On the $L^p$-theory of vector-valued elliptic operators

TL;DR

This paper develops an $L^p$-theory for vector-valued elliptic operators, proving their generation of analytic semigroups under certain conditions, and explores regularity, domain, and positivity properties.

Contribution

It introduces new $L^p$-theoretic results for vector-valued elliptic operators with coupling, including semigroup generation and regularity analysis.

Findings

Operators generate analytic semigroups in $L^p$ spaces.

Established local elliptic regularity results.

Characterized positivity of the semigroup.

Abstract

In this paper, we study vector--valued elliptic operators of the form $\mathcal{L}f:=\mathrm{div}(Q\nabla f)-F\cdot\nabla f+\mathrm{div}(Cf)-Vf$ acting on vector-valued functions $f:\mathbb{R}^d\to\mathbb{R}^m$ and involving coupling at zero and first order terms. We prove that $\mathcal{L}$ admits realizations in $L^p(\mathbb{R}^d,\mathbb{R}^m)$, for $1<p<\infty$, that generate analytic strongly continuous semigroups provided that $V=(v_{ij})_{1\le i,j\le m}$ is a matrix potential with locally integrable entries satisfying a sectoriality condition, the diffusion matrix $Q$ is symmetric and uniformly elliptic and the drift coefficients $F=(F_{ij})_{1\le i,j\le m}$ and $C=(C_{ij})_{1\le i,j\le m}$ are such that $F_{ij},C_{ij}:\mathbb{R}^d\to\mathbb{R}^d$ are bounded. We also establish a result of local elliptic regularity for the operator $\mathcal{L}$, we investigate on the $L^p$-maximal domain of $\mathcal{L}$ and we characterize the positivity of the associated semigroup.

Equations

$$\mathcal{L}f=\mathrm{div}(Q\nabla f)-F\cdot \nabla f+\mathrm{div}(Cf)-Vf$$

$$(\mathcal{L}f{)}_i=\mathrm{div}(Q\nabla f_i)-\sum _{j=1}^m{F}_{ij}\cdot \nabla f_j+\sum _{j=1}^m\mathrm{div}(f_j{C}_{ij})-\sum _{j=1}^m{v}_{ij}{f}_j$$

$${\Delta }_Q|f|\ge \frac{1}{|f|}\sum _{j=1}^m{f}_j{\Delta }_Q{f}_j{\chi }_{\{f\ne 0\}},$$

$$|\mathrm{Im}⟨V(x)\xi ,\xi ⟩|\le M\mathrm{Re}⟨V(x)\xi ,\xi ⟩,$$

$$D_{p,\max }(\mathcal{L}):=\{f\in L^p(\text{mathds}{R}^d,\text{mathds}{R}^m)\cap W_{\mathrm{loc}}^{2,p}(\text{mathds}{R}^d;\text{mathds}{R}^m):\mathcal{L}f\in L^p(\text{mathds}{R}^d;\text{mathds}{R}^m)\}.$$

$$\Vert \cdot {\Vert }_p:f=(f_1,\dots ,f_m)\mapsto \Vert f{\Vert }_p:={\left({\int }_{\text{mathds}{R}^d}(\sum _{j=1}^m|f_j{|}^2{)}^{\frac{p}{2}}dx\right)}^{\frac{1}{p}}.$$

$$\mathcal{L}f=\mathrm{div}(Q\nabla f)-F\cdot \nabla f+\mathrm{div}(Cf)-Vf,$$

$$(\mathrm{div}(Q\nabla f){)}_i=\mathrm{div}(Q\nabla f_i),$$

$$(F\cdot \nabla f{)}_i=\sum _{j=1}^m⟨F_{ij},\nabla f_j⟩$$

$$(\mathrm{div}(Cf){)}_i=\sum _{j=1}^m\mathrm{div}(f_j{C}_{ij})$$

$$(Vf{)}_i=\sum _{j=1}^m{v}_{ij}{f}_j$$

$$(\mathrm{div}(Q\nabla f),\varphi )=-{\int }_{\text{mathds}{R}^d}\sum _{i=1}^m⟨Q\nabla f_i,\nabla {\varphi }_i⟩dx,$$

$$(F\cdot \nabla f,\varphi )=\sum _{j=1}^m{\int }_{\text{mathds}{R}^d}(F_{ij}\cdot \nabla f_j){\bar{\varphi }}_{i}dx,$$

$$(\mathrm{div}(Cf),\varphi )=-\sum _{j=1}^m{\int }_{\text{mathds}{R}^d}{f}_j⟨C_{ij},\nabla {\varphi }_i⟩dx$$

$${\eta }_1|\xi {|}^2\le ⟨Q(x)\xi ,\xi ⟩\le {\eta }_2|\xi {|}^2,$$

$$|\mathrm{Im}⟨V(x)\xi ,\xi ⟩|\le M\mathrm{Re}⟨V(x)\xi ,\xi ⟩,$$

$$|⟨V(x){\xi }_1,{\xi }_2⟩|\le (1+M)⟨V_s(x){\xi }_1,{\xi }_1{⟩}^{1/2}⟨V_s(x){\xi }_2,{\xi }_2{⟩}^{1/2}$$

$$\left|{\int }_{\text{mathds}{R}^d}⟨V_{as}(x)f(x),g(x)⟩dx\right|\le (1+M)\Vert V_s^{1/2}f{\Vert }_{L^2(\text{mathds}{R}^d,\text{mathds}{C}^m)}\Vert V_s^{1/2}g{\Vert }_{L^2(\text{mathds}{R}^d,\text{mathds}{C}^m)}.$$

$$\mathrm{a}(f,g)$$

$$+\sum _{i,j=1}^m{\int }_{\text{mathds}{R}^d}{f}_j⟨C_{ij},\nabla g_i⟩dx+{\int }_{\text{mathds}{R}^d}⟨Vf,g⟩dx,$$

$$D(\mathrm{a})=\{f\in H^1(\text{mathds}{R}^d,\text{mathds}{C}^m):{\int }_{\text{mathds}{R}^d}⟨V_{s}f,f⟩dx<\infty \}:=D({\mathrm{a}}_0),$$

$${\mathrm{a}}_0(f,g)=\sum _{j=1}^m⟨Q\nabla f_j,\nabla g_j{⟩}_2+{\int }_{\text{mathds}{R}^d}⟨V_s(x)f(x),g(x)⟩dx.$$

$$\mathrm{Re}\mathrm{a}(f)$$

$$+\sum _{i,j=1}^m{\int }_{\text{mathds}{R}^d}\mathrm{Re}\left(f_j⟨C_{ij},\nabla f_i⟩\right)dx+{\int }_{\text{mathds}{R}^d}\mathrm{Re}⟨Vf,f⟩dx$$

$$\ge {\eta }_1\sum _{i=1}^m{\int }_{\text{mathds}{R}^d}|\nabla f_i{|}^{2}dx-(\Vert F{\Vert }_{\infty }+\Vert C{\Vert }_{\infty }){\int }_{\text{mathds}{R}^d}\sum _{i=1}^m|f_i|\sum _{i=1}^m|\nabla f_i|dx$$

$$\ge ({\eta }_1-\epsilon ){\int }_{\text{mathds}{R}^d}\sum _{i=1}^m|\nabla f_i{|}^{2}dx-c_{\epsilon }{\int }_{\text{mathds}{R}^d}\sum _{i=1}^m|f_i{|}^{2}dx.$$

$$\Vert \cdot {\Vert }_{{\mathrm{a}}_0}:=\sqrt{\Vert \cdot {\Vert }_2^2+{\mathrm{a}}_0(\cdot )}.$$

$$\Vert \cdot {\Vert }_{\mathrm{a}}:=\sqrt{(1+\omega )\Vert \cdot {\Vert }_2^2+\mathrm{Re}a(\cdot )}.$$

$$\mathrm{b}(f):=\sum _{i,j=1}^m{\int }_{\text{mathds}{R}^d}(F_{ij}\cdot \nabla f_j){\bar{f}}_{i}dx+\sum _{i,j=1}^m{\int }_{\text{mathds}{R}^d}{f}_j⟨C_{ij},\nabla f_i⟩dx.$$

$$|\mathrm{a}(f,g)|\le c\Vert f{\Vert }_{{\mathrm{a}}_0}\Vert g{\Vert }_{{\mathrm{a}}_0},\forall f,g\in D(\mathrm{a}).$$

$$|\mathrm{a}(f,g)|$$

$$\le c_1\Vert f{\Vert }_{{\mathrm{a}}_0}\Vert g{\Vert }_{{\mathrm{a}}_0}+c_2\Vert f{\Vert }_{H^1(\text{mathds}{R}^d,\text{mathds}{C}^m)}\Vert g{\Vert }_{H^1(\text{mathds}{R}^d,\text{mathds}{C}^m)}+c_3\Vert V_s^{1/2}f{\Vert }_2\Vert V_s^{1/2}g{\Vert }_2$$

$$\le c\Vert f{\Vert }_{{\mathrm{a}}_0}\Vert g{\Vert }_{{\mathrm{a}}_0}.$$

Full text

1. Introduction

The present paper deals with a class of vector–valued elliptic operators of the form

[TABLE]

acting on smooth functions $f:\mathds{R}^{d}\to\mathds{R}^{m}$, for some integers $d,m\geq 1$, and involving coupling through the first and zero order terms. More precisely, for $f=(f_{1},\dots,f_{m}):\mathds{R}^{d}\to\mathds{R}^{m}$, one has

[TABLE]

for each $i\in\{1,\dots,m\}$.

We point out that the operator $\mathcal{L}$ appears in the study of Navier-Stokes equations. More precisely, in [25, 26], H. Triebel used a reduced form of Navier–Stokes type equations on $\mathds{R}^{n}$ (where $d=m=n$ in such case) that matches vector–valued semilinear parabolic evolution equations via the Leray/Helmoltz projector, see [25, Chapter 6] for details. Moreover, a similar reduction method were applied in [11, 12] to convert Navier-Stokes equation to a semilinear parabolic system. The linear operator in [11, 12] is more appropriate to our situation. Besides, parabolic systems appear also in the study of Nash equilibrium for stochastic differential games, see [7, 8, 19] and [1, Section 6].

In the scalar case, the theory of elliptic operators, is by now well understood, see [21] and [16] for bounded and unbounded coefficients respectively. However, the situation is quite different in the vector–valued case. Indeed, the interest into operators as in (1.1) in the whole space with possibly unbounded coefficients has started only in 2009 by Hieber et al. [10] with coupling through the lower order term of the elliptic operator and the motivation were the Navier-Stokes equation. Afterwards, few papers appeared, see [1, 3, 6, 14, 15, 17, 18]. In [1, 3, 6] the authors studied the associated parabolic equation in $C_{b}$-spaces, assuming, among others, that the coefficients of the elliptic operator are Hölder continuous. In [6], solution to the parabolic system has been extrapolated to the $L^{p}$-scale provided the uniqueness.

In what concerns a Schrödinger type operator $\mathcal{A}=\mathrm{div}(Q\nabla\cdot)-V$, which corresponds to $F=C=0$ in (1.1), and its associated semigroup, a comprehensive study in $L^{p}$-spaces can be find in [14, 15, 17, 18]. Indeed, in [17], it has been associated a sesquilinear form to $\mathcal{A}$, for symmetric potential $V$, and it has been established a consistent $C_{0}$-semigroup in $L^{p}(\mathds{R}^{d},\mathds{R}^{m})$, $p\geq 1$, which is analytic for $p\neq 1$. This is done by assuming that $V$ is pointwisely semi-definite positive with locally integrable entries and $Q$ is symmetric, bounded and satisfies the well-known ellipticity condition. Moreover, the author investigated on compactness and positivity of the semigroup. In [15], the authors associated a $C_{0}$-semigroup, in $L^{p}$-spaces, which is not necessarily analytic, to the Schrödinger operator with typically nonsymmetric potential, provided that the diffusion matrix $Q$ is, in addition to the ellipticity condition, differentiable, bounded together with its first derivatives, $V$ is semi–definite positive and its entries are locally bounded. Here, the authors followed the approach adopted by Kato in [13] for scalar Schrödinger operators with complex potential. The main tool has been local elliptic regularity and a Kato’s type inequality for vector–valued functions, i.e.,

[TABLE]

for smooth functions $f:\mathds{R}^{d}\to\mathds{R}^{m}$, where $\Delta_{Q}:=\mathrm{div}(Q\nabla\cdot)$, see [15, Proposition 2.3]. Further properties such as maximal domain and others have been also investigated. The papers [14, 18] focused on the domain of the operator and further regularity properties. So that, under growth and smoothness assumptions on $V$, the authors coincide the domain of $\mathcal{A}$ with its natural domain $W^{2,p}(\mathds{R}^{d},\mathds{R}^{m})\cap D(V_{p})$, for $p\in(1,\infty)$, where $D(V_{p})$ refers to the domain of multiplication by $V$ in $L^{p}(\mathds{R}^{d},\mathds{R}^{m})$. Furthermore, ultracontractivity, kernel estimates and, in the case of a symmetric potential, asymptotic behavior of the eigenvalues have been considered in [18].

In this article, using form methods and extrapolation techniques, we give a general framework of existence of analytic strongly continuous semigroup $\{S_{p}(t)\}_{t\geq 0}$ associated to suitable realizations of $\mathcal{L}$ in $L^{p}$-spaces, for $1<p<\infty$, under mild assumptions on the coefficients of $\mathcal{L}$. Namely, we assume that $Q$ is bounded and elliptic, $F$ and $C$ are bounded with a semi–boundedness condition on their divergences and $V$ has locally integrable entries and satisfies the following pointwise sectoriality condition

[TABLE]

for all $x\in\mathds{R}^{d}$ and all $\xi\in\mathds{C}^{m}$. For further regularity, we assume that the entries of $Q$ are in $C_{b}^{1}(\mathds{R}^{d})$ and $V$ is locally bounded. Note that, in [15, Proposition 5.4], see also [18, Proposition 4.5], the above inequality has been stated as a sufficient condition for the analyticity of the semigroup generated by realizations of $\mathcal{A}$ in $L^{p}(\mathds{R}^{d},\mathds{R}^{m})$, $p\in(1,\infty)$. Moreover, by [14, Example 4.3], one can see that without such a condition one may not have an analytic semigroup. Note also that, even in the scalar case, the existence of a semigroup in $L^{p}$-spaces associated to elliptic operators with unbounded drift and/or diffusion terms is not a general fact, see [24] and [20, Propostion 3.4 and Proposition 3.5]. Furthermore, we point out that coupling through the diffusion (second order) term does not lead to $L^{p}$-contractive semigroups, see [5].

On the other hand, we establish a result of local elliptic regularity for solutions to elliptic systems, see Theorem 4.2. Namely, for given two vector–valued locally $p$–integrable functions $f,g\in L^{p}_{\mathrm{loc}}(\mathds{R}^{d},\mathds{R}^{m})$ satisfying $\mathcal{L}f=g$ in a weak sense (distribution sense). Then $f$ belongs to $W^{2,p}_{loc}(\mathds{R}^{d},\mathds{R}^{m})$, for $p\in(1,\infty)$. This result generalizes [2, Theorem 7.1] to the vector–valued case. Thanks to this result we prove that the domain $D(L_{p})$ of $L_{p}$, for $p\in(1,\infty)$, coincides with the maximal domain :

[TABLE]

We also characterize the positivity of the semigroup $\{S_{p}(t)\}_{t\geq 0}$. We prove that $\{S_{p}(t)\}_{t\geq 0}$ is positive if, and only if, the operator $\mathcal{L}$ is coupled only through the potential term and the coupling coefficients $v_{ij}$, $i\neq j$, are negative or null.

The organization of this paper is as follows: in Section 2, we associate a sesquilinear form to the operator $\mathcal{L}$ in $L^{2}(\mathds{R}^{d},\mathds{C}^{m})$ and we deduce the existence of an analytic $C_{0}$–semigroup $\{S_{2}(t)\}_{t\geq 0}$ associated to $\mathcal{L}$. In Section 3, we prove that $\{S_{2}(t)\}_{t\geq 0}$ is quasi $L^{\infty}$–contractive and we extend $\{S_{2}(t)\}_{t\geq 0}$ to an analytic $C_{0}$–semigroup in $L^{p}(\mathds{R}^{d},\mathds{C}^{m})$ by extrapolation techniques. In Section 4, we establish a local elliptic regularity result and we show that the domain of the generator of $\{S_{2}(t)\}_{t\geq 0}$ coincides with the maximal domain of $\mathcal{L}$ in $L^{p}(\mathds{R}^{d},\mathds{R}^{m})$, for $p\in(1,\infty)$. Section 5 is devoted to determine the positivity of $\{S_{2}(t)\}_{t\geq 0}$.

Notation

Let $\mathds{K}$ denotes the fields $\mathds{R}$ or $\mathds{C}$, $d,m\geq 1$ any integers, $\langle\cdot,\cdot\rangle$ the inner-product of $\mathds{K}^{N}$, $N=d,m$. So that, for $x=(x_{1},\dots,x_{N})$, $y=(y_{1},\dots,y_{N})$ in $\mathds{R}^{N}$, $\langle x,y\rangle=\displaystyle\sum_{i=1}^{N}x_{i}\bar{y}_{i}$ and $x\cdot y=\displaystyle\sum_{i=1}^{N}x_{i}y_{i}$.

The space $L^{p}(\mathds{R}^{d},\mathds{K}^{m})$, $1<p<\infty$, is the vector–valued Lebesgue space endowed with the norm

[TABLE]

We denote by $\langle\cdot,\cdot\rangle_{p,p^{\prime}}$ the duality product between $L^{p}(\mathds{R}^{d},\mathds{K}^{m})$ and $L^{p^{\prime}}(\mathds{R}^{d},\mathds{K}^{m})$ for $1<p<\infty$ where $p^{\prime}=\frac{p}{p-1}$. For $p=2$, we denote it simply by $\langle\cdot,\cdot\rangle_{2}$.

We write $f\in L^{p}_{\mathrm{loc}}(\mathds{R}^{d},\mathds{K}^{m})$ if $\chi_{B}f$ belongs to $L^{p}(\mathds{R}^{d},\mathds{K}^{m})$ for every bounded $B\subset\mathds{R}^{d}$, with $\chi_{B}$ is the indicator function of $B$.

For $k\in\mathds{N}$, $W^{k,p}(\mathds{R}^{d},\mathds{K}^{m})$ denotes the vector–valued Sobolev space constituted of vector–valued functions $f=(f_{1},\dots,f_{m})$ such that $f_{j}\in W^{k,p}(\mathds{R}^{d})$, for all $j\in\{1,\dots,m\}$, where $W^{k,p}(\mathds{R}^{d})$ is the classical Sobolev space of order $k$ over $L^{p}(\mathds{R}^{d})$. Note that all the derivatives are considered in the distribution sense. $W_{loc}^{k,p}(\mathds{R}^{d},\mathds{K}^{m})$ is the set of all measurable functions $f$ such that the distributional derivative $\partial^{\alpha}f$ belongs to $L_{loc}^{p}(\mathds{R}^{d},\mathds{K}^{m})$, for all $\alpha\in\mathds{N}^{d}$ such that $|\alpha|\leq k$. For $y=(y_{1},\dots,y_{m})\in\mathds{R}^{m}$, we write $y\geq 0$ if $y_{j}\geq 0$ for all $j\in\{1,\dots,m\}$.

2. The sesquilinear form and the semigroup in $L^{2}(\mathds{R}^{d},\mathds{C}^{m})$

We consider the following differential expression

[TABLE]

where $f:\mathds{R}^{d}\to\mathds{R}^{m}$ and the derivatives are considered in the sense of distributions. Here, $Q=(q_{ij})_{1\leq i,j\leq d}$ and $V=(v_{ij})_{1\leq i,j\leq m}$ are matrices where the entries are scalar functions: $v_{ij},q_{ij}:\mathds{R}^{d}\to\mathds{R}$, and $F=(F_{ij})_{1\leq i,j\leq m}$ and $C=(C_{ij})_{1\leq i,j\leq m}$ are matrix functions with vector–valued entries: $F_{ij},C_{ij}:\mathds{R}^{d}\to\mathds{R}^{d}$. So that

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

for each $i\in\{1,\dots,m\}$.

Actually, for $f=(f_{1},\dots,f_{m})\in W^{1,p}_{\mathrm{loc}}(\mathds{R}^{d},\mathds{C}^{m})$ for some $1<p<\infty$, $\mathrm{div}(Q\nabla f)$, $F\cdot\nabla f$ and $\mathrm{div}(Cf)$ are vector–valued distributions and are defined as follow

[TABLE]

[TABLE]

and

[TABLE]

for every $\phi=(\phi_{1},\dots,\phi_{m})\in C_{c}^{\infty}(\mathds{R}^{d},\mathds{C}^{m})$.

Throughout this paper we make the following assumptions

Hypotheses (H1):

• •

$Q:\mathds{R}^{d}\to\mathds{R}^{d\times d}$ is measurable such that, for every $x\in\mathds{R}^{d}$, $Q(x)$ is symmetric and there exist $\eta_{1},\eta_{2}>0$ such that

[TABLE]

for all $x,\xi\in\mathds{R}^{d}$.

• •

$F_{ij},C_{ij}\in L^{\infty}(\mathds{R}^{d},\mathds{R}^{d})$, for all $i,j\in\{1,\dots,d\}$.

• •

$v_{ij}\in L^{1}_{\mathrm{loc}}(\mathds{R}^{d})$, for every $i\in\{1,\dots,m\}$ and there exists $M>0$ such that

[TABLE]

for all $x\in\mathds{R}^{d}$ and all $\xi\in\mathds{C}^{m}$.

Let us define, for every $x\in\mathds{R}^{d}$, $V_{s}(x):=\frac{1}{2}(V(x)+V^{*}(x))$ to be the symmetric part of $V(x)$, where $V^{*}(x)$ is the conjugate matrix of $V(x)$. $V_{as}(x):=V(x)-V_{s}(x)$ denotes the antisymmetric part of $V(x)$.

We start by a technical lemma

Lemma 2.1**.**

Let $x\in\mathds{R}^{d}$ and assume $V$ satisfying (2.3). Then

[TABLE]

*for every $\xi_{1},\xi_{2}\in\mathds{C}^{m}$. Moreover, the inequality holds true also when substituting $V$ by $V_{as}$.

In particular,*

[TABLE]

for every measurable $f$ and $g$ such that $V_{s}^{1/2}f,\,V_{s}^{1/2}g\in L^{2}(\mathds{R}^{d},\mathds{C}^{m})$.

Proof.

For $x\in\mathds{R}^{d}$, $\langle V(x)\cdot,\cdot\rangle$ is a sesquilinear form over $\mathds{C}^{m}$. Taking into the account that, for every $\xi\in\mathds{C}^{m}$, ${\rm Re}\,\langle V(x)\xi,\xi\rangle=\langle V_{s}(x)\xi,\xi\rangle.$ Then, (2.4) follows by (2.3) and [21, Proposition 1.8]. Moreover, (2.4) holds true also when taking $V_{as}$ instead of $V$ in the left hand side of the inequality. Now, Cauchy Schwartz inequality yields (2.5). ∎

Let us now consider the sesquilinear form $\mathrm{a}$ given by

[TABLE]

with domain

[TABLE]

where

[TABLE]

The form $\mathrm{a}$ satisfies the following properties

Proposition 2.2**.**

Assume Hypotheses (H1) are satisfied. Then,

• •

$\mathrm{a}$* is densely defined;*

• •

there exists $\omega>0$ such that $\mathrm{a}_{\omega}:=\mathrm{a}+\omega$ is accretive: ${\rm Re}\,\mathrm{a}(f)+\omega\|f\|_{2}^{2}\geq 0$, for all $f\in D(\mathrm{a})$;

• •

$\mathrm{a}$* is continuous;*

• •

$\mathrm{a}$* is closed on $D(\mathrm{a})$.*

Proof.

Clearly, $C_{c}^{\infty}(\mathds{R}^{d},\mathds{C}^{m})\subseteq D(\mathrm{a})$ and thus, $\mathrm{a}$ is densely defined. Moreover, by application of Young’s inequality, one obtains, for every $f\in D(\mathrm{a})$ and every $\varepsilon>0$,

[TABLE]

So by choosing $\varepsilon=\eta_{1}/2$ and $\omega\geq c_{\eta_{1}/2}$, one obtains ${\rm Re}\,\mathrm{a}(f)+\omega\|f\|_{2}^{2}\geq 0$, which shows that $\mathrm{a}_{\omega}$ is accretive.

On the other hand, according to [17, Proposition 2.1], $(D(\mathrm{a}),\|\cdot\|_{\mathrm{a}_{0}})$ is a Banach space, where

[TABLE]

It is then enough to show that $\|\cdot\|_{\mathrm{a}}$ is equivalent to $\|\cdot\|_{\mathrm{a}_{0}}$ to conclude the closedness of $\mathrm{a}$, where $\|\cdot\|_{\mathrm{a}}$ is the graph norm associated to $\mathrm{a}$ and it is given by

[TABLE]

Here $\omega$ is such that $\mathrm{a}_{\omega}$ is accretive. Let us first prove that $\|\cdot\|_{\mathrm{a}}\lesssim\|\cdot\|_{\mathrm{a}_{0}}$. Let $f\in D(A)$, one has $\\ A(f)=\mathrm{a}_{0}(f)+b(f)$, where

[TABLE]

The claim then follows by application of Young’s inequality when estimating $\mathrm{b}$ as in the align above. Conversely, since $\mathrm{a}_{0}(f)=\mathrm{a}(f)-b(f)$, in a similar way one deduces that $\|\cdot\|_{\mathrm{a}_{0}}\lesssim\|\cdot\|_{\mathrm{a}}$.

It remains to show that $\mathrm{a}$ is continuous in $(D(A),\|\cdot\|_{\mathrm{a}_{0}})$, that is

[TABLE]

In view of (2.5), Cauchy-Schwartz inequality and the continuity of $\mathrm{a}_{0}$, c.f. [17, Proposition 2.1 (iii)], one gets

[TABLE]

∎

We, finally, conclude the main theorem of this section as an immediate consequence of [21, Proposition 1.51 Theorem 1.52] and Proposition 2.2

Theorem 2.3**.**

Assume Hypotheses (H1) are satisfied. Then, $\mathcal{L}$ admits a realization $L=L_{2}$ in $L^{2}(\mathds{R}^{d},\mathds{C}^{m})$ that generates an analytic $C_{0}$-semigroup $\{S_{2}(t)\}_{t\geq 0}$. Moreover, there exists $\omega\geq 0$ such that

[TABLE]

3. Extrapolation of the semigroup to the $L^{p}$–scale

In this section we extrapolate $\{S_{2}(t)\}_{t\geq 0}$ to an analytic strongly continuous semigroup in $L^{p}(\mathds{R}^{d},\mathds{R}^{m})$. For that purpose, it suffices to prove that there exists $\tilde{\omega}\in\mathds{R}$ such that $\{S^{\tilde{\omega}}_{2}(t):=\exp(-\tilde{\omega}t)S_{2}(t)\}_{t\geq 0}$ satisfies the following $L^{\infty}$-contractivity property:

[TABLE]

From now on, we use the following notation:

[TABLE]

and

[TABLE]

for every $x,y,z$ in $\mathds{R}^{d}$. We also drop the $x$ and denotes simply $\langle\cdot,\cdot\rangle_{Q}$ and $|\cdot|_{Q}$ for the ease of notation.

In this section we make use of the following hypotheses

Hypotheses (H2):

• •

$F_{ij},C_{ij}\in W^{1,\infty}_{\mathrm{loc}}(\mathds{R}^{d})$, for all $i,j\in\{1,\dots,m\}$, and there exists $\gamma\in\mathds{R}$ such that

[TABLE]

and

[TABLE]

for every $\xi\in\mathds{R}^{m}$ and $x\in\mathds{R}^{d}$.

We state, now, the first result of this section

Proposition 3.1**.**

Assume Hypotheses (H1) and (H2). Then there exists $\tilde{\omega}\in\mathds{R}$ such that $\{S^{\tilde{\omega}}_{2}(t):=\exp(-\tilde{\omega}t)S_{2}(t)\}_{t\geq 0}$ is $L^{\infty}$-contractive.

Proof.

According to the characterization of $L^{\infty}$-contractivity property given by [23, Theorem 1], it suffices to prove that: for $\tilde{\omega}\geq 0$ such that $\mathrm{a}_{\tilde{\omega}}$ is accretive, the following statements hold:

1. (1)

$f\in D(\mathrm{a})$ implies $(1\wedge|f|)\mathrm{sign}(f)\in D(\mathrm{a})$, 2. (2)

${\rm Re}\,\mathrm{a}_{\tilde{\omega}}\left(f,f-(1\wedge|f|)\mathrm{sign}(f)\right)\geq 0,\qquad\forall f\in D(\mathrm{a}),$

where $\mathrm{sign}(f):=\frac{f}{|f|}\chi_{\{f\neq 0\}}$. The first item follows by [17, Lemma 3.2]. Let us show (2). Set $\mathcal{P}_{f}:=(1\wedge|f|)\mathrm{sign}(f)$ and let $\tilde{\omega}$ be bigger enough, so that $\mathrm{a}_{\tilde{\omega}}$ is accretive and $\tilde{\omega}\geq\gamma$. According to [17, Lemma 3.2], we claim that

[TABLE]

for every $i\in\{1,...,m\}$. Therefore,

[TABLE]

where

[TABLE]

and

[TABLE]

Now, one has

[TABLE]

Consequently, since by (2.3), ${\rm Re}\,\langle Vf,f\rangle\geq 0$ a.e., it follows that

[TABLE]

On the other hand,

[TABLE]

Applying an integration by part, one obtains

[TABLE]

where $\mathrm{div}(C^{*}):=(\mathrm{div}(C))^{*}$ is (pointwisely) the conjugate matrix of $\mathrm{div}(C)=(\mathrm{div}(C_{ij}))_{1\leq i,j\leq m}$.

Summing up one obtains

[TABLE]

where

[TABLE]

and

[TABLE]

Since by [18, Lemma 2.4], one has

[TABLE]

Then,

[TABLE]

Therefore,

[TABLE]

Moreover, according to (3.3) that holds true also for $C^{*}$, one gets

[TABLE]

Now, taking in consideration (3.5), (3.7) and (3), one obtains

[TABLE]

Moreover, in view of Young’s inequality, for every $\varepsilon>0$ there exists $c_{\varepsilon}>0$ such that

[TABLE]

Consequently, for $\varepsilon$ being such that $\eta_{1}>\varepsilon$, say $\varepsilon=\eta_{1}/2$, and $\tilde{\omega}>c_{\eta_{1}/2}+\gamma$, one gets

[TABLE]

and this ends the proof.

∎

Hence, we have the following main result of this section.

Theorem 3.2**.**

Let $1<p<\infty$ and assume Hypotheses (H1) and (H2). Then, $\mathcal{L}$ has a realization $L_{p}$ in $L^{p}(\mathds{R}^{d},\mathds{C}^{m})$ that generates an analytic $C_{0}$-semigroup $\{S_{p}(t)\}_{t\geq 0}$.

Proof.

Let $2<p<\infty$. Instead of considering $\min(\omega,\tilde{\omega})$, we assume $\omega>\tilde{\omega}$. In view of Theorem 2.3 and Proposition 3.1, the semigroup $\{S_{2}^{\omega}(t)\}_{t\geq 0}$ is analytic in $L^{2}(\mathds{R}^{d},\mathds{C}^{m})$ and $L^{\infty}$–contactive. Therefore, using the Riesz-Thorin interpolation Theorem, $\{S_{2}^{\omega}(t)\}_{t\geq 0}$ has a unique analytic bounded extension $\{S_{p}^{\omega}(t)\}_{t\geq 0}$ to $L^{p}(\mathds{R}^{d},\mathds{C}^{m})$. Moreover, for every $f\in L^{2}(\mathds{R}^{d},\mathds{C}^{m})\cap L^{\infty}(\mathds{R}^{d},\mathds{C}^{m})$, one claims

[TABLE]

where $\theta=\frac{2}{p}$. Since by Theorem 2.3, the semigroup $\{S_{2}^{\omega}(t)\}_{t\geq 0}$ is strongly continuous in $L^{2}(\mathds{R}^{d},\mathds{C}^{m})$, it follows directly from (3) that $\{S^{\omega}_{p}(t)\}_{t\geq 0}$ is strongly continuous in $L^{p}(\mathds{R}^{d},\mathds{C}^{m})$.

For the case $1<p<2$, we argue by duality. Indeed, the adjoint semigroup $\{S^{*}(t)\}_{t\geq 0}$ is associated to $\mathcal{L}^{*}$, the formal adjoint of $\mathcal{L}$, where

[TABLE]

Since the coefficients of $\mathcal{L}^{*}$ satisfy Hypotheses (H1) and (H2), similarly to $\mathcal{L}$, then $\{S^{*}(t)\}_{t\geq 0}$ is an analytic $C_{0}$-semigroup in $L^{2}(\mathds{R}^{d},\mathds{C}^{m})$ which is quasi $L^{\infty}$-contractive. Consequently, $\{S(t)\}_{t\geq 0}$ is quasi contractive in $L^{1}(\mathds{R}^{d},\mathds{C}^{m})$. So, the same interpolation arguments yield an extrapolation of $\{S(t)\}_{t\geq 0}$ to a holomorpic $C_{0}$-semigroup in $L^{p}(\mathds{R}^{d},\mathds{C}^{m})$, for $1<p<2$. ∎

Remarks 3.3*.*

a) The semigroups $\{S_{p}(t)\}_{t\geq 0}$, $1<p\leq 2$, can be extrapolated to a strongly continuous semigroup in $L^{1}(\mathds{R}^{d},\mathds{C}^{m})$. This follows, according to [27], as a consequence of the consistency and the quasi-contractivity of $\{S_{p}(t)\}_{t\geq 0}$, $1<p\leq 2$.

b) If there exists a nonnegative locally bounded function $\mu:\mathds{R}^{d}\to\mathds{R}^{+}$ such that $\displaystyle\lim_{|x|\to\infty}\mu(x)=+\infty$ and

[TABLE]

Then, for every $1<p<\infty$, $L_{p}$ has a compact resolvent and thus $\{S_{p}(t)\}_{t\geq 0}$ is compact. The proof of this claim is identical to [17, Proposition 4.3].

4. Local elliptic regularity and maximal domain of $L_{p}$

Since the coefficients of $\mathcal{L}$ are real, from now on, we consider vector–valued functions with real components. Thus, $L_{p}$ acts on $D(L_{p})\subset L^{p}(\mathds{R}^{d},\mathds{R}^{m})$, for every $p\in(1,\infty)$ and its associated semigroup $\{S_{p}(t)\}_{t\geq 0}$ acts on $L^{p}(\mathds{R}^{d},\mathds{R}^{m})$. Moreover, we assume that $C\equiv 0$ and thus

[TABLE]

Throughout this section, we use the notation $\Delta_{Q}:=\mathrm{div}(Q\nabla\cdot)$ and, in addition to Hypotheses (H1), we assume the following

Hypotheses (H3):

• •

$q_{ij}\in C_{b}^{1}(\mathds{R}^{d})$, for all $i,j\in\{1,\dots,d\}$.

• •

$v_{ij}\in L^{\infty}_{\mathrm{loc}}(\mathds{R}^{d})$, for all $i,j\in\{1,\dots,m\}$.

Remark 4.1*.*

The assumption $C\equiv 0$ is actually without loss of generalities. Indeed, for every $f\in C_{c}^{\infty}(\mathds{R}^{d},\mathds{R}^{m})$, one has

[TABLE]

Hence, $\tilde{\mathcal{L}}-\gamma$ has the same expression of (4.1) and the matrices $Q$, $\tilde{F}:=F-C$ and $\tilde{V}:=V-\mathrm{div}(C)-\gamma I_{m}$ satisfy Hypotheses (H1) and (H2).

4.1. Local elliptic regularity

Here we give a regularity result for weak solutions to systems of elliptic equations. The following theorem generalizes [2, Theorem 7.1] to the vector valued case.

Theorem 4.2**.**

Let $p\in(1,\infty)$ and assume Hypotheses (H1)–(H3). Let $f$ and $g$ belong to $L^{p}_{\mathrm{loc}}(\mathds{R}^{d},\mathds{R}^{m})$ such that $\mathcal{L}f=g$ in the distribution sense. Then, $f\in W^{2,p}_{\mathrm{loc}}(\mathds{R}^{d},\mathds{R}^{m})$.

Proof.

Let $f=(f_{1},\dots,f_{m})$ and $g=(g_{1},\dots,g_{m})$ belong to $L^{p}_{\mathrm{loc}}(\mathds{R}^{d},\mathds{R}^{m})$ and assume that $\mathcal{L}f=g$ in the sense of distributions. Hence,

[TABLE]

for each $i\in\{1,\dots,m\}$. Now, let $\varphi\in C_{c}^{2}(\mathds{R}^{d})$ and $i\in\{1,\dots,m\}$. A straightforward computation yields

[TABLE]

Then, by (4.2) one gets

[TABLE]

Actually, $\tilde{g}_{i}\in W^{-1,p}(\mathds{R}^{d}):=(W^{1,p^{\prime}}(\mathds{R}^{d}))^{{}^{\prime}}$. Indeed, since $g_{i}$ and $f_{j}$ belong to $L^{p}_{\mathrm{loc}}(\mathds{R}^{d},\mathds{R}^{m})$, then $\varphi g_{i}$, $(\Delta_{Q}\varphi)f_{i}$ and $v_{ij}f_{j}\varphi$ lie in $L^{p}(\mathds{R}^{d})$ and thus in $W^{-1,p}(\mathds{R}^{d})$, for every $j\in\{1,\dots,m\}$. On the other hand, for every $\psi\in C_{c}^{\infty}(\mathds{R}^{d})$, one has

[TABLE]

which shows that $\varphi F_{ij}\cdot\nabla f_{j}\in W^{-1,p}(\mathds{R}^{d})$, for every $j\in\{1,\dots,m\}$. Similarly, we get the claim for $(Q\nabla\varphi)\cdot\nabla f_{i}$. Therefore, for all $\lambda>0$,

[TABLE]

Thus, according to [4, Proposition 2.2], $\varphi f_{i}\in W^{1,p}(\mathds{R}^{d})$ and this is true for every $\varphi\in C_{c}^{\infty}(\mathds{R}^{d})$, which implies that $f_{i}\in W^{1,p}_{\mathrm{loc}}(\mathds{R}^{d})$.

Now, coming back to (4.2), one obtains $\Delta_{Q}f_{i}\in L^{p}_{\mathrm{loc}}(\mathds{R}^{d})$. We then conclude by [2, Theorem 7.1] that $f_{i}$ belongs to $W^{2,p}_{\mathrm{loc}}(\mathds{R}^{d})$. ∎

4.2. $L^{p}$-maximal domain

The aim of this section is to coincide the domain $D(L_{p})$ of the generator of $\{S_{p}(t)\}_{t\geq 0}$ with its maximal domain in $L^{p}(\mathds{R}^{d},\mathds{R}^{m})$. We start by showing that $C_{c}^{\infty}(\mathds{R}^{d};\mathds{C}^{m})\subset D(L_{p})$.

Lemma 4.3**.**

Let $p\geq 1$ and assume Hypotheses (H1)–(H3). Then, $C_{c}^{\infty}(\mathds{R}^{d},\mathds{R}^{m})\subset D(L_{p})$ and $L_{p}f=\mathcal{L}f$, for all $f\in C_{c}^{\infty}(\mathds{R}^{d},\mathds{R}^{m})$.

Proof.

Let $f\in C_{c}^{\infty}(\mathds{R}^{d},\mathds{C}^{m})$. One has $\mathcal{L}f\in L^{2}(\mathds{R}^{d},\mathds{R}^{m})$ and integrating by parts, one claims $\langle-\mathcal{L}f,g\rangle_{2}=\mathrm{a}(f,g)$, for all $g\in D(\mathrm{a})$. Therefore, $f\in D(L_{2})$ and $L_{2}f=\mathcal{L}f$. Moreover, one has

[TABLE]

Since $\mathcal{L}f\in L^{p}(\mathds{R}^{d},\mathds{R}^{m})$, for all $p\geq 1$, and by consistency of the semigroups $\{S_{p}(t)\}_{t\geq 0}$, $p\in[1,\infty)$, Equation (4.3) holds true in $L^{p}(\mathds{R}^{d},\mathds{R}^{m})$, that is

[TABLE]

By consequence, $f\in D(L_{p})$ and $L_{p}f=\mathcal{L}f$ for all $p\geq 1$. ∎

We next show that the space of test functions is a core for $L_{p}$, for $p\in(1,\infty)$. That is, $C_{c}^{\infty}(\mathds{R}^{d};\mathds{R}^{m})$ is dense in $D(L_{p})$ by the graph norm.

Proposition 4.4**.**

Let $1<p<\infty$ and assume Hypotheses (H1)–(H3). Then, the set of test functions $C_{c}^{\infty}(\mathds{R}^{d};\mathds{R}^{m})$ is a core for $L_{p}$.

Proof.

Fix $1<p<\infty$ and let $\lambda>\gamma$ be bigger enough so that it belongs to $\rho(L_{p})$. It suffices to prove that $(\lambda-L_{p})C_{c}^{\infty}(\mathds{R}^{d},\mathds{R}^{m})$ is dense in $L^{p}(\mathds{R}^{d},\mathds{R}^{m})$. For this purpose, let $f\in L^{p^{\prime}}(\mathds{R}^{d};\mathds{R}^{m})$ be such that $\langle(\lambda-\mathcal{L})\varphi,f\rangle_{p,p^{\prime}}=0$, for all $\varphi\in C_{c}^{\infty}(\mathds{R}^{d};\mathds{R}^{m})$. Then,

[TABLE]

in the sense of distributions. By Theorem 4.2, one obtains $f_{j}\in W^{2,p^{\prime}}_{\mathrm{loc}}(\mathds{R}^{d})$ for all $j\in\{1,\dots,m\}$. Then, (4.4) holds true almost everywhere on $\mathds{R}^{d}$.

Now, consider $\zeta\in C_{c}^{\infty}(\mathds{R}^{d})$ such that $\chi_{B(1)}\leq\zeta\leq\chi_{B(2)}$ and define $\zeta_{n}(\cdot)=\zeta(\cdot/n)$ for $n\in\mathds{N}$. Assume $p^{\prime}<2$ and multiply (4.4) by $\zeta_{n}(|f|^{2}+\varepsilon^{2})^{\frac{p^{\prime}-2}{2}}f\in L^{p}(\mathds{R}^{d},\mathds{R}^{m})$ for $\varepsilon>0,\,n\in\mathds{N}$. Integrating by parts, one obtains

[TABLE]

for all $\delta>0$ and some $C_{\delta}>0$. Moreover, according to [18, Lemma 2.4], one has

[TABLE]

So that, choosing $\delta=\delta_{p}<p^{\prime}-1$ and $\lambda>\gamma+C_{\delta_{p}}$, one gets

[TABLE]

Upon $\varepsilon\to 0$, one obtains

[TABLE]

A straightforward computation yields

[TABLE]

So that $\|\Delta_{Q}\zeta_{n}\|_{\infty}$ tends to [math] as $n\to\infty$. Therefore, upon $n\to\infty$, one claims

[TABLE]

Hence, $f=0$.

On the other hand, if $p^{\prime}\geq 2$, multiplying (4.4) by $\zeta_{n}|f|^{p^{\prime}-2}f$, in a similar way, one gets

[TABLE]

It thus follows that $f=0$ by letting $n$ tends to $\infty$.

∎

We show in the next that the domain $D(L_{p})$ is equal to the $L^{p}$-maximal domain of $\mathcal{L}$.

Proposition 4.5**.**

Let $1<p<\infty$ and assume Hypotheses (H1)–(H3). Then

[TABLE]

Proof.

We first show that $D(L_{p})\subseteq D_{p,\max}(\mathcal{L})$. Let $f\in D(L_{p})$ and $(f_{n})_{n}\subset C_{c}^{\infty}(\mathds{R}^{d},\mathds{R}^{m})$ such that $f_{n}\to f$ and $\mathcal{L}f_{n}\to L_{p}f$ in $L^{p}(\mathds{R}^{d},\mathds{R}^{m})$. Let $\Omega$ be a bounded domain of $\mathds{R}^{d}$ and $\phi\in C_{c}^{2}(\Omega)$. Consider, on $\Omega$, the differential operator

[TABLE]

A straightforward computation yields

[TABLE]

Thus, $(\Lambda(\phi f_{n}))_{n}$ converges in $L^{p}(\Omega,\mathds{R}^{m})$. Taking into the account that $\Lambda$ is an elliptic operator with bounded coefficients on $\Omega$, thus the domain of $\Lambda$, with Dirichlet boundary condition, coincides with $W^{2,p}(\Omega,\mathds{R}^{m})\cap W^{1,p}_{0}(\Omega,\mathds{R}^{m})$. In particular, $(\phi f_{n})_{n}$ converges in $W^{2,p}(\Omega,\mathds{R}^{m})$, which implies that $\phi f\in W^{2,p}(\Omega,\mathds{R}^{m})$. Now, the arbitrariness of $\Omega$ and $\phi$ yields $f\in W^{2,p}_{\mathrm{loc}}(\mathds{R}^{d},\mathds{R}^{m})$. Furthermore, $(\mathcal{L}f_{n})_{n}$ converges locally in $L^{p}(\mathds{R}^{d},\mathds{R}^{m})$ to $\mathcal{L}f$ and by pointwise convergence of subsequences, one claims $L_{p}f=\mathcal{L}f$.

In order to prove the other inclusion it suffices to show that $\lambda-\mathcal{L}$ is one to one on $D_{p,\max}(\mathcal{L})$, for some $\lambda>0$. Indeed, this implies that $\lambda\in\rho(\mathcal{L}_{p,\max})\cap\rho(L_{p})$, where $\mathcal{L}_{p,\max}$ is the realization of $\mathcal{L}$ on $D_{p,\max}(\mathcal{L})$. Since $D_{p,\max}(\mathcal{L})\subset D(L_{p})$, thus $L_{p}=\mathcal{L}_{p,\max}$. Now, let $f\in D_{p,\max}(\mathcal{L})$ be such that $(\lambda-\mathcal{L})f=0$. Arguing similarly as in the proof of Proposition 4.4, one obtains $f=0$ and this ends the proof.

∎

Remark 4.6*.*

It is relevant to have $D(L_{p})\subset W^{2,p}(\mathds{R}^{d},\mathds{R}^{m})$, for $1<p<\infty$, which is equivalent to the coincidence of domains $D(L_{p})=W^{2,p}(\mathds{R}^{d},\mathds{R}^{m})\cap D(V_{p})$, where $D(V_{p})$ refers to the maximal domain of multiplication by $V$ in $L^{p}(\mathds{R}^{d},\mathds{R}^{m})$. Actually, in [18, Section 3], it has been shown the following

[TABLE]

for all $f\in W^{2,p}(\mathds{R}^{d},\mathds{R}^{m})\cap D(V_{p})$, provided that $V=\hat{V}+vI_{m}$, with $0\leq v\in W^{1,p}_{\mathrm{loc}}(\mathds{R}^{d})$ such that $|\nabla v|\leq Cv$ and $\hat{V}$ satisfies

[TABLE]

for some $\gamma\in[0,1/2)$. Now, taking into the account, the Landau’s inequality

[TABLE]

for every $\varepsilon>0$, one claims

[TABLE]

Therefore, $D(L_{p})=W^{2,p}(\mathds{R}^{d},\mathds{R}^{m})\cap D(V_{p})$.

5. Positivity

In this section we characterize the positivity of the semigroup $\{S_{p}(t)\}_{t\geq 0}$ for $1<p<\infty$. Since the family of semigroups $\{S_{p}(t)\}_{t\geq 0}$, $p\in[1,\infty)$, is consistent, i. e., $S_{p}(t)f=S_{q}(t)f$, for every $t\geq 0$, $1\leq p,q<\infty$ and all $f\in L^{p}(\mathds{R}^{d},\mathds{R}^{m})\cap L^{q}(\mathds{R}^{d},\mathds{R}^{m})$, it suffices to characterize the positivity of $\{S_{2}(t)\}_{t\geq 0}$. For this purpose, we endow $\mathds{R}^{m}$ with the usual partial order: $x\geq y$ if and only if, $x_{i}\geq y_{i}$, for all $i\in\{1,\dots,m\}$. As in Section 4, we assume that $C\equiv 0$. By positivity of $\{S_{2}(t)\}_{t\geq 0}$ we mean $S_{2}(t)f\geq 0$ a.e., for every $t\geq 0$ and all $f\in L^{2}(\mathds{R}^{d},\mathds{R}^{m})$ such that $f\geq 0$ a.e.

We apply the Ouhabaz’ criterion for invariance of closed convex subsets by semigroups, c.f. [23, Theorem 3] and [22]. We then get the following result

Theorem 5.1**.**

Assume Hypotheses (H1). Then, the semigroup $\{S_{2}(t)\}_{t\geq 0}$ is positive, if and only if, $F_{ij}=0$ and $v_{ij}\leq 0$ almost everywhere and for every $i\neq j\in\{1,\cdots,m\}$.

Proof.

Let $\mathcal{C}=\{f\in L^{2}(\mathds{R}^{d},\mathds{R}^{m}):f\geq 0\;\mathrm{a.e.}\}$ and $\mathcal{P}_{+}f=f^{+}=(f_{i}^{+})_{1\leq i\leq m}$, where $f_{i}^{+}=\max(0,f_{i})$. Then, $\mathcal{C}$ is a closed convex subset of $L^{2}(\mathds{R}^{d},\mathds{R}^{m})$ and $\mathcal{P}_{+}$ is the corresponding projection. Now, let $\omega\geq 0$ such that $\mathrm{a}_{\omega}$ is accretive. According to [23, Theorem 3 (iii)], $\{S_{2}(t)\}_{t\geq 0}$ is positive if, and only if, the form $\mathrm{a}$ satisfies the following

• •

$f\in D(\mathrm{a})$ implies $f^{+}\in D(\mathrm{a})$,

• •

$\mathrm{a}_{\omega}(f^{+},f^{-})\leq 0$, for all $f\in D(\mathrm{a})$, where $f^{-}=f-f^{+}$.

Now, assume that $\{S_{2}(t)\}_{t\geq 0}$ is positive. Let $i\neq j\in\{1,\dots,m\}$, $n\in\mathds{N}$ and $0\leq\varphi\in C_{c}^{\infty}(\mathds{R}^{d})$. Set $f=\zeta_{n}e_{i}-\varphi e_{j}$. One has

[TABLE]

Letting $n\to\infty$, by dominated convergence theorem, one gets $\int_{\mathds{R}^{d}}v_{ij}\varphi\,dx\leq 0$ for every $0\leq\varphi\in C_{c}^{\infty}(\mathds{R}^{d})$, which implies that $v_{ij}\leq 0$ almost everywhere. On the other hand, considering, for every $n\in\mathds{N}$,

[TABLE]

where $x_{k}$ is the k-th component of $x\in\mathds{R}^{d}$, for every $k\in\{1,\cdots,d\}$. Then,

[TABLE]

Therefore,

[TABLE]

where $F_{ij}^{(k)}$ indicates the $k$-th component of $F_{ij}$. So, by letting $n\to\infty$, one deduces that $F_{ij}^{(k)}\leq 0$ almost everywhere and for each $k\in\{1,\cdots,d\}$. In a similar way, one gets $F_{ij}^{(k)}\geq 0$ a.e. by considering $\tilde{g}$ instead of $g$, where

[TABLE]

So that $F_{ij}=0$ almost everywhere.

Conversely, assume $F_{ij}=0$ and $v_{ij}\leq 0$ for all $i\neq j\in\{1,\dots,m\}$. Let $f\in D(\mathrm{a})$, then, by [17, Theorem 4.2], one gets $f^{+}\in D(a)$. Furthermore, it follows, by [9, Theorem 7.9], that $\nabla f^{+}_{i}=\chi_{\{f_{i}>0\}}\nabla f_{i}$ and $\nabla f^{-}_{i}=\chi_{\{f_{i}<0\}}\nabla f_{i}$. Let us now prove that $\mathrm{a}_{\omega}(f^{+},f^{-})\leq 0$. One has

[TABLE]

This ends the proof. ∎