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Optimal regular differential operators with variable coefficients and applications
Fixed Point Theory and Applications volume 2013, Article number: 42 (2013)
Abstract
In this paper, maximal regularity properties for linear and nonlinear highorder elliptic differentialoperator equations with VMO coefficients are studied. For the linear case, the uniform coercivity property of parameterdependent 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 highorder nonlinear equation are established. In application, the maximal regularity properties of the anisotropic elliptic equation and the system of equations with VMO coefficients are derived.
AMS Subject Classification:58I10, 58I20, 35Bxx, 35Dxx, 47Hxx, 47Dxx.
1 Introduction
The goal of the present paper is to study the nonlocal boundary value problems (BVPs) for parameterdependent linear differentialoperator equations (DOEs) with discontinuous toporder coefficients
and the nonlinear equation
where a is a complexvalued function, s is a positive and λ is a complex parameter; A=A(x), {A}_{k}={A}_{k}(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 (vanishing mean oscillation). Sarason class VMO was at first defined in [1]. In the recent years, there has been considerable interest to elliptic and parabolic equations with VMO coefficients. This is mainly due to the fact that VMO spaces contain 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., [2–11]). 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 [3], where excellent presentation and relations with 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 (differential equations) and integro DEs can be expressed in the form of DOEs. Many researchers (see, e.g., [12–24]) 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 [12, 14, 18, 19].
Here the equation with toporder VMOoperator coefficients is considered in abstract spaces. We will 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 p, E, m and A such that
Note that the principal part of a corresponding differential operator is non selfadjoint. Nevertheless, the sharp uniform coercive estimate for the resolvent and Fredholmness are established. Then, the existence and uniqueness of the above nonlinear problem are derived. In application, we study maximal regularity properties of anisotropic elliptic equations in mixed {L}^{p} spaces and 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 the 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 semigroup 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 main assertions.
2 Notations and background
Throughout the paper, we set E 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 [25, 26]) is the space of all Evalued local integrable functions with the norm
where B ranges in the class of the balls in {R}^{n}, B is the Lebesgue measure of B 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 \eta (r) the VMO modulus of f.
Note that if E=C, where C is the set of complex numbers, then \mathit{BMO}(E) and \mathit{VMO}(E) coincide with JohnNirenberg class BMO and Sarason class VMO, respectively.
The Banach space E is called a UMDspace if the Hilbert operator
is bounded in {L}_{p}(R,E), p\in (1,\mathrm{\infty}) (see, e.g., [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 φpositive (or positive) in a Banach space E with 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 A+\lambda I will be written as A+\lambda and denoted by {A}_{\lambda}. It is known [[28], §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 [14, 23]) 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 Schwartz 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 a 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
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) is 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 to see that
Let s be a positive parameter. We define in {W}^{m,p}(\mathrm{\Omega};{E}_{0},E) the following parameterized norm:
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 equation (1) a.e. on (0,1) is said to be a solution of the problem (1) on (0,1).
From [21] we have the following theorem.
Theorem A1 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 ntuples of nonnegative integer numbers such that
\varkappa =\frac{\alpha }{m}\le 1\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}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 A2 Suppose all 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 [[2], Theorem 2.1], we have the following result.
Lemma A1 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
Proof Indeed, we observe that if f\in \mathit{VMO}(E) with VMO modulus η, there exists a constant C such that {\parallel f(xy)f(x)\parallel}_{\ast ,E}\le C\eta (r) for \parallel y\parallel \le r so that the Evalued usual mollifiers converge to f in the BMO norm. More precisely, given f\in \mathit{VMO}(E) with VMO modulus \eta (r), we can find a sequence of Evalued {C}^{\mathrm{\infty}} functions \{{f}_{h}\} converging to f in Evalued BMO spaces as h\to 0 with VMO moduli {\eta}_{h} such that {\eta}_{h}\le \eta (r). In a similar way, other cases are derived. □
From [[26], Theorem 1] and [[29], Corollary 2.7], we have the following.
Theorem A3 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:
Theorem A4 Assume all 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 A5 Let E be a UMD space, p\in (1,\mathrm{\infty}) 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 following commutator operator is bounded in {L}^{p}(R;E):
Note that singular integral operators in Evalued {L}^{p} spaces were studied, e.g., in [30].
Theorem A6 Assume all 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 parameterdependent DOE with constant coefficients
where {\nu}_{k}\in \{0,1,\dots ,2m1\}, a, {\alpha}_{ki}, {\beta}_{ki} are complex numbers, {\mu}_{i}=\frac{i}{2m}+\frac{1}{2mp}, {\theta}_{k}=\frac{{\nu}_{k}}{2m}+\frac{1}{2mp}, s is a positive and λ is a complex parameter; {A}_{\lambda}=A+\lambda and A is a linear operator in E. Let {\omega}_{1},{\omega}_{2},\dots ,{\omega}_{2m} be roots of the equation a{\omega}^{2m}+1=0, [{\upsilon}_{ij}] be a 2mdimensional matrix and \eta =[{\upsilon}_{ij}] be a determinant of the matrix [{\upsilon}_{ij}], where
It is known that (see, e.g., [[24], §1.15]) if the operator A is φpositive in E, then operators {\omega}_{k}{s}^{\frac{1}{2m}}{A}_{\lambda}^{\frac{1}{2m}}, k=1,2,\dots ,2m generate the following analytic semigroups:
Let
From [[19], Theorem 1] and [[22], Theorem 3.2], we obtain the following.
Theorem A7 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 \eta \ne 0;

(3)
arg{\omega}_{j}\pi \le \frac{\pi}{2}\phi, j=1,2,\dots ,m, arg{\omega}_{j}\le \frac{\pi}{2}\phi, j=m+1,\dots ,2m and \frac{\lambda}{{\omega}_{j}}\in S(\phi ).
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}^{2m,p}(0,1;E(A),E). Moreover, the following coercive uniform estimate holds:
\sum _{i=0}^{2m}{s}^{\frac{i}{2m}}\lambda {}^{1\frac{i}{2m}}{\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}^{2m}{\parallel {f}_{k}\parallel}_{{E}_{k}}]. 
(2)
For {f}_{k}=0, the solution is represented as
\begin{array}{rcl}u(x)& =& {\int}_{0}^{1}{G}_{\lambda s}(x,y)f(y)\phantom{\rule{0.2em}{0ex}}dy,{G}_{\lambda s}(x,y)\\ =& \sum _{k=1}^{2m}\sum _{j=1}^{2m}\sum _{i=0}^{{\nu}_{k}}[{B}_{kij}(\lambda ){\left({s}^{1}{A}_{\lambda}\right)}^{\frac{1}{2m}(2m+{\nu}_{k}i1)}{U}_{j\lambda s}(x){U}_{k\lambda s}(1y)]+{U}_{0\lambda s}(xy),\end{array}(3)
where {B}_{kij}(\lambda ) are uniformly bounded operators in E and
where
Consider the BVP for DOE with variable coefficients
where a=a(x) is a complexvalued function, {m}_{k}\in \{0,1,\dots ,2m1\}, {\alpha}_{ki}, {\beta}_{ki} are complex numbers, s is a positive and λ is a complex parameter, {\theta}_{k}=\frac{{\nu}_{k}}{2m}+\frac{1}{2mp} and A(x) is a linear operator in E.
Let {\omega}_{1}={\omega}_{1}(x), {\omega}_{2}={\omega}_{2}(x),\dots ,{\omega}_{2m}={\omega}_{2m}(x) be roots of the equation a(x){\omega}^{2m}+1=0, [{\upsilon}_{ij}] be a 2mdimensional matrix and \eta (x)=[{\upsilon}_{ij}] be a determinant of the function matrix [{\upsilon}_{ij}], where
In the next theorem, we consider the case when principal coefficients are continuous. The wellposedness of this problem occurs in studying of equations with VMO coefficients. From [[19], Theorem 3] and [[22], Theorem 3.2], we get the following.
Theorem A8 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)
arg{\omega}_{j}\pi \le \frac{\pi}{2}\phi, j=1,2,\dots ,m, arg{\omega}_{j}\le \frac{\pi}{2}\phi, j=m+1,\dots ,2m and \frac{\lambda}{{\omega}_{j}}\in S(\phi ) a.e. x\in (0,1);

(3)
a\in C[0,1], a(0)=a(1) and \eta (x)\ne 0 for 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),\phantom{\rule{2em}{0ex}}A(0)=A(1).
Then, for 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;

(3)
arg{\omega}_{j}\pi \le \frac{\pi}{2}\phi, j=1,2,\dots ,m, arg{\omega}_{j}\le \frac{\pi}{2}\phi, j=m+1,\dots ,2m and \frac{\lambda}{{\omega}_{j}}\in S(\phi ) for 0\le \phi <\pi, \eta (x)\ne 0 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}^{2m,p}(0,1;E(A),E), we have
where
Here {B}_{kij}^{\mathrm{\prime}}(\lambda ) are uniformly bounded operators and
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}^{(2m)}([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 the above, we get (6) for u\in {C}^{(2m)}([0,1];E(A)). Then a density argument and Theorem A_{3} give 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 following uniform coercive estimate holds:
for u\in {W}^{2m,p}(0,b;E(A),E), \lambda \in S(\phi ) with large enough \lambda .
Proof By Lemma 1, for any solution u\in {W}^{2m,p}(0,b;E(A),E) of the problem (8), we have
where
here {B}_{kij}(\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 s}(x) in a similar way as in [[20], 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}^{2m,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 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 [18, 19]), 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}^{2m,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 to see that
By Theorem A_{1} for u\in {W}^{2m,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}^{2m,p}(0,1;E(A),E) and sufficiently large \lambda  that there is a small ε and C(\epsilon ) such that
In view of (18) for u\in {W}^{2m,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 that the rank space coincides with the all space {L}^{p}(0,1;E). It suffices to prove that there is a solution u\in {W}^{2m,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 [18, 19]. More precisely, by virtue of [[23], Theorem 3.4], UMD spaces satisfy the multiplier condition. Moreover, by 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 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 an analytic semigroup for \phi \in (\frac{\pi}{2},\pi ) (see, e.g., [[29], §1.14.5]).
Remark 1 Conditions a(0)=a(1), A(0)=A(1) arise due to nonlocality of the boundary conditions (4). If 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 the differential operator generated by the problem (1). We will show the separability and Fredholmness of (1).
Theorem 4 Assume the following:

(1)
Condition 1 holds;

(2)
for any \epsilon >0, there is C(\epsilon )>0 such that for a.e. x\in (0,1) and
{\parallel {A}_{k}(x)u\parallel}_{E}\le \epsilon {\parallel u\parallel}_{{(E(A),E)}_{\frac{k}{2m},\mathrm{\infty}}}+C(\epsilon )\parallel u\parallel ,\phantom{\rule{1em}{0ex}}u\in {(E(A),E)}_{\frac{1}{2},\mathrm{\infty}}.
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}^{2m,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}^{2m,p}(0,1;E(A),E). Put {O}_{s}u={Q}_{s}u+{Q}_{1}u, where
By the second assumption and Theorem A_{1}, there is a small ε and C(\epsilon ) such that
By Theorem 3, the operator {Q}_{s}+\lambda has a bounded inverse {({Q}_{s}+\lambda )}^{1} from {L}^{p}(0,1;E) to {W}^{2m,p}(0,1;E(A),E) for sufficiently large \lambda . So, (22) implies the following uniform estimate:
It is clear to see that
Then, by the above relation and by virtue of Theorem 3, we get the assertion. □
Theorem 4 implies the following result.
Result 2 Suppose all 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 conditions of Theorem 4 hold and {A}^{1}\in {\sigma}_{\mathrm{\infty}}(E). Then the problem (23) is Fredholm from {W}^{2m,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}^{2m,p}(0,1;E(A),E) for large enough \lambda ; that is, the operator {O}_{s}+\lambda is Fredholm from {W}^{2m,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 complexvalued function and A=A(x), {A}_{k}={A}_{k}(x) are linear operators in a Banach space E, where b(s) is a positive continuous function independent of u.
Theorem 4 implies the following.
Result 3 Let all conditions of Theorem 4 be satisfied. Then the problem (24) has a unique solution u\in {W}^{2m,p}(0,b;E(A),E) for f\in {L}^{p}(0,b(s);E), p\in (1,\mathrm{\infty}), \lambda \in {S}_{\phi} with large enough \lambda , and the following coercive uniform estimate holds:
Proof Really, under the substitution \tau =xb(s), the moving boundary problem (24) maps to the following BVP with parameter in the fixed domain (0,1):
where
Then, by virtue of Theorem 4, we obtain the assertion. □
Consider the following nonlinear problem:
where {\nu}_{k}\in \{0,1,\dots ,2m1\}, {\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 (25). Assume A is a φpositive operator in a Banach space E. Let
Remark 2 By using [[28], §1.8.], we obtain that the embedding {D}^{j}Y\in {E}_{j} is continuous and there exists a constant {C}_{1} such that for w\in Y, W=\{{w}_{0,}{w}_{1},\dots ,{w}_{2m1}\}, {w}_{j}={D}^{j}w(\cdot ),
Condition 2 Assume the following are satisfied:

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

(2)
a\in \mathit{VMO}\cap {L}^{\mathrm{\infty}}(R), a(0)=a(b);

(3)
arg{\omega}_{j}\pi \le \frac{\pi}{2}\phi, j=1,2,\dots ,m, arg{\omega}_{j}\le \frac{\pi}{2}\phi, j=m+1,\dots ,2m and \frac{\lambda}{{\omega}_{j}}\in S(\phi ) for \lambda \in S(\phi ), 0\le \phi <\pi, \eta (x)\ne 0 a.e. x\in [0,1];

(3)
F(x,{\upsilon}_{0},{\upsilon}_{1},\dots ,{\upsilon}_{2m1}):[0,b]\times {X}_{0}\to E is a measurable function for each {\upsilon}_{i}\in {E}_{i}, i=0,1,\dots ,2m1; 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},\dots ,{u}_{2m1}\} and \overline{U}=\{{\overline{u}}_{0},{\overline{u}}_{1},\dots ,{\overline{u}}_{2m1}\} for a.a. x\in [0,b], {u}_{i},{\overline{u}}_{i}\in {E}_{i} and

(4)
for U\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 {L}_{\mathrm{\infty}}(0,1;L(E))\cap \mathit{VMO}(L(E)), {x}_{0}\in (0,1), where 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},\dots ,{w}_{2m1}\}\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 2 hold. Then there is b\in (0,{b}_{0}] such that the problem (26) has a unique solution belonging to space {W}^{2m,p}(0,b;E(A),E).
Proof
Consider the linear problem
where
By virtue of Result 3, the problem (26) 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 (25) locally by means of maximal regularity of the linear problem (26) 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 (27). 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
Bear 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 [14, 24, 28]; also, for nonsmooth domains, these questions were investigated, e.g., in [31].
Let \mathrm{\Omega}\subset {R}^{n} be an open connected set with compact {C}^{2l}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 complexvalued functions, {\alpha}_{ki} and {\beta}_{ki} are complex numbers,
For 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}^{2m,2l,\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);

(2)
arg{\omega}_{j}\pi \le \frac{\pi}{2}\phi, j=1,2,\dots ,m, arg{\omega}_{j}\le \frac{\pi}{2}\phi, j=m+1,\dots ,2m and \frac{\lambda}{{\omega}_{j}}\in S(\phi ) for \lambda \in S(\phi ), 0\le \phi <\pi, \eta (x)\ne 0, a.e. x\in [0,1];

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

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

(5)
{b}_{j\beta}\in {C}^{2l{l}_{j}}(\partial \mathrm{\Omega}) for each j, β and {m}_{j}<2l, {\sum}_{j=1}^{l}{b}_{j\beta}({y}^{\mathrm{\prime}}){\sigma}_{j}\ne 0 for \beta ={l}_{j}, {y}^{\mathrm{\prime}}\in \partial G, 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 =2l}{a}_{\alpha}(y){\xi}^{\alpha}\ne 0;

(7)
for each {y}_{0}\in \partial \mathrm{\Omega}, the 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}^{\mathrm{\prime}}\in {R}^{n1} with {\xi}^{\mathrm{\prime}}+\nu \ne 0.
Then

(a)
for all f\in {L}^{\mathbf{p}}(G), \lambda \in S(\phi ) and sufficiently large \lambda , the problem (28)(30) has a unique solution u belonging to {W}^{2m,2l,\mathbf{p}}(G) and the following coercive uniform estimate holds:
\sum _{i=0}^{2m}{s}^{\frac{i}{2m}}\lambda {}^{1\frac{i}{2m}}{\parallel \frac{{\partial}^{i}u}{{\partial}^{i}x}\parallel}_{{L}^{\mathbf{p}}(G)}+\sum _{\beta =2l}{\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 (28)(30) is Fredholm in {L}^{\mathbf{p}}(G).
Proof Let E={L}^{{p}_{1}}(\mathrm{\Omega}). Then by virtue of [27], 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 (28)(30) 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 [[14], Theorem 8.2], the problem
has a unique solution for f\in {L}^{{p}_{1}}(\mathrm{\Omega}) and arg\nu \in S(\phi ), \nu \to \mathrm{\infty}, and the operator A is Rpositive in {L}^{{p}_{1}}(\mathrm{\Omega}), i.e., Condition 1 holds. Moreover, it is known that the embedding {W}^{2l,{p}_{1}}(\mathrm{\Omega})\subset {L}^{{p}_{1}}(\mathrm{\Omega}) is compact (see, e.g., [[28], Theorem 3.2.5]). Then, by using interpolation properties of Sobolev spaces (see, e.g., [[28], §4]), it is clear to see that 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 complexvalued functions, N is finite or infinite natural number, {\alpha}_{ki} and {\beta}_{ki} are complex numbers, {\mu}_{i}=\frac{i}{2m}+\frac{1}{2mp}.
Let {a}_{ij}(x) be a real function and
From Theorem 4, we obtain the following.
Theorem 8 Suppose the following conditions are satisfied:

(1)
a\in \mathit{VMO}\cap {L}^{\mathrm{\infty}}(R), a(0)=a(1);

(2)
arg{\omega}_{j}\pi \le \frac{\pi}{2}\phi, j=1,2,\dots ,m, arg{\omega}_{j}\le \frac{\pi}{2}\phi, j=m+1,\dots ,2m and \frac{\lambda}{{\omega}_{j}}\in S(\phi ) for 0\le \phi <\pi, \eta (x)\ne 0 a.e. x\in (0,1);

(3)
{a}_{ij}\ne 0 and {a}_{ij}={a}_{ji}, {a}_{ij}\in \mathit{VMO}\cap {L}^{\mathrm{\infty}}(R), p\in (1,\mathrm{\infty}).
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 , problem (32)(33) has a unique solution u={\{{u}_{m}(x)\}}_{1}^{\mathrm{\infty}} belonging to {W}^{2m,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 matrices such that
It is clear to see that the operator A is Rpositive in {l}_{q}. Therefore, by Theorem 4, the problem (31)(32) has a unique solution u\in {W}^{2m,p}((0,1);{l}_{q}(A),{l}_{q}) for all f\in {L}^{p}((0,1);{l}_{q}), \lambda \in S(\phi ) and the estimate (33) 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 Theorems 4 and 5, we can obtain a different class of maximal regular BVPs for partial differential or pseudodifferential equations or its finite and infinite systems with VMO coefficients.
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Dedicated to Professor Hari M Srivastava.
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Shakhmurov, V. Optimal regular differential operators with variable coefficients and applications. Fixed Point Theory Appl 2013, 42 (2013). https://doi.org/10.1186/16871812201342
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DOI: https://doi.org/10.1186/16871812201342