An Interior Regularity Property for the Solution to a Linear Elliptic System with Singular Coefficients in the Lower-Order Term

Abstract

This paper deals with the interior higher differentiability of the solution u to the Dirichlet problem related to system $-\mathrm{div}\left(A\right(x\left)Du\right)+B(x,u)=f$ on a bounded Lipschitz domain $\mathsf{\Omega }$ in ${\mathbb{R}}^n$. The matrix $A\left(x\right)$ lies in the John and Nirenberg space $BMO$. The lower-order term $B(x,u)$ is controlled with respect to the spatial variable by a function $b\left(x\right)$ belonging to the Marcinkiewicz space $L^{n,\infty }$. The novelty here is the presence of a singular coefficient in the lower-order term.

1. Introduction

We investigate the higher integrability of the gradient of the weak solution of the elliptic system

$$-\mathrm{div}\left(A\right(x\left)Du\right)+B(x,u)=f$$

(1)

in a bounded domain $\mathsf{\Omega }$ of ${\mathbb{R}}^n$, $n>2$ where $A\left(x\right)=\left(A_{ij}\left(x\right)\right)$ is a symmetric, positive-definite matrix with measurable coefficients and f a vector field in $L^2(\mathsf{\Omega },{\mathbb{R}}^n)$.

Let $B:(x,u)\in \mathsf{\Omega }\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ be a Carathéodory vector field such that

$$|B(x,u)-B(x,u^{\prime })|⩽b\left(x\right)|u-u^{\prime }|$$

(2)

$$⟨B(x,u)-B(x,u^{\prime }),u-u^{\prime }⟩⩾0$$

(3)

$$B(x,0)=0$$

(4)

where $b\left(x\right)$ is a nonnegative real function belonging to the Marcinkiewicz space $L^{n,\infty }\left(\mathsf{\Omega }\right)$.

The matrix is elliptic, i.e.,

$$⟨A\left(x\right)Y,Y⟩⩾{\parallel Y\parallel }^2$$

(5)

for every matrix $Y\in {\mathbb{R}}^{n\times n}$.

We emphasize that no extra differentiability can be obtained for solutions even if the data $f\left(x\right)$ and $b\left(x\right)$ are smooth, without any differentiability assumption on the operator $A\left(x\right)$.

Here, we assume that the entries of the matrix A lie in the John–Nirenberg space $BMO$ and there exists a nonnegative function $K\left(x\right)$ belonging to $L^{n,\infty }\left(\mathsf{\Omega }\right)$ such that

$$|A(x+h{e}_i)-A\left(x\right)|⩽K\left(x\right)\left|h\right|,i=1,...,n$$

(6)

for a.e. $x\in \mathsf{\Omega }$ and $h\in \mathbb{R}$ such that $x+h{e}_i\in \mathsf{\Omega }$.

Condition (6) states that the derivative $D_{x_i}A$ has the same integrability as $K\left(x\right)$.

The functions

$$A\left(x\right)=\left({\displaystyle \frac{e^{-\left|x\right|}}{\Lambda }}-\Lambda log\left|x\right|\right)\mathrm{Id},$$

$$K\left(x\right)={\displaystyle \frac{e^{-\left|x\right|}}{\Lambda }}+\Lambda {\displaystyle \frac{1}{\left|x\right|}},$$

(7)

$$b\left(x\right)={\displaystyle \frac{\lambda }{\left|x\right|}}$$

where $\Lambda$ and $\lambda$ are positive constants, satisfying, for example, assumptions (2)–(6).

Early contributions on the second-order regularity of solutions of equations with discontinuity in the coefficients are due to Miranda [1,2] and consider coefficients in the Sobolev class $W^{1,n}$. Chiarenza, Frasca, Longo [3] and Chiarenza [4] established regularity results for equations in a nondivergence form with $VMO$ coefficients.

In 2018, Giannetti and Moscariello [5] studied the Dirichlet problem for the second-order elliptic equation

$$\sum _{i,j=1}^n{a}_{ij}\left(x\right){\displaystyle \frac{{\partial }^{2}u}{\partial x_i\partial x_j}}=f$$

under the assumption that f is in $L^2$ and that the coefficients $a_{ij}$ are measurable and bounded functions with the first derivatives in the Marcinkiewicz class $L^{n,\infty }$ and have a sufficiently small distance to $L^{\infty }$. They proved the solvability of the problem in $W^{2,2}\cap W_0^{1,2^{🟉}}$, where $2^{🟉}={\displaystyle \frac{2n}{n-2}}$. A higher integrability result for the gradient of the solution is proved when $f\in L^p$, $p>2$.

We underline that the role of the weak $L^n$ structure is crucial to ensure the higher integrability of $Du$ as shown in example $1.2$ of [5]. Reference [6] assumed the smallness of the BMO norm.

Reference [7] studied the Dirichlet problem for a uniformly elliptic equation of type (1) introducing a hypothesis that relates the coefficient of the lower-order term $b\left(x\right)$ with the right-hand side f. A regularizing effect on the solution of the Dirichlet problem in the case of uniformly elliptic equations follows (for more details, see [8,9]).

Reference [10] established a higher differentiability result also for an equation of the type in (1). A recent significant development is due to Stroffolini [11] who studied the Dirichlet problem for a linear system with coefficients in BMO and obtained regularity results for the minimizers of functionals of the Calculus of Variations applied to linear and nonlinear systems with a principal part as in (1).

In [12], Moscariello and Pascale established a higher differentiability result for this class of systems. They extended their result in [13] for some nonlinear elliptic systems with growth coefficients in $BMO$.

In the present paper, the novelty consists of studying local regularity properties of solutions to linear systems with the presence of a lower-order term as in (1).

Definition 1.

A vector field u in the Sobolev space $W_0^{1,2}(\mathsf{\Omega },{\mathbb{R}}^n)$ is a weak solution of the Dirichlet problem

$$\left\{\begin{array}{cc}-\mathrm{div}\left(A\right(x\left)Du\right)+B(x,u)=f\hfill & \mathit{in}\mathsf{\Omega }\hfill \\ u=0\hfill & \mathit{on}\partial \mathsf{\Omega }\hfill \end{array}\right.$$

(8)

if it is verified that

$${\int }_{\mathsf{\Omega }}⟨A\left(x\right)Du\left(x\right),D\phi \left(x\right)⟩dx+{\int }_{\mathsf{\Omega }}B(x,u)\phi \left(x\right)dx={\int }_{\mathsf{\Omega }}f\left(x\right)\phi \left(x\right)dx,$$

(9)

$\forall \phi \in C_0^{\infty }(\mathsf{\Omega },{\mathbb{R}}^n)$.

It is useful to set

$$K\left(x\right)=K_1\left(x\right)+K_2\left(x\right)$$

(10)

with $K_1\in L^{\infty }$ and $K_2\in L^{n,\infty }$. Our main result is the following:

Theorem 1.

Let Ω be a Lipschitz domain. Let us assume (2)–(6). Then, the problem (8) admits a unique solution $u\in W_0^{1,2}(\mathsf{\Omega },{\mathbb{R}}^n)$. Moreover, there exists $\epsilon >0$ depending on n such that if

$$\parallel K_2{\parallel }_{L^{n,\infty }}<\epsilon ,$$

(11)

then $u\in W_{loc}^{2,2}(\mathsf{\Omega },{\mathbb{R}}^n)$ and

$${\int }_{B_R}{\left|D^{2}u\right|}^{2}dx⩽C\left[{\int }_{B_{2R}}\left({\displaystyle \frac{1}{R^2}}+1\right)\left({\left|Du\right|}^2+{\left|u\right|}^2+{\left|f\right|}^2\right)dx\right]$$

for every ball $B_{2R}\subset \subset \mathsf{\Omega }$ and for C depending on n, $\parallel K_2{\parallel }_{L^{n,\infty }}$, the $BMO$-norm of A and ${\parallel b\parallel }_{L^{n,\infty }}$.

Let us discuss condition (11). First of all, we underline that condition (11) does not imply that the norm of $K\left(x\right)$ is small in $L^{n,\infty }$. Indeed, by easy calculations (for more details, see [12]), if $K\left(x\right)$ is the function in the example (7), condition (11) reduces to $\Lambda <\epsilon {\omega }_n^{-{\displaystyle \frac{1}{n}}}$.

On the other hand,

$${\parallel K\parallel }_{L^{n,\infty }}⩾{\displaystyle \frac{C}{\Lambda }}$$

where $C=C\left(n\right)$, and so the norm of K will be large for the small $\epsilon$.

As is known, $L^{\infty }$ is not dense in $L^{n,\infty }$. Notice that condition (11) is equivalent to saying that there exists a function $K_1\in L^{\infty }\left(\mathsf{\Omega }\right)$ such that

$$\parallel K-K_1{\parallel }_{L^{n,\infty }}<\epsilon .$$

The condition (11) measures how far $K\left(x\right)$ is from the bounded functions and it is satisfied whenever $K\left(x\right)$ lies in the Lorentz space $L^{n,q}$, $1<q<\infty$. Finally, we point out that our Theorem applies also for $A\left(x\right)\in W^{1,n}$ since by embedding $W^{1,n}\subset BMO$ (see Theorem 2).

A similar condition occurs in the study of nonlinear elliptic and parabolic equations with conventional terms [14,15]. See also [16]. We point out that Theorem 1 is new even in the case that $A\left(x\right)$ is a bounded matrix.

2. Notations and Preliminary Results

2.1. $BMO$ Spaces

Definition 2

([17,18]). Let Ω be a cube or the entire space ${\mathbb{R}}^n$. The $BMO\left(\mathsf{\Omega }\right)$ space consists of all functions g that are integrable on every cube $Q\subset \mathsf{\Omega }$ with sides parallel to those of Ω and satisfy

$${\displaystyle {\parallel g\parallel }_{🟉}=\underset{Q}{sup}\left\{\frac{1}{\left|Q\right|}{\int }_Q|g-g_Q|dx\right\}<\infty ,}$$

where ${\displaystyle g_Q=\frac{1}{\left|Q\right|}{\int }_{Q}g\left(y\right)dy}$ and $\left|Q\right|$ denotes the Lebesgue measure of Q.

The functional ${\parallel ·\parallel }_{🟉}$ does not define a norm since it vanishes on constant functions. $BMO$ becomes a Banach space if we identify functions which differ a.e. by a constant.

Therefore, the function ${\parallel ·\parallel }_{🟉}$ is properly a norm on the quotient space of $BMO$ functions modulo the space of constant functions on the domain considered.

Bounded functions are in $BMO$. On the other hand, $BMO$ contains unbounded functions. An example of $BMO$ functions is given by the following:

$$g\left(x\right)=log\left|x\right|,x\in B(0,1).$$

We also recall that $L^{\infty }$ is not dense in $BMO$.

Another example of the BMO function is contained in [19].

Lemma 1.

Let $M\mu$ denote the Hardy–Littlewood maximal function of a Radon measure μ in ${\mathbb{R}}^n$, and suppose $M\mu \left(x\right)$ is finite and positive at some point x, then $log\left(M\mu \right)$ belongs to $BMO\left({\mathbb{R}}^n\right)$.

Theorem 2

([17]). For any cube $Q\subset {\mathbb{R}}^n$, the following inclusion holds with continuous embedding:

$$W^{1,n}\left(Q\right)\hookrightarrow BMO\left(Q\right).$$

(12)

2.2. Lorentz Spaces

Let $\mathsf{\Omega }$ be a bounded domain in ${\mathbb{R}}^n$. Given $1<p,q<\infty$, the Lorentz space $L^{p,q}\left(\mathsf{\Omega }\right)$ consists of all measurable functions g defined on $\mathsf{\Omega }$ for which the quantity

$${\parallel g\parallel }_{L^{p,q}}^q=p{\int }_0^{\infty }{\left|{\mathsf{\Omega }}_t\left(g\right)\right|}^{{\displaystyle \frac{q}{p}}}{t}^{q-1}dt$$

is finite, where ${\mathsf{\Omega }}_t\left(g\right)=\{x\in \mathsf{\Omega }:|g\left(x\right)|>t\}$ and $|{\mathsf{\Omega }}_t|$ is the Lebesgue measure of ${\mathsf{\Omega }}_t$. Remark that ${\parallel ·\parallel }_{L^{p,q}}$ is equivalent to a norm. Endowed with this norm, $L^{p,q}$ becomes a Banach space.

For $p=q$, the Lorentz space $L^{p,p}$ coincides with the Lebesgue space $L^p$. For $q=\infty$, the class $L^{p,\infty }$ consists of all measurable functions g defined on $\mathsf{\Omega }$ such that

$${\displaystyle {\parallel g\parallel }_{L^{p,\infty }}^p=\underset{t>0}{sup}{t}^p\left|{\mathsf{\Omega }}_t\left(g\right)\right|<\infty .}$$

$L^{p,\infty }$ coincides with the Marcinkiewicz class, weak $L^p$. The following inclusions for Lorentz spaces hold

$$L^r\left(\mathsf{\Omega }\right)\subset L^{p,q}\left(\mathsf{\Omega }\right)\subset L^{p,r}\left(\mathsf{\Omega }\right)\subset L^{p,\infty }\left(\mathsf{\Omega }\right)\subset L^q\left(\mathsf{\Omega }\right),$$

whenever $1⩽q<p<r⩽\infty .$

Let us recall the following Hölder inequality in Lorentz spaces.

Theorem 3

([20]). If $0<p_1,p_2,p<\infty$ and $0<q_1,q_2,q⩽\infty$ obey ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{p_1}}+{\displaystyle \frac{1}{p_2}}$ and ${\displaystyle \frac{1}{q}}={\displaystyle \frac{1}{q_1}}+{\displaystyle \frac{1}{q_2}}$, then

$${\parallel fg\parallel }_{L^{p,q}}⩽{\parallel f\parallel }_{L^{p_1,q_1}}{\parallel g\parallel }_{L^{p_2,q_2}}$$

(13)

whenever the right-hand side norms are finite.

The Sobolev embedding theorem in Lorentz spaces will be fundamental.

Theorem 4

([21]). Let us assume that $1<p<n$, $1⩽q⩽p$, and then any function $u\in W_0^{1,1}\left(\mathsf{\Omega }\right)$ such that $|\nabla u|\in L^{p,q}\left(\mathsf{\Omega }\right)$ actually belongs to $L^{p^{🟉},q}\left(\mathsf{\Omega }\right)$ and

$${\parallel u\parallel }_{L^{p^{🟉},q}}⩽C_{n,p}{\parallel \nabla u\parallel }_{L^{p,q}},$$

(14)

where $p^{🟉}={\displaystyle \frac{np}{n-p}}$ and $C_{n.p}={\omega }_n^{-{\displaystyle \frac{1}{n}}}{\displaystyle \frac{p}{n-p}}$. Here, ${\omega }_n$ is the Lebesgue measure of the unit ball in ${\mathbb{R}}^n$.

By using Theorem 4 and the Hölder inequality, we immediately obtain

$${\int }_D|k\left(x\right)\phi \nabla \psi |dx⩽C_{n,2}{\parallel k\parallel }_{L^{n,\infty }\left(D\right)}{\parallel \nabla \phi \parallel }_{L^2\left(D\right)}{\parallel \nabla \psi \parallel }_{L^2\left(D\right)}$$

(15)

for any $\phi \in W_0^{1,2}\left(D\right)$ and $\psi$ such that $\nabla \psi \in L^2\left(D\right)$.

We define the distance of the given $f\in L^{p,\infty }$ to $L^{\infty }$ as

$${\displaystyle {\mathrm{dist}}_{L^{p,\infty }}(f,L^{\infty })=\underset{g\in L^{\infty }}{inf}{\parallel f-g\parallel }_{L^{p,\infty }}.}$$

We underline that assuming $\parallel K_2{\parallel }_{L^{n,\infty }}<\epsilon$ is equivalent to ${\displaystyle \underset{K_1\in L^{\infty }}{inf}{\parallel K-K_1\parallel }_{L^{n,\infty }}<\epsilon }$.

2.3. Difference Quotient

To prove a higher differentiability result for solutions to (1), we will introduce the finite difference operator in order to apply the difference quotient method.

Definition 3

([22]). For every vector valued function $F:{\mathbb{R}}^n\to {\mathbb{R}}^N$, the finite difference operator and the difference quotient are, respectively, defined by

$${\tau }_{i,h}F\left(x\right)=F(x+h{e}_i)-F\left(x\right),$$

$${\Delta }_{h}F\left(x\right)={\displaystyle \frac{{\tau }_{i,h}F\left(x\right)}{h}},h\in \mathbb{R}\setminus \left\{0\right\}$$

where $h\in \mathbb{R}$, $e_i$ is the unit vector in the $x_i$ direction and $i\in \{1,\dots ,n\}$.

The function ${\Delta }_{h}f$ is defined in the set

$${\Delta }_h\mathsf{\Omega }=\{x\in \mathsf{\Omega }:x+h{e}_i\in \mathsf{\Omega }\}.$$

We list some elementary properties of the finite difference operator.

Proposition 1

([22]). Let f and g be two functions such that $F,G\in W^{1,p}(\mathsf{\Omega };{\mathbb{R}}^N)$, with $p\ge 1$, and let us consider

$${\mathsf{\Omega }}_{\left|h\right|}:=\left\{x\in \mathsf{\Omega }:dist(x,\partial \mathsf{\Omega })>\left|h\right|\right\}.$$

Then, we have the following:

$\left(d1\right)$

${\tau }_{h}F\in W^{1,p}\left(\mathsf{\Omega }\right)$ and

$$D_i\left({\tau }_{h}F\right)={\tau }_h\left(D_{i}F\right).$$

$\left(d2\right)$

If at least one of the functions F or G has support contained in ${\mathsf{\Omega }}_{\left|h\right|}$, then

$${\int }_{\mathsf{\Omega }}F{\tau }_{h}Gdx=-{\int }_{\mathsf{\Omega }}G{\tau }_{-h}Fdx.$$

$\left(d3\right)$

The following equality holds:

$${\tau }_h\left(FG\right)\left(x\right)=F(x+h{e}_s){\tau }_{h}G\left(x\right)+G\left(x\right){\tau }_{h}F\left(x\right).$$

The next Lemma is a kind of integral version of the Lagrange Theorem. The following results will be useful in the sequel.

Lemma 2

([22]). If $0<\rho <R$, $\left|h\right|<{\displaystyle \frac{R-\rho }{2}}$, $1<p<+\infty$, $i\in \{1,\dots ,n\}$ and $F,D_{i}F\in L^p\left(B_R\right)$, then

$${\int }_{B_{\rho }}|{\tau }_h{F\left(x\right)|}^{p}dx\le {\left|h\right|}^p{\int }_{B_R}{\left|D_{i}F\left(x\right)\right|}^{p}dx.$$

Moreover,

$${\int }_{B_{\rho }}\left|F\right(x+h{e}_i){|}^{p}dx\le {\int }_{B_R}{\left|F\left(x\right)\right|}^{p}dx.$$

Let us recall the Sobolev embedding property.

Lemma 3

(see Lemma 8.2 in [22]). Let $F:{\mathbb{R}}^n\to {\mathbb{R}}^N$, $F\in L^p\left(B_R\right)$ with $1<p<+\infty$. Suppose that there exist $\rho \in (0,R)$ and $C>0$ such that

$$\sum _{s=1}^n{\int }_{B_{\rho }}|{\tau }_h{F\left(x\right)|}^{p}dx\le C^p{\left|h\right|}^p,$$

for every h with $\left|h\right|<{\displaystyle \frac{R-\rho }{2}}$. Then, $F\in W^{1,p}(B_{\rho };{\mathbb{R}}^N)\cap L^{{\displaystyle \frac{np}{n-p}}}(B_{\rho };{\mathbb{R}}^N)$. Moreover,

$${\left|\right|DF\left|\right|}_{L^p\left(B_{\rho }\right)}\le C$$

and

$${\left|\right|F\left|\right|}_{L^{{\displaystyle \frac{np}{n-p}}}\left(B_{\rho }\right)}\le c\left(C+{\left|\right|F\left|\right|}_{L^p\left(B_R\right)}\right),$$

with $c\equiv c(n,N,p)$.

We recall an iteration Lemma. It finds a remarkable application in the so-called hole-filling method.

Lemma 4

(see Lemma 6.1 [22]). Let $h:[\rho ,R_0]\to \mathbb{R}$ be a nonnegative bounded function and $0<\vartheta <1$, $A,B\ge 0$ and $\beta >0$. Suppose that

$$h\left(r\right)\le \vartheta h\left(d\right)+{\displaystyle \frac{A}{{(d-r)}^{\beta }}}+B,$$

for all $\rho \le r<d\le R_0$. Then,

$$h\left(\rho \right)\le {\displaystyle \frac{cA}{{(R_0-\rho )}^{\beta }}}+B,$$

where $c=c(\vartheta ,\beta )>0$.

3. Proof of Theorem 1

In this section, we prove our main result: Theorem 1. In the first part of the proof, we show the uniqueness of the solution. In the second part, we establish an a priori estimate for the second derivatives of the solutions. In the third part, we construct the suitable approximating problems proving that the a priori estimate is preserved when we pass to the limit.

• Step 1. The uniqueness

Let u and v be two solutions of problem (8). We use $w=u-v$ as the test function for the solution u and the solution v, respectively:

$${\int }_{\mathsf{\Omega }}⟨A\left(x\right)Du,Du-Dv⟩+B(x,u)(u-v)dx={\int }_{\mathsf{\Omega }}f(u-v)dx$$

(16)

and

$${\int }_{\mathsf{\Omega }}⟨A\left(x\right)Dv,Du-Dv⟩+B(x,v)(u-v)dx={\int }_{\mathsf{\Omega }}f(u-v)dx$$

(17)

By subtracting (17) from (16), we obtain

$${\int }_{\mathsf{\Omega }}⟨A\left(x\right)(Du-Dv),Du-Dv⟩+\left[B(x,u)-B(x,v)\right](u-v)dx=0$$

Then, from (5) and (3), we obtain

$${\parallel Du-Dv\parallel }_{L^2}^2⩽{\int }_{\mathsf{\Omega }}⟨A\left(x\right)(Du-Dv),Du-Dv⟩dx⩽0$$

and so $u=v$ a.e.

• Step 2. The a priori estimate

Theorem 5.

Let Ω be a Lipschitz domain. If u is the solution of (8) in $W_{loc}^{2,2}(\mathsf{\Omega },{\mathbb{R}}^n)$, then there exists ε, depending only on n, such that if $\parallel K_2{\parallel }_{L^{n,\infty }}<\epsilon$, the following estimate holds

$${\int }_{B_R}{\left|D^{2}u\right|}^{2}dx⩽C\left[{\int }_{B_{2R}}\left({\displaystyle \frac{1}{R^2}}+1\right)\left({\left|Du\right|}^2+{\left|u\right|}^2+{\left|f\right|}^2\right)dx\right]$$

(18)

for every ball $B_{2R}\subset \subset \mathsf{\Omega }$ and for a constant C depending on n, $\parallel K_2{\parallel }_{L^{n,\infty }}$, the $BMO$-norm of A and ${\parallel b\parallel }_{L^{n,\infty }}$.

Proof. 

Let us fix a ball $B_R\subset \mathsf{\Omega }$ and arbitrary radii $R<s<t<2R$, with R small enough. Let $u\in W_0^{1,2}(\mathsf{\Omega },{\mathbb{R}}^n)$ be a solution of (8) and then in particular for any $\phi \in C_0^{\infty }(B_t,{\mathbb{R}}^n)$

$${\int }_{B_t}⟨A\left(x\right)Du\left(x\right),D\phi \left(x\right)⟩dx+{\int }_{B_t}B(x,t)\phi \left(x\right)dx={\int }_{B_t}f\left(x\right)\phi \left(x\right)dx$$

i.e., u solves the equation

$$-\mathrm{div}\left(A\left(x\right)Du\right)+B(x,u)=f\mathrm{in}{B}_t.$$

(19)

Since $u\in W^{2,2}(B_t,{\mathbb{R}}^n)$, $A\left(x\right)Du\in L^2(B_t,{\mathbb{R}}^{n\times n})$. Now, by Hodge decomposition, we decompose uniquely

$$ADu=D\Psi +H$$

(20)

with H a divergence-free vector field and $\Psi \in W_0^{1,2}(B_t,{\mathbb{R}}^{n\times n})$ (see [23] (p. 148), relation (2.8) and line 17 of p. 149).

By (19) and (20),

$$\mathrm{div}D\Psi =\Delta \Psi =B(x,u)-f\mathrm{in}{B}_t$$

and, by the classical theory,

$${\parallel D\Psi \parallel }_{L^2\left(B_t\right)}⩽{\parallel B(·,u)\parallel }_{L^2\left(B_t\right)}+{\parallel f\parallel }_{L^2\left(B_t\right)}.$$

(21)

Let T be the projection operator of a vector field onto a gradient field. Notice that $H=ADu-D\Psi$ where $D\Psi =T\left(ADu\right)$, i.e.,

$$H=ATDu-T\left(ADu\right).$$

Then, using Theorem 3.1 and Lemma 3.3 of [11], we conclude that H is in $L^2(B_t,{\mathbb{R}}^n)$ and

$${\parallel H\parallel }_{L^2\left(B_t\right)}⩽{c\parallel A\parallel }_{🟉}{\parallel Du\parallel }_{L^2\left(B_t\right)}$$

(22)

where $c=c\left(n\right)$. Finally, from (19) and relations (20), (21) and (22), we obtain

$${\parallel ADu\parallel }_{L^2\left(B_t\right)}⩽c\left(n\right)\left({\parallel A\parallel }_{🟉}{\parallel Du\parallel }_{L^2\left(B_t\right)}+{\parallel f\parallel }_{L^2\left(B_t\right)}+{\parallel B(·,u)\parallel }_{L^2\left(B_t\right)}\right).$$

(23)

Let $\eta \in C_0^{\infty }\left(B_t\right)$, $0⩽\eta ⩽1$ be a cut-off function such that $\eta \equiv 1$ on $B_s$ and $\left|D\eta \right|\le {\displaystyle \frac{c}{t-s}}$. Since u is a weak solution of (1), we are able to use $\phi ={\tau }_{-h}\left({\eta }^2{\tau }_{h}u\right)$ as a test function in (9) obtaining

$$\begin{array}{ccc}\hfill {\int }_{B_t}⟨A\left(x\right)Du,D\left({\tau }_{-h}\left({\eta }^2{\tau }_{h}u\right)\right)⟩dx& +& {\int }_{B_t}⟨B(x,u),{\tau }_{-h}\left({\eta }^2{\tau }_{h}u\right)⟩dx\hfill \\ \\ & =& {\int }_{B_t}⟨f\left(x\right),{\tau }_{-h}\left({\eta }^2{\tau }_{h}u\right)⟩dx\hfill \end{array}$$

and using the properties of difference quotients

$$\begin{array}{ccc}\hfill {\int }_{B_t}⟨{\tau }_h\left(A\left(x\right)Du\right),D\left({\eta }^2{\tau }_{h}u\right)⟩dx& =& -{\int }_{B_t}⟨B(x,u),{\tau }_{-h}\left({\eta }^2{\tau }_{h}u\right)⟩dx\hfill \\ \\ & +& {\int }_{B_t}⟨f\left(x\right),{\tau }_{-h}\left({\eta }^2{\tau }_{h}u\right)⟩dx.\hfill \end{array}$$

It follows that

$$\begin{array}{ccc}\hfill & & {\int }_{B_t}{\eta }^2⟨{\tau }_h\left(A\left(x\right)Du\right),D{\tau }_{h}u⟩dx+2{\int }_{B_t}\eta ⟨{\tau }_h\left(A\left(x\right)Du\right),\nabla \xi \otimes {\tau }_{h}u⟩\hfill \\ & =& -{\int }_{B_t}⟨B(x,u),{\tau }_{-h}\left({\eta }^2{\tau }_{h}u\right)⟩dx+{\int }_{B_t}⟨f\left(x\right),{\tau }_{-h}\left({\eta }^2{\tau }_{h}u\right)⟩dx.\hfill \end{array}$$

(24)

We remark that

$${\tau }_h\left(A\left(x\right)Du\right)=A(x+h{e}_i)D{\tau }_{h}u+\left({\tau }_{h}A\left(x\right)\right)Du.$$

Then, from (24), we obtain

$$\begin{array}{ccc}\hfill {\int }_{B_t}{\eta }^2⟨A(x+h{e}_i)D{\tau }_{h}u,D{\tau }_{h}u⟩dx=& -& {\int }_{B_t}{\eta }^2⟨\left({\tau }_{h}A\left(x\right)\right)Du,D{\tau }_{h}u⟩dx\hfill \\ & -& 2{\int }_{B_t}\eta ⟨A(x+h{e}_i)D{\tau }_{h}u,\nabla \eta \otimes {\tau }_{h}u⟩dx\hfill \\ & -& 2{\int }_{B_t}\eta ⟨{\tau }_{h}A\left(x\right))Du\left(x\right),\nabla \eta \otimes {\tau }_{h}u⟩dx\hfill \\ & -& {\int }_{B_t}\left[B(x,u)\right]{\tau }_{-h}\left({\eta }^2{\tau }_{h}u\right)dx\hfill \\ & +& {\int }_{B_t}\left[f\left(x\right)\right]{\tau }_{-h}\left({\eta }^2{\tau }_{h}u\right)dx\hfill \\ & =& I_1+I_2+I_3+I_4+I_5.\hfill \end{array}$$

(25)

Estimate of $I_1$

To estimate $I_1$, we first use property $\left(d1\right)$ and assumption (6). In the same spirit of (15), by the Hölder inequality in Lorentz spaces (Theorem 3) and Young’s inequality with a constant $\nu \in (0,1)$,

$$\begin{array}{ccc}\hfill |I_1|& ⩽& {\int }_{B_t}{\eta }^2\left|h\right|K\left(x\right)\left|Du\right||D{\tau }_{h}u|dx\hfill \\ \\ & ⩽& {\displaystyle \frac{\nu }{2}}{\int }_{B_t}{\eta }^2|D{\tau }_h{u|}^{2}dx+{\displaystyle \frac{1}{2\nu }}{\int }_{B_t}{\left|h\right|}^2{\left|K_2\right|}^2{\left|\eta Du\right|}^{2}dx\hfill \\ \\ & +& {\int }_{B_t}{\eta }^2\parallel K_1{\parallel }_{L^{\infty }}\left|h\right|\left|Du\right||D{\tau }_{h}u|dx\hfill \\ \\ & ⩽& {\displaystyle \frac{\nu }{2}}{\int }_{B_t}{\eta }^2|D{\tau }_h{u|}^{2}dx+{\displaystyle \frac{{\left|h\right|}^2}{2\nu }}\parallel K_2{\parallel }_{L^{n,\infty }\left(B_t\right)}^2{\parallel \eta Du\parallel }_{L^{2^{🟉},2}\left(B_t\right)}^2\hfill \\ \\ & +& {\displaystyle \frac{\nu }{2}}{\int }_{B_t}{\eta }^2|D{\tau }_h{u|}^{2}dx+{\displaystyle \frac{{\left|h\right|}^2}{2\nu }}{\int }_{B_t}{\eta }^2\parallel K_1{\parallel }_{L^{\infty }}^2{\left|Du\right|}^{2}dx.\hfill \end{array}$$

By Theorem 4, we obtain the following:

$$\begin{array}{ccc}\hfill |I_1|& ⩽& \nu {\int }_{B_t}{\eta }^2{\left|D{\tau }_{h}u\right|}^{2}dx\hfill \\ & +& {\displaystyle \frac{{\left|h\right|}^2}{2\nu }}{C}_{2,n}^2\parallel K_2{\parallel }_{L^{n,\infty }}^2{\parallel D\left(\eta Du\right)\parallel }_{L^2\left(B_t\right)}^2\hfill \\ & +& {\displaystyle \frac{{\left|h\right|}^2}{2\nu }}{\int }_{B_t}{\eta }^2\parallel K_1{\parallel }_{L^{\infty }}^2{\left|Du\right|}^{2}dx.\hfill \end{array}$$

(26)

Estimate of $I_2$

$$\begin{array}{ccc}\hfill |I_2|& ⩽& 2{\int }_{B_t-B_s}\left|\eta \right||A(x+h{e}_i){\tau }_{h}u\left|\right|D{\tau }_{h}u\left|\right|D\eta |dx\hfill \\ \\ & ⩽& {\int }_{B_t-B_s}{\eta }^2|D{\tau }_h{u|}^{2}dx+{\int }_{B_t-B_s}{\left|D\eta \right|}^2{\left|A(x+h{e}_i){\tau }_{h}u\right|}^{2}dx.\hfill \end{array}$$

Since $\left|D\eta \right|\le {\displaystyle \frac{1}{t-s}}$, we obtain

$$\left|I_2\right|⩽{\int }_{B_t-B_s}{\eta }^2|D{\tau }_h{\left(u\right)|}^{2}dx+{\displaystyle \frac{1}{{(t-s)}^2}}{\int }_{B_t-B_s}{\left|A(x+h{e}_i){\tau }_{h}u\right|}^{2}dx.$$

(27)

Estimate of $I_3$

By Young’s inequality, assumption (6), Lemma 3 and Hölder inequality (13)

$$\begin{array}{ccc}\hfill |I_3|& ⩽& 2{\int }_{B_t}|\eta ||A(x+h{e}_i)-A\left(x\right)||Du||D\eta ||{\tau }_{h}u|dx\hfill \\ \\ & ⩽& {\int }_{B_t}{|\eta |}^2|A(x+h{e}_i)-A\left(x\right){|}^2{|Du|}^{2}dx+{\int }_{B_t-B_s}{|D\eta |}^2{|{\tau }_{h}u|}^{2}dx\hfill \\ \\ & ⩽& 2{\int }_{B_t}{|h|}^2{|\eta |}^2|K_2{|}^2{|Du|}^{2}dx+2{\int }_{B_t}{|h|}^2{|\eta |}^2\parallel K_1{\parallel }_{L^{\infty }}^2{|Du|}^{2}dx\hfill \\ \\ & +& {\int }_{B_t-B_s}{|D\eta |}^2{|{\tau }_{h}u|}^{2}dx\hfill \\ \\ & ⩽& {2|h|}^2\parallel K_2{\parallel }_{L^{n,\infty }}^2{\parallel \eta Du\parallel }_{L^{2^{🟉},2}}^2+{2|h|}^2\parallel K_1{\parallel }_{L^{\infty }}^2{\int }_{B_t}{|\eta |}^2{|Du|}^{2}dx\hfill \\ \\ & +& {\displaystyle \frac{1}{{(t-s)}^2}}{\int }_{B_t-B_s}{|{\tau }_{h}u|}^{2}dx.\hfill \end{array}$$

Applying Theorem 4, we obtain the following:

$$\begin{array}{ccc}\hfill |I_3|& ⩽& 4C_{n,2}^2{|h|}^2\parallel K_2{\parallel }_{L^{n,\infty }}^2{\parallel \eta D^{2}u\parallel }_{L^2}^2\hfill \\ & +& {2|h|}^2\parallel K_1{\parallel }_{L^{\infty }}^2{\int }_{B_t}{\eta }^2{|Du|}^{2}dx\hfill \\ & +& {\displaystyle \frac{1}{{(t-s)}^2}}{\int }_{B_t-B_s}{|{\tau }_{h}u|}^{2}dx\hfill \\ & +& {4|h|}^2{C}_{n,2}^2{\displaystyle \frac{1}{{(t-s)}^2}}\parallel K_2{\parallel }_{L^{n,\infty }}^2{\int }_{B_t-B_s}{|Du|}^{2}dx.\hfill \end{array}$$

(28)

Estimate of $I_4$

By using Young’s inequality, assumption (2) and Hölder inequality (13)

$$\begin{array}{ccc}\hfill |I_4|& ⩽& {|h|}^2{\int }_{B_t}|B(x,u)||{\Delta }_{-h}\left({\eta }^2{\Delta }_{h}u\right)|dx\hfill \\ & ⩽& {2|h|}^2{\int }_{B_t}b|\eta (x-h{e}_i)u||{\Delta }_{-h}\eta {\Delta }_{h}u+\eta (x-h{e}_i){\Delta }_{-h}{\Delta }_{h}u|dx\hfill \\ & ⩽& {\displaystyle \frac{{|h|}^2}{\nu }}{\int }_{B_t}{b}^2|\eta (x-h{e}_i){u|}^{2}dx+{2\nu |h|}^2{\int }_{B_t}|{\Delta }_{-h}{\eta |}^2{|{\Delta }_{h}u|}^{2}dx\hfill \\ & +& {2\nu |h|}^2{\int }_{B_t}{\eta }^2(x-h{e}_i){|{\Delta }_{-h}{\Delta }_{h}u|}^{2}dx\hfill \\ & ⩽& 2{\displaystyle \frac{{|h|}^2}{\nu }}{\parallel b\parallel }_{L^{n,\infty }}^2\parallel \eta (x-h{e}_i){u\parallel }_{L^{2^{🟉},2}}^2+{2\nu |h|}^2{\int }_{B_t}|{\Delta }_{-h}{\eta |}^2{|{\Delta }_{h}u|}^{2}dx\hfill \\ & +& {2\nu |h|}^2{\int }_{B_t}{\eta }^2(x-h{e}_i){|{\Delta }_{-h}{\Delta }_{h}u|}^{2}dx.\hfill \end{array}$$

(29)

By using Theorem 4 in the first integral in the right-hand side, we obtain

$$\begin{array}{ccc}\hfill |I_4|& ⩽& 2{\displaystyle \frac{{\left|h\right|}^2}{\nu }}{C}_{n,2}^2{\parallel b\parallel }_{L^{n,\infty }}^2{\parallel D\left(\eta u\right)\parallel }_{L^2}^2+{2\nu \left|h\right|}^2{\displaystyle \frac{1}{{(t-s)}^2}}{\int }_{B_t}{\left|{\Delta }_{h}u\right|}^{2}dx\hfill \\ \\ & +& {2\nu \left|h\right|}^2{\int }_{B_t}{\eta }^2(x-h{e}_i){\left|{\Delta }_{-h}{\Delta }_{h}u\right|}^{2}dx\hfill \end{array}$$

and so

$$\begin{array}{ccc}\hfill |I_4|& ⩽& 4{\displaystyle \frac{{\left|h\right|}^2}{\nu }}{C}_{n,2}^2{\parallel b\parallel }_{L^{n,\infty }}{\parallel \eta Du\parallel }_{L^2}^2+4{\displaystyle \frac{{\left|h\right|}^2}{\nu }}{C}_{n,2}^2{\parallel b\parallel }_{L^{n,\infty }}{\parallel uD\eta \parallel }_{L^2}^2\hfill \\ \\ & +& {2\nu \left|h\right|}^2{\displaystyle \frac{1}{{(t-s)}^2}}{\int }_{B_t}{\left|{\Delta }_{h}u\right|}^{2}dx\hfill \\ \\ & +& {2\nu \left|h\right|}^2{\int }_{B_t}{\eta }^2(x-h{e}_i){\left|{\Delta }_{-h}{\Delta }_{h}u\right|}^{2}dx.\hfill \end{array}$$

Estimate of $I_5$

$$\begin{array}{ccc}\hfill |I_5|& ⩽& {|h|}^2{\int }_{B_t}|f||{\Delta }_{-h}\left({\eta }^2{\Delta }_{h}u\right)|dx\hfill \\ & ⩽& {\displaystyle \frac{{\nu |h|}^2}{2}}{\int }_{B_t}|{\Delta }_{-h}\left({\eta }^2{\Delta }_{h}u\right){|}^{2}dx+{\displaystyle \frac{{|h|}^2}{\nu }}{\int }_{B_t}{|f|}^{2}dx.\hfill \end{array}$$

(30)

By arguing as above, we obtain

$$\begin{array}{ccc}\hfill |I_5|& ⩽& {\displaystyle \frac{\nu }{2}}{|h|}^2{\int }_{B_t}{\eta }^2|{\Delta }_{-h}{\Delta }_h{u|}^{2}dx+{\nu |h|}^2{\displaystyle \frac{1}{{(t-s)}^2}}{\int }_{B_t-B_s}{|Du|}^{2}dx\hfill \\ \\ & +& {\displaystyle \frac{{|h|}^2}{\nu }}{\int }_{B_t}{|f|}^{2}dx.\hfill \end{array}$$

By combining (26)–(30), dividing by ${\left|h\right|}^2$, as $h\to 0$, we obtain the following:

$$\begin{array}{ccc}\hfill {\int }_{B_t}{\eta }^2{\left|D^{2}u\right|}^{2}dx& ⩽& {\displaystyle \frac{7}{2}}\nu {\int }_{B_t}{\eta }^2{\left|D^{2}u\right|}^{2}dx\hfill \\ & +& \left({\displaystyle \frac{1}{\nu }}+4\right)C_{n,2}^2\parallel K_2{\parallel }_{L^{n,\infty }}^2{\int }_{B_t}{\left|\eta D^{2}u\right|}^{2}dx\hfill \\ & +& {\int }_{B_t-B_s}{\eta }^2{\left|D^{2}u\right|}^{2}dx\hfill \\ & +& {\displaystyle \frac{1}{{(t-s)}^2}}{\int }_{B_t-B_s}{\left|A\left(x\right)Du\right|}^{2}dx\hfill \\ & +& \left({\displaystyle \frac{1}{\nu }}+4\right)\parallel K_2{\parallel }_{L^{n,\infty }}^2{C}_{n,2}^2{\displaystyle \frac{1}{{(t-s)}^2}}{\int }_{B_t-B_s}{\left|Du\right|}^{2}dx\hfill \\ & +& {\displaystyle \frac{1}{{(t-s)}^2}}{\int }_{B_t-B_s}{\left|Du\right|}^{2}dx\hfill \\ & +& {\displaystyle \frac{4}{\nu }}{C}_{n,2}^2{\parallel b\parallel }_{L^{n,\infty }}{\displaystyle \frac{1}{{(t-s)}^2}}{\int }_{B_t-B_s}{\left|u\right|}^{2}dx\hfill \\ & +& {\displaystyle \frac{2\nu }{{(t-s)}^2}}{\int }_{B_t-B_s}{\left|Du\right|}^{2}dx\hfill \\ & +& \left[\left(2+{\displaystyle \frac{1}{2\nu }}\right)\parallel K_1{\parallel }_{L^{\infty }}^2+{\displaystyle \frac{4}{\nu }}{C}_{n,2}^2{\parallel b\parallel }_{L^{n,\infty }}\right]{\int }_{B_t}{\eta }^2{\left|Du\right|}^{2}dx.\hfill \\ & +& {\displaystyle \frac{1}{\nu }}{\int }_{B_t}{\left|f\right|}^{2}dx.\hfill \end{array}$$

(31)

We can estimate ${\parallel ADu\parallel }_{L^2\left(B_t\right)}$ by using (23). Indeed, from inequality (13) and Theorem 4,

$$\begin{array}{ccc}\hfill {\parallel bu\parallel }_{L^2\left(B_t\right)}^2& ⩽& {\parallel b\parallel }_{L^{n,\infty }\left(B_t\right)}^2{\parallel u\parallel }_{L^{2^{🟉},2}\left(B_t\right)}^2\hfill \\ \\ & ⩽& {c\parallel b\parallel }_{L^{n,\infty }\left(B_t\right)}^2\left({\parallel u\parallel }_{L^2\left(B_t\right)}^2+{\parallel Du\parallel }_{L^2\left(B_t\right)}^2\right).\hfill \end{array}$$

Then, by assumptions (2) and (4),

$$\begin{array}{ccc}\hfill {\parallel ADu\parallel }_{L^2\left(B_t\right)}^2& ⩽& c\left[{\parallel A\parallel }_{🟉}^2{\parallel Du\parallel }_{L^2\left(B_t\right)}^2+{\parallel b\parallel }_{L^{n,\infty }\left(B_t\right)}^2\left({\parallel u\parallel }_{L^2\left(B_t\right)}^2+{\parallel Du\parallel }_{L^2\left(B_t\right)}^2\right)\right]\hfill \\ & +& {c\parallel f\parallel }_{L^2\left(B_t\right)}^2\hfill \end{array}$$

(32)

where $c=c\left(n\right)$.

Here, we first choose the number $\nu$, such that $1-{\displaystyle \frac{7}{2}}\nu >0$, and then $\parallel K_2{\parallel }_{L^{n,\infty }}$ is not large enough so that we are able to reabsorb the integrals on the right-hand side, including those related to $\parallel \eta D^2{u\parallel }_{L^2\left(B_t\right)}$, into the integrals on the left-hand side. Obviously, from the definition of the distance, this is equivalent to requiring that $\parallel K_2{\parallel }_{L^{n,\infty }}<\epsilon$ for a suitable $\epsilon >0$. More precisely, if $\nu$ is such that $\left(1-{\displaystyle \frac{7\nu }{2}}\right)>0$, since $\eta =1$ on $B_s$, on account of (32), for

$$\parallel K_2{\parallel }_{L^{n,\infty }}^2<{\displaystyle \frac{(1-\frac{7}{2}\nu )}{C_{n,2}^2\left(\frac{1}{\nu }+4\right)}}$$

we obtain from (31)

$$\begin{array}{ccc}\hfill C{\int }_{B_s}{\left|D^{2}u\right|}^{2}dx& ⩽& {\int }_{B_t-B_s}{\left|D^{2}u\right|}^{2}dx\hfill \\ \\ & +& {\displaystyle \frac{c}{{(t-s)}^2}}\left({\parallel A\parallel }_{🟉}^2+{\parallel b\parallel }_{L^{n,\infty }}^2+{\parallel K_2\parallel }_{L^{n,\infty }}^2\right)\hfill \\ \\ & ·& {\int }_{B_t}\left({\left|Du\right|}^2+{\left|u\right|}^2+{\left|f\right|}^2\right)dx\hfill \\ \\ & +& c\left(\parallel K_1{\parallel }_{L^{\infty }}^2+{\parallel b\parallel }_{L^{n,\infty }}+1\right)\hfill \\ \\ & ·& {\int }_{B_t}\left({\left|Du\right|}^2+{\left|u\right|}^2+{\left|f\right|}^2\right)dx\hfill \end{array}$$

where $C=\left(1-{\displaystyle \frac{7}{2}}\nu \right)-\left({\displaystyle \frac{1}{\nu }}+4\right)C_{n,2}^2{\parallel K_2\parallel }_{L^{n,\infty }}^2$ and $c=c(n,\nu )$.

Now, we fill the hole:

$$\begin{array}{ccc}\hfill {\int }_{B_s}{\left|D^{2}u\right|}^{2}dx& ⩽& {\displaystyle \frac{1}{C+1}}{\int }_{B_t}{\left|D^{2}u\right|}^{2}dx\hfill \\ \\ & +& {\displaystyle \frac{\mathcal{A}}{{(t-s)}^2}}{\int }_{B_t}\left({\left|Du\right|}^2+{\left|u\right|}^2+{\left|f\right|}^2\right)dx\hfill \\ \\ & +& \mathcal{B}{\int }_{B_t}\left({\left|Du\right|}^2+{\left|u\right|}^2+{\left|f\right|}^2\right)dx\hfill \end{array}$$

where

$$\mathcal{A}={\displaystyle \frac{c}{C+1}}\left({\parallel A\parallel }_{🟉}^2+{\parallel b\parallel }_{L^{n,\infty }}^2+{\parallel K_2\parallel }_{L^{n,\infty }}^2\right)$$

and

$$\mathcal{B}={\displaystyle \frac{c}{C+1}}\left(\parallel K_1{\parallel }_{L^{\infty }}^2+{\parallel b\parallel }_{L^{n,\infty }}+1\right).$$

By applying Lemma 4, we obtain the estimate (18).

• Step 3. The approximation

We first extend the matrices $A\left(x\right)$ and $b\left(x\right)$ to ${\mathbb{R}}^n$. We put zero outside of $\mathsf{\Omega }$. Then, we take $\rho \in C_0^{\infty }\left({\mathbb{R}}^n\right)$ such that $\mathrm{supp}\rho \subset \overline{B_1\left(0\right)}$, $\rho ⩾0$, $\rho \ne 0$ and ∞ and we consider the convolution $A_N=A🟉{\rho }_N$, with ${\rho }_N={\displaystyle \frac{N^n\rho \left(Nx\right)}{\int \rho }}$.

$$A_N\left(x\right)={\int }_{{\mathbb{R}}^n}A\left(y\right){\rho }_N(x-y)dy,x\in \overline{\mathsf{\Omega }}.$$

We notice the following:

• $A_N\in C^{\infty }(\overline{\mathsf{\Omega }},{\mathbb{R}}^{n\times n})\cap L^{\infty }(\mathsf{\Omega },{\mathbb{R}}^{n\times n})$;
• $⟨A_{N}Y,Y⟩⩾{\parallel Y\parallel }^2$;
• $|A_N(x+h{e}_i)-A_N\left(x\right)|⩽K_N\left(x\right)\left|h\right|$, where $K_N\left(x\right)=(K🟉{\rho }_N)\left(x\right)$;
• $K_N\left(x\right)\in L^{n,\infty }$;
• $\parallel A_N{\parallel }_{🟉}⩽{\parallel A\parallel }_{🟉}$;
• $A_N$ converges to A in $L^2$.

For all positive integers N, let $u_N\in W_0^{1,2}(\mathsf{\Omega },{\mathbb{R}}^n)$ be the solution of the Dirichlet problems:

$$\left\{\begin{array}{c}-\mathrm{div}\left(A_{N}D{u}_N\right)+B(x,u_N)=f\mathrm{in}\mathsf{\Omega }\hfill \\ u_N=0\mathrm{on}\partial \mathsf{\Omega }\hfill \end{array}\right.$$

that converge weakly in $W_0^{1,2}$ and strongly in $L^2$ to u (see [11]). A result contained in [24] guarantees that $u_N\in W_{loc}^{2,2}(\mathsf{\Omega },{\mathbb{R}}^n)$. Let us consider as the test function $\phi =u_N$

$${\int }_{\mathsf{\Omega }}⟨A_N\left(x\right)D{u}_N,D{u}_N⟩dx+{\int }_{\mathsf{\Omega }}B(x,u_N)u_{N}dx={\int }_{\mathsf{\Omega }}f{u}_{N}dx$$

since by (3) $⟨B(x,u_N),u_N⟩⩾0$, we obtain

$${\int }_{\mathsf{\Omega }}{\left|D{u}_N\right|}^{2}dx⩽{\int }_{\mathsf{\Omega }}⟨A_N\left(x\right)D{u}_N,D{u}_N⟩dx⩽{\int }_{\mathsf{\Omega }}f{u}_{N}dx$$

$\left\{u_N\right\}$ is bounded in $W_0^{1,2}\left(\mathsf{\Omega }\right)$ and there exists a function $u\in W_0^{1,2}\left(\mathsf{\Omega }\right)$ and a subsequence, still denoted $\left\{u_N\right\}$ such that $u_N$ converge weakly to a function $u\in W_0^{1,2}\left(\mathsf{\Omega }\right)$ and $u_N\to u$ in $L^2\left(\mathsf{\Omega }\right)$.

$$\begin{array}{ccc}\hfill \left|{\int }_{\mathsf{\Omega }}⟨A_N\left(x\right)D{u}_N-A\left(x\right)Du,D\phi ⟩dx\right|& =& \left|{\int }_{\mathsf{\Omega }}(f-B(x,u_N))\phi dx-{\int }_{\mathsf{\Omega }}(f-B(x,u))\phi dx\right|\hfill \\ & ⩽& {\int }_{\mathsf{\Omega }}|B(x,u)-B(x,u_N)\left|\right|\phi |dx\hfill \\ & ⩽& {\parallel \phi \parallel }_{\infty }{\int }_{\mathsf{\Omega }}\left|b\right||u_N-u|dx.\hfill \end{array}$$

(33)

Since $b\in L^{n,\infty }\subset L^2$ and $u_N\to u$ in $L^2$,

$${\int }_{\mathsf{\Omega }}\left|b\right||u_N{-u|dx⩽\parallel b\parallel }_{L^2}{\parallel u-u_N\parallel }_{L^2}\to 0.$$

The last relation in (33) also implies

$${\int }_{\mathsf{\Omega }}B(x,u_N)\phi dx\to {\int }_{\mathsf{\Omega }}B(x,u)\phi dx$$

and then u solves problem (8).

Let $\epsilon >0$ be the number fixed in (18), and let us assume

$$\parallel K_2{\parallel }_{L^{n,\infty }}<\epsilon .$$

We notice that from Lemma A.4 in [25], we have

$$\parallel K_N-K_1{\parallel }_{L^{n,\infty }}⩽{\parallel K_2\parallel }_{L^{n,\infty }}<\epsilon .$$

More precisely, if $K_1\in L^{\infty }\left(\mathsf{\Omega }\right)$ is a function such that $\parallel K_2{\parallel }_{L^{n,\infty }}<\epsilon$, we obtain

$$\parallel K_N-K_1{\parallel }_{L^{n,\infty }}⩽\parallel K-K_1{\parallel }_{L^{n,\infty }}+{\parallel {\left(K_1\right)}_N-K_1\parallel }_{L^{n,\infty }}.$$

Since $K_1\in L^p\left(\mathsf{\Omega }\right)$ for every $p⩾n$, thanks to Theorem 3 the second term on the right-hand side of the previous inequality goes to 0 as $N\to +\infty$. Then, we can assume

$$\parallel K_N-K_1{\parallel }_{L^{n,\infty }}<\epsilon$$

for N sufficiently large.

Now, arguing as in Theorem 5,

$$\begin{array}{ccc}\hfill {\int }_{B_s}{|D^2{u}_N|}^{2}dx& ⩽& {\displaystyle \frac{1}{C+1}}{\int }_{B_t}{|D^2{u}_N|}^{2}dx\hfill \\ \\ & +& {\displaystyle \frac{\mathcal{A}}{{(t-s)}^2}}{\int }_{B_t}\left(|D{u}_N{|}^2+|u_N{|}^2+{|f|}^2\right)dx\hfill \\ \\ & +& \mathcal{B}{\int }_{B_t}\left(|D{u}_N{|}^2+|u_N{|}^2+{|f|}^2\right)dx\hfill \end{array}$$

where

$$\mathcal{A}={\displaystyle \frac{c}{C+1}}\left({\parallel A\parallel }_{🟉}^2+{\parallel b\parallel }_{L^{n,\infty }}^2+{\parallel K_2\parallel }_{L^{n,\infty }}^2\right)$$

and

$$\mathcal{B}={\displaystyle \frac{c}{C+1}}\left(\parallel K_1{\parallel }_{L^{\infty }}^2+{\parallel b\parallel }_{L^{n,\infty }}+1\right).$$

Applying Lemma 4, we obtain

$$\begin{array}{ccc}\hfill {\int }_{B_R}{|D^2{u}_N|}^{2}dx& ⩽& C\left[\left({\displaystyle \frac{1}{R^2}}+1\right)\left({\int }_{B_{2R}}\left(|D{u}_N{|}^2+|u_N{|}^2+{|f|}^2\right)dx\right)\right].\hfill \end{array}$$

We deduce that $|D^2{u}_N|$ is a bounded sequence in $L^2\left(B_R\right)$. Then, by compactness, up to a sequence, $|D^2{u}_N|$ converges weakly to $|D^{2}u|$ in $L^2(B_R,{\mathbb{R}}^n)$ and, by the semicontinuity of the norm with respect to weak convergence, we obtain

$$\begin{array}{ccc}\hfill {\int }_{B_R}{\left|D^{2}u\right|}^{2}dx& ⩽& {\displaystyle \underset{N}{lim\; inf}{\int }_{B_R}{\left|D^2{u}_N\right|}^{2}dx}\hfill \\ \\ & ⩽& c\left[\left({\displaystyle \frac{1}{R^2}}+1\right)\left({\int }_{B_{2R}}\left({\left|Du\right|}^2+{\left|u\right|}^2+{\left|f\right|}^2\right)dx\right)\right]\hfill \end{array}$$

where $c=c(n,\parallel K_2{\parallel }_{L^{n,\infty }},\parallel A{\parallel }_{🟉},\parallel b{\parallel }_{L^{n,\infty }})$. □