Free boundary value problems for abstract elliptic equations and applications

TL;DR

This paper investigates free boundary value problems for elliptic differential-operator equations with variable coefficients, establishing uniform maximal regularity and Fredholm properties in vector-valued Hölder spaces.

Contribution

It introduces new results on the regularity and Fredholmness of free boundary problems for elliptic equations with variable coefficients.

Findings

Established uniform maximal regularity in vector-valued Hölder spaces

Proved Fredholmness of the free boundary value problem

Extended theory to variable coefficient elliptic equations

Abstract

Free bondary value problem for elliptic differential-operator equations with variable coefficients is studied. The uniform maximal regularity properties and Fredholmness of this problem are obtained in vector-valued Holder spaces.

Equations

$${\Omega }_f=\left\{\left(x,y\right)\in \mathbb{R}\times {\mathbb{R}}_{+},f\in C_b\left(\mathbb{R};E\right),0<y<\Vert f\left(x\right)\Vert \right\}.$$

$${\Gamma }_0=\mathbb{R}\times 0,{\Gamma }_f=\left\{\left(x,y\right)\in \mathbb{R}\times {\mathbb{R}}_{+},y=\Vert f\left(x\right)\Vert \right\}.$$

$$f\in C_b^{\left(1\right)}\left([0,T;C_b^{\left(1\right)}\left(\mathbb{R};E\right)\right),$$

$$u\left(t,.\right)\in W_p^2\left({\Omega }_{f\left(t\right)};E\right),t\in J=[0,T).$$

$$-\Delta u\left(t,z\right)+A\left(x\right)u\left(t,z\right)=0\text{, }t\in J,z\in {\Omega }_{f\left(t\right)}$$

$$\frac{\partial u}{\partial y}=0,t\in J,z\in {\Omega }_{f\left(t\right)},$$

$$u\left(t,z\right)=f\left(t,x\right)\text{, }t\in J,z\in {\Gamma }_{f\left(t\right)},$$

$$\underset{z\to \infty }{\lim }u\left(t,z\right)=\nu \left(t\right),t\in [0,T),$$

$$f_t\left(t,x\right)+\sqrt{1+f_x^2\left(t,x\right)}\frac{\partial }{\partial n}u\left(t,z\right)=0,t\in \left(0,T\right),z\in {\Gamma }_{f\left(t\right)},$$

$$f\left(0,x\right)=f_0\left(x\right),x\in \mathbb{R},$$

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

$$L_{1}u=u{\mid }_{{}_{\Gamma }}=0,$$

$$\sum _{i,j=1}^n{\Vert \frac{{\partial }^{2}u}{\partial x_i\partial x_j}\Vert }_{C^{\gamma }\left(G;E\right)}+{\Vert Au\Vert }_{C^{\gamma }\left(G;E\right)}\le C{\Vert F\Vert }_{C^{\gamma }\left(G;E\right)}.$$

$$D\left(G\right)=C_0^{2,\gamma }\left(G;E\left(A\right),E\right)=\{u\in C^{2,\gamma }\left(G;E\right)\cap C\left(G;E\left(A\right)\right),$$

$$u{\mid }_{{}_{\Gamma }}=0\},Gu=Lu.$$

$$S\left(\phi \right)=\left\{\lambda \in \mathbb{C}\text{, }\left|\arg \lambda \right|\le \phi \right\}\cup \left\{0\right\},0\le \phi <\pi .$$

$${\Vert {\left(A+\lambda I\right)}^{-1}\Vert }_{B\left(E\right)}\le M{\left(1+\left|\lambda \right|\right)}^{-1}$$

$${\Vert u\Vert }_{E\left(A^{\theta }\right)}={\left({\Vert u\Vert }^p+{\Vert A^{\theta }u\Vert }^p\right)}^{\frac{1}{p}},1\le p<\infty ,-\infty <\theta <\infty .$$

$$\Vert {\left(A\left(x\right)+\lambda I\right)}^{-1}\Vert \le M{\left(1+\left|\lambda \right|\right)}^{-1}$$

$${\Vert f\Vert }_{C^{\gamma }\left(\Omega ;E\right)}={\Vert f\Vert }_{C^{\gamma }\left(\Omega ;E\right)}+{\left[f\right]}^{\gamma }\left(E\right),$$

$${\left[f\right]}^{\gamma }\left(E\right)=\underset{x\ne y,x,y\in \Omega }{\sup }\frac{{\Vert f\left(x\right)-f\left(y\right)\Vert }_E}{{\left|x-y\right|}^{\gamma }}.$$

$$C^{\gamma ,m}\left(\Omega ;E\right)=\left\{f\in C^m\left(\Omega ;E\right),f^{\left(m\right)}\in C^{\gamma }\left(\Omega ;E\right)\right\},$$

$${\Vert f\Vert }_{C^{\gamma ,m}\left(\Omega ;E\right)}={\Vert f\Vert }_{C^m\left(\Omega ;E\right)}+{\Vert f^{\left(m\right)}\Vert }_{C^{\gamma }\left(\Omega ;E\right)}<\infty .$$

$$D^{\alpha }=\frac{{\partial }^{|\alpha |}}{\partial x_1^{{\alpha }_1}\partial x_2^{{\alpha }_2}...\partial x_n^{{\alpha }_n}}.$$

$${\Vert f\Vert }_{C^{m,\gamma }\left(\Omega ;E_0,E\right)}={\Vert f\Vert }_{C^{m,\gamma }\left(\Omega ;E\right)}+{\Vert f\Vert }_{C^{\gamma }\left(\Omega ;E_0\right)}.$$

$$D_A\left(\theta ,p\right)=\left\{u\in E,{\Vert u\Vert }_{D_A\left(\theta ,p\right)}={\Vert t^{1-\theta -\frac{1}{p}}AU\left(t\right)u\Vert }_{L^p\left(0,1;E\right)}<\infty \right\}$$

$$\text{for }1\le p<\infty \text{ and }\theta \in \left(0,1\right);$$

$$D_A\left(\theta ,\infty \right)=\left\{u\in E,{\Vert u\Vert }_{D_A\left(\theta ,p\right)}={\Vert t^{1-\theta }AU\left(t\right)u\Vert }_{L^{\infty }\left(0,1;E\right)}<\infty \right\}$$

$$\text{for }p=\infty \text{ and }0<\theta \le 1;$$

$$D_A\left(\theta \right)=\left\{u\in D_A\left(\theta ,\infty \right),\underset{t\to 0}{\lim }{t}^{1-\theta }AU\left(t\right)u=0\right\}.$$

Full text

**Free boundary value problems for abstract elliptic equations and applications **

Veli B. Shakhmurov

Department of Mechanical engineering, Okan University, Akfirat, Tuzla 34959 Istanbul, Turkey,

E-mail: veli.sahmurov@okan.edu.tr

Abstract

Free bondary value problem for elliptic differential-operator equations with variable coefficients is studied. The uniform maximal regularity properties and Fredholmness of this problem are obtained in vector-valued Holder spaces.

**MSC 2010: 35xx, 47Fxx, 47Hxx, 35Pxx **

**Key words: Free **boundary value problems, Differential-operator equations, Banach-valued function spaces, Operator-valued multipliers, Interpolation of Banach spaces, Semigroup of operators.

**1. Introduction, notations and background **

In last years, the maximal regularity properties of boundary value problems (BVPs) for differential-operator equations (DOEs) have found many applications in PDE, psedo DE and in the different physical process (see for references $\left[1-4\right]$, $\left[6\right],$ $\left[8\right]$, $\left[10\right]$, $\left[12-23\right]$, $\left[27-28\right]$ ).

Let $\Omega$ be a domin in $R^{n}$ and $E$ is a Banach space. $C_{b}^{\left(m\right)}\left(\Omega;E\right)$ will denote the spaces of $E$-valued bounded uniformly stongly continuous and $m$-times continuously differentiable functions on $\Omega$. For $m=0$ it denotes by $C_{b}\left(\Omega;E\right).$ Let $\mathbb{C}$ denote the set of complex numbers. For $E=\mathbb{C}$ the space $C^{\left(m\right)}\left(\Omega;E\right)$ will be denoted by $C_{b}^{\left(m\right)}\left(\Omega\right).$ Moreover, $C_{b}^{\infty}\left(\Omega;E\right)$ denotes spaces of $E$-valued bounded strongly continiously differentiable functions of arbitrary order. We put $\mathbb{R=}\left(-\infty,\infty\right)$ and $\mathbb{R}_{+}=\left(0,\infty\right).$ Let $f\left(x\right)$ is a $E-$valued function and $f\left(x\right)\neq 0$. Consider

[TABLE]

The boundaries of $\Omega_{f}$ are given by

[TABLE]

Consider the following  problem: Given $f_{0},$ $\nu\in C_{b}^{\left(2\right)}\left(\mathbb{R};E\right).$ Find a pair of functions $(u,f)$ possessing the regularity

[TABLE]

[TABLE]

and satisfying the following equations a.e.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $A$ is a linear operator in a Banach space $E$ and $z=\left(x,y\right)$ represents a generic point in $\bar{\Omega}_{f}$. Moreover, $\Delta$ denotes the Laplace operator with respect to the Euclidean metric, $\frac{\partial}{\partial n}$ denotes the derivative in direction of the outer unit normal $n$ at $\Gamma_{f\left(t\right)}$.

Maximal regularity properties of partial DOEs in $L_{p}$ spaces were studied in $\left[1\right]$, $\left[4\right]$, $\left[7\right]$, $\left[18-23\right].$ The results in $\left[4\right]$ and $\left[18-23\right]$ were restricted to rectangular domain and equations that were not contained mixed derivatives in leading part. Moreover, problems investigated in $\left[1\right]$ and $\left[8\right]$ involve only bounded operator coefficients. In $\left[18\right]$ the Dirichlet problem for the elliptic differential-operator equation of the second order in general domain was studied.

In contrast to all above we study general BVP $\left(1.1\right)$ for equation with unbounded operator coefficients in the general domain.

Consider the BVP

[TABLE]

[TABLE]

where $\Gamma$ is a boundary of region $G\subset R^{n}$ and $a_{ij}$ are real-valued functions on $\bar{G}.$

We say that the problem $\left(1.3\right)$ is maximal $H$-regular (or separable in Holder space $C^{\gamma}$) if:

(1) for all $F\in C^{\gamma}\left(G;E\right)$ there exists a unique solution $u\in$ $C^{2,\gamma}\left(G;E\left(A\right),E\right)$ satisfying $\left(1.3\right)$ a.e. on $G;$

(2) there exists a positive constant $C$ independent of $F$ such that

[TABLE]

Let $G$ denote the operator generated by the problem $\left(1.3\right)$ for $\lambda=0$, i.e.,

[TABLE]

[TABLE]

The paper is organized as follows: Section1 collects definitions and background materials, embedding theorems of Sobolev-Lions spaces.

Let $\mathbb{C}$ be the set of complex numbers and

[TABLE]

Let $E_{1}$ and $E_{2}$ be two Banach spaces. $L\left(E_{1},E_{2}\right)$ denotes the space of all bounded linear operators from $E_{1}$ to $E_{2}.$ For $E_{1}$ $=E_{2}=E$ it will be doneted by $L\left(E\right).$

A linear operator $A$ is said to be $\varphi-$positive in a Banach space $E$ with bound $M>0$ if $D\left(A\right)$ is dense on $E$ and

[TABLE]

with $\lambda\in S\left(\varphi\right),$ $\varphi\in\left[0,\right.\left.\pi\right)$, where $I$ is an identity operator in $E$.

Sometimes instead of $A+\lambda I$ will be written $A+\lambda$ and will denoted by $A_{\lambda}.$ It is known that ($\left[\text{25, \S 1.15.1}\right])$ there exist fractional powers $A^{\theta}$ of positive operator $A.$ Let $E\left(A^{\theta}\right)$ denote the space $D\left(A^{\theta}\right)$ with graphical norm

[TABLE]

A linear operator $A\left(x\right)$ is said to be positive in $E$ uniformly in $x$ if $D\left(A\left(x\right)\right)$ is independent of$\ x$, $D\left(A\left(x\right)\right)$ is dense in $E$ and

[TABLE]

for all $\lambda\in S\left(\varphi\right)$ and $\varphi\in\left[0,\right.\left.\pi\right)$.

Let $\Omega$ be a domain in $R^{n}$. $C(\Omega,E)$ and $C^{m}(\Omega;E)$ will denote the spaces of $E$-valued bounded uniformly strongly continuous and $m$-times continuously differentiable functions on $\Omega,$ respectively.

Let $0<\gamma\leq 1.$ $C^{\gamma}\left(\Omega;E\right)$ denotes the space of $E$-valued strongly bounded continiuous functions that are defined on $\Omega\subset R^{n}$ with the norm

[TABLE]

where

[TABLE]

$C^{\gamma,m}\left(\Omega;E\right)$ denotes the space of $E$-valued strongly bounded continiuous functions that are defined on $\Omega\subset R^{n}$ with the norm

[TABLE]

[TABLE]

Let $E_{0}$ and $E$ be two Banach spaces and $E_{0}$ is continuously and densely embedded into $E.$ Let $m$ be a natural number.

Let $\alpha=\left(\alpha_{1},\alpha_{2},...,\alpha_{n}\right)$ are $n$ tuples of nonnegative integer numbers and

[TABLE]

$C^{m,\gamma}\left(\Omega;E_{0},E\right)$ denote the space of $E_{0}$-valued bounded uniformly stongly continuous and $m$-times continuously differentiable functions on $\Omega$ with norm

[TABLE]

For $E_{0}=$ $E$ the space $C^{m,\gamma}\left(\Omega;E_{0},E\right)$ will denoted by $C^{m,\gamma}\left(\Omega;E\right).$

Let $A$ be a linear operator in a Banach space $E$ so that is a generator of analytic semigroup $U\left(t\right)=U_{A}\left(t\right).$ Let

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From $\left[\text{15, Proposition 2.2.2}\right]$ we obtain the following

[TABLE]

[TABLE]

[TABLE]

$H\left(E_{0},E\right)$ denotes the class of linear operators that are isomorphisim from $E_{0}$ onto $E$ and are negative generators of stronge continious and analytic semigroups.

Let $S^{{}^{\prime}}\left(R^{n};E\right)$ denote the space of all continuous linear operators $L:S\left(R^{n};E\right)\rightarrow E,$ equipped with the bounded convergence topology. Recall $S\left(R^{n};E\right)$ is norm dense in $L^{p}\left(R^{n};E\right)$ when $1\leq p<\infty.$

Let $L_{q}^{\ast}\left(E\right)$ denote the space of all $E-$valued functions $u\left(t\right)$ such that

[TABLE]

Let $F$ denote the Fourier transform. Fourier-analytic definition of $E-$valued Besov space on $R^{n}$ are defined as in $\left[\text{25 \S\ 3}\right]$, i.e.,

[TABLE]

[TABLE]

[TABLE]

For appropriate domain $\Omega\subset R^{n}$ the space $B_{p,q}^{s}\left(\Omega;E\right)$ is defined as usulal restriction of the space $B_{p,q}^{s}\left(R^{n};E\right).$

For $E=\mathbb{C}$ the space $B_{p,q}^{s}\left(\Omega;\mathbb{C}\right)$ will be denoted by $B_{p,q}^{s}\left(\Omega\right)$.

Let $h^{s}=h^{s}\left(R^{n};E\right)$ denote the closure of $S\left(R^{n};E\right)$ in $B_{\infty,\infty}^{s}\left(R^{n};E\right)$. Assume that $\Omega$ is an open subset of $R^{n}$ and let $r_{\Omega}$ denote the restriction operator with respect to $\Omega$, i.e., $r_{\Omega}u=u\mid_{\Omega}$for $B_{\infty,\infty}^{s}\left(R^{n};E\right).$ Here, $h^{s}\left(\Omega;E\right)$ is defined as the closure of $r_{\Omega}\left(S\left(R^{n};E\right)\right)$ in $B_{\infty,\infty}^{\gamma}\left(\Omega;E\right)$ and $h^{m,\gamma}\left(\Omega;E\right)$ is defined as the closure of $r_{\Omega}\left(S\left(\Omega;E\right)\right)$ in $C^{m,\gamma}\left(\Omega;E\right).$ For $E=\mathbb{C}$ the spaces $h^{s}\left(\Omega;E\right),$ $h^{m,\gamma}\left(\Omega;E\right)$ will be denoted by $h^{s}$ and $h^{m,\gamma}$, respectively. Moreover, let $C_{0}^{s}\left(\Omega;E\right)$ denote the closure of $C^{\infty}\left(\Omega;E\right)$ in $C^{s}\left(\Omega;E\right).$

Let

[TABLE]

[TABLE]

[TABLE]

with the norm

[TABLE]

Here,

[TABLE]

and

[TABLE]

**Remark 1.1. **In order to formulate our result, let

[TABLE]

and $f\in h_{\nu}^{2,\alpha}$ given. Let $u_{f}$ denote the unique solution of the BVP

[TABLE]

where $A$ is a linear operator in a Banach space $E.$

[TABLE]

and define

[TABLE]

It is clear to see that $u_{\nu}\equiv\nu.$ Hence, $\nu\in V_{\nu}.$ More precisely, by following $\left[\text{9, Lemma 5.10}\right]$ it can be shown that $V_{\nu}$ is a open neighborhood of $\nu$ in $h_{\nu}^{s,2}\left(E\right)$ and that

[TABLE]

Suppose now that$(u,f)$ is a classical solution of $\left(1.1\right)-\left(1.2\right)$. We call $(u,f)$ a classical Hölder solution on $J$ if it possesses the additional regularity

[TABLE]

We will prove the following main result

**Theorem 1. **Given $f_{0}\in V_{\nu}$, there exist $t^{+}=t^{+}\left(f_{0}\right)$ and a unique maximal classical Hölder solution$(u,f)$ of problem $\left(1.1\right)-\left(1.2\right)$ on $[0,t^{+})$. Moreover, the mapping $(t,f_{0})\rightarrow f$ defines a local $C^{\infty}$-semiflow on $V_{\nu}$. If $t^{+}<\infty$ and $f:$ $\left[\left.0,t^{+}\right)\right.\rightarrow V_{\nu}$ is uniformly continuous then either

[TABLE]

In the first stage, we transform problem $\left(1.1\right)-\left(1.2\right)$ into a nonlinear problem on a fixed domain

[TABLE]

with respect to only the unknown function $f$, which determines the free boundary $\Gamma_{f},$ where $O$ is a nonlinear operator in $E.$

Then, by using the solution of the above problem we will show the exsistence of regular solution of the free BVP $\left(1.1\right)-\left(1.2\right).$

2. Transformed problem

Let $\nu=\nu\left(t\right)>0$ be fixed. Define

[TABLE]

where $I$ is an identity element in the Banach space $E.$

Consider the following transformation

[TABLE]

It is easily verified that $\varphi_{g}$ is a diffeomorphism of class $C^{2}$ which maps $\Omega_{g}$ onto the strip $Q=\mathbb{R}\times\left(0,1\right)$. Moreover,

[TABLE]

[TABLE]

Let $u$ be an $E$-valued function defined on $Q$. Here, $u_{\mid_{\Gamma_{i}}}$denote the restriction of $u$ on $\Gamma_{i}$, where

[TABLE]

**Lemma 2.1. **Given $g\in\Phi_{\eta}$ and $\upsilon\in C^{2,\gamma}\left(Q;E\left(A\right),E\right);$ under the the maps $\left(2.2\right)$ the operators in $\left(1.1\right)$ are transformed into the following:

[TABLE]

[TABLE]

where $n_{0}=\left(-g_{x},1\right)$, $n_{1}=\left(0,-1\right)$ denote the outer normals according to $\Gamma_{f}$ and $\Gamma_{0}$, i.e.,

[TABLE]

[TABLE]

**Lemma 2.2. **Given $g\in\Phi_{\nu}$ and $\upsilon\in C^{2,\gamma}\left(Q;E\left(A\right),E\right).$ Under the map $\left(2.2\right)$ the problem $\left(1.1\right)$ is transformed into the following:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

A pair $\left(\upsilon,g\right)$ is called a solution of the problem $\left(2.5\right)$ if

[TABLE]

[TABLE]

and $\left(\upsilon,g\right)$ satisfies $\left(2.5\right)$ a.e. on $\left[0,\right.\left.T\right)\times Q.$

**Condition 2.1. **Assume the following conditions are satisfied:

(1) $\mathop{\displaystyle\sum}\limits_{i,j=1}^{2}a_{ij}\xi_{i}\xi_{j}\geq C\left|\xi\right|^{2}$, for $\xi=\left(\xi_{1},\xi_{2},...,\xi_{n}\right)\in R^{n}$ and $C>0;$

(2) operator $A$ is a positive operator in a Banach space $E$ for some $\varphi\in\left(0\right.,\left.\pi\right].$

In a similar way as in $\left[\text{9, Lemma 2.2}\right]$ we obtain

**Lemma 2.3. **Assume the Condition 2.1 are satisfied. Then for given $g\in\Phi,$ we have

[TABLE]

[TABLE]

and

[TABLE]

where $\alpha\left(g\right)>0$, $\gamma_{i}$ are trace operators from $Q$ to $\Gamma_{i}$, $i=0,1,$

[TABLE]

[TABLE]

[TABLE]

3. Abstract elliptic equation in the fixed domain

In this section we study the elliptic BVP

[TABLE]

[TABLE]

where $B\left(g\right)$ and $B_{i}\left(g\right)$ are differential operators defined by $\left(2.6\right).$

We will derive a priori estimates as well as isomorphism properties in framwork of abstract Holder spaces.

**Condition 3.1. **Assume the following conditions are satisfied:

(1) $a_{ij}\in C^{0,\alpha}\left(\bar{G}\right),$ $a_{ij}=a_{ji};$

(2) $\mathop{\displaystyle\sum}\limits_{i,j=1}^{2}a_{ij}\left(g\right)\xi_{i}\xi_{j}\geq C\left(g\right)\left|\xi\right|^{2}$, for $\xi=\left(\xi_{1},\xi_{2},...,\xi_{n}\right)\in R^{n}$ and $C\left(g\right)>0;$

(3) operator $A\left(g\right)$ is uniformly positive in a Banach algebra $E$ for some $\varphi\in\left(0\right.,\left.\pi\right].$

Here $\partial B\left(g\right)\left[\psi,.\upsilon\right]$ denotes the Gateaux derivative of operator function $B\left(g\right)$ at $\psi$ in the direction of $\upsilon.$

**Lemma 3.1. **Suppose the Condition 3.1 is satisfied and $A\left(g\right)$ is Gateaux differentiable for $g\in h_{\Phi}^{2,\alpha}$, $\alpha\in\left(0,1\right)$. Then the map $g\rightarrow O_{0}\left(g\right)=\left\{B\left(g\right),\text{ }B_{0}\left(g\right)\right\}$ is bounded linear operator-function from $Y$ into $X\times h^{1,\alpha}\left(A\right)$ and have continious derivatives of all order with respect to $g\in h_{\Phi}^{2,\alpha}$, i.e.

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

for $g\in h_{+}^{2,\alpha}$and $u,$ $\upsilon\in Y.$

**Proof.   **It is clear to see that $\left(\varphi,\upsilon\right)\rightarrow\varphi\upsilon$ is bilinear and continuous from $h^{s}$ into $Y.$ Moreover, the mapping

[TABLE]

are continious and are infinitely many differentible from $h^{s}$ into $Y.$ By using the definition of the space $Y$ and Lemma 2.3 we get that for all fixed $g\in h_{\Phi}^{2,\alpha}$ the operator $u\rightarrow B\left(g\right)$ is bounded linear operator from $Y$ into $X.$ So, we obtain that

[TABLE]

Hence, in view of Lemma 2.3 we obtain

[TABLE]

By using $\left[\text{4, Theorem 2}\right]$ we obtain the following:

**Theorem 3.1. **Suppose the Condition 3.1 is satisfied and $\alpha\in\left(0,1\right)$. Then for $\lambda\in S\left(\varphi\right)$ and for sufficiently large $\left|\lambda\right|:$

(a) the operator $u\rightarrow\tilde{O}\left(g\right)u=\left\{\left(B\left(g\right)+\lambda\right)u,\text{ }\gamma_{0}u,\text{ }\left(\eta+g\right)B_{1}\left(g\right)\right\}$ is isomorphism from $Y$ onto $X\times h^{2,\alpha}\left(A\right)\times h^{1,\alpha}\left(A\right);$

(b) for $\mu>0$ the operator $u\rightarrow\left\{\left(B\left(g\right)+\lambda\right)u,\text{ }\mu\gamma_{0}u,\text{ }\left(\eta+g\right)B_{1}\left(g\right)\right\}$ is isomorphism from $Y$ onto $X\times h^{2,\alpha}\left(A\right)\times h^{1,\alpha}\left(A\right);$

(c) For $u\in Y$ there exists a positive constant $C$, depending only on $g$, $\eta$, $p$ and $E$ such that the coercive estimate holds:

[TABLE]

Proof. Indeed, since the domain $Q$ is a strip, the functios $g$, $\eta$ are fixed smooth functions, by virtue of trace theorem in Hoder space $\left[\text{15, \S\ 2}\right]$ we obtain the assertion.

Consider now, the following BVPs

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Consider the operators $\tilde{S}\left(g\right)$ and $\tilde{K}\left(g\right)$ generated by problems $\left(3.4\right)-\left(3.5\right)$ and $\left(3.6\right)-\left(3.7\right)$, respectively, i.e.

[TABLE]

[TABLE]

and

[TABLE]

From the Theorem 3.5 we obtain that the inverse operators $\tilde{O}^{-1}\left(g\right),$ $\tilde{S}^{-1}\left(g\right),$ $\tilde{K}^{-1}\left(g\right)$ are bounded from $X\times h^{2,\alpha}\left(A\right)\times h^{1,\alpha}\left(A\right)$, $X,$ $h^{2,\alpha}\left(E\right)$ into $Y,$ respectively.

Here,

[TABLE]

Assume $g\in h_{+}^{2,\alpha}\left(\mathbb{R};E\right),$ $\psi\in D_{A}\left(\alpha\right)$, and put $u=$ $K\left(g\right)\upsilon.$ Then $u$ is the solution of the BVP $\left(3.6\right)-\left(3.7\right).$

**Condition 3.1. **Assume $A\left(g\right)$ is Gateaux differentiable for $g\in h_{+}^{2,\alpha}$, $\alpha\in\left(0,1\right)$ and the operator $\partial A\left(g\right)$ is uniformly $R$-positive in UMD (see e.g $\left[8\right]$ for definitions) Banach algebra $E.$

**Lemma 3.2. **Suppose the conditions 3.1 and 3.2 are satisfied. Then we have

[TABLE]

and

[TABLE]

for $g\in h_{+}^{2,\alpha}$, $\psi\in E$ and $\upsilon\in Y.$

**Proof. **It follows from Lemma 3.1 and Theorem 3.1 that the map $g\rightarrow\tilde{O}\left(g\right)$ is isomorphism from $Y$ onto $X\times h^{2,\alpha}\left(A\right)\times h^{1,\alpha}\left(A\right)$ and have continious derivatives of all order with respect to $g\in h_{+}^{2,\alpha}$, i.e.

[TABLE]

and

[TABLE]

Then by reasoning as the Lemma 3.4 in $\left[6\right]$ we obtain the assertion.

4. The nonlinear operator for free BVPs

In this section we introduce the basic nonlinear operator and we derive some properties of it.

Moreover, we show that the corresponding evolution problem involving this operator is equivalent to the original problem $\left(1.2\right).$ Given $g\in h_{+}^{2,\alpha},$ we define the following operator

[TABLE]

From lemmas $3.1$ and $3.2$ we get that

[TABLE]

Assume that $g_{0}\in h_{+}^{2,\alpha}$ , and let $\sigma=\left[0,\right.\left.T\right).$ A function $\sigma\rightarrow B_{10}$ is a classical solution of

[TABLE]

iff $g\in C\left(\sigma;h_{+}^{2,\alpha}\right)\cap$ $C^{1}\left(\sigma;h^{2,\alpha}\left(E\right)\right)$ and $g$ satisfies $\left(4.2\right)$ pointwise.

**Lemma 4.1. **Suppose the Condition 3.1 is satisfied. Then for $g_{0}\in h_{+}^{2,\alpha}$:

(a) Suppose that $g$ is a classical solution of problem $\left(4.2\right)$ on $\sigma$ and let $\upsilon\left(t,.\right)=K\left(g\left(t,.\right)\right)g\left(t,.\right).$ Then the pair $(\upsilon,$ $g)$ is a classical solution of $\left(2.5\right)$ on $\sigma$, having the additional regularity

[TABLE]

[TABLE]

(b) Suppose that $(\upsilon,$ $g)$ is a classical solution of $\left(2.5\right)$ on $\sigma$, having the regularity $\left(4.3\right).$ Then $g$ is a classical solution of $\left(4.2\right)$ on $\sigma$.

**Proof. **The proof is obtained from Lemma 2.2 and definitions of the spaces $B_{ip},$ $i=0,1$ and $Y$.

For fixed $g\in h_{\Phi}$ consider the operator

[TABLE]

In view of Theorem 3.1 we obtain that

[TABLE]

**Lemma 4.2. **Suppose the conditions 3.1 and 3.2 are satisfied. Then $O\in C^{\infty}\left(h_{+}^{2,\alpha},h^{1,\alpha}\left(A\right)\right)$ and

[TABLE]

[TABLE]

for $g\in h_{+}^{2,\alpha}$, $\psi\in h^{2,\alpha}\left(A\right)$ and $\upsilon\in Y.$

**Proof. **The assertion is obtained from Lemma 3.1 and Lemma 3.2.

5. Linear equation with constant coefficients

We put

[TABLE]

Consider the problem

[TABLE]

[TABLE]

where $A=A\left(g\right)\left(x_{0},0\right),$ $a_{ij}=a_{ij}\left(g\right)\left(x_{0},0\right)$ and $a_{ij}\left(g\right)$ are defined by $\left(2.7\right)$ and $\psi\in h^{2,\alpha}\left(A\right)$. By applying the Fourier transform to the problem $\left(5.1\right)-\left(5.2\right)$ with respect to $x_{1}$ we get

[TABLE]

[TABLE]

where $x_{2}$ and $\hat{u}\left(\eta,x_{2}\right)$ are denoted by $y$ and $\hat{u}=\hat{u}\left(\eta,y\right),$ respectively.

Let

[TABLE]

There exists an $\alpha$ $>0$ with

[TABLE]

The condition $\left(5.5\right)$ implies that

[TABLE]

Moreover, we define

[TABLE]

**Remark 5.1. **Given $\eta\in\mathbb{R}$ and $z\in\mathbb{C},$ then in view of $\left(5.5\right)$ there is exactly one root of the equation $\left(5.7\right)$ with positive real part. It is given by

[TABLE]

where

[TABLE]

We put

[TABLE]

[TABLE]

The main result of this section is the following:

**Theorem 5.1. **Assume the Condition 3.1 is satisfied. Then problem $\left(5.1\right)-\left(5.2\right)$ has a unique solution $u\in\tilde{Y}$ for $\psi\in h^{2,\alpha}\left(A\right)$ and $u$ is represented by

[TABLE]

Moreover, the estimate holds

[TABLE]

For the proof we need some preparation. Here,

[TABLE]

[TABLE]

We need the following lemmas:

**Lemma 5.1. **Assume the Condition 3.1 is satisfied. Then there exists a unique solution $\hat{u}\left(\eta,y\right)$ of $\left(5.3\right)-\left(5.4\right)$ expressing as

[TABLE]

Moreover, the following uniformly in $\eta$ estimate holds

[TABLE]

**Proof. **In view of positivity of operator $A$ we know that $-A_{\mu}^{\frac{1}{2}}\left(\eta\right)$ is analytic semigroup in $E$ (see e.g. $\left[\text{25, \S\ 1.18}\right]$). The the equation $\left(5.3\right)$ have a solution $\hat{u}=U_{\eta,\mu}\left(y\right)\hat{\psi}\left(\eta\right)$ on $\left(0,\infty\right),$ where

[TABLE]

and $\lambda\left(\eta,\mu\right)$ is a root of $\left(5.7\right)$ with positive real part. Then from the above expresion and the properties of analytic semigroups we get the uniform estimate

[TABLE]

[TABLE]

where $U\left(y\right)$ is a semigroup generated by $-A.$

**Lemma 5.2. **Assume the Condition 3.1 is satisfied. Then operator functions $q_{0}\left(\eta,y\right)$ and $q_{j}\left(\eta,y\right)$ are Fourier multipliears in h^{\alpha}\left(\mathbb{R};E\right)\uniformly with respect to $y\in\mathbb{R}_{+}.$

**Proof. ** In view of $\left(5.6\right)$ and the Remark 5.1 we get that

[TABLE]

[TABLE]

Then by using the multiplier results in $\left[\text{5}\right]$ we can prove that the operator function $q_{0}\left(.,y\right)$, $q_{j}\left(.,y\right)$ are multipliers in $h^{\alpha}\left(\mathbb{R};E\right)$ uniformly in $y\in\mathbb{R}_{+}$ and $\mu\in\mathbb{R}.$

Let

[TABLE]

Lemma 5.3. Assume the Condition 3.1 is satisfied. Then

[TABLE]

**Proof. **Indeed, by properties of Fourier multiplier operators from in $h^{\alpha}\left(\mathbb{R};E\right)$ and the theory of analytic semigroups there exists $\omega>0$ such that we have

[TABLE]

[TABLE]

By estimates $\left(5.12\right)$ and Remark 5.1 we obtain the assertion.

**Proof of Theorem 5.1. ** By Lemma 5.1 the problem $\left(5.1\right)-\left(5.2\right)$ has a solution

[TABLE]

By Lemmas 5.2, 5.3 the operator-functions $q_{0}\left(\eta,y\right)$ and $q_{j}\left(\eta,y\right)$ are Fourier multipliears in h^{\alpha}\left(\mathbb{R};E\right)\uniformly with respect to $y\in\mathbb{R}_{+}.$ Then from the estimate 5.11 we obtain trhe assertion.

Here $K_{0\mu}=K_{0\mu}\left(x_{0}\right)$ denotes the inverse of operator $\tilde{K}_{0}\left(x_{0}\right)+\mu^{2}$, i.e.

[TABLE]

**Result 5.1. **From Theorem 5.1 we obtain that the opertor $u\rightarrow K_{0\mu}u$ is bounded from $h^{2,\alpha}\left(A\right)^{\text{\ }}$into $\tilde{Y}$ and the following estimate holds

[TABLE]

[TABLE]

Consider now the BVP

[TABLE]

[TABLE]

where

[TABLE]

and $a_{ij}\left(g\right)$ are defined by $\left(2.7\right)$.

Here,

[TABLE]

6. Regularity properties of abstract elliptic operator with constant coefficients

Consider the operator

[TABLE]

Here, $O_{10}$ denotes the constant coefficients version of $O_{1}$ fixed in $\left(x_{0},0\right)$ by $\left(5.13\right),$ i.e.

[TABLE]

Here,

[TABLE]

[TABLE]

We show the following result:

**Theorem 6.1. **Assume the Condition 3.1 is satisfied. Then the operator $u\rightarrow\left(O_{10}+\mu^{2}\right)u$ is an isomorphisim from $h^{2,\alpha}\left(A\right)$ onto $h^{1,\alpha}\left(A\right).$ Moreover, the following uniform estimates hold

[TABLE]

[TABLE]

for all $u\in h^{2,\alpha}\left(A\right)$ and $\mu>0$ with sufficiently large $\mu_{0}.$

**Proof. ** In view of  Lemma 3.1 and Theorem 5.1 it is clear to see that the solution of the problem

[TABLE]

has a unique solution $u$ exspressed as

[TABLE]

By lemmas 5.1-5.3 and by multiplier result in $\left[5\right]$ we get that the operator functions $\left|\mu\right|^{2-i}\eta^{i}K\left(\eta,x_{2}\right)$ and $AK\left(\eta,x_{2}\right)$ are multipliers in h^{\alpha}\left(\mathbb{R};E\right)\uniformly with respect to $x_{2}\in\mathbb{R}_{+},$ where

[TABLE]

Hence, we obtain the estimate

[TABLE]

Moreover, by definitions of the space $h^{2,\alpha}\left(A\right)$ we get $\left(6.1\right)$ and the estimate

[TABLE]

Then in view of the above estimate, by reasoning as in the Theorem 5.1 we get the assertion and corresponding esimates.

By reasoning as in Theorem 6.1 we obtain

**Theorem 6.2. **Assume the Condition 3.1 is satisfied. Then the operator $O_{10}$ is positive and $-O_{10}$ is a generator of an analytic semigroup in $h^{1,\alpha}\left(A\right)$.

Proof.  Indeed, for positivity of the operator $O_{10}$ in $h^{1,\alpha}\left(A\right)$ we need to show the estimate

[TABLE]

i.e. we have to prove the estimate

[TABLE]

for $u\in h^{2,\alpha}\left(A\right)$. By reasoning as the above we get that the function $\mu^{2}\eta a_{2}\left(\eta,\mu\right)$ is a multiplier in h^{\alpha}\left(\mathbb{R};E\right)\uniformly with respect to $x_{2}\in\mathbb{R}_{+},$ where

[TABLE]

So, the operator $O_{10}$ is positive in $h^{1,\alpha}\left(A\right)$. Then $-O_{10}$ is a generator of an analytic semigroup in $h^{1,\alpha}\left(A\right)$.

Here

[TABLE]

for $g\in h_{+}^{2+\alpha}$ and $u\in h^{2,\alpha}\left(A\right).$

Consider the operator $O_{20}$ that is a constant coefficients version of $O_{2}$ with fixed in $\left(x_{0},0\right)$ by $\left(5.15\right)$, i.e.

[TABLE]

for $g\in h^{2,\alpha}\left(A\right).$

We prove the following result:

**Theorem 6.3. **Assume the Conditions 3.1 and 3.2 are satisfied. Then the operator $u\rightarrow\left(O_{20}+\mu^{2}\right)u$ is an isomorphisim from $h^{2,\alpha}\left(A\right)$ onto $h^{1,\alpha}\left(A\right).$ Moreover, the following estimate holds

[TABLE]

[TABLE]

for all $u\in h^{2,\alpha}\left(A\right)$, for sufficiently large $\mu_{0}$ and $\mu>0.$

**Proof. **By Lemma 3.1 we have

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

By using lemmas 5.1-5.3 we get that the operator function

[TABLE]

is a multiplier in $h^{\alpha}\left(\mathbb{R};E\right)$ uniformly with respect to $x_{2}\in\mathbb{R}_{+},$ where

[TABLE]

Then by reasoning as in the Theorem 6.1 we obtain the assertion.

From Theorem 6.3 we obtain

**Result 6.1. **Assume the conditions 3.1 and 3.2 are satisfied. Then the operator $O_{20}$ is positive and $-O_{20}$ is a generator of an analityc semigroup in $h^{1,\alpha}\left(A\right)$.

Consider first of all, the BVP

[TABLE]

[TABLE]

where $A=A\left(g\right)\left(x_{0},0\right),$ $a_{kj}=a_{kj}\left(g\right)\left(x_{0},0\right)$ and $a_{kj}\left(g\right)$ are defined by $\left(5.15\right)$. ** **

Let $S_{0}=S\left(g\right)\left(x_{0},0\right)$ denotes the realizasion operator in $\tilde{X}$ generated by $\left(6.7\right)-\left(6.8\right)$ for $\mu=0$, i.e.

[TABLE]

From $\left[\text{4, Theorem 2}\right]$ we obtain the following:

**Result 6.2. **Assume the Condition 3.1 is satisfied. Then;

(1) problem $\left(6.7\right)-\left(6.8\right)$ for sufficientli large $\mu>0$ has a unique solution $u\in\tilde{Y}$ for $V\in\tilde{X};$

(2) the uniform coercive estimate holds

[TABLE]

(3) the operator $S_{0}$ is a positive and $-S_{0}$ is a generator of an analytic semigrop in $h^{1,\alpha}\left(A\right).$

The estimate $\left(6.9\right)$ particularly, implies that $\left(S_{0}+\mu^{2}\right)^{-1}\in B\left(\tilde{X},\tilde{Y}\right).$

Consider the inhomogenous problem

[TABLE]

[TABLE]

Theorem 6.4. Assume the conditions 3.1 and 3.2 are satisfied. Then the operator $u\rightarrow G_{0}u=\left\{L_{0}u,\text{ }\gamma u\right\}$ is an isomorphisim from $\tilde{Y}$ onto $\tilde{X}\times h^{2,\alpha}\left(A\right).$

**Proof. **From definition of $\tilde{X}$, $\tilde{Y}$, $h^{2,\alpha}\left(A\right)$, from expresion of $L_{0}$ and by virtue of trace result in $\tilde{Y}$ $\left[\text{15, Ch.2}\right]$ we get that

[TABLE]

i.e. the operator $G_{0}$ is bounded linear from $\tilde{Y}$ into $\tilde{X}\times h^{2,\alpha}\left(A\right).$ Hence, in view of Banach theorem it is sufficient to show that the operator $G_{0}$ is inective and surjective from $\tilde{Y}$ onto $\tilde{X}\times h^{2,\alpha}\left(A\right)$. From Theorem 5.1 we obtain that the corresponding homogenous problem $L_{0}u=0,$ $\gamma u=0$ has a zero solution, i.e. the operator $G_{0}$ is inective. So, it remain to show that this operator is surjective. By Theorem 5.1 we obtain that problem $L_{0}u=0,$ \gamma u=\psi\has a solution $u_{1}\in$ $\tilde{Y}$ for all $\psi\in h^{2,\alpha}\left(A\right).$ Moreover, from the Result 6.2 we get that problem $L_{0}u=V,$ $\gamma u=0$ has a solution $u_{2}\in$ $\tilde{Y}$ for all $V\in\tilde{X}.$ Then $u=u_{1}+u_{2}$ is a solution of $\left(6.10\right)-\left(6.11\right)$ that belongs to $\tilde{Y},$ i.e. the operator $G_{0}$ is surgective from $\tilde{Y}$ onto $\tilde{X}\times h^{2,\alpha}\left(A\right).$

From Theorems 5.1 and 6.4 we obtain the following

**Result 6.3. **The soluion $u$ of the problem $\left(6.10\right)-\left(6.11\right)$ is exspressed as

[TABLE]

where

[TABLE]

here $r_{+}$ is the restriction operator from $R^{2}$ into $R_{+}^{2}$ and $\tilde{V}=\tilde{V}\left(x_{1},x_{2}\right)$ is an exstension of $V\left(x_{1},x_{2}\right)$ on $R^{2}$, i.e

[TABLE]

$N_{\mu}\left(\xi_{1},x_{2}\right)$-is operator function defined by $\left(5.8\right)$ and $u_{1}$ is a solution of the equation

[TABLE]

Let $O_{3}=O_{3}\left(g\right)$ be the operator in $\left(4.6\right)$ defined by

[TABLE]

for $g\in h^{2,\alpha}\left(A\right)$ and $u\in Y.$

In view of Lemma 3.1 and Lemma 3.2 we get

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Consider the operator $O_{30}$ that is the constant coefficients version of $O_{3}$ fixed in $\left(x_{0},0\right)$ which defined by $\left(5.16\right)$, i.e. from the above equality and from $\left(6.13\right)$ we get $O_{30k}=B_{0}\left(g\right)S\left(g\right)G_{k}\left(g\right)\left(x_{0},0\right),$

[TABLE]

where

[TABLE]

[TABLE]

For $u\in h^{2,\alpha}\left(A\right)$ define the operator by

[TABLE]

For later purposes we need the following technical lemmas:

Lemma 6.1. Assume the conditions 3.1 and 3.2 are satisfied. Then:

(a) $P_{1}\in B\left(h^{2,\alpha}\left(A\right),\tilde{X}\right);$

(b) There exists a positive constant $C=C(g)$ such that

[TABLE]

for all $u\in h^{2,\alpha}\left(A\right)$, $r\in\left(0,1\right),$ where

[TABLE]

here, $U_{r}\left(x_{1}^{0},0\right)$ denotes the two dimensional ball with radius $r$ centered at $\left(x_{1}^{0},0\right).$

**Proof.  **Indeed, from the expression $\left(6.14\right)$ in view of the Theorem 5.1 and by virtue of trace theorem in $Y$ we get (a); Then by using the integral mean value theorem and the trace theorem we obtain

[TABLE]

Let

[TABLE]

Lemma 6.2. Assume the conditions 3.1 and 3.2 are satisfied. Then the operator $u\rightarrow\left(O_{304}+\mu^{2}\right)u$ is an isomorphisim from $h^{2,\alpha}\left(A\right)$ onto $h^{1,\alpha}\left(A\right).$ Moreover, the following estimate holds

[TABLE]

[TABLE]

for all $u\in h^{2,\alpha}\left(A\right)$ and for sufficiently large $\mu_{0}$, $\mu>0,.$

**Proof. **From the expression $\left(6.14\right)$ and Lemma 6.1 by reasoning as in $\left[\text{9, lemma 5.3}\right]$ we get that the operator $u\rightarrow\left(P_{1}+\mu^{2}\right)u$ is an isomorphism from $\tilde{Y}$ onto $\tilde{X}.$ The Theorem 6.4 implies that the operator $u\rightarrow\left[S_{x_{0}}\left(g\right)+\mu^{2}\right]G_{40}u$ is an isomorphism from $\tilde{Y}$ onto $\tilde{Y}$ for sufficiently large $\mu>0.$ Then in view of trace theorem in $\tilde{Y}$ we get that the operator $u\rightarrow\left[O_{304}+\mu^{2}\right]u$ is an isomorphism from $\tilde{Y}$ onto $h^{1,\alpha}\left(A\right).$ Moreover, in view of Result 6.3 by reasoning as in the Theorem 6.1 we obtain the estimates $\left(6.16\right)$ for all $u\in h^{2,\alpha}\left(A\right)$ and for $\mu>0$ with sufficiently large $\mu_{0}.$

Lemma 6.3. Assume the conditions 3.1 and 3.2 are satisfied. Then the operator $u\rightarrow\left(O_{303}+\mu^{2}\right)u$ is an isomorphisim from $h^{2,\alpha}\left(A\right)$ onto $h^{1,\alpha}\left(A\right).$ Moreover, the following estimate holds

[TABLE]

[TABLE]

for all $u\in h^{2,\alpha}\left(A\right)^{\text{\ \ }}$and for sufficiently large $\mu_{0}$, $\mu>0.$

**Proof. **From the expression $\left(6.14\right)$ by properties of positive operators we get that the map $u\rightarrow\left(G_{30}+\mu^{2}\right)u$ is an isomorphism from $\tilde{Y}$ onto $\tilde{X}.$ The Theorem 6.4 implies that the operator $u\rightarrow\left[S_{x_{0}}\left(g\right)+\mu^{2}\right]G_{30}u$ is an isomorphism from $\tilde{Y}$ onto $\tilde{Y}$ for sufficiently large $\mu>0.$ Then in view of trace theorem in $\tilde{Y}$ we get that the operator $u\rightarrow\left[O_{303}+\mu^{2}\right]u$ is an isomorphism from $\tilde{Y}$ onto $h^{1,\alpha}\left(A\right).$ Moreover, in view of Result 6.3 by reasoning as in the Theorem 6.1 we obtain the estimates $\left(6.17\right)$ for all $u\in h^{2,\alpha}\left(A\right)$ and for $\mu>0$ with sufficiently large $\mu_{0}.$

In a similar way as Lemma 6.1 and by reasoning as in $\left[\text{9},\text{lemma 5.4}\right]$ we obtain

Lemma 6.4. Assume the conditions 3.1 and 3.2 are satisfied. Then the operator $u\rightarrow\left(O_{30k}+\mu^{2}\right)u$ is an isomorphisim from $h^{2,\alpha}\left(A\right)$ onto $h^{1,\alpha}\left(A\right).$ Moreover, the following estimate holds

[TABLE]

[TABLE]

for all $u\in h^{2,\alpha}\left(A\right)$ and $\mu>0.$

**Theorem 6.5. **Assume the conditions 3.1 and 3.2 are satisfied. Then the operator $u\rightarrow\left(Q_{30}+\mu^{2}\right)u$ is an isomorphisim from $h^{2,\alpha}\left(A\right)$ onto $h^{1,\alpha}\left(A\right).$ Moreover, the following estimate holds

[TABLE]

[TABLE]

for all $u\in h^{2,\alpha}\left(A\right)$ and $\mu>0.$

**Proof. **From the expressions $\left(6.13\right)-\left(6.14\right)$ and from lemmas 6.2-6.4 we get that the operator $u\rightarrow K_{1x_{0}}\left(g\right)u$ is an isomorphism from $\tilde{Y}$ onto $\tilde{X}.$ The Theorem 6.4 implies that the operator $u\rightarrow\left[S_{x_{0}}\left(g\right)+\mu^{2}\right]K_{1x_{0}}\left(g\right)u$ is an isomorphism from $\tilde{Y}$ onto $\tilde{Y}$ for sufficiently large $\mu>0.$ Then in view of trace theorem in $\tilde{Y}$ we get that the operator $u\rightarrow\left[O_{30}+\mu^{2}\right]u$ is an isomorphism from $\tilde{Y}$ onto $h^{1,\alpha}\left(A\right)$ for sufficiently large $\mu_{0}$ and $\mu>0.$

From Theorem 6.5 we obtain

**Result 6.4. **Assume the Conditions 3.1 and 3.2 are satisfied. Then the operator $O_{30}$ is positive and $-O_{30}$ is a generator of an analytic semigroup in $h^{1,\alpha}\left(A\right)$.

From Theorems 6.1, 6.3 and 6.5 we obtain

**Result 6.5.  **Assume the Conditions 3.1 and 3.2 are satisfied. Then

[TABLE]

for sufficiently large $\mu_{0}>0.$

In view of theorems 6.3, 6.5  and by lemmas 2.3, 3.1 and Theorem 3.1 we obtain

**Result 6.6. **Assume the Conditions 3.1 and 3.2 are satisfied. The the following estimate holds

[TABLE]

for all

[TABLE]

Let

[TABLE]

Now, we will show that

[TABLE]

**Theorem 6.6. **Assume the Conditions 3.1 and 3.2 are satisfied. Suppose

[TABLE]

Then the operator $u\rightarrow\left(Q_{0}+\mu^{2}\right)u$ is an isomorphisim from $h^{2,\alpha}\left(A\right)$ onto $h^{1,\alpha}\left(A\right).$ Moreover, the following estimate holds

[TABLE]

[TABLE]

for all $u\in h^{2,\alpha}\left(A\right)$ and for sufficiently large $\mu_{0}$, $\mu>0$. Particularly, the operator $O_{0}$ is positive and $-O_{0}$ is a generator of an analytic semigroup in $h^{1,\alpha}\left(A\right)$.

**Proof. **Indeed, by using Theorems 6.1, 6.3, 6.5, the Results 6.3, 6.5, by reasoning as in lemma 5.4, Theorem 5.6, Corollary 5.7 in $\left[2\right]$ and the perturbation results for space $H\left(h^{2,\alpha}\left(A\right),h^{1,\alpha}\left(A\right)\right)$ we obtain the assertion.

Let

[TABLE]

[TABLE]

**Remark 6.1. **Suppose that $g\in W_{1}.$ Then

[TABLE]

From Theorem 5.1 we know that there is a constant $M>0$ such that

[TABLE]

for all $g\in h_{+}^{2,\beta}$ satisfying $\left\|g\right\|_{h^{2,\beta}}\leq\chi.$ Now define

[TABLE]

7. Coercive estimates for the linearization

It is clear that $O_{0}=O_{10}+$ $O_{20}+$ $O_{30}$ may be viewed as a principal part of the linearization of $\partial O\left(g\right)$ with coefficients fixed in $\left(x^{0},0\right)$.  Our main goal in this section is to prove that the operator $O$ belongs to the class

[TABLE]

We use the estimates of local operators in the preceding section to derive coercive estimates for the linearization operator $\partial O\left(g\right)$. Let

[TABLE]

where

[TABLE]

[TABLE]

Given $g\in h_{+}^{2,\alpha}$ and $t\in\left[0,1\right],$ set

[TABLE]

and observe that

[TABLE]

Let $\delta>0$ be given and let $\left\{V_{j},\varphi_{j}\text{, }j\in\mathbb{N}\right\}$ denote $\delta$-localizasion sequence for $S=\mathbb{R\times}\left(-\frac{\delta}{2},\frac{\delta}{2}\right)$, the covering $\left\{V_{j}\text{, }j\in\mathbb{N}\right\}$ has finite multiplicity, diam $U_{j}$ $<\delta$, and $\left\{V_{j},\varphi_{j}\text{, }j\in\mathbb{N}\right\}$ is a partition of unity on $S$ with $\mathop{\displaystyle\sum}\limits_{j\in\mathbb{N}}\varphi_{j}\left(x\right)\equiv 1$. Moreover, we fix $x_{1j}\in\mathbb{R}$ such that $\left(x_{1j},0\right)\in V_{j},$ $j\in\mathbb{N}.$

Here, we will prove the following result

**Theorem 7.1. **Assume the conditions 3.1 and 3.2 are satisfied. Suppose that $W_{0}\subset W_{1}$ is compact, $\beta\in\left(0,\alpha\right)$ and that $k>0.$ Then there exist $\delta\in$ $(0,1]$, a $\delta$-localization sequence $\left\{V_{j},\varphi_{j}\text{, }j\in\mathbb{N}\right\}$, $\beta\in\left(0,\alpha\right)$ and a positive constant $C=C(W_{0},M,\delta)$ such that

[TABLE]

for all $\upsilon\in h^{2,\alpha}\left(A\right),$ $j\in\mathbb{N},$ $t\in\left[0,1\right]$ and $g\in W_{0}.$

For proving Theorem 7.1 we need some preparation. Let

[TABLE]

Consider the following equation

[TABLE]

for

[TABLE]

From $\left(7.2\right)$ for $u_{j}=u\varphi_{j}$, $u\in D_{A}\left(\alpha\right)$ we get

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

By freezing in $\left(7.3\right)$ coefficients at points $\left(x_{1j},0\right)$ we have localized equations

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

here

[TABLE]

and $O_{3ij}$, $G_{i}$ are defined as in $\left(6.14\right)$; moreover,

[TABLE]

[TABLE]

[TABLE]

$O_{k}\left(x_{1j}\right)$ are local operators fixed at points $\left(x_{1j},0\right)$ defined by equalities $\left(6.3\right),$ $\left(6.5\right),$ $\left(6.14\right)$, $\left(6.15\right)$, respectively.  From expressions of $F_{j},$ $K\left(g\right)\left(x_{1j},0\right)$ by using $\left(6.3\right),$ $\left(6.5\right),$ $\left(6.14\right)$, $\left(6.15\right)$ we get that $F_{j}\in h^{1,\alpha}\left(A\right).$

For proving the Theorem 7.1 we need the following lemmas:

**Lemma 7.1. ** The operator $u\rightarrow\left(O\left(x_{1j}\right)+\mu^{2}\right)u$ is an isomorphisim from $h^{2,\alpha}\left(A\right)$ onto $h^{1,\alpha}\left(A\right).$ Moreover, the following estimate holds

[TABLE]

for all $u\in h^{2,\alpha}\left(A\right)$ and for sufficiently large $\mu>0$.

Proof. Consider the equation

[TABLE]

Then, by virtue of Theorem 6.6 we obtain that the operator $u\rightarrow\left(O\left(x_{1j}\right)+\mu^{2}\right)u$ is an isomorphisim from $h^{2,\alpha}\left(A\right)$ onto $h^{1,\alpha}\left(A\right)$ and the estimate $\left(7.7\right)$ holds.

**Lemma 7.2. **There is a positive $\varepsilon\in\left(0,1\right)$ such that the following local estimate holds

[TABLE]

**Proof. ** From $\left(7.6\right)$ we get

[TABLE]

Then by taking into account of expressions $B_{0}\left(g\right)$, $B_{0j}$, $K\left(g\right)$, $K_{j}$ and in view of smoothness of coefficients, choosing $\delta$ sufficiently small we have

[TABLE]

[TABLE]

[TABLE]

**Lemma 7.3. **There is a positive $\varepsilon\in\left(0,1\right)$ such that the following local estimate holds

[TABLE]

**Proof. ** From the $\left(7.6\right)$ we get

[TABLE]

[TABLE]

Then, by using the smoothness of coefficients and choosing $\delta$ sufficiently small, we get the estimate $\left(7.9\right).$

**Lemma 7.3. **There is a positive $\varepsilon\in\left(0,1\right)$ such that the following local estimate holds

[TABLE]

**Proof. **From expressions $\left(6.13\right)-\left(6.14\right)$ we get

[TABLE]

Moreover, from expressions $O_{31}$ and $G_{1}$ in $\left(6.14\right)$ by boundedness of functions $g$, $\nu$, $\beta$ we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Then by boundedness of operator $B_{0}\left(g\right)S\left(g\right),$ smoothness of coefficients, choosing $\delta$ sufficiently we obtain from the above

[TABLE]

In a similar way, from expressions $O_{32}$ and $G_{2}$ in $\left(6.14\right)$ we have

[TABLE]

In view of the condition on the operator $\partial A\left(g\right),$ boundedness of operator $B_{0}\left(g\right)S\left(g\right),$ smoothness of coefficients, choosing $\delta$ sufficiently we obtain

[TABLE]

Finally, from expressions $O_{34}$ an $G_{4}$ in $\left(6.14\right)$ by boundedness of functions $g$, $\nu$, $\beta$ we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Then the estiamate $\left(7.10\right)$ is obtained from $\left(7.12\right)-\left(7.15\right).$

Now, we can prove the Theorem 7.1.

Proof of Theorem 7.1. By virtue of Theorem 6.6 the following estimate holds

[TABLE]

for all solution $u_{j}\in h^{2,\alpha}\left(A\right)$ of the equation $\left(7.5\right).$

Whence, using smoothness of coefficients of equations $\left(7.3\right)-\left(7.6\right),$ in view of Lemmas 7.1-7.3 for $\mu$ with sufficiently large Re$\mu>0$ we get

[TABLE]

Moreover, by appling the microlocal analysis reasoning as in theorems 1, 2 in $\left[21\right]$ and in theorem 6.2 and as in $\left[9\text{, Lemma 6. 5-6.7}\right]$ we obtain the same estimates for corresponding commutators operators. Then from this and from $\left(7.16\right)$, $\left(7.17\right)$ we get the assertion.

From the Theorem 7.1 by microlocal analysis reasoning as in theorem 6.2 and corollary 6.3 in $\left[\text{9}\right]$ we obtain

**Corollary 7.1. **Assume the conditions 3.1, 3.2 are satisfied and $K\subset W$ is compact. Then there exist positive constants $\mu_{0}$ and $C=C(K)$ such that

[TABLE]

for all $\upsilon\in h^{2,\alpha}\left(A\right),$ $g\in K,$ $t\in\left[0,1\right]$ and $\mu\in\left\{z:\mathop{\mathrm{R}e}z>\mu_{0}\right\}.$

**Corollary 7.2. **Let $g\in W$ and $t\in\left[0,1\right]$ be given. Then

[TABLE]

Now, by using Theorem 7.1 we can prove Theorem 1.

**Proof of Theorem 1: **Let $f_{0}\in V_{\nu}$ be given and set $g_{0}=f_{0}-\nu.$ Observe that $g_{0}\in W_{\alpha,1}=W.$ It follows from Lemma 4.1 that we only have to prove that there exist $t_{+}>0$ and a unique maximal classical solution of $\left(1.1\right)-\left(1.2\right)$ on $[0,\left.t_{+}\right)$ satisfying

[TABLE]

if $t_{+}<\infty$ and $g\in C_{b}\left([0,\left.t_{+}\right);W\right).$

By reasoning as $\left[\text{9, Lemma 5.10}\right]$ It follows that $W$ is an open subset of $h_{+}^{2,\alpha}$. Hence, thanks to Lemma 4.2 and Corollary 7.2, we know that $O\in H\left(W,h^{1,\alpha}\right)$ and that

[TABLE]

Let now $\beta\in\left(0,\alpha\right)$ be fixed and observe that $W\subset W_{\beta,1}$. Thus the very same arguments as above also ensure that

[TABLE]

It is not difficult to see that the maximal $h^{1,\alpha}$-realization of $\partial O\left(g\right)\in B\left(h^{2,\beta},h^{1,\beta}\right)$ for $g\in W$, is just the linear operator in $\left(7.19\right)$. Note that

[TABLE]

where $\left(.,.\right)_{\alpha-\beta,\infty}$ denotes the real interpolation. Consequently, invoking Theorem 2.3 in $\left[24\right]$, we find that

[TABLE]

where $M\left(E_{1},E_{2}\right)$ denotes the class of all operators in s having the property of maximal regularity in the sense of Da Prato and Grisvard $\left[6\right]$. The existence of a unique maximal classical solution of (E)g o and the property of a smooth semiflow on $W$ can now be obtained along the lines of the proofs of Proposition 3.5 and Theorem 3.2 in $[24]$.

Finally suppose that $t_{+}<\infty$, $g\in C_{b}\left([0,\left.t_{+}\right);W\right)$ and that $\left(7.18\right)$ is not true. Then $g_{1}=$ $\lim\limits_{t\rightarrow t_{+}}$ $g\left(t\right)$ exists in $W$. Hence taking    $g_{1}$ as initial value in $\left(4.2\right)$ one easily constructs a solution $\bar{g}$ of $\left(4.2\right)$ for initial date $g_{1}$ extending $g$. This contradicts the maximality of $g$.

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