Weak maximum principle for Dirichlet problems with convection or drift terms

: In this paper, dedicated to Italo Capuzzo Dolcetta, a maximum principle for some linear boundary value problems with lower order terms of order one is proved: the aim of this paper is the proof that the solutions can be zero at most in a zero measure set, if we assume that the data are greater or equal than zero (but not identically zero).


Introduction
During the period 1968-1970, in the "Dipartimento di Matematica dell'Università di Roma", I recall two persons: Guido Stampacchia, my teacher for the last two university years, and Italo Capuzzo Dolcetta, to whom this paper is dedicated.
A common point connecting them is the maximum principle: one of the scientific interests of Italo (see [1][2][3][4][5][6],) and one of the subjects of the courses taught by Guido Stampacchia, in the classical framework in the first course and in the Sobolev framework (see [15]) in the second one. For recent results on maximum principle see also [13,16].
In this paper, I study the positivity, up to a zero measure set, of the solutions of Dirichlet problems having a first order term (either of convection type or of drift type).

Setting
Let Ω be a bounded, open subset of R N , N > 2 and M : Ω → R N 2 , be a bounded and measurable matrix such that α|ξ| 2 ≤ M(x)ξ · ξ, |M(x)| ≤ β, a.e. x ∈ Ω, ∀ ξ ∈ R N . (2.1) We assume that E(x) is a vector field and f (x), g(x) are functions such that E ∈ (L N (Ω)) N , and we consider the following boundary value problems with a lower order convection term or with a lower order drift term: We will also consider the two above boundary value problems if a zero order term is present; that is For the existence and properties of solutions we refer to [7] and to the references therein (see also [11], [14]) for ψ, Ψ, to [9] (and to the references therein) for u and to [10] for z; see also the references of [7][8][9][10] for bibliographic informations. We only point out that in [10] the existence of a weak, bounded solution z of (2.7) is proved assuming on E no more than E ∈ (L 2 (Ω)) N , instead of (2.2).

Positivity of solutions
We recall that positivity (that is, greater or equal than zero, but not identically zero) of solutions of the above boundary value problems holds in the case of data f or g positive (that is, greater or equal than zero, but not identically zero): see [7,9].

Positivity of solutions up a zero measure set
The aim of this paper is the proof that the solutions of the above boundary value problems can be zero at most in a zero measure set, if we assume that the data f or g are greater or equal than zero, but not identically zero.

Dirichlet problems with convection terms
This section deals with the boundary value problem (2.4). Of course the solution ψ (or Ψ) is understood in weak (or distributional) sense.
for every v ∈ W 1,2 0 (Ω) (or v smooth); ii moreover it is proved that ψ ≥ 0, if g ≥ 0 (of course not zero a.e.); where T k is the Stampacchia truncation: The next theorem improves the statement (ii); the proof hinges on the approach of [12].
3) with m ≥ 2N N+2 and g ≥ 0 (of course not zero a.e.). Then the solution ψ ∈ W 1,2 0 (Ω) is positive and it is zero at most on a set of zero Lebesgue measure. Proof. In subsection 2.1 we recalled that ψ(x) ≥ 0; thus we can use as test function in (3.1). Then we have Now we use twice the Young inequality (with 0 < B < α 2 ) and we deduce that (recalling that 0 ≤ φ ≤ 1) The last inequality implies that By contradiction, we assume that ψ(x) = 0 on a subset of positive measure of Ω.
Since ψ(x) = 0 on a subset of positive measure of ω, we can use Poincaré inequality in ω so that, for Thus we proved that ψ ≡ 0 in ω. Since ω is an arbitrary subset of Ω , we conclude that ψ = 0 a.e. in Ω and then g(x) should be equal to zero a.e. in Ω; which is a contradiction.
Remark 3.2. Note that in (3.6) we only need E ∈ (L 2 (Ω)) N , which is a weaker demand with respect to assumption (2.2). Now we discuss the case of infinite energy solutions, which appears if we assume 1 < m < 2N N+2 in (2.3).

Remark 3.3.
Since the existence of a solution ψ is proved in [7] as limit of a sequence {ψ n } (every ψ n ≥ 0) of solutions of approximating problems, a possible approach is to repeat the proof of the previous theorem on the sequence {ψ n } in order to prove that ψ n satisfies inequality (3.4).
Similarly, we prove that ∀ k > 0, the sequence {T k (ψ n )} converges weakly to T k (ψ) in W 1,2 0 (Ω). (3.11) Now we prove that the sequence { ∇ψ n 1+ψ n } converges weakly to ∇ψ 1+ψ in L 2 . (3.12) Indeed, for every Φ ∈ (L 2 (Ω)) N , Observe that, in the first integral, ∇T k (ψ n ) converges weakly in L 2 and Φ 1+ψ n converges strongly in L 2 ; moreover The estimate (3.10) says that the last integral is uniformly (with respect to n) small for k large. Thus (3.12) is proved, so that we pass to the limit, by weak L 2 lower semicontinuity in the inequality (3.4) for the sequence {ψ n }, that is and we have Thus it is possible to prove the following theorems, where ψ is a solution obtained as said above. (Ω) which is positive and it is zero at most on a set of zero measure.
In [8], it is proved the existence of a very weak solution (entropy solution) if the data are very singular (e.g., E ∈ L 2 ). Even for this solution it is possible to prove the maximum principle, thanks to the above discussion of Remark 3.3, as stated in the following theorem.
Theorem 3.5. Assume (2.1), E ∈ (L 2 (Ω)) N , g ∈ L 1 (Ω), g ≥ 0 (of course not zero a.e.). Then the entropy solution ψ is positive and it is zero at most on a set of zero measure.

Dirichlet problems with drift terms
This section deals with the boundary value problem (2.5). Of course the solution u (or z) is understood in weak (or distributional) sense.

A duality approach
Here we give a different proof of Theorem 4.1. In this section we studied the Dirichlet problem whereas, in the previous section, the Dirichlet problems studied include If the measure {u(x) = 0} is zero, then χ {u(x)=0} = 0 and in [7] is proved that ψ = 0. Thus (by contradiction) assume that the measure of {u(x) = 0} is strictly positive. Then the duality (i.e., use ψ as test function in the first problem and u in the second problem) gives

|E| ∈ L 2
Remark 4.3. If we assume (2.1), E ∈ (L 2 (Ω)) N , f ∈ L ∞ (Ω), in [10], it is proved the existence of a weak, bounded solution z of (2.7), that is for every v ∈ W 1,2 0 (Ω). If we consider the boundary value problem (2.7), (in place of (2.5)) with the same test function used in the proof of Theorem 4.1, the new proof changes slightly: instead of (4.2), we have (since 0 ≤ z h+z ≤ 1) Ω |∇z| 2 (h + z) 2 φ 2 and the conclusion on the solution z is again the positivity up, at most, a set of zero measure.