 and u∈Lp⁢(Ω) with u≥0 and u≠0.
For all t > 0 define T t = S t | C ⁢ ( Ω ¯ ) : C ⁢ ( Ω ¯ ) → C ⁢ ( Ω ¯ ) . Then the semigroup T is irreducible on C⁢(Ω¯).

Proof.

a This is a special case of Theorem 3.1.

b This follows immediately from the characterisation of closed ideals in C⁢(Ω¯) and statement a. ∎

Note that Condition II in Corollary 3.2 is satisfied if Condition I is valid and T is a C0-semigroup on C⁢(Ω¯), where T is as in statement b. It is also satisfied if there exists a t>0 such that St⁢𝟙Ω=𝟙Ω¯.

It is a consequence of Corollary 3.2 that the semigroup has a strictly positive kernel if Ω is bounded.

Corollary 3.3.

Let Ω⊂Rd be a bounded open set and S a semigroup on L2⁢(Ω). Let p∈[2,∞). Suppose that

S t ⁢ L p ⁢ ( Ω ) ⊂ C ⁢ ( Ω ¯ ) and S t * ⁢ L p ⁢ ( Ω ) ⊂ C ⁢ ( Ω ¯ ) for all t > 0 ,
for all x ∈ Ω ¯ there are t > 0 and w ∈ L p ⁢ ( Ω ) such that ( S t ⁢ w ) ⁢ ( x ) ≠ 0 , and
for all x ∈ Ω ¯ there are t > 0 and w ∈ L p ⁢ ( Ω ) such that ( S t * ⁢ w ) ⁢ ( x ) ≠ 0 .

Further suppose that (St|Lp⁢(Ω))t>0 is a positive irreducible holomorphic C0-semigroup on Lp⁢(Ω). Then for t>0 there exists a function Kt∈C⁢(Ω¯×Ω¯) such that

( S t ⁢ u ) ⁢ ( x ) = ∫ Ω K t ⁢ ( x , y ) ⁢ u ⁢ ( y ) ⁢ 𝑑 y

for all u∈L2⁢(Ω) and x∈Ω¯. Moreover, Kt⁢(x,y)>0 for all x,y∈Ω¯ and t>0.

Proof.

We would like to apply [8, Theorem 2.1]. To do so, we need a semigroup on an L2-space over Ω¯, which needs a bit of care since the boundary ∂⁡Ω might have non-zero Lebesgue measure. Let λ denote the Lebesgue measure on Ω and define the Borel measure μ on Ω¯ given by

μ ⁢ ( B ) = λ ⁢ ( B ∩ Ω )

for each Borel set B⊂Ω¯. Then μ is strictly positive on each non-empty open subset of Ω¯. Moreover, for each q∈[1,∞), the embedding Lq⁢(Ω)↪Lq⁢(Ω¯,μ), given by extending functions on Ω by 0 on ∂⁡Ω, is an isomorphism. Hence we can transport the semigroup S on L2⁢(Ω) to a semigroup T on L2⁢(Ω¯,μ). Then assumption I implies that Tt⁢Lp⁢(Ω¯,μ)⊂C⁢(Ω¯) and Tt*⁢Lp⁢(Ω¯,μ)⊂C⁢(Ω¯) for all t>0. It follows from [8, Theorem 2.1] that the operator Tt has a continuous kernel Kt∈C⁢(Ω¯×Ω¯) for all t>0. Then

( S t ⁢ u ) ⁢ ( x ) = ∫ Ω K t ⁢ ( x , y ) ⁢ u ⁢ ( y ) ⁢ 𝑑 y

for all t>0, u∈L2⁢(Ω) and x∈Ω¯. If t>0, then Kt≥0 almost everywhere on Ω×Ω since St is a positive operator. Hence by continuity Kt⁢(x,y)≥0 for all t>0 and x,y∈Ω¯.

Finally, let t>0 and x,y∈Ω¯. By assumption III there exist s>0 and w∈Lp⁢(Ω) such that (Ss*⁢w)⁢(y)≠0. There are t1,t2∈(0,∞) such that t=t1+t2 and t1<s. Define v:Ω¯→ℝ by v⁢(z)=Kt1⁢(z,y). Then v≠0 since

0 ≠ ( S s * ⁢ w ) ⁢ ( y ) = ( S t 1 * ⁢ S s - t 1 * ⁢ w ) ⁢ ( y ) = ∫ Ω v ⁢ ( z ) ⁢ ( S s - t 1 * ⁢ w ) ⁢ ( z ) ⁢ 𝑑 z .

Therefore, Kt⁢(x,y)=(St2⁢v)⁢(x)>0 by Corollary 3.2 a, where we use assumptions II and I. ∎

3.3 Irreducibility on C0⁢(Ω)

Analogously to Corollary 3.2 one can use Theorem 3.1 to derive irreducibility for semigroups on C0⁢(Ω). This yields the following corollary. Note that Ω does not have to be bounded in this subsection.

Corollary 3.4.

Let Ω⊂Rd be an open set and p∈[1,∞). Let S be a positive irreducible holomorphic C0-semigroup on Lp⁢(Ω). Suppose

S t ⁢ L p ⁢ ( Ω ) ⊂ C 0 ⁢ ( Ω ) for all t > 0 and
for all x ∈ Ω there are t > 0 and w ∈ L p ⁢ ( Ω ) such that ( S t ⁢ w ) ⁢ ( x ) ≠ 0 .

Then one has the following:

For all t > 0 the operator S t : L p ⁢ ( Ω ) → C 0 ⁢ ( Ω ) is positivity improving. This means ( S t ⁢ u ) ⁢ ( x ) > 0 for all x ∈ Ω , t>0 and u∈Lp⁢(Ω) with u≥0 and u≠0.
Suppose that for all t > 0 the operator S t | C 0 ⁢ ( Ω ) ∩ L p ⁢ ( Ω ) extends to a continuous operator T t from C 0 ⁢ ( Ω ) into C 0 ⁢ ( Ω ) . Then the semigroup T is irreducible on C 0 ⁢ ( Ω ) .

Note that if Ω is bounded, then C0⁢(Ω)⊂Lp⁢(Ω) and the operator St|C0⁢(Ω) is indeed a continuous operator from C0⁢(Ω) into C0⁢(Ω). Moreover, condition II in Corollary 3.4 is satisfied if Ω is bounded, condition I is valid and T is a C0-semigroup on C0⁢(Ω), where T is defined as in b.

Similarly as in the proof of Corollary 3.3 we obtain a kernel for the semigroup in case Ω is bounded.

Corollary 3.5.

Let Ω⊂Rd be a bounded open set and S a semigroup on L2⁢(Ω). Let p∈[2,∞). Suppose that

S t ⁢ L p ⁢ ( Ω ) ⊂ C 0 ⁢ ( Ω ) and S t * ⁢ L p ⁢ ( Ω ) ⊂ C 0 ⁢ ( Ω ) for all t > 0 ,
for all x ∈ Ω there are t > 0 and w ∈ L p ⁢ ( Ω ) such that ( S t ⁢ w ) ⁢ ( x ) ≠ 0 , and
for all x ∈ Ω there are t > 0 and w ∈ L p ⁢ ( Ω ) such that ( S t * ⁢ w ) ⁢ ( x ) ≠ 0 .

Further suppose that (St|Lp⁢(Ω))t>0 is a positive irreducible holomorphic C0-semigroup on Lp⁢(Ω). Then for t>0 there exists a function Kt∈C0⁢(Ω×Ω) such that

( S t ⁢ u ) ⁢ ( x ) = ∫ Ω K t ⁢ ( x , y ) ⁢ u ⁢ ( y ) ⁢ 𝑑 y

for all u∈L2⁢(Ω) and x∈Ω. Moreover, Kt⁢(x,y)>0 for all x,y∈Ω and t>0.

Proof.

All is similar as in the proof of Corollary 3.3, but one obtains that Kt∈C⁢(Ω¯×Ω¯). It remains to show that Kt∈C0⁢(Ω×Ω). Let t>0 and x∈∂⁡Ω. Since St⁢Lp⁢(Ω)⊂C0⁢(Ω), it follows that

0 = ( S t ⁢ u ) ⁢ ( x ) = ∫ Ω K t ⁢ ( x , z ) ⁢ u ⁢ ( z ) ⁢ 𝑑 z

for all u∈Cc⁢(Ω). Hence Kt⁢(x,z)=0 for almost all z∈Ω and by continuity for all z∈Ω. By duality Kt⁢(z,y)=0 for all z∈Ω and y∈∂⁡Ω. Because

K 2 ⁢ t ⁢ ( x , y ) = ∫ Ω K t ⁢ ( x , z ) ⁢ K t ⁢ ( z , y ) ⁢ 𝑑 z ,

one deduces that Kt⁢(x,y)=0 if x∈∂⁡Ω or y∈∂⁡Ω. So K2⁢t∈C0⁢(Ω×Ω). ∎

3.4 A Note on the Long-Time Behaviour

It is worthwhile to say a few sentences on how the properties that we discussed above are related to the long-time behaviour of the semigroup.

Remark 3.6.

In the situation of Corollary 3.2, and for bounded Ω in the situation of Corollary 3.4, the semigroup on Lp⁢(Ω) consists of compact operators. Let -A be the generator and let λ1 be the principal eigenvalue of A. Then by [7, Corollary C-III.3.16] there is a spectral gap in the sense that there exists an ε>0 such that {λ∈σ⁢(A):Re⁡λ≤λ1+ε}={λ1}. Moreover, if λ1=0, then St converges in ℒ⁢(Lp⁢(Ω)) to a rank-one projection if t→∞ (see [7, Proposition C-III.3.5]).

4 Strict Positivity of Principal Eigenvectors and Other Applications

In this section we use the theorems from Section 3 to establish strict positivity of the principal eigenfunction of an elliptic operator for three types of boundary conditions: Dirichlet (Subsection 4.1), Robin (Subsection 4.2) and mixed (Subsection 4.4). For each of these boundary conditions we prove, besides strict positivity of the principal eigenvector, also irreducibility of the corresponding semigroup on a suitable space of continuous functions and a positivity improving property for the corresponding elliptic problem. Moreover, in Subsection 4.3 we shall show that our results have a surprising consequence for elliptic problems with complex Robin boundary conditions.

Throughout this section, let Ω⊂ℝd be a bounded non-empty, open and connected set with boundary Γ=∂⁡Ω. For all k,l∈{1,…,d} let ak⁢l,bk,ck,c0∈L∞⁢(Ω,ℝ). We assume that the coefficients ak⁢l satisfy a uniform ellipticity condition, namely that there exists a μ>0 such that, for almost all x∈Ω, the inequality

Re ⁢ ∑ k , l = 1 d a k ⁢ l ⁢ ( x ) ⁢ ξ k ⁢ ξ l ¯ ≥ μ ⁢ | ξ | 2

holds for all ξ∈ℂd. In the following subsections we will define elliptic operators with the coefficients ak⁢l,bk,ck,c0 by means of form methods. Loosely speaking, the operator is equal to

u ↦ - ∑ k , l = 1 d ∂ l ⁡ a k ⁢ l ⁢ ∂ k ⁡ u - ∑ k = 1 d ∂ k ⁡ b k ⁢ u + ∑ k = 1 d c k ⁢ ∂ k ⁡ u + c 0 ⁢ u

with boundary conditions. Moreover, depending on the boundary conditions, we will impose different regularity assumptions on the boundary of Ω in each subsection.

Most results in this section are a combination of theorems from the literature with Theorem 3.1 and its corollaries. For each type of boundary conditions we state a theorem which describes a positivity improving property of the parabolic equation, and a corollary which yields a similar result for the corresponding elliptic equation.

4.1 Dirichlet Boundary Conditions

In this subsection we assume that Ω is Wiener regular. This means that for all φ∈C⁢(Γ) there exists a function u∈C⁢(Ω¯)∩C2⁢(Ω) such that Δ⁢u=0 on Ω and u|Γ=φ. For instance, Ω is Wiener regular if it has Lipschitz boundary.

Define the form 𝔞:H01⁢(Ω)×H01⁢(Ω)→ℂ by

𝔞 ⁢ ( u , v ) = ∫ Ω ∑ k , l = 1 d a k ⁢ l ⁢ ( ∂ k ⁡ u ) ⁢ ∂ l ⁡ v ¯ + ∫ Ω ∑ k = 1 d b k ⁢ u ⁢ ∂ k ⁡ v ¯ + ∫ Ω ∑ k = 1 d c k ⁢ ( ∂ k ⁡ u ) ⁢ v ¯ + ∫ Ω c 0 ⁢ u ⁢ v ¯ .

Then 𝔞 is a closed sectorial form. Let A be the m-sectorial operator on L2⁢(Ω) associated with 𝔞 and let S be the semigroup generated by -A on L2⁢(Ω). Then S is a positive semigroup by [25, Theorem 2.6 or Corollary 4.3] and irreducibility of S follows from [25, Theorem 4.5]. Since the embedding H01⁢(Ω)⊂L2⁢(Ω) is compact, the operator A has compact resolvent. If t>0, then St⁢L2⁢(Ω)⊂C0⁢(Ω) by [9, (8)], where we used that Ω is Wiener regular. For all t>0 let Tt=St|C0⁢(Ω):C0⁢(Ω)→C0⁢(Ω). Then T is a holomorphic C0-semigroup by [9, Theorem 1.3]. Clearly T is positive. The following result shows that T is also irreducible.

Theorem 4.1.

The operator A on L2⁢(Ω) and the semigroup T on C0⁢(Ω) have the following properties:

For all t > 0 the operator T t is positivity improving. In particular, the semigroup T is irreducible.
Let u be the principal eigenfunction of A . Then u ∈ C 0 ⁢ ( Ω ) and u ⁢ ( x ) > 0 for all x ∈ Ω .

Proof.

Statement a follows from Corollary 3.4 and statement b from Proposition 2.4 c. ∎

Remark 4.2.

Theorem 4.1 can also be derived from known, but much less elementary results from PDE. Statement b follows from the Harnack inequality (see for instance [18, Theorem 8.20]). Next, let t>0 and let Kt be the kernel of the operator St. The De Giorgi–Nash theorem implies that Kt is continuous. Then the Harnack inequality shows that Kt is strictly positive on Ω×Ω. Therefore Tt is positivity improving and T is irreducible.

Remark 4.3.

For the special case of the Laplacian, the strict parabolic maximum principle of Evans [17, Section 2.3.3] was used in [4, Theorem 3.3] to prove Theorem 4.1. The proof in Evans, however, is based on a mean value property which is not valid for operators with variable coefficients.

We conclude this subsection with a positivity improving property for the corresponding elliptic problem. Since the semigroup S has Gaussian kernel bounds it follows that the semigroup S extends to a C0-semigroup on Lp⁢(Ω) for all p∈[1,∞). We denote its generator by -Ap. As A has compact resolvent, it follows that Ap has compact resolvent too and that the spectrum of Ap coincides with the spectrum of A by [27]. We obtain from Theorem 4.1 the following corollary about regularity of the corresponding elliptic problem.

Corollary 4.4.

Let λ∈R be smaller than the first eigenvalue of A; let p∈(d2,∞). If u∈D⁢(Ap) and (-λ⁢I+Ap)⁢u>0, then u∈C0⁢(Ω) and u⁢(x)>0 for all x∈Ω.

Proof.

Denote the generator of T by -Ac and choose μ∈ℝ such that μ<λ and μ<inf⁡{Re⁡ν:ν∈σ⁢(Ac)}. The semigroup T is irreducible according to Theorem 4.1 a. Hence it follows from [7, Definition C-III.3.1] that the resolvent (-μ⁢I+Ac)-1 is positivity improving on C0⁢(Ω). Note that the operator (-μ⁢I+Ac)-1 coincides with the restriction of (-μ⁢I+Ap)-1 to C0⁢(Ω).

One deduces from [9, Corollary 2.10] that the range of the resolvents (-λ+Ap)-1 and (-μ+Ap)-1 are contained in C0⁢(Ω), where we use that p>d2. Set f=(-λ⁢I+Ap)⁢u. Then the resolvent identity implies that

u = ( - λ ⁢ I + A p ) - 1 ⁢ f

= ( λ - μ ) ⁢ ( - μ ⁢ I + A p ) - 1 ⁢ ( - λ ⁢ I + A p ) - 1 ⁢ f + ( - μ ⁢ I + A p ) - 1 ⁢ f

≥ ( λ - μ ) ⁢ ( - μ ⁢ I + A p ) - 1 ⁢ ( - λ ⁢ I + A p ) - 1 ⁢ f

= ( λ - μ ) ⁢ ( - μ ⁢ I + A c ) - 1 ⁢ ( - λ ⁢ I + A p ) - 1 ⁢ f ≫ 0 ,

where ≫ is to be understood in C0⁢(Ω). This proves the corollary. ∎

Note that σ⁢(Ac)=σ⁢(A2) by [5, Proposition 3.10.3].

4.2 Robin Boundary Conditions

In this subsection we assume in addition that Ω has Lipschitz boundary. Further let β∈L∞⁢(Γ,ℝ). Define the form 𝔞:H1⁢(Ω)×H1⁢(Ω)→ℂ by

𝔞 ⁢ ( u , v ) = ∫ Ω ∑ k , l = 1 d a k ⁢ l ⁢ ( ∂ k ⁡ u ) ⁢ ∂ l ⁡ v ¯ + ∫ Ω ∑ k = 1 d b k ⁢ u ⁢ ∂ k ⁡ v ¯ + ∫ Ω ∑ k = 1 d c k ⁢ ( ∂ k ⁡ u ) ⁢ v ¯ + ∫ Ω c 0 ⁢ u ⁢ v ¯ + ∫ Γ β ⁢ ( Tr ⁡ u ) ⁢ Tr ⁡ v ¯ .

Then 𝔞 is a closed sectorial form. Let A be the m-sectorial operator on L2⁢(Ω) associated with 𝔞 and let S be the semigroup generated by -A on L2⁢(Ω). Then S is a positive semigroup by [25, Theorem 2.6]. Moreover, S is irreducible on L2⁢(Ω) by [25, Corollary 2.11] together with the discussion in [25, p. 106]. Since Ω is bounded and Lipschitz, the operator A has compact resolvent. If t>0, then St⁢L2⁢(Ω)⊂C⁢(Ω¯) by [8, Remark 6.2]. Let Tt=St|C⁢(Ω¯):C⁢(Ω¯)→C⁢(Ω¯) for all t>0. Then T is a C0-semigroup by [8, Remark 6.2].

Theorem 4.5.

The operator A on L2⁢(Ω) and the semigroup T on C⁢(Ω¯) have the following properties:

For all t > 0 the operator T t is positivity improving. In particular, the semigroup T is irreducible.
Let u be the principal eigenvalue of A . Then u ∈ C ⁢ ( Ω ¯ ) and u ⁢ ( x ) > 0 for all x ∈ Ω ¯ .

Proof.

Statement a follows from Corollary 3.2 and statement b from Proposition 2.4 c. ∎

Note that it follows again from the Harnack inequality that u⁢(x)>0 for all x∈Ω. The above theorem, however, says much more, namely that u is also strictly positive on the boundary of Ω and hence, bounded away from 0. This is of interest in the study of non-linear equations, and is new under such general conditions as we have here. Under much stronger regularity conditions, for instance if Ω has a C2-boundary and all coefficients are smooth, one can of course deduce Theorem 4.5 from Hopf’s minimum principle, see for example [22, Theorem 1.2].

Again, we also derive a corresponding elliptic result. By the Gaussian kernel bounds of [13, Theorem 2.2] and [12] the semigroup S on L2⁢(Ω) extrapolates to a C0-semigroup on Lp⁢(Ω) for all p∈[1,∞), whose generator we denote by -Ap. If p>d2, the resolvent operators of Ap map Lp⁢(Ω) into C⁢(Ω¯) by [24, Theorem 3.14 (iv)]. Hence by exactly the same arguments as in the proof of Corollary 4.4 we can obtain the following consequence of Theorem 4.5.

Corollary 4.6.

Let λ∈R be smaller than the first eigenvalue of A, let p∈(d2,∞) and u∈D⁢(Ap). Suppose that (-λ⁢I+Ap)⁢u>0. Then u∈C⁢(Ω¯) and u⁢(x)>0 for all x∈Ω¯.

4.3 The Bottom of the Spectrum for Complex Robin Boundary Conditions

In this subsection we consider complex Robin boundary conditions and show that Theorem 4.5 has surprising consequences for this situation. Note that in Theorem 4.5 the function β is real valued.

As in Subsection 4.2 we assume that Ω has Lipschitz boundary Γ. For the coefficients of the differential operator we assume that ak⁢l=al⁢k and bk=ck for all k,l∈{1,…,d}.

For all β∈L∞⁢(Γ) define the form 𝔞β:H1⁢(Ω)×H1⁢(Ω)→ℂ by

𝔞 β ⁢ ( u , v ) = ∫ Ω ∑ k , l = 1 d a k ⁢ l ⁢ ( ∂ k ⁡ u ) ⁢ ∂ l ⁡ v ¯ + ∫ Ω ∑ k = 1 d b k ⁢ u ⁢ ∂ k ⁡ v ¯ + ∫ Ω ∑ k = 1 d b k ⁢ ( ∂ k ⁡ u ) ⁢ v ¯ + ∫ Ω c 0 ⁢ u ⁢ v ¯ + ∫ Γ β ⁢ ( Tr ⁡ u ) ⁢ Tr ⁡ v ¯ .

Then 𝔞β is a closed sectorial form. Let Aβ be the m-sectorial operator associated with 𝔞. Since Ω is bounded and Lipschitz, the operator Aβ has compact resolvent. Note that 𝔞β is symmetric if β is real valued.

Proposition 4.7.

Let β∈L∞⁢(Γ) with Im⁡β≠0. Then min⁡{Re⁡λ:λ∈σ⁢(Aβ)}>min⁡σ⁢(ARe⁡β).

Proof.

Let λ∈σ⁢(Aβ). There exists a u∈D⁢(Aβ) such that Aβ⁢u=λ⁢u and ∥u∥L2⁢(Ω)=1. Then

Re λ = Re ( A β u , u ) L 2 ⁢ ( Ω ) = Re 𝔞 β ( u ) = 𝔞 Re ⁡ β ( u ) ≥ min σ ( A Re ⁡ β ) ,

where we used that 𝔞Re⁡β is symmetric. If Re⁡λ=min⁡σ⁢(ARe⁡β), then 𝔞Re⁡β⁢(u)=min⁡σ⁢(ARe⁡β). So u∈D⁢(ARe⁡β) and ARe⁡β⁢u=λ1⁢u, where λ1=min⁡σ⁢(ARe⁡β) and we used Proposition 2.4 d. Using Theorem 4.5, one deduces that u∈C⁢(Ω¯) and u⁢(x)≠0 for all x∈Γ (even for all x∈Ω¯). Let ∂ν⁡u denote the (weak) co-normal derivative of u. Then ∂ν⁡u+β⁢u|Γ=0 in L2⁢(Γ) since u∈D⁢(Aβ). But also ∂ν⁡u+(Re⁡β)⁢u|Γ=0 in L2⁢(Γ) since u∈D⁢(ARe⁡β). So (Im⁡β)⁢u|Γ=0 in L2⁢(Γ) and hence Im⁡β=0 almost everywhere. This is a contradiction. ∎

4.4 Mixed Boundary Conditions

In this subsection we assume that Ω has Lipschitz boundary. Further, let D⊂∂⁡Ω be a closed set and define N=∂⁡Ω∖D. We consider elliptic differential operators with mixed boundary conditions where, roughly speaking, we wish to have Dirichlet boundary conditions on D and Neumann boundary conditions on N. This yields an example where we apply Theorem 3.1 with a set X such that Ω⫋X⫋Ω¯.

In contrast to the previous sections, we restrict ourselves to differential operators with second order coefficients only, i.e. we assume that bk=ck=c0=0 for all k∈{1,…,d}.

Since the pure Dirichlet and pure Neumann case have been considered in the previous subsections, we assume that D≠∅ and N≠∅. Let ∂⁡D be the boundary of D in the relative topology of ∂⁡Ω. We need a technical assumption which states that the set of points from the Dirichlet boundary part is large enough with respect to the boundary measure (see [15]). Precisely, we suppose that there exists a δ>0 such that for all x∈∂⁡D and r∈(0,1] there exists a y∈D∩B⁢(x,r) such that

(4.1) N ∩ B ⁢ ( y , δ ⁢ r ) = ∅ .

Next we introduce the generator. Let

C D ∞ ⁢ ( Ω ) = { χ | Ω : χ ∈ C c ∞ ⁢ ( ℝ d ) ⁢ and ⁢ D ∩ supp ⁡ χ = ∅ }

and let WD1,2⁢(Ω) be the closure of CD∞⁢(Ω) in W1,2⁢(Ω). Define the form 𝔞:WD1,2⁢(Ω)×WD1,2⁢(Ω)→ℂ by

𝔞 ⁢ ( u , v ) = ∫ Ω ∑ k , l = 1 d a k ⁢ l ⁢ ( ∂ k ⁡ u ) ⁢ ∂ l ⁡ v ¯ .

Then 𝔞 is a closed sectorial form.

Let A be the operator associated with 𝔞 on L2⁢(Ω) and let S be the semigroup generated by -A on L2⁢(Ω). Finally, let CD⁢(Ω¯)={u∈C⁢(Ω¯):u|D=0}. We shall first show that S leaves the space CD⁢(Ω¯) invariant and that the restriction to CD⁢(Ω¯) is a C0-semigroup. Note that in Subsections 4.1 and 4.2 we could quote the literature to have a C0-semigroup on C0⁢(Ω) and C⁢(Ω¯).

Theorem 4.8.

Adopt the above notation and assumptions.

The semigroup S is positive and irreducible.
If t > 0 , then S t ⁢ L 2 ⁢ ( Ω ) ⊂ C D ⁢ ( Ω ¯ ) . In particular, the semigroup S leaves C D ⁢ ( Ω ¯ ) invariant.

For all t>0 define Tt=St|CD⁢(Ω¯):CD⁢(Ω¯)→CD⁢(Ω¯).

The semigroup T is a C 0 - semigroup on C D ⁢ ( Ω ¯ ) .

Proof.

a This follows from [25, Corollary 4.3 and Theorem 4.5].

b We first show that C⁢(Ω¯)∩WD1,2⁢(Ω)⊂CD⁢(Ω¯). Let v∈C⁢(Ω¯)∩WD1,2⁢(Ω). Since (Tr⁡w)|D=0ℋd-1-almost everywhere for all w∈CD∞⁢(Ω), it follows by density that (Tr⁡w)|D=0ℋd-1-almost everywhere for all functions w∈WD1,2⁢(Ω). In particular, (Tr⁡v)|D=0ℋd-1-almost everywhere. Let z∈D. Suppose that v⁢(z)≠0. Since v is continuous, there exists an s∈(0,1) such that v⁢(x)≠0 for all x∈Ω¯∩B⁢(z,s). If z is in the interior of D in the relative topology of ∂⁡Ω, then this contradicts (Tr⁡v)|D=0ℋd-1-almost everywhere. Alternatively, if z∈∂⁡D, then (4.1) gives a contradiction. So v⁢(z)=0. Therefore v∈CD⁢(Ω¯) and the inclusion C⁢(Ω¯)∩WD1,2⁢(Ω)⊂CD⁢(Ω¯) follows.

It is a consequence of [15, Theorem 1.1] that S maps into the (globally) Hölder continuous functions on Ω¯. Let t>0 and u∈L2⁢(Ω). Then St⁢u∈C⁢(Ω¯)∩WD1,2⁢(Ω)⊂CD⁢(Ω¯) and statement b follows.

D ⁢ ( A C D ) = { u ∈ C D ⁢ ( Ω ¯ ) ∩ D ⁢ ( A ) : A ⁢ u ∈ C D ⁢ ( Ω ¯ ) }

and ACD⁢u=A⁢u for all u∈D⁢(ACD).

First we shall show that D⁢(ACD) is dense in CD⁢(Ω¯). We shall do this in two steps. If u∈CD⁢(Ω¯)+ and ε>0, then D∩supp⁡((u-ε)+)=∅. Regularising (u-ε)+ it follows that CD⁢(Ω¯)+ is contained in the CD⁢(Ω¯)-closure of {χ|Ω¯:χ∈Cc∞⁢(ℝd)⁢ and ⁢D∩supp⁡χ=∅}. Hence by linearity {χ|Ω¯:χ∈Cc∞⁢(ℝd)⁢ and ⁢D∩supp⁡χ=∅} is dense in CD⁢(Ω¯).

By Proposition 4.9 b below there exists a c>0 such that u∈C⁢(Ω¯) and ∥u∥C⁢(Ω¯)≤c⁢∑k=1d∥fk∥Ld+1⁢(Ω) for all u∈WD1,2⁢(Ω) and f1,…,fd∈Ld+1⁢(Ω) such that

𝔞 ⁢ ( u , v ) = ∑ k = 1 d ( f k , ∂ k ⁡ v ) L 2 ⁢ ( Ω )

for all v∈WD1,2⁢(Ω). Now let χ∈Cc∞⁢(ℝd) and suppose that D∩supp⁡χ=∅. Let ε>0. Since Cc∞⁢(Ω) is dense in Ld+1⁢(Ω), for all k∈{1,…,d} there exists a wk∈Cc∞⁢(Ω) such that

∥ w k - ∑ l = 1 d a l ⁢ k ⁢ ∂ l ⁡ χ ∥ L d + 1 ⁢ ( Ω ) < ε .

Define f=-∑k=1d∂k⁡wk∈Cc∞⁢(Ω). There exists a unique u∈WD1,2⁢(Ω) such that 𝔞⁢(u,v)=(f,v)L2⁢(Ω) for all v∈WD1,2⁢(Ω). Then u∈C⁢(Ω¯) by Proposition 4.9 a below. So u∈C⁢(Ω¯)∩WD1,2⁢(Ω)⊂CD⁢(Ω¯) by the first step in the proof of statement b. Clearly u∈D⁢(A) and A⁢u=f. Obviously, f∈CD⁢(Ω¯). Hence u∈D⁢(ACD). Moreover, if v∈WD1,2⁢(Ω), then

𝔞 ⁢ ( u - χ | Ω , v ) = ∑ k , l = 1 d ∫ Ω a k ⁢ l ⁢ ( ∂ k ⁡ u - ∂ k ⁡ χ ) ⁢ ∂ l ⁡ v ¯

= ( f , v ) L 2 ⁢ ( Ω ) - ∑ k , l = 1 d ∫ Ω a k ⁢ l ⁢ ( ∂ k ⁡ χ ) ⁢ ∂ l ⁡ v ¯

= - ∑ k = 1 d ( ∂ k ⁡ w k , v ) L 2 ⁢ ( Ω ) - ∑ k = 1 d ∑ l = 1 d ( a l ⁢ k ⁢ ∂ l ⁡ χ , ∂ k ⁡ v ) L 2 ⁢ ( Ω )

= ∑ k = 1 d ( w k - ∑ l = 1 d a l ⁢ k ⁢ ∂ l ⁡ χ , ∂ k ⁡ v ) L 2 ⁢ ( Ω ) .

So u-χ|Ω¯∈C⁢(Ω¯) and

∥ u - χ | Ω ¯ ∥ C ⁢ ( Ω ¯ ) ≤ c ⁢ ∑ k = 1 d ∥ w k - ∑ l = 1 d a l ⁢ k ⁢ ∂ l ⁡ χ ∥ L d + 1 ⁢ ( Ω ) ≤ c ⁢ d ⁢ ε .

We showed that χ|Ω¯ belongs to the closure of D⁢(ACD) in CD⁢(Ω¯). Hence D⁢(ACD) is dense in CD⁢(Ω¯).

Now we are able to complete the proof of statement (c). By [15, Theorem 7.5] the semigroup S has Gaussian kernel bounds. Hence there exists an M>0 such that ∥St∥∞→∞≤M for all t∈(0,1]. Then ∥Tt∥∞→∞≤M for all t∈(0,1]. If u∈D⁢(ACD), then

∥ ( I - T t ) ⁢ u ∥ C D ⁢ ( Ω ¯ ) ≤ ∫ 0 t ∥ S s ⁢ A C D ⁢ u ∥ ∞ ≤ M ⁢ t ⁢ ∥ A C D ⁢ u ∥ ∞

for all t∈(0,1]. Hence we have limt↓0⁡Tt⁢u=u in CD⁢(Ω¯). Since D⁢(ACD) is dense in CD⁢(Ω¯) the semigroup T is a C0-semigroup. ∎

In the proof Theorem 4.8 we needed the following regularity results of [15].

Proposition 4.9.

Let p∈(d,∞).

If u ∈ W D 1 , 2 ⁢ ( Ω ) and f ∈ L p ⁢ ( Ω ) with 𝔞 ⁢ ( u , v ) = ( f , v ) L 2 ⁢ ( Ω ) for all v ∈ W D 1 , 2 ⁢ ( Ω ) , then u ∈ C ⁢ ( Ω ¯ ) .
There exists a constant c > 0 such that ∥ u ∥ C ⁢ ( Ω ¯ ) ≤ c ⁢ ∑ k = 1 d ∥ f k ∥ L p ⁢ ( Ω ) for all u ∈ W D 1 , 2 ⁢ ( Ω ) and f 1 , … , f d ∈ L p ⁢ ( Ω ) such that 𝔞 ⁢ ( u , v ) = ∑ k = 1 d ( f k , ∂ k ⁡ v ) L 2 ⁢ ( Ω ) for all v ∈ W D 1 , 2 ⁢ ( Ω ) .

Proof.

This follows as in [15, proof of Theorem 6.8]. Since ∅≠D≠∂⁡Ω, the form 𝔞 is coercive. Hence the identity operator in [15, Theorem 6.8] is not needed. ∎

Similar to the case of Dirichlet and Robin boundary conditions, we obtain irreducibility of the semigroup on the space CD⁢(Ω¯).

Theorem 4.10.

The operator A on L2⁢(Ω) and the semigroup T on CD⁢(Ω¯) have the following properties:

For all t > 0 the operator T t is positivity improving. In particular, the semigroup T is irreducible.
Let u be the principal eigenvector of A . Then u ⁢ ( x ) > 0 for all x ∈ Ω ∪ N .

Proof.

a Choose p=2. Let X=Ω∪N. Then Ω⊂X⊂Ω¯. It follows from Theorem 4.8 b that condition I in Theorem 3.1 is valid and condition II follows from Theorem 4.8 (c) for every x∈X. Hence Theorem 3.1 implies that (St⁢u)⁢(x)>0 for all x∈X, t>0 and u∈L2⁢(Ω) with u≥0 and u≠0. So T is positivity improving and consequently irreducible.

b This follows immediately from the proof of statement a. ∎

Remark 4.11.

Note that CD⁢(Ω¯)=C0⁢(Ω∪N), the closure of Cc⁢(Ω∪N) in C⁢(Ω¯). It follows that u∈C0⁢(Ω∪N) is a quasi-interior point if and only if u⁢(x)>0 for all x∈Ω∪N.

By [15, Theorem 7.5] the semigroup S has Gaussian kernel bounds. Hence the semigroup extends consistently to Lp⁢(Ω) for all p∈[1,∞). We denote the generator by -Ap. If p∈(d2,∞), then a Laplace transform gives that the resolvent of Ap maps Lp⁢(Ω) into CD⁢(Ω¯). By the same arguments as in the proof of Corollary 4.4 we obtain the following consequence of Theorem 4.10.

Corollary 4.12.

Let λ∈R be smaller than the first eigenvalue of A and let p∈(d2,∞). If u∈D⁢(Ap) and (-λ⁢I+Ap)⁢u>0, then u∈CD⁢(Ω¯) and u⁢(x)>0 for all x∈Ω∪N.

5 The Strong Maximum Principle for Parabolic Equations

In this section we show how our results, in particular Corollary 3.4 and Theorem 4.1, can be employed to prove strong minimum and maximum principles for parabolic and elliptic differential operators. Throughout this section let Ω⊂ℝd be a bounded non-empty open set with boundary Γ. For all k,l∈{1,…,d} let ak⁢l,bk,ck,c0∈L∞⁢(Ω,ℝ). We assume that there exists a μ>0 such that

Re ⁢ ∑ k , l = 1 d a k ⁢ l ⁢ ( x ) ⁢ ξ k ⁢ ξ l ¯ ≥ μ ⁢ | ξ | 2

for all ξ∈ℂd and almost every x∈Ω. Define 𝒜:Hloc1⁢(Ω)→𝒟′⁢(Ω) by

〈 𝒜 ⁢ u , v 〉 𝒟 ′ ⁢ ( Ω ) × 𝒟 ⁢ ( Ω ) = ∑ k , l = 1 d ∫ Ω a k ⁢ l ⁢ ( ∂ k ⁡ u ) ⁢ ∂ l ⁡ v ¯ + ∑ k = 1 d ∫ Ω b k ⁢ u ⁢ ∂ k ⁡ v ¯ + ∑ k = 1 d ∫ Ω c k ⁢ ( ∂ k ⁡ u ) ⁢ v ¯ + ∫ Ω c 0 ⁢ u ⁢ v ¯

for all u∈Hloc1⁢(Ω) and v∈Cc∞⁢(Ω). Define the operator Ac,max in C⁢(Ω¯) by

D ⁢ ( A c , max ) = { u ∈ H loc 1 ⁢ ( Ω ) ∩ C ⁢ ( Ω ¯ ) : 𝒜 ⁢ u ∈ C ⁢ ( Ω ¯ ) }

and Ac,max=𝒜|D⁢(Ac,max). In this section we shall prove a maximum principle for parabolic equations involving the operator Ac,max.

The maximum and minimum principles in this section are not completely novel. For operators in non-divergence form they are classical. For operators in divergence form as we consider them here, there are results in [20, Theorem 6.25], with a slightly different notion of solution and domain of the operator. Still, we find it worthwhile to include this section since it shows that our approach from the previous sections yields a new short and elementary proof for strong parabolic and elliptic maximum principles under very general assumptions on the coefficients of the operator.

5.1 The Strong Maximum Principle for Mild Solutions

In this subsection, we assume in addition that Ω is connected and Wiener regular (see the beginning of Subsection 4.1 for a definition). Moreover, we assume that the coefficients satisfy

∫ Ω c 0 ⁢ v + ∑ k = 1 d ∫ Ω b k ⁢ ∂ k ⁡ v ≥ 0

for all v∈Cc∞⁢(Ω)+. Fix T∈(0,∞). Let φ∈C⁢([0,T],C⁢(Γ)) and u0∈C⁢(Ω¯). Formally we consider the problem

(5.1) [ u ˙ ⁢ ( t ) = - A c , max ⁢ u ⁢ ( t ) for all ⁢ t ∈ [ 0 , T ] , u ⁢ ( t ) | Γ = φ ⁢ ( t ) for all ⁢ t ∈ [ 0 , T ] , u ⁢ ( 0 ) = u 0 .

As in [3] we say that u∈C⁢([0,T],C⁢(Ω¯)) is a mild solution of (5.1) if

∫ 0 t u ⁢ ( s ) ⁢ 𝑑 s ∈ D ⁢ ( A c , max ) , u ⁢ ( t ) = u 0 - A c , max ⁢ ∫ 0 t u ⁢ ( s ) ⁢ 𝑑 s and u ⁢ ( t ) | Γ = φ ⁢ ( t )

for all t∈[0,T]. Arendt [3, Theorem 6.5] proved the following theorem.

Theorem 5.1.

Let φ∈C⁢([0,T],C⁢(Γ)) and u0∈C⁢(Ω¯) with u0|Γ=φ⁢(0). Then there exists a unique function u∈C⁢([0,T],C⁢(Ω¯)) such that u is a mild solution of (5.1). Moreover, if φ≥0 and u0≥0, then u≥0.

The last part can be improved with the aid of Corollary 3.4. This is the main result of this subsection.

Theorem 5.2.

Let φ∈C⁢([0,T],C⁢(Γ)) and u0∈C⁢(Ω¯) with u0|Γ=φ⁢(0), φ≥0 and u0≥0. Let u∈C⁢([0,T],C⁢(Ω¯)) be the mild solution of (5.1).

If u 0 ≠ 0 , then u ⁢ ( t , x ) > 0 for all t ∈ ( 0 , T ] and x ∈ Ω .
If t 0 ∈ [ 0 , T ) and φ ⁢ ( t 0 ) ≠ 0 , then u⁢(t,x)>0 for all t∈(t0,T] and x∈Ω.

Proof.

a There exists an x0∈Ω such that u0⁢(x0)≠0. Let χ∈Cc∞⁢(Ω) be such that 0≤χ≤𝟙 and χ⁢(x0)=1. Consider v0=χ⁢u0∈C0⁢(Ω). By Theorem 5.1 there exists a unique v∈C⁢([0,T],C⁢(Ω¯)) such that

(5.2) ∫ 0 t v ⁢ ( s ) ⁢ 𝑑 s ∈ D ⁢ ( A c , max ) , v ⁢ ( t ) = v 0 - A c , max ⁢ ∫ 0 t v ⁢ ( s ) ⁢ 𝑑 s and v ⁢ ( t ) | Γ = 0

for all t∈[0,T]. The function v can be described via a semigroup. Let Ac be the part of Ac,max in C0⁢(Ω). So

D ⁢ ( A c ) = { u ∈ H loc 1 ⁢ ( Ω ) ∩ C 0 ⁢ ( Ω ) : 𝒜 ⁢ u ∈ C 0 ⁢ ( Ω ) }

and Ac=𝒜|D⁢(Ac). Then -Ac generates a C0-semigroup on C0⁢(Ω) by [9, Theorem 1.3] (see also Subsection 4.1 or [6, Section 4]). Let T be the semigroup generated by -Ac. Then T is positive and irreducible by Theorem 4.1. Define w:[0,T]→C⁢(Ω¯) by w⁢(t)=Tt⁢v0. Then it is easy to see that w∈C⁢([0,T],C⁢(Ω¯)) and that w satisfies (5.2) with v replaced by w. So v⁢(t)=w⁢(t)=Tt⁢v0 for all t∈(0,T] by the uniqueness property. The semigroup T also extends to a positive irreducible holomorphic C0-semigroup on Ld⁢(Ω) and this semigroup maps Ld⁢(Ω) into C0⁢(Ω). Hence we can apply Corollary 3.4 a and conclude that v⁢(t,x)>0 for all t∈(0,T] and x∈Ω.

Finally, consider u-v∈C⁢([0,T],C⁢(Ω¯)). Then

∫ 0 t ( u - v ) ⁢ ( s ) ⁢ 𝑑 s ∈ D ⁢ ( A c , max ) , ( u - v ) ⁢ ( t ) = ( u 0 - v 0 ) - A c , max ⁢ ∫ 0 t ( u - v ) ⁢ ( s ) ⁢ 𝑑 s and ( u - v ) ⁢ ( t ) | Γ = φ ⁢ ( t )

for all t∈[0,T]. So u-v≥0 by the last part of Theorem 5.1. In particular, u⁢(t,x)≥v⁢(t,x)>0 for all t∈(0,T] and x∈Ω.

b Apply statement a to u~0=u⁢(t0,⋅), u~⁢(t,x)=u⁢(t-t0,x) and φ~⁢(t)=φ⁢(t-t0), where t∈[0,T-t0] and x∈Ω¯. ∎

In the following corollary we show how a strong parabolic maximum principle can be derived from Theorem 5.2.

Corollary 5.3.

Assume that A⁢1Ω=0. Let φ∈C⁢([0,T],C⁢(Γ)) and u0∈C⁢(Ω¯). Let u∈C⁢([0,T],C⁢(Ω¯)) be the mild solution of (5.1). Moreover, let t0∈(0,T] and x0∈Ω. If u⁢(t0,x0)≥u⁢(t,x) for all t∈[0,t0] and x∈Ω¯, then u is constant on [0,t0]×Ω¯.

Proof.

Define v∈C⁢([0,t0],C⁢(Ω¯)) by v⁢(t,x)=u⁢(t0,x0)-u⁢(t,x). Define v0∈C⁢(Ω¯) by v0⁢(x)=u⁢(t0,x0)-u0⁢(x) and define ψ∈C⁢([0,t0],C⁢(Γ)) by ψ⁢(t,x)=u⁢(t0,x0)-φ⁢(t,x). Then v0|Γ=ψ⁢(0) and v≥0. So ψ≥0 and v0≥0. Also

∫ 0 t v ⁢ ( s ) ⁢ 𝑑 s ∈ D ⁢ ( A c , max ) , v ⁢ ( t ) = v 0 - A c , max ⁢ ∫ 0 t v ⁢ ( s ) ⁢ 𝑑 s and v ⁢ ( t ) | Γ = ψ ⁢ ( t )

for all t∈[0,t0]. Since v⁢(t0,x0)=0, it follows from Theorem 5.2 that v0=0 and ψ=0. Then the uniqueness part of Theorem 5.1 implies that v=0. Hence u is constant on [0,t0]×Ω¯. ∎

In the next two corollaries we deduces a strong elliptic maximum principle from the parabolic result in Corollary 5.3.

Corollary 5.4.

Let u∈Hloc1⁢(Ω)∩C⁢(Ω¯) and suppose that A⁢u=0. If u≥0 and u|Γ≠0, then u⁢(x)>0 for all x∈Ω.

Proof.

Define v0=u, φ⁢(t)=u|Γ and v⁢(t)=u for all t∈[0,T]. Then v is a mild solution of (5.1) with u0 replaced by v0. Now apply Theorem 5.2 a. ∎

Corollary 5.5.

Suppose that A⁢1Ω=0. Let u∈Hloc1⁢(Ω)∩C⁢(Ω¯,R) and suppose that A⁢u=0. If there exists an x0∈Ω such that u⁢(x0)=maxΩ⁡u, then u is constant.

Proof.

Consider v=u⁢(x0)⁢𝟙Ω¯-u. Then v≥0 and 𝒜⁢v=0. Since v⁢(x0)=0, it follows from Corollary 5.4 that v=0. Hence u=u⁢(x0)⁢𝟙Ω¯ and u is constant. ∎

5.2 Mild and Very Weak Solutions

Theorem 5.2 and the parabolic maximum principle in Corollary 5.3 used the concept of a mild solution of (5.1). In this subsection we show under a differentiability condition that mild solutions are the same as very weak solutions.

Throughout this subsection we assume that the coefficients ak⁢l,ck satisfy ak⁢l,ck∈L∞⁢(Ω,ℝ)∩C1⁢(Ω) for all k,l∈{1,…,d}.

Fix a time T∈(0,∞). For all u∈C⁢([0,T]×Ω¯) define u~∈C⁢([0,T],C⁢(Ω¯)) by (u~⁢(t))⁢(x)=u⁢(t,x). If ψ∈Cc∞⁢(Ω), define 𝒜*⁢ψ∈Cc⁢(Ω) by

𝒜 * ⁢ ψ = - ∑ k , l = 1 d ∂ k ⁡ ( a k ⁢ l ⁢ ∂ l ⁡ ψ ) + ∑ k = 1 d b k ⁢ ∂ k ⁡ ψ - ∑ k = 1 d ∂ k ⁡ ( c k ⁢ ψ ) + c 0 ⁢ ψ .

If φ∈Cc∞⁢((0,T)×Ω) define 𝒜*⁢φ∈C⁢([0,T]×Ω¯) by

( 𝒜 * ⁢ φ ) ⁢ ( t , x ) = ( 𝒜 * ⁢ ( φ ~ ⁢ ( t ) ) ) ⁢ ( x ) .

Moreover, we define φt∈Cc∞⁢((0,T)×Ω) by

φ t ⁢ ( t , x ) = ( ∂ ∂ ⁡ t ⁢ φ ) ⁢ ( t , x ) .

The second condition in the next theorem states that u is a very weak solution.

Theorem 5.6.

Let u∈C⁢([0,T]×Ω¯). Then the following are equivalent:

∫ 0 t u ~ ⁢ ( s ) ⁢ 𝑑 s ∈ D ⁢ ( A c , max ) and u ~ ⁢ ( t ) = u ~ ⁢ ( 0 ) - A c , max ⁢ ∫ 0 t u ~ ⁢ ( s ) ⁢ 𝑑 s for all t ∈ [ 0 , T ] .
If φ ∈ C c ∞ ⁢ ( ( 0 , T ) × Ω ) , then ∫ 0 T ∫ Ω u ⁢ ( t , x ) ⁢ ( φ t - 𝒜 * ⁢ φ ) ⁢ ( t , x ) ⁢ 𝑑 x ⁢ 𝑑 t = 0 .

Proof.

i ⇒ ii Let φ∈Cc∞⁢((0,T)×Ω). Define v∈C⁢([0,T]×Ω¯) by v⁢(t,x)=∫0tu⁢(s,x)⁢𝑑s. Then v~⁢(t)=∫0tu~⁢(s)⁢𝑑s for all t∈[0,T]. Therefore

∫ 0 T ∫ Ω u ⁢ ( t , x ) ⁢ ( 𝒜 * ⁢ φ ) ⁢ ( t , x ) ⁢ 𝑑 x ⁢ 𝑑 t = ∫ Ω ∫ 0 T ( ∂ ∂ ⁡ t ⁢ v ⁢ ( t , x ) ) ⁢ ( 𝒜 * ⁢ φ ) ⁢ ( t , x ) ⁢ 𝑑 t ⁢ 𝑑 x

= - ∫ Ω ∫ 0 T v ⁢ ( t , x ) ⁢ ( 𝒜 * ⁢ φ t ) ⁢ ( t , x ) ⁢ 𝑑 t ⁢ 𝑑 x

= - ∫ 0 T ( v ~ ⁢ ( t ) , 𝒜 * ⁢ φ ~ t ⁢ ( t ) ¯ ) L 2 ⁢ ( Ω ) ⁢ 𝑑 t

= - ∫ 0 T ( A c , max ⁢ v ~ ⁢ ( t ) , φ ~ t ⁢ ( t ) ¯ ) L 2 ⁢ ( Ω ) ⁢ 𝑑 t

= ∫ 0 T ( u ~ ⁢ ( t ) - u ~ ⁢ ( 0 ) , φ ~ t ⁢ ( t ) ¯ ) L 2 ⁢ ( Ω ) ⁢ 𝑑 t

= ∫ 0 T ∫ Ω u ⁢ ( t , x ) ⁢ φ t ⁢ ( t , x ) ⁢ 𝑑 x ⁢ 𝑑 t - ∫ Ω u ⁢ ( 0 , x ) ⁢ ∫ 0 T φ t ⁢ ( t , x ) ⁢ 𝑑 t ⁢ 𝑑 x .

Since ∫0Tφt⁢(t,x)⁢𝑑t=φ⁢(T,x)-φ⁢(0,x)=0 for all x∈Ω, the implication follows.

ii ⇒ i Let ψ∈Cc∞⁢(Ω), and define f,g∈C⁢[0,T] by f⁢(t)=(u~⁢(t),ψ)L2⁢(Ω) and g⁢(t)=(u~⁢(t),𝒜*⁢ψ)L2⁢(Ω). Let τ∈Cc∞⁢((0,T)). Then τ⊗ψ¯∈Cc∞⁢((0,T)×Ω). So by assumption

∫ 0 T ∫ Ω u ⁢ ( t , x ) ⁢ ( ( τ ⊗ ψ ¯ ) t - 𝒜 * ⁢ ( τ ⊗ ψ ¯ ) ) ⁢ ( t , x ) ⁢ 𝑑 x ⁢ 𝑑 t = 0 .

Hence

∫ 0 T τ ′ ⁢ ( t ) ⁢ f ⁢ ( t ) ⁢ 𝑑 t = ∫ 0 T ∫ Ω u ⁢ ( t , x ) ⁢ τ ′ ⁢ ( t ) ⁢ ψ ¯ ⁢ ( x ) ⁢ 𝑑 x ⁢ 𝑑 t

= ∫ 0 T ∫ Ω u ⁢ ( t , x ) ⁢ τ ⁢ ( t ) ⁢ ( 𝒜 * ⁢ ψ ¯ ) ⁢ ( x ) ⁢ 𝑑 x ⁢ 𝑑 t

= ∫ 0 T τ ⁢ ( t ) ⁢ g ⁢ ( t ) ⁢ 𝑑 t

and f′=-g weakly. Since f and g are continuous the lemma of du Bois–Reymond implies that f is differentiable and f′=-g in the classical sense. Let t∈[0,T]. Then

( u ~ ⁢ ( t ) , ψ ) L 2 ⁢ ( Ω ) = f ⁢ ( t ) = f ⁢ ( 0 ) - ∫ 0 t g ⁢ ( s ) ⁢ 𝑑 s = ( u ~ ⁢ ( 0 ) , ψ ) L 2 ⁢ ( Ω ) - ∫ 0 t ( u ~ ⁢ ( s ) , 𝒜 * ⁢ ψ ) L 2 ⁢ ( Ω ) ⁢ 𝑑 s .

( ∫ 0 t u ~ ⁢ ( s ) ⁢ 𝑑 s , 𝒜 * ⁢ ψ ) L 2 ⁢ ( Ω ) = ( u ~ ⁢ ( 0 ) - u ~ ⁢ ( t ) , ψ ) L 2 ⁢ ( Ω ) .

This is for all ψ∈Cc∞⁢(Ω). It follows from elliptic regularity, see Proposition A.1 in the appendix, that ∫0tu~⁢(s)⁢𝑑s∈Hloc1⁢(Ω). Hence i is valid. ∎

Now we can reformulate the results of the previous subsection using the notion of very weak solutions. We consider the parabolic cylinder ΩT=(0,T)×Ω with parabolic boundary ∂*⁡ΩT=({0}×Ω¯)∪((0,T]×∂⁡Ω). Given ψ∈C⁢(∂*⁡ΩT), we formally consider the Dirichlet problem for the heat equation

($D(\psi)$) { u ∈ C ⁢ ( Ω T ¯ ) , ∂ t ⁡ u - 𝒜 ⁢ u = 0 on ⁢ Ω T , u | ∂ * ⁡ Ω T = ψ .

We say that u∈C⁢(ΩT¯) is a very weak solution of D⁢(ψ) if

∫ 0 T ∫ Ω u ⁢ ( t , x ) ⁢ ( φ t - 𝒜 * ⁢ φ ) ⁢ ( t , x ) ⁢ 𝑑 x ⁢ 𝑑 t = 0

for all φ∈Cc∞⁢((0,T)×Ω) and u|∂*⁡ΩT=ψ. Then the following holds.

Theorem 5.7.

Assume Ω is connected and Wiener regular. Then for all ψ∈C⁢(∂*⁡ΩT) there exists a unique very weak solution of D⁢(ψ).

For this solution of D⁢(ψ) the following maximum principles are valid.

Theorem 5.8.

Assume Ω is connected and Wiener regular. Let ψ∈C⁢(∂T*⁡Ω) and let u∈C⁢(ΩT¯) be the very weak solution D⁢(ψ). Then one has the following:

If ψ ≥ 0 , then u ≥ 0 .
Let t 0 ∈ [ 0 , T ) and x 0 ∈ Ω ¯ . Suppose that u ⁢ ( t 0 , x 0 ) > 0 and ψ ≥ 0 . Then u ⁢ ( t , x ) > 0 for all t ∈ ( t 0 , T ] and x ∈ Ω .
Suppose 𝒜 ⁢ 𝟙 Ω = 0 . Let t 0 ∈ ( 0 , T ] and x 0 ∈ Ω . If u ⁢ ( t , x ) ≤ u ⁢ ( t 0 , x 0 ) for all t ∈ [ 0 , t 0 ] and x ∈ Ω ¯ , then u is constant on [ 0 , t 0 ] × Ω ¯ .

The maximum principle for elliptic operators has been proved before in [18, Theorem 8.19].

Communicated by Julián López-Gómez

Funding statement: Part of this work is supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand.

A Regularity

In the proof of Theorem 5.6 we used the following regularity result for very weak solutions.

Proposition A.1.

Let Ω⊂Rd be open. For all k,l∈{1,…,d} let ak⁢l,ck∈W1,∞⁢(Ω) and bk,c0∈L∞⁢(Ω). Assume that there exists a μ>0 such that

(A.1) Re ⁢ ∑ k , l = 1 d a k ⁢ l ⁢ ( x ) ⁢ ξ k ⁢ ξ l ¯ ≥ μ ⁢ | ξ | 2

for all ξ∈Cd and almost every x∈Ω. For all φ∈Cc∞⁢(Ω) define A*⁢φ∈L∞,c⁢(Ω) by

𝒜 * ⁢ φ = - ∑ k , l = 1 d ∂ k ⁡ ( a k ⁢ l ¯ ⁢ ∂ l ⁡ φ ) + ∑ k = 1 d b k ¯ ⁢ ∂ k ⁡ φ - ∑ k = 1 d ∂ k ⁡ ( c k ¯ ⁢ φ ) + c 0 ¯ ⁢ φ .

Let u,f,f1,…,fd∈L2⁢(Ω) and suppose that

( u , 𝒜 * ⁢ φ ) L 2 ⁢ ( Ω ) = ( f , φ ) L 2 ⁢ ( Ω ) - ∑ j = 1 d ( f j , ∂ j ⁡ φ ) L 2 ⁢ ( Ω )

for all φ∈Cc∞⁢(Ω). Then u∈Wloc1,2⁢(Ω).

Proof.

Fix χ∈Cc∞⁢(Ω). We shall show that χ⁢u∈W1,2⁢(Ω). Without loss of generality we may assume that ak⁢l,ck∈W1,∞⁢(ℝd) and bk,c0∈L∞⁢(ℝd), and that (A.1) is valid for all ξ∈ℂd and almost every x∈ℝd. Define the form 𝔞:W1,2⁢(ℝd)×W1,2⁢(ℝd)→ℂ by

𝔞 ⁢ ( v , w ) = ∫ ℝ d ∑ k , l = 1 d a k ⁢ l ⁢ ( ∂ k ⁡ v ) ⁢ ∂ l ⁡ w ¯ + ∫ ℝ d ∑ k = 1 d b k ⁢ v ⁢ ∂ k ⁡ w ¯ + ∫ ℝ d ∑ k = 1 d c k ⁢ ( ∂ k ⁡ v ) ⁢ w ¯ + ∫ ℝ d c 0 ⁢ v ⁢ w ¯ .

Then 𝔞 is a closed sectorial form. Let A be the m-sectorial operator associated with 𝔞. Note that we have Cc∞⁢(ℝd)⊂D⁢(A*) and that A*⁢φ=𝒜*⁢φ for all φ∈Cc∞⁢(Ω).

Define

g = χ ⁢ f - ∑ j = 1 d ( ∂ j ⁡ χ ) ⁢ f j + ∑ k , l = 1 d u ⁢ ∂ k ⁡ ( a k ⁢ l ⁢ ∂ l ⁡ χ ) - ∑ k = 1 d b k ⁢ u ⁢ ∂ k ⁡ χ + ∑ k = 1 d c k ⁢ u ⁢ ∂ k ⁡ χ

and for all j∈{1,…,d} define

g j = χ ⁢ f j - ∑ l = 1 d a j ⁢ l ⁢ u ⁢ ∂ l ⁡ χ - ∑ k = 1 d a k ⁢ j ⁢ u ⁢ ∂ k ⁡ χ .

Then g,g2∈L2,c⁢(Ω)⊂L2⁢(ℝd). It is a tedious elementary exercise to show that

(A.2) ( χ ⁢ u , A * ⁢ φ ) L 2 ⁢ ( ℝ d ) = ( g , φ ) L 2 ⁢ ( ℝ d ) - ∑ j = 1 d ( g j , ∂ j ⁡ φ ) L 2 ⁢ ( ℝ d )

for all φ∈Cc∞⁢(ℝd). It follows from [1, Theorem 9.8] that Cc∞⁢(ℝd) is a core for A*. Obviously, D⁢(A*)⊂W1,2⁢(ℝd). Hence (A.2) is valid for all φ∈D⁢(A*).

Without loss of generality we may assume that that Re⁡c0 is large enough such that A* is invertible. Then

( χ u , w ) L 2 ⁢ ( ℝ d ) = ( g , ( A * ) - 1 w ) L 2 ⁢ ( ℝ d ) - ∑ j = 1 d ( g j , ∂ j ( A * ) - 1 w ) L 2 ⁢ ( ℝ d )

for all w∈L2⁢(ℝd). Let m∈{1,…,d}. Then

( χ u , ∂ m v ) L 2 ⁢ ( ℝ d ) = ( g , ( A * ) - 1 ∂ m v ) L 2 ⁢ ( ℝ d ) - ∑ j = 1 d ( g j , ∂ j ( A * ) - 1 ∂ m v ) L 2 ⁢ ( ℝ d )

for all v∈W1,2⁢(ℝd). It follows from the ellipticity condition that the operator ∂j(A*)-1∂m from W1,2⁢(ℝd) into L2⁢(ℝd) has a bounded extension from L2⁢(ℝd) into L2⁢(ℝd) for all j∈{1,…,d}. Since D⁢(A)⊂W1,2⁢(ℝd), it follows by duality that there is an M>0 such that |(χ⁢u,(A*)-1⁢∂m⁡v)L2⁢(ℝd)|≤M⁢∥v∥L2⁢(ℝd) for all v∈W1,2⁢(ℝd). Hence χ⁢u∈W1,2⁢(ℝd), as required. ∎

We emphasise that all the coefficients of the operator in Proposition A.1 may be complex valued, including the second-order coefficients.

Remark A.2.

Suppose in addition to the conditions of the coefficients in Proposition A.1 that bk∈W1,∞⁢(ℝd) for all k∈{1,…,d}. Let p∈(1,∞), u,f,f1,…,fd∈Lp⁢(Ω) and suppose

∫ Ω u ⁢ 𝒜 * ⁢ φ ¯ = ∫ Ω f ⁢ φ ¯ - ∑ j = 1 d ∫ Ω f j ⁢ ∂ j ⁡ φ ¯

for all φ∈Cc∞⁢(Ω). Then u∈Wloc1,p⁢(Ω). The proof is almost the same. The operator A is consistent with an operator Ap in Lp⁢(ℝd) such that the semigroups generated by -A and -Ap are consistent. Let q∈(1,∞) be the dual exponent of p. The inclusions D⁢(Ap*)⊂W1,q⁢(ℝd) and D⁢(Ap)⊂W1,p⁢(ℝd) are in [16, Corollary 3.8]. The bounded extension of ∂j(Ap*)-1∂m follows from [16, Theorem 1.4].

If in addition f1=…=fd=0, then one can deduce as in the proof of Proposition A.1 that χ⁢u∈D⁢(Ap**). Since D⁢(Ap)=W2,p⁢(ℝd) (see [16, Proposition 5.1]), one establishes that u∈Wloc2,p⁢(Ω).

For real coefficients a slightly more generally version of Remark A.2 has been proved by Zhang and Bao in [29, Theorem 1.5], where f is even allowed to be an element of a larger Lq-space if d≥3 and a Lorentz space if d=2. We refer to [29] for an account on known regularity results for very weak solutions of elliptic operators.

Acknowledgements

We are grateful to Daniel Daners for illuminating discussions on semilinear problems, the weak Harnack inequality and Hopf’s maximum principle. We wish to thank the referee for the comments and suggestions. The second-named author is most grateful for the hospitality extended to him during a fruitful stay at the University of Ulm. He wishes to thank the University of Ulm for financial support.

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Received: 2019-09-26

Revised: 2020-04-18

Accepted: 2020-04-19

Published Online: 2020-05-12

Published in Print: 2020-08-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2020-2091

Keywords for this article

Maximum Principle; Irreducible Semigroup; Elliptic Operator

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