 Research
 Open Access
 Published:
Linear and nonlinear abstract elliptic equations with VMO coefficients and applications
Fixed Point Theory and Applications volume 2013, Article number: 6 (2013)
Abstract
In this paper, maximal regularity properties for linear and nonlinear elliptic differentialoperator equations with VMO (vanishing mean oscillation) coefficients are studied. For linear case, the uniform separability properties for parameter dependent boundary value problems is obtained in ${L}^{p}$ spaces. Then the existence and uniqueness of a strong solution of the boundary value problem for a nonlinear equation is established. In application, the maximal regularity properties of the anisotropic elliptic equation and the system of equations with VMO coefficients are derived.
MSC: 58I10, 58I20, 35Bxx, 35Dxx, 47Hxx, 47Dxx.
1 Introduction
The goal of the present paper is to study the nonlocal boundary value problems (BVPs) for parameter dependent linear differentialoperator equations (DOEs) with discontinuous toporder coefficients,
and the nonlinear equation
where a is a complex valued function, s is a positive, λ is a complex parameter, and $A=A(x)$, ${A}_{i}={A}_{i}(x)$ are linear and B is a nonlinear operator in a Banach space E. Here, the principal coefficients a and A may be discontinuous. More precisely, we assume that a and $A(\cdot ){A}^{1}({x}_{0})$ belong to the operatorvalued Sarason class VMO. The Sarason class VMO was first defined in [1]. In the recent years, there has been considerable interest in elliptic and parabolic equations with VMO coefficients. This is mainly due to the fact that the VMO class contains as a subspace $C(\overline{\mathrm{\Omega}})$ that ensures the extension of ${L}_{p}$theory of operators with continuous coefficients to discontinuous coefficients (see, e.g., [2–7]). On the other hand, the Sobolev spaces ${W}^{1,n}(\mathrm{\Omega})$ and ${W}^{\sigma ,\frac{\sigma}{n}}(\mathrm{\Omega})$, $0<\sigma <1$, are also contained in VMO. Global regularity of the Dirichlet problem for elliptic equations with VMO coefficients has been studied in [2–4]. We refer to the survey [4], where an excellent presentation and relations with another similar results can be found concerning the regularizing properties of these operators in the framework of Sobolev spaces.
It is known that many classes of PDEs (partial differential equations), pseudo DEs and integro DEs can be expressed in the form of DOEs. Many researchers (see, e.g., [8–24]) investigated similar spaces of functions and classes of PDEs under a single DOE. Moreover, the maximal regularity properties of DOEs with continuous coefficients were studied, e.g., in [10, 19, 20].
Here, the equation with toporder VMOoperator coefficients is considered in abstract spaces. We shall prove the uniform separability of the problem (1), i.e., we show that, for each $f\in {L}^{p}(0,1;E)$, there exists a unique strong solution u of the problem (1) and a positive constant C depending only on p, γ, E and A such that
Note that the principal part of a corresponding differential operator is nonselfadjoint. Nevertheless, the sharp uniform coercive estimate for the resolvent and Fredholmness are established. Then the existence and uniqueness of the above nonlinear problem is derived. In application, we study maximal regularity properties of anisotropic elliptic equations in mixed ${L}^{\mathbf{p}}$ spaces and the systems (finite or infinite) of differential equations with VMO coefficients in the scalar ${L}^{p}$ space.
Since (1) involves unbounded operators, it is not easy to get representation for the Green function and estimate of solutions. Therefore, we use the modern harmonic analysis elements, e.g., the Hilbert operators and the commutator estimates in Evalued ${L}^{p}$ spaces, embedding theorems of SobolevLions spaces and some semigroups estimates to overcome these difficulties. Moreover, we also use our previous results on equations with continuous leading coefficients and the perturbation theory of linear operators to obtain our main assertions.
2 Notations and background
Throughout the paper, we set E to be a Banach space and $\mathrm{\Omega}\subset {R}^{n}$. ${L}^{p}(\mathrm{\Omega};E)$ denotes the space of all strongly measurable Evalued functions that are defined on Ω with the norm
$\mathit{BMO}(E)$ (bounded mean oscillation) (see [21, 25]) is the space of all Evalued local integrable functions with the norm
where B ranges in the class of the balls in ${R}^{n}$ and ${f}_{B}$ is the average $\frac{1}{B}{\int}_{B}f(x)\phantom{\rule{0.2em}{0ex}}dx$.
For $f\in \mathit{BMO}(E)$ and $r>0$, we set
where B ranges in the class of balls with radius ρ.
We will say that a function $f\in \mathit{BMO}(E)$ is in $\mathit{VMO}(E)$ if ${lim}_{r\to +0}\eta (r)=0$. We will call the $\eta (r)$ a VMO modulus of f.
Note that, if $E=\mathbf{C}$, where C is the set of complex numbers, then $\mathit{BMO}(E)$ and $\mathit{VMO}(E)$ coincide with the JohnNirenberg class BMO and Sarason class VMO, respectively.
The Banach space E is called a UMD (unconditional martingale difference)space if the Hilbert operator
is bounded in ${L}_{p}(R,E)$, $p\in (1,\mathrm{\infty})$ (see, e.g., [26, 27]). UMD spaces include, e.g., ${L}_{p}$, ${l}_{p}$ spaces and Lorentz spaces ${L}_{pq}$, $p,q\in (1,\mathrm{\infty})$.
Let
A linear operator A is said to be a φpositive (or positive) in a Banach space E with the bound $M>0$ if $D(A)$ is dense on E and
for $\lambda \in {S}_{\phi}$, $\phi \in (0,\pi ]$, I is an identity operator in E and $L(E)$ is the space of bounded linear operators in E. Sometimes instead of $A+\lambda I$, it will be written as $A+\lambda $ and denoted by ${A}_{\lambda}$. It is known [[22], §1.15.1] that there exist fractional powers ${A}^{\theta}$ of the positive operator A. Let $E({A}^{\theta})$ denote the space $D({A}^{\theta})$ with the graphical norm
Let ${E}_{1}$ and ${E}_{2}$ be two Banach spaces. A set $W\subset L({E}_{1},{E}_{2})$ is called Rbounded (see [24, 26]) if there is a positive constant C such that for all ${T}_{1},{T}_{2},\dots ,{T}_{m}\in W$ and ${u}_{1,}{u}_{2},\dots ,{u}_{m}\in {E}_{1}$, $m\in N$
where $\{{r}_{j}\}$ is a sequence of independent symmetric $\{1,1\}$valued random variables on $[0,1]$.
Let $S({R}^{n};E)$ denote the Schwarz class, i.e., the space of all Evalued rapidly decreasing smooth functions on ${R}^{n}$. Let F denote the Fourier transformation. A function $\mathrm{\Psi}\in {L}^{\mathrm{\infty}}({R}^{n};B({E}_{1},{E}_{2}))$ is called a Fourier multiplier from ${L}_{p}({R}^{n};{E}_{1})$ to ${L}_{p}({R}^{n};{E}_{2})$ if the map $u\to {\mathrm{\Lambda}}_{\mathrm{\Psi}}u={F}^{1}\mathrm{\Psi}(\xi )Fu$, $u\in S({R}^{n};{E}_{1})$ is well defined and extends to a bounded linear operator
The set of all multipliers from ${L}_{p}({R}^{n};{E}_{1})$ to ${L}_{p}({R}^{n};{E}_{2})$ will be denoted by ${M}_{p}^{p}({E}_{1},{E}_{2})$. For ${E}_{1}={E}_{2}=E$, it will be denoted by ${M}_{p}^{p}(E)$.
Let
Definition 1 A Banach space E is said to be a space satisfying the multiplier condition if, for any $\mathrm{\Psi}\in {C}^{(n)}({R}^{n};L(E))$, the Rboundedness of the set $\{{\xi}^{\beta}{D}_{\xi}^{\beta}\mathrm{\Psi}(\xi ):\xi \in {R}^{n}\mathrm{\setminus}0,\beta \in {U}_{n}\}$ implies that Ψ is a Fourier multiplier in ${L}_{p}({R}^{n};E)$, i.e., $\mathrm{\Psi}\in {M}_{p}^{p}(E)$ for any $p\in (1,\mathrm{\infty})$.
Definition 2 The φpositive operator A is said to be an Rpositive in a Banach space E if there exists $\phi \in [0,\pi )$ such that the set ${L}_{A}=\{A{(A+\lambda )}^{1}:\lambda \in {S}_{\phi}\}$ is Rbounded.
A linear operator $A(x)$ is said to be positive in E uniformly in x if $D(A(x))$ is independent of x, $D(A(x))$ is dense in E and
for all $\lambda \in S(\phi )$, $\phi \in [0,\pi )$.
Let ${\sigma}_{\mathrm{\infty}}({E}_{1},{E}_{2})$ denote the space of all compact operators from ${E}_{1}$ to ${E}_{2}$. For ${E}_{1}={E}_{2}=E$, it is denoted by ${\sigma}_{\mathrm{\infty}}(E)$.
Assume ${E}_{0}$ and E are two Banach spaces and ${E}_{0}$ is continuously and densely embedded into E. Let m be a natural number. ${W}^{m,p}(\mathrm{\Omega};{E}_{0},E)$ (the socalled SobolevLions type space) denotes a space of all functions $u\in {L}^{p}(\mathrm{\Omega};{E}_{0})$ possessing the generalized derivatives ${D}_{k}^{m}u=\frac{{\partial}^{m}u}{\partial {x}_{k}^{m}}$ such that ${D}_{k}^{m}u\in {L}^{p}(\mathrm{\Omega};E)$ endowed with the norm
For ${E}_{0}=E$, the space ${W}^{m,p}(\mathrm{\Omega};{E}_{0},E)$ will be denoted by ${W}^{m,p}(\mathrm{\Omega};E)$. It is clear that
Let s be a positive parameter. We define in ${W}^{m,p}(\mathrm{\Omega};{E}_{0},E)$ the following parameterdependent norm:
The space ${B}_{p,p}^{s}(\mathrm{\Gamma};{E}_{0},E)$ is defined as a trace space for ${W}^{m,p}(\mathrm{\Omega};{E}_{0},E)$ as in a scalar case (see, e.g., [[22], §3.6.1]).
Function $u\in {W}^{2,p}(0,1;E(A),E,{L}_{k})=\{u\in {W}^{2,p}(0,1;E(A),E),{L}_{k}u=0\}$ satisfying the equation (1) a.e. on $(0,1)$ is said to be a solution of the problem (1) on $(0,1)$.
From [28] we have
Theorem A_{ 1 } Suppose the following conditions are satisfied:

(1)
E is a Banach space satisfying the multiplier condition with respect to $p\in (1,\mathrm{\infty})$ and A is an Rpositive operator in E;

(2)
$\alpha =({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n})$ are n tuples of nonnegative integer numbers such that $\varkappa =\frac{\alpha }{m}\le 1$ and $0<\mu \le 1\varkappa $;

(3)
$\mathrm{\Omega}\in {R}^{n}$ is a region such that there exists a bounded linear extension operator from ${W}^{m,p}(\mathrm{\Omega};E(A),E)$ to ${W}^{m,p}({R}^{n};E(A),E)$.
Then the embedding
is continuous and there exists a positive constant ${C}_{\mu}$ such that
for all $u\in {W}^{m,p}(\mathrm{\Omega};E(A),E)$ and $0<h\le {h}_{0}<\mathrm{\infty}$.
Theorem A_{ 2 } Suppose all the conditions of Theorem A_{1} are satisfied. Assume Ω is a bounded region in ${R}^{n}$ and ${A}^{1}\in {\sigma}_{\mathrm{\infty}}(E)$. Then, for $0<\mu \le 1\varkappa $, the embedding
is compact.
In a similar way as in [[3], Theorem 2.1], we have the following result.
Lemma A_{ 1 } Let E be a Banach space and $f\in \mathit{VMO}(E)$. The following conditions are equivalent:

(1)
$f\in \mathit{VMO}(E)$;

(2)
f is in the BMO closure of the set of uniformly continuous functions which belong to VMO;

(3)
${lim}_{y\to 0}{\parallel f(xy)f(x)\parallel}_{\ast ,E}=0$.
For $f\in {L}^{p}(\mathrm{\Omega};E)$, $p\in (1,\mathrm{\infty})$, $a\in {L}^{\mathrm{\infty}}({R}^{n})$, consider the commutator operator
From [[21], Theorem 1] and [[29], Corollary 2.7], we have
Theorem A_{ 3 } Let E be a UMD space and $a\in \mathit{VMO}\cap {L}^{\mathrm{\infty}}({R}^{n})$. Then $H[a,f]$ is a bounded operator in ${L}^{p}(R;E)$, $p\in (1,\mathrm{\infty})$.
From Theorem A_{3} and the property (2) of Lemma A_{1} we obtain, respectively, the following.
Theorem A_{ 4 } Assume all the conditions of Theorem A_{3} are satisfied. Also, let $a\in \mathit{VMO}\cap {L}^{\mathrm{\infty}}({R}^{n})$ and let η be the VMO modulus of a. Then, for any $\epsilon >0$, there exists a positive number $\delta =\delta (\epsilon ,\eta )$ such that
Theorem A_{ 5 } Let E be a UMD space and $A(\cdot )$ uniformly Rpositive in E. Moreover, let $A(\cdot ){A}^{1}({x}_{0})\in {L}_{\mathrm{\infty}}(R;L(E))\cap \mathit{BMO}(L(E))$, ${x}_{0}\in R$. Then the commutator operator
is bounded in ${L}^{p}(R;E)$, $p\in (1,\mathrm{\infty})$.
Theorem A_{ 6 } Assume all the conditions of Theorem A_{5} are satisfied and η is a VMO modulus of $A(\cdot ){A}^{1}({x}_{0})$.
Then, for any $\epsilon >0$, there exists a positive number $\delta =\delta (\epsilon ,\eta )$ such that
Consider the nonlocal BVP for a parameter dependent DOE with constant coefficients
where ${m}_{k}\in \{0,1\}$, a, ${\alpha}_{ki}$, ${\beta}_{ki}$ are complex numbers, s is a positive parameter, λ is a complex spectral parameter, ${\mu}_{i}=\frac{i}{2}+\frac{1}{2p}$, ${\theta}_{k}=\frac{{m}_{k}}{2}+\frac{1}{2p}$, ${A}_{\lambda}=A+\lambda $ and A is a linear operator in E. Let ${\omega}_{1}$, ${\omega}_{2}$ be roots of the equation $a{\omega}^{2}+1=0$ and let
It is known that if the operator A is φpositive in E, then operators ${\omega}_{k}{A}_{\lambda}^{\frac{1}{2}}$, $k=1,2$ generate analytic semigroups
Let
From [[19], Theorem 2] and [[30], Theorem 3.1], we obtain
Theorem A_{ 7 } Assume the following conditions are satisfied:

(1)
E is a Banach space satisfying the multiplier condition with respect to $p\in (1,\mathrm{\infty})$;

(2)
A is an Rpositive operator in E for $0\le \phi <\pi $ and $\mu \ne 0$;

(3)
$Re{\omega}_{k}\ne 0$ and $\frac{\lambda}{{\omega}_{k}}\in S(\phi )$ for $\lambda \in S(\phi )$, $k=1,2$.
Then

(1)
for $f\in {L}_{p}(0,1;E)$, ${f}_{k}\in {E}_{k}$, $\lambda \in S(\phi )$ and for sufficiently large $\lambda $, the problem (2) has a unique solution $u\in {W}^{2,p}(0,1;E(A),E)$. Moreover, the coercive uniform estimate holds
$$\sum _{i=0}^{2}{s}^{\frac{i}{2}}\lambda {}^{1\frac{i}{2}}{\parallel {u}^{(i)}\parallel}_{{L}^{p}(0,1;E)}+{\parallel Au\parallel}_{{L}^{p}(0,1;E)}\le C[{\parallel f\parallel}_{{L}^{p}(0,1;E)}+\sum _{k=1}^{2}{\parallel {f}_{k}\parallel}_{{E}_{k}}].$$ 
(2)
For ${f}_{k}=0$, the solution is represented as
(3)
where ${B}_{ij}(\lambda )$ and ${D}_{ij}(\lambda )$ are uniformly bounded operators in E and
Consider the BVP for a DOE with variable coefficients
where $a=a(x)$ is a complex valued function, s is a positive parameter, ${m}_{k}\in \{0,1\}$, ${\alpha}_{ki}$, ${\beta}_{ki}$ are complex numbers, λ is a spectral parameter, ${\theta}_{k}=\frac{{m}_{k}}{2}+\frac{1}{2p}$ and $A(x)$ is a linear operator in E.
Let ${\omega}_{1}={\omega}_{1}(x)$, ${\omega}_{2}={\omega}_{2}(x)$ be roots of the equation $a(x){\omega}^{2}+1=0$ and let
In the next theorem, let us consider the case that principal coefficients are continuous. The wellposedness of this problem occurs in the studying of equations with VMO coefficients. From [[19], Theorem 3] and [[21], Theorem 5.1], we get
Theorem A_{ 8 } Suppose the following conditions are satisfied:

(1)
E is a Banach space satisfying the multiplier condition with respect to $p\in (1,\mathrm{\infty})$;

(2)
$a\in C[0,1]$, $a\in S(\phi )$, $a(0)=a(1)$, and $\mu (x)\ne 0$ for a.e. $x\in [0,1]$;

(3)
$Re{\omega}_{k}(x)\ne 0$ and $\frac{\lambda}{{\omega}_{k}}\in S(\phi )$ for $\lambda \in S(\phi )$, $k=1,2$. a.e. $x\in [0,1]$;

(4)
$A(x)$ is a uniformly Rpositive operator in E and
$$A(\cdot ){A}^{1}({x}_{0})\in C([0,1];L(E)),\phantom{\rule{1em}{0ex}}{x}_{0}\in (0,1),A(0)=A(1).$$
Then, for all $f\in {L}^{p}(0,1;E)$, $\lambda \in S(\phi )$ and for sufficiently large $\lambda $, there is a unique solution $u\in {W}^{2,p}(0,1;E(A),E)$ of the problem (4). Moreover, the following coercive uniform estimate holds:
3 DOEs with VMO coefficients
Consider the principal part of the problem (1)
Condition 1 Assume the following conditions are satisfied:

(1)
E is a UMD space, $p\in (1,\mathrm{\infty})$;

(2)
$a\in \mathit{VMO}\cap {L}^{\mathrm{\infty}}(R)$, ${\eta}_{1}$ is a VMO modulus of a, $a\in S(\phi )$, $\mu (x)\ne 0$;

(3)
$Re{\omega}_{k}(x)\ne 0$ and $\frac{\lambda}{{\omega}_{k}}\in S(\phi )$ for $\lambda \in S(\phi )$, $0\le \phi <\pi $, $k=1,2$. a.e. $x\in [0,1]$;

(4)
$A(x)$ is a uniformly Rpositive operator in E and
$$A(\cdot ){A}^{1}({x}_{0})\in {L}_{\mathrm{\infty}}(0,1;L(E))\cap \mathit{VMO}(L(E)),\phantom{\rule{1em}{0ex}}{x}_{0}\in (0,1);$$ 
(5)
$a(0)=a(1)$, $A(0)=A(1)$ and ${\eta}_{2}$ is a VMO modulus of $A(\cdot ){A}^{1}({x}_{0})$.
First, we obtain an integral representation formula for solutions.
Lemma 1 Let Condition 1 hold and $f\in {L}^{p}(0,1;E)$. Then, for all solutions u of the problem (5) belonging to ${W}^{2,p}(0,1;E(A),E)$, we have
where
here ${B}_{ij}^{\mathrm{\prime}}(\lambda )$, ${D}_{ij}^{\mathrm{\prime}}(\lambda )$ are uniformly bounded operators,
and the expression ${\mathrm{\Gamma}}_{2\lambda}^{\mathrm{\prime}}(x,xy)$ is a scalar multiple of ${\mathrm{\Gamma}}_{2\lambda}(x,xy)$.
Proof Consider the problem (5) for $a(x)=a({x}_{0})$ and $A(x)=A({x}_{0})$, i.e.,
Let $u\in {C}^{(2)}([0,1];E(A))$ be a solution of the problem (7). Taking into account the equality ${L}_{0}u=({L}_{0}L)u+Lu$ and Theorem A_{7}, we get
Setting $x={x}_{0}$ in above, we get (6) for $u\in {C}^{(2)}([0,1];E(A))$. Then the density argument, using Theorem A_{3}, gives the conclusion for
Consider the problem (5) on $(0,b)$, i.e.,
□
Theorem 1 Suppose Condition 1 is satisfied. Then there exists a number $b\in (0,1)$ such that the uniform coercive estimate
holds for large enough $\lambda $ and $u\in {W}^{2,p}(0,b;E(A),E)$, ${L}_{bk}u=0$, $\lambda \in S(\phi )$, where C is a positive constant depending only on p, ${M}_{0}$, ${\eta}_{1}$, ${\eta}_{2}$.
Proof By Lemma 1, for any solution $u\in {W}^{2,p}(0,b;E(A),E)$ of the problem (8), we have
where
here ${B}_{ij}^{\mathrm{\prime}}(\lambda )$, ${D}_{ij}^{\mathrm{\prime}}(\lambda )$ are uniformly bounded operators,
and
Moreover, from (10) and (11), clearly, we get
where the expression ${\mathrm{\Gamma}}_{b\lambda}^{\mathrm{\prime}}(x,xy)$ differs from ${\mathrm{\Gamma}}_{2b\lambda}(x,xy)$ only by a constant.
Consider the operators
Since the operators ${B}_{0\lambda}$ and ${B}_{1\lambda}$ are regular on ${L}^{p}(0,b;E)$, by using the positivity properties of A and the analyticity of semigroups ${U}_{k\lambda}(x)$ in a similar way as in [[30], Theorem 3.1], we get
Since the Hilbert operator is bounded in ${L}^{p}(R;E)$ for a UMD space E, we have
Thus, by virtue of Theorems A_{4}, A_{6} and in view of (10)(12) for any $\epsilon >0$, there exists a positive number $b=b(\epsilon ,{\eta}_{1},{\eta}_{2})$ such that
Hence, the estimates (13)(15) imply (9). □
Theorem 2 Assume Condition 1 holds. Let $u\in {W}^{2,p}(0,1;E(A),E)$ be a solution of (4). Then, for sufficiently large $\lambda $, $\lambda \in S(\phi )$, the following coercive uniform estimate holds:
Proof This fact is shown by a covering and flattening argument, in a similar way as in Theorem A_{8}. Particularly, by partition of unity, the problem is localized. Choosing diameters of supports for corresponding finite functions, by using Theorem 1, Theorems A_{4}, A_{6}, A_{7} and embedding Theorem A_{1} (see the same technique for DOEs with continuous coefficients [19, 20]), we obtain the assertion.
Let ${Q}_{s}$ denote the operator in ${L}^{p}(0,1;E)$ generated by the problem (4) for $\lambda =0$, i.e.,
□
Theorem 3 Assume Condition 1 holds. Then, for all $f\in {L}^{p}(0,1;E)$, $\lambda \in S(\phi )$ and for large enough $\lambda $, the problem (5) has a unique solution $u\in {W}^{2,p}(0,1;E(A),E)$. Moreover, the following coercive uniform estimate holds:
Proof First, let us show that the operator $Q+\lambda $ has a left inverse. Really, it is clear that
By Theorem A_{1} for $u\in {W}^{2,p}(0,1;E(A),E)$, we have
Then, by virtue of (16) and in view of the above relations, we infer, for all $u\in {W}^{2,p}(0,1;E(A),E)$ and sufficiently large $\lambda $, that there is a small ε and $C(\epsilon )$ such that
From the estimate (18) for $u\in {W}^{2,p}(0,1;E(A),E)$, we get
The estimate (19) implies that (4) has a unique solution and the operator ${Q}_{s}+\lambda $ has a bounded inverse in its rank space. We need to show the rank space coincides with all the space ${L}^{p}(0,1;E)$. It suffices to prove that there is a solution $u\in {W}^{2,p}(0,1;E(A),E)$ for all $f\in {L}^{p}(0,1;E)$. This fact can be derived in a standard way, approximating the equation with a similar one with smooth coefficients [19, 20]. More precisely, by virtue of [[23], Theorem 3.4], UMD spaces satisfy the multiplier condition. Moreover, by the part (2) of Lemma A_{1}, given $a\in \mathit{VMO}$ with VMO modules $\eta (r)$, we can find a sequence of mollifiers functions $\{{a}_{h}\}$ converging to a in BMO as $h\to 0$ with a VMO modulus ${\eta}_{h}$ such that ${\eta}_{h}(r)\le \eta (r)$. In a similar way, it can be derived for the operator function $A(x){A}^{1}({x}_{0})\in \mathit{VMO}(L(E))$. □
Result 1 Theorem 3 implies that the resolvent ${({Q}_{s}+\lambda )}^{1}$ satisfies the following sharp uniform estimate:
for $arg\lambda \le \phi $, $\phi \in (0,\pi )$ and $s>0$.
The estimate (20) particularly implies that the operator Q is uniformly positive in ${L}^{p}(0,1;E)$ and generates analytic semigroups for $\phi \in (\frac{\pi}{2},\pi )$ (see, e.g., [[22], §1.14.5]).
Remark 1 Conditions $a(0)=a(1)$, $A(0)=A(1)$ arise due to nonlocality of the boundary conditions (4). If the boundary conditions are local, then the conditions mentioned above are not required any more.
Consider the problem (1), where ${L}_{k}u$ is the same boundary condition as in (4). Let ${O}_{s}$ denote a differential operator generated by the problem (1). We will show the separability and Fredholmness of (1).
Theorem 4 Assume

(1)
Condition 1 holds;

(2)
for any $\epsilon >0$, there is $C(\epsilon )>0$ such that for a.e. $x\in (0,1)$ and for $0<{\nu}_{0}<1$, $0<{\nu}_{1}<\frac{1}{2}$,
Then, for all $f\in {L}^{p}(0,1;E)$ and for large enough $\lambda $, $\lambda \in S(\phi )$, there is a unique solution $u\in {W}^{2,p}(0,1;E(A),E)$ of the problem (1) and the following coercive uniform estimate holds:
Proof It is sufficient to show that the operator ${O}_{s}+\lambda $ has a bounded inverse ${({O}_{s}+\lambda )}^{1}$ from ${L}^{p}(0,1;E)$ to ${W}^{2,p}(0,1;E(A),E)$. Put ${O}_{s}u={Q}_{s}u+{Q}_{0}u$, where
By the second assumption and Theorem A_{1}, there is a small ε and $C(\epsilon )$ such that
In view of estimates (17) and (22), we have
for $u\in {W}^{2,p}(0,1;E(A),E)$ and ${\delta}_{k}<1$. By Theorem 3, the operator ${Q}_{s}+\lambda $ has a bounded inverse ${({Q}_{s}+\lambda )}^{1}$ from ${L}^{p}(0,1;E)$ to ${W}^{2,p}(0,1;E(A),E)$ for sufficiently large $\lambda $. So, (23) implies the following uniform estimate:
It is clear that
Then, by above relation and by virtue of Theorem 3, we get the assertion. □
Theorem 4 implies the following result.
Result 2 Suppose all the conditions of Theorem 4 are satisfied. Then the resolvent ${({O}_{s}+\lambda )}^{1}$ of the operator ${O}_{s}$ satisfies the following sharp uniform estimate:
for $arg\lambda \le \phi $, $\phi \in [0,\pi )$ and $s>0$.
Consider the problem (1) for $\lambda =0$, i.e.,
Theorem 5 Assume all the conditions of Theorem 4 hold and ${A}^{1}\in {\sigma}_{\mathrm{\infty}}(E)$. Then the problem (24) is Fredholm from ${W}^{2,p}(0,1;E(A),E)$ into ${L}^{p}(0,1;E)$.
Proof Theorem 4 implies that the operator ${O}_{s}+\lambda $ has a bounded inverse ${({O}_{s}+\lambda )}^{1}$ from ${L}^{p}(0,1;E)$ to ${W}^{2,p}(0,1;E(A),E)$ for large enough $\lambda $; that is, the operator ${O}_{s}+\lambda $ is Fredholm from ${W}^{2,p}(0,1;E(A),E)$ into ${L}^{p}(0,1;E)$. Then, by virtue of Theorem A_{2} and by perturbation theory of linear operators, we obtain the assertion. □
4 Nonlinear DOEs with VMO coefficients
Let, at first, consider the linear BVP in a moving domain $(0,b(s))$,
where a is a complex valued function, and $A=A(x)$, ${A}_{i}={A}_{i}(x)$ are possible operators in a Banach space E, where $b(s)$ is a positive continuous independent of u.
Theorem 4 implies the following result.
Result 3 Let all the conditions of Theorem 4 be satisfied. Then the problem (25), for $f\in {L}^{p}(0,b(s);E)$, $p\in (1,\mathrm{\infty})$, $\lambda \in {S}_{\phi}$ and for large enough $\lambda $, has a unique solution $u\in {W}^{2,p}(0,b;E(A),E)$ and the following coercive uniform estimate holds:
Proof Really, under the substitution $\tau =xb(s)$, the moving boundary problem (25) maps to the following BVP with a parameter in a fixed domain $(0,1)$:
where
Then, by virtue of Theorem 4, we obtain the assertion. □
Consider the nonlinear problem
where ${m}_{k}\in \{0,1\}$, ${\alpha}_{ki}$, ${\beta}_{ki}$ are complex numbers, $x\in (0,b)$, where b is a positive number in $(0,{b}_{0}]$.
In this section, we will prove the existence and uniqueness of a maximal regular solution of the nonlinear problem (26). Assume A is a φpositive operator in a Banach space E. Let
Remark 2 By using [[22], §1.8], we obtain that the embedding ${D}^{j}Y\in {E}_{j}$ is continuous and there exists the constant ${C}_{1}$ such that for $w\in Y$, $W=\{{w}_{0,}{w}_{1}\}$, ${w}_{j}={D}^{j}w(\cdot )$, $j=0,1$,
Condition 2 Assume the following are satisfied:

(1)
$\eta ={(1)}^{{m}_{1}}{\alpha}_{1}{\beta}_{2}{(1)}^{{m}_{2}}{\alpha}_{2}{\beta}_{1}\ne 0$ and $a(x)$ is a positive continuous function on $[0,b]$, $a(0)=a(b)$;

(2)
E is a UMD space and $p\in (1,\mathrm{\infty})$;

(3)
$F(x,{\upsilon}_{0},{\upsilon}_{1}):[0,b]\times {X}_{0}\to E$ is a measurable function for each ${\upsilon}_{i}\in {E}_{i}$, $i=0,1$; $F(x,\cdot ,\cdot )$ is continuous with respect to $x\in [0,b]$ and $f(x)=F(x,0)\in X$. Moreover, for each $R>0$, there exists ${\mu}_{R}$ such that
$${\parallel F(x,U)F(x,\overline{U})\parallel}_{E}\le {\mu}_{R}{\parallel U\overline{U}\parallel}_{{X}_{0}},$$where $U=\{{u}_{0},{u}_{1}\}$ and $\overline{U}=\{{\overline{u}}_{0},{\overline{u}}_{1}\}$ for a.a. $x\in [0,b]$, ${u}_{i},{\overline{u}}_{i}\in {E}_{i}$ and
$${\parallel U\parallel}_{{X}_{0}}\le R,\phantom{\rule{2em}{0ex}}{\parallel \overline{U}\parallel}_{{X}_{0}}\le R.$$ 
(4)
for $U=\{{u}_{0},{u}_{1}\}\in {X}_{0}$, the operator $B(x,U)$ is Rpositive in E uniformly with respect to $x\in [0,b]$; $B(x,U){B}^{1}({x}^{0},U)\in C([0,b];B(E))$, where the domain definition $D(B(x,U))$ does not depend on x and U; $B(x,W):(0,b)\times {X}_{0}\to B(E(A),E)$ is continuous, where $A=A(x)=B(x,W)$ for fixed $W=\{{w}_{0},{w}_{1}\}\in {X}_{0}$;

(5)
for each $R>0$, there is a positive constant $L(R)$ such that ${\parallel [B(x,U)B(x,\overline{U})]\upsilon \parallel}_{E}\le L(R){\parallel U\overline{U}\parallel}_{{X}_{0}}{\parallel A\upsilon \parallel}_{E}$ for $x\in (0,b)$, $U,\overline{U}\in {X}_{0}$, ${\parallel U\parallel}_{{X}_{0}},{\parallel \overline{U}\parallel}_{{X}_{0}}\le R$ and $\upsilon \in D(A)$ and $A(0)=A(b)$.
Theorem 6 Let Condition 1 hold. Then there is $b\in (0,{b}_{0}]$ such that the problem (26) has a unique solution that belongs to the space ${W}_{p}^{2}(0,b;E(A),E)$.
Proof Consider the linear problem
where
By virtue of Result 3, the problem (27) has a unique solution for all $f\in X$ and for sufficiently large $d>0$ that satisfies the following
where the constant C does not depend on $f\in X$ and $b\in (0,{b}_{0}]$. We want to solve the problem (26) locally by means of maximal regularity of the linear problem (27) via the contraction mapping theorem. For this purpose, let w be a solution of the linear BVP (27). Consider a ball
For $\upsilon \in {B}_{r}$, consider the linear problem
where
Define a map Q on ${B}_{r}$ by $Q\upsilon =u$, where u is a solution of the problem (28). We want to show that $Q({B}_{r})\subset {B}_{r}$ and that Q is a contraction operator provided b is sufficiently small and r is chosen properly. For this aim, by using maximal regularity properties of the problem (28), we have
By assumption (5), we have
where
Bearing in mind
where $R={C}_{1}r+{\parallel w\parallel}_{Y}$ is a fixed number. In view of the above estimates, by a suitable choice of ${\mu}_{R}$, ${L}_{R}$ and for sufficiently small $b\in [0;{b}_{0})$, we have
i.e.,
Moreover, in a similar way, we obtain
By a suitable choice of ${\mu}_{R}$, ${L}_{R}$ and for sufficiently small $b\in (0,{b}_{0})$, we obtain ${\parallel Q\upsilon Q\overline{\upsilon}\parallel}_{Y}<\eta {\parallel \upsilon \overline{\upsilon}\parallel}_{Y}$, $\eta <1$, i.e., Q is a contraction operator. Eventually, the contraction mapping principle implies a unique fixed point of Q in ${B}_{r}$ which is the unique strong solution $u\in Y$. □
5 Boundary value problems for anisotropic elliptic equations with VMO coefficients
The Fredholm property of BVPs for elliptic equations with parameters in smooth domains were studied, e.g., in [8, 10], also, for nonsmooth domains, these questions were investigated, e.g., in [31].
Let $\mathrm{\Omega}\subset {R}^{n}$ be an open connected set with a compact ${C}^{2m}$boundary ∂ Ω. Let us consider the nonlocal boundary value problems on a cylindrical domain $G=(0,1)\times \mathrm{\Omega}$ for the following anisotropic elliptic equation with VMO toporder coefficients:
where s is a positive parameter, a, ${d}_{i}$ are complex valued functions, ${\alpha}_{ki}$ and ${\beta}_{ki}$ are complex numbers,
If $G=(0,1)\times \mathrm{\Omega}$, $\mathbf{p}=({p}_{1},p)$, ${L}^{\mathbf{p}}(G)$ will denote the space of all psummable scalarvalued functions with a mixed norm (see, e.g., [[32], §1]), i.e., the space of all measurable functions f defined on G, for which
Analogously, ${W}^{2,2m,\mathbf{p}}(G)$ denotes the anisotropic Sobolev space with the corresponding mixed norm [[32], §10].
Theorem 7 Let the following conditions be satisfied;

(1)
$a,{d}_{0}\in \mathit{VMO}\cap {L}^{\mathrm{\infty}}(R)$, $a(0)=a(1)$, $a\in S(\phi )$, $\mu (x)\ne 0$, a.e. $x\in [0,1]$;

(2)
$Re{\omega}_{k}\ne 0$ and $\frac{\lambda}{{\omega}_{k}}\in S(\phi )$ for $\lambda \in S(\phi )$, $0\le \phi <\pi $, $k=1,2$ a.e. $x\in [0,1]$;

(3)
${d}_{1}\in {L}^{\mathrm{\infty}}$, ${d}_{1}(\cdot ,y){d}_{0}^{\frac{1}{2}\nu}(\cdot )\in {L}^{\mathrm{\infty}}(0,1)$ for a.e. $y\in \mathrm{\Omega}$ and $0<\nu <\frac{1}{2}$;

(4)
${a}_{\alpha}\in \mathit{VMO}\cap {L}^{\mathrm{\infty}}({R}^{n})$ for each $\alpha =2m$ and ${a}_{\alpha}\in [{L}^{\mathrm{\infty}}+{L}^{{\gamma}_{k}}](\mathrm{\Omega})$ for each $\alpha =k<2m$ with ${r}_{k}\ge q$ and $2mk>\frac{l}{{r}_{k}}$;

(5)
${b}_{j\beta}\in {C}^{2m{m}_{j}}(\partial \mathrm{\Omega})$ for each j, β and ${m}_{j}<2m$, , for $\beta ={m}_{j}$, , where $\sigma =({\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{n})\in {R}^{n}$ is a normal to ∂G;

(6)
for $y\in \overline{\mathrm{\Omega}}$, $\xi \in {R}^{n}$, $\nu \in S(\phi )$, $\phi \in (0,\pi )$, $\xi +\nu \ne 0$ let $\nu +{\sum}_{\alpha =2m}{a}_{\alpha}(y){\xi}^{\alpha}\ne 0$;

(7)
for each ${y}_{0}\in \partial \mathrm{\Omega}$, a local BVP in local coordinates corresponding to ${y}_{0}$
has a unique solution $\vartheta \in {C}_{0}({R}_{+})$ for all $h=({h}_{1},{h}_{2},\dots ,{h}_{n})\in {R}^{n}$, and for ${\xi}^{\mid}\in {R}^{n1}$ with ${\xi}^{\mid}+\nu \ne 0$.
Then

(a)
for all $f\in {L}^{\mathbf{p}}(G)$, $\lambda \in S(\phi )$ and sufficiently large $\lambda $, the problem (29)(31) has a unique solution u belonging to ${W}^{2,2m,\mathbf{p}}(G)$ and the following coercive uniform estimate holds:
$$\sum _{i=0}^{2}{s}^{\frac{i}{2}}\lambda {}^{1\frac{i}{2}}{\parallel \frac{{\partial}^{i}u}{{\partial}^{i}x}\parallel}_{{L}^{\mathbf{p}}(G)}+\sum _{\beta =2m}{\parallel {D}_{y}^{\beta}u\parallel}_{{L}^{\mathbf{p}}(G)}\le C{\parallel f\parallel}_{{L}^{\mathbf{p}}(G)};$$ 
(b)
for $\lambda =0$, the problem (29)(31) is Fredholm in ${L}^{\mathbf{p}}(G)$.
Proof Let $E={L}^{{p}_{1}}(\mathrm{\Omega})$. Then by virtue of [26], the part (1) of Condition 1 is satisfied. Consider the operator A acting in ${L}^{{p}_{1}}(\mathrm{\Omega})$ defined by
For $x\in \mathrm{\Omega}$, also consider operators in ${L}^{{p}_{1}}(\mathrm{\Omega})$
The problem (29)(31) can be rewritten in the form (1), where $u(x)=u(x,\cdot )$, $f(x)=f(x,\cdot )$ are functions with values in $E={L}^{{p}_{1}}(\mathrm{\Omega})$. By virtue of [13], the problem
has a unique solution for $f\in {L}^{{p}_{1}}(\mathrm{\Omega})$ and arg $\nu \in S(\phi )$, $\nu \to \mathrm{\infty}$. Moreover, in view of [[10], Theorem 8.2], the operator A is Rpositive in ${L}^{{p}_{1}}(\mathrm{\Omega})$, i.e., Condition 1 holds. Moreover, it is known that the embedding ${W}^{2m,{p}_{1}}(\mathrm{\Omega})\subset {L}^{{p}_{1}}(\mathrm{\Omega})$ is compact (see, e.g., [[22], Theorem 3.2.5]). Then, by using interpolation properties of Sobolev spaces (see, e.g., [[22], §4]), it is clear that the condition (2) of Theorem 4 is fulfilled too. Then from Theorems 4, 5, the assertions are obtained. □
6 Systems of differential equations with VMO coefficients
Consider the nonlocal BVPs for infinity systems of parameterdifferential equations with principal VMO coefficients,
where s is a positive parameter, a, ${b}_{mj}$, ${d}_{mj}$ are complex valued functions, N is a finite or infinite natural number, ${\alpha}_{ki}$ and ${\beta}_{ki}$ are complex numbers, ${\mu}_{i}=\frac{i}{2}+\frac{1}{2p}$.
Let
Let Q denote the operator in ${L}^{p}(0,1;{l}_{q})$ generated by the problem (32)(33). Let
Theorem 8 Suppose the following conditions are satisfied:

(1)
$a\in \mathit{VMO}\cap {L}^{\mathrm{\infty}}(R)$, $a(0)=a(1)$, $a\in S(\phi )$, $\mu (x)\ne 0$ a.e. $x\in (0,1)$;

(2)
$Re{\omega}_{k}(x)\ne 0$ and $\frac{\lambda}{{\omega}_{k}}\in S(\phi )$ for $\lambda \in S(\phi )$, a.e. $x\in (0,1)$, $0\le \phi <\pi $, $k=1,2$;

(3)
${d}_{j}\in \mathit{VMO}\cap {L}^{\mathrm{\infty}}(R)$, ${b}_{mj},{d}_{mj}\in {L}^{\mathrm{\infty}}(0,1)$, $p\in (1,\mathrm{\infty})$;

(4)
there are $0<{\nu}_{0}<1$, $0<{\nu}_{1}<\frac{1}{2}$ such that
Then, for all $f(x)={\{{f}_{m}(x)\}}_{1}^{N}\in {L}^{p}(0,1;{l}_{q})$, $\lambda \in S(\phi )$ and for sufficiently large $\lambda $, the problem (32)(33) has a unique solution $u={\{{u}_{m}(x)\}}_{1}^{\mathrm{\infty}}$ belonging to ${W}^{2,p}((0,1),{l}_{q}(D),{l}_{q})$ and the following coercive estimate holds:
Proof Really, let $E={l}_{q}$, A and ${A}_{k}(x)$ be infinite matrices such that
It is clear that the operator A is Rpositive in ${l}_{q}$. Therefore, by Theorem 4, the problem (32)(33) has a unique solution $u\in {W}^{2,p}((0,1);{l}_{q}(D),{l}_{q})$ for all $f\in {L}^{p}((0,1);{l}_{q})$, $\lambda \in S(\phi )$ the estimate (34) holds. □
Remark 3 There are many positive operators in different concrete Banach spaces. Therefore, putting concrete Banach spaces and concrete positive operators (i.e., pseudodifferential operators or finite or infinite matrices for instance) instead of E and A, respectively, by virtue of Theorem 4, 5, we can obtain a different class of maximal regular BVPs for partial differential or pseudodifferential equations or their finite and infinite systems with VMO coefficients.
References
 1.
Sarason D: On functions of vanishing mean oscillation. Trans. Am. Math. Soc. 1975, 207: 391–405.
 2.
Chiarenza F, Frasca M, Longo P: Interior ${W}^{2,p}$ estimates for non divergence elliptic equations with discontinuous coefficients. Ric. Mat. 1991, 40: 149–168.
 3.
Chiarenza F, Frasca M, Longo P: ${W}^{2,p}$solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Am. Math. Soc. 1993, 336(2):841–853.
 4.
Chiarenza F: ${L}^{p}$ regularity for systems of PDEs with coefficients in VMO. 5. In Nonlinear Analysis, Function Spaces and Applications. Prometheus, Prague; 1994.
 5.
Miranda C: Partial Differential Equations of Elliptic Type. Springer, Berlin; 1970.
 6.
Maugeri A, Palagachev DK, Softova L: Elliptic and Parabolic Equations with Discontinuous Coefficients. Wiley, Berlin; 2000.
 7.
Krylov NV: Parabolic and elliptic equations with VMO coefficients. Commun. Partial Differ. Equ. 2007, 32(3):453–475. 10.1080/03605300600781626
 8.
Amann H 1. In Linear and QuasiLinear Problems. Birkhäuser, Boston; 1995.
 9.
Ashyralyev A: On wellposedeness of the nonlocal boundary value problem for elliptic equations. Numer. Funct. Anal. Optim. 2003, 24(1–2):1–15. 10.1081/NFA120020240
 10.
Denk R, Hieber M, Prüss J: R boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 2003., 166: Article ID 788
 11.
Favini A, Shakhmurov V, Yakubov Y: Regular boundary value problems for complete second order elliptic differentialoperator equations in UMD Banach spaces. Semigroup Forum 2009, 79(1):22–54. 10.1007/s0023300991380
 12.
Gorbachuk VI, Gorbachuk ML: Boundary Value Problems for DifferentialOperator Equations. Naukova Dumka, Kiev; 1984.
 13.
Heck H, Hieber M: Maximal ${L}^{p}$ regularity for elliptic operators with VMO coefficients. J. Evol. Equ. 2003, 3: 62–88.
 14.
Lions JL, Peetre J: Sur une classe d’espaces d’interpolation. Publ. Math. 1964, 19: 5–68.
 15.
Ragusa MA: Necessary and sufficient condition for VMO function. Appl. Math. Comput. 2012, 218(24):11952–11958. 10.1016/j.amc.2012.06.005
 16.
Ragusa MA: Embeddings for MorreyLorentz spaces. J. Optim. Theory Appl. 2012, 154(2):491–499. 10.1007/s109570120012y
 17.
Sobolevskii PE: Inequalities coerciveness for abstract parabolic equations. Dokl. Akad. Nauk SSSR 1964, 57(1):27–40.
 18.
Shakhmurov VB: Coercive boundary value problems for regular degenerate differentialoperator equations. J. Math. Anal. Appl. 2004, 292(2):605–620. 10.1016/j.jmaa.2003.12.032
 19.
Shakhmurov VB: Separable anisotropic differential operators and applications. J. Math. Anal. Appl. 2007, 327(2):1182–1201. 10.1016/j.jmaa.2006.05.007
 20.
Shakhmurov VB: Degenerate differential operators with parameters. Abstr. Appl. Anal. 2007, 2006: 1–27.
 21.
Segovia C, Torrea JL: Vectorvalued commutators and applications. Indiana Univ. Math. J. 1989, 38(4):959–971. 10.1512/iumj.1989.38.38044
 22.
Triebel H: Interpolation Theory. Function Spaces. Differential Operators. NorthHolland, Amsterdam; 1978.
 23.
Weis L: Operatorvalued Fourier multiplier theorems and maximal ${L}_{p}$ regularity. Math. Ann. 2001, 319: 735–758. 10.1007/PL00004457
 24.
Yakubov S, Yakubov Ya: DifferentialOperator Equations. Ordinary and Partial Differential Equations. Chapman & Hall/CRC, Boca Raton; 2000.
 25.
John F, Nirenberg L: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 1961, 14: 415–476. 10.1002/cpa.3160140317
 26.
Burkholder DL: A geometric condition that implies the existence of certain singular integrals of Banachspacevalued functions. In Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II. Wadsworth, Belmont; 1983:270–286. Chicago, Ill., 1981
 27.
Hytönen T, Weis L: A T 1 theorem for integral transformations with operatorvalued kernel. J. Reine Angew. Math. 2006, 599: 155–200.
 28.
Shakhmurov VB: Embedding theorems and maximal regular differential operator equations in Banachvalued function spaces. J. Inequal. Appl. 2005, 4: 605–620.
 29.
Blasco O: Operator valued BMO commutators. Publ. Mat. 2009, 53(1):231–244.
 30.
Shakhmurov VB, Shahmurova A: Nonlinear abstract boundary value problems atmospheric dispersion of pollutants. Nonlinear Anal., Real World Appl. 2010, 11: 932–951. 10.1016/j.nonrwa.2009.01.037
 31.
Grisvard P: Elliptic Problems in Nonsmooth Domains. Pitman, London; 1985.
 32.
Besov OV, Ilin VP, Nikolskii SM: Integral Representations of Functions and Embedding Theorems. Nauka, Moscow; 1975.
Author information
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Shakhmurov, V.B. Linear and nonlinear abstract elliptic equations with VMO coefficients and applications. Fixed Point Theory Appl 2013, 6 (2013). https://doi.org/10.1186/1687181220136
Received:
Accepted:
Published:
Keywords
 differential equations with VMO coefficients
 boundary value problems
 differentialoperator equations
 maximal ${L}^{p}$ regularity
 abstract function spaces
 nonlinear elliptic equations