Maximal regularity, analytic semigroups, and dynamic and general Wentzell boundary conditions with a diffusion term on the boundary

Abstract

We show maximal regularity results concerning parabolic systems with dynamic boundary conditions and a diffusion theorem on the boundary in the framework of $L^p$ spaces, $1<p<\infty$. Analyticity results can be derived for the semigroups generated by suitable classes of uniformly elliptic operators with general Wentzell boundary conditions having diffusion terms on the boundary.

1 Introduction

The main aim of this paper is the study of parabolic systems with dynamic boundary conditions in the form

$$\begin{aligned} \left\{ \begin{array}{ll} D_t u(t,x) = \mathcal{A}u(t,x) + f(t,x), &{} (t,x) \in (0, T) \times \Omega , \\ D_t \gamma u(t,\cdot ) = L \gamma u(t,\cdot ) + \gamma Eu(t,\cdot ) + h(t,\cdot ), &{} t \in (0, T) \\ u(0,x) = u_0(x), &{} x \in \Omega , \\ (\gamma u)(0) = v_0. \end{array}\right. \end{aligned}$$

(1.1)

Here $\mathcal{A}$ is a linear, strongly elliptic, second-order differential operator in the open bounded subset $\Omega$ of $\mathbb {R}^n$, L is a second-order strongly elliptic operator in $\partial \Omega$, E is a first-order differential operator and $\gamma$ is the trace operator in $\partial \Omega$. A typical example of (1.1) is

$$\begin{aligned} \left\{ \begin{array}{ll} D_t u(t,x) = \alpha (x) \Delta u(t,x) + f(t,x), &{} \qquad (t,x) \in (0, T) \times \Omega , \\ D_t u(t,x') - a(x') \Delta _{\mathrm{LB}} u(t,x') + b(x') \frac{\partial u}{\partial \nu }(t,x') \\ \quad - c(x') u(t,x') = h(t,x'), &{} \quad (t,x') \in (0, T) \times \partial \Omega , \\ u(0,x) = u_0(x), &{} \quad x \in \Omega \end{array}\right. \end{aligned}$$

(1.2)

where we have indicated with $\Delta _{\mathrm{LB}}$ the Laplace–Beltrami operator in $\partial \Omega$, with $\frac{\partial \cdot }{\partial \nu }$ the unit normal derivative, pointing outside $\Omega$, and $\alpha$ and a are positively valued. Strictly connected with (1.1) and (1.2) are, respectively,

$$\begin{aligned} \begin{array}{ll} D_t u(t,x) = \mathcal{A}u(t,x) + f(t,x), &{} (t,x) \in (0, T) \times \Omega , \\ \mathcal{A}u(t,\cdot ) - L \gamma u(t,\cdot ) - \gamma Eu(t,\cdot ) = h(t,\cdot ), &{} t \in (0, T) \\ u(0,x) = u_0(x), &{} x \in \Omega , \\ (\gamma u)(0) = v_0. \end{array} \end{aligned}$$

(1.3)

and

$$\begin{aligned}&D_t u(t,x) = \alpha (x) \Delta u(t,x) + f(t,x), \qquad (t,x) \in (0, T) \times \Omega , \nonumber \\&\alpha (x') \Delta u(t,x') - a(x') \Delta _{\mathrm{LB}} u(t,x') + b(x') \frac{\partial u}{\partial \nu }(t,x') \nonumber \\&\quad - c(x') u(t,x') = h(t,x'), \qquad (t,x') \in (0, T) \times \partial \Omega , \nonumber \\&u(0, x) = u_0(x), \qquad x \in \Omega \end{aligned}$$

(1.4)

in the framework of $L^p$ spaces, both in $\Omega$ and in $\partial \Omega$. Here $\Omega$ is an open, bounded subset of $\mathbb {R}^n$, with suitably smooth boundary $\partial \Omega$, $\alpha$ and a are positively valued and $\Delta _{\mathrm{LB}}$ is the Laplace–Beltrami operator in $\partial \Omega$. We shall call boundary conditions in the form of (1.3) general Wentzell boundary conditions.

In our knowledge, problems (1.2) and (1.4) seem to have been introduced and discussed (from the physical point of view) in [18]. These systems contain a diffusion term of the boundary, given by a strongly elliptic operator in $\partial \Omega$ (for example, the Laplace-Beltrami operator). Similar systems without this term were studied, in different functional settings, in [1, 4, 6, 9, 10, 12,13,14].

Systems in the form (1.1)–(1.2) seem to have been considered only recently. The first paper where a problem in the form (1.2) is really studied seems to be [7]. In it, it was considered the system

$$\begin{aligned} \begin{array}{ll} D_t u(t,x) = Au(t,x) = \nabla \cdot (a(x) \nabla u) (t,x), &{} (t,x) \in [0, T]\times \Omega , \\ Au(t,x') + \beta (x') D_{\nu _A} u(t,x') + \gamma (x') -q \Delta _{\mathrm{LB}}u(t,x') = 0, &{} (t,x') \in [0, T]\times \partial \Omega , \\ u(0,x) = u_0(x), &{} x \in \Omega , \end{array} \end{aligned}$$

(1.5)

with A strongly elliptic in divergence form, $\beta (x') > 0$ in $\partial \Omega$, $D_{\nu _A}$ conormal derivative, $q \in [0, \infty )$. It is proved that, if $1 \le p \le \infty$, then the closure of a suitable realisation of the problem in the space $L^p(\Omega \times \partial \Omega )$ ($1 \le p \le \infty$) gives rise to an analytic semigroup (not strongly continuous if $p = \infty )$. The continuous dependence on the coefficients had already been considered in [3]. The case of a non-symmetric elliptic operator has been recently discussed in [8].

In [23], the author considered the case of a domain $\Omega$ with merely Lipschitz boundary, with a strongly elliptic operator A (independent of t). It was shown that a realisation of A with the general boundary condition $(Au)_{|\partial \Omega } - \gamma \Delta _{\mathrm{LB}} u + D_{\nu _A} u + \beta u = g$ in $\partial \Omega$ generates a strongly continuous compact semigroup in $C(\overline{\Omega })$.

In the paper [22], the authors treated (1.2) in the particular case $A(t,x,D_x) = \Delta _x$, $f \equiv 0$, $h \equiv 0$, $L(t) = l \Delta _{\mathrm{LB}}$ with $l > 0$ and $B(t,x',D_x) = k D_{\nu }$, where k may be negative (in contrast with the previously quoted literature). They showed that, if the initial datum $u_0$ is in $H^1(\Omega )$ and $u_{0|\partial \Omega } \in H^1(\partial \Omega )$, then (1.2) has a unique solution u in $C([0, \infty ); H^1(\Omega )) \cap C^1((0, \infty ); H^1(\Omega )) \cap C((0, \infty ); H^3(\Omega ))$, with $u_{|\partial \Omega }$ in $C([0, \infty ); H^1(\partial \Omega )) \cap C^1((0, \infty ); H^1(\partial \Omega )) \cap C((0, \infty ); H^3(\partial \Omega ))$.

In [11], (1.1) and (1.2) are studied in the setting of spaces of Hölder continuous functions. Results of maximal regularity are proved. Here also the operator E may be essentially arbitrary in the class of linear partial differential operators of order not exceeding one (apart some regularity of the coefficients).

Finally, we discuss some content of [4]. In this paper, the authors prove maximal regularity results for very general classes of mixed parabolic problems. Even systems in the form (1.1) are considered. In this particular case, they find necessary and sufficient conditions in order that there exists a unique solution $(u, \rho )$, with $\rho = \gamma u$, with $u \in W^{1,p}(0, T; L^p(\Omega )) \cap L^{p}(0, T; W^{2,p}(\Omega ))$, $\rho \in W^{\frac{3}{2} - \frac{1}{2p}, p}(0, T; L^p(\partial \Omega )) \cap L^p(0, T; W^{3-\frac{1}{p},p}(\partial \Omega ))$.

In the present paper, we discuss (1.1) from several points of view. We begin (Sect. 2) by considering the strongly elliptic problem depending on the complex parameter $\lambda$

$$\begin{aligned} \lambda g - Lg = h \end{aligned}$$

in a compact smooth manifold $\Gamma$ (without boundary) and the corresponding parabolic problem

$$\begin{aligned} \left\{ \begin{array}{l} D_t v(t,x') = Lv(t,x') + h(t,x'), \\ v(0,x') = v_0(x') \end{array} \right. \end{aligned}$$

We find necessary and sufficient conditions on h and $v_0$, in order that there exists a unique solution v in $W^{1,p}(0, T; L^p(\Gamma )) \cap L^{p}(0, T; W^{2,p}(\Gamma ))$ ($p \in (1, \infty )$). These results are essentially well known, but we are not aware of an exposition of them fitting our needs.

In Sect. 3, we prove a theorem of maximal regularity for (1.1), giving necessary and sufficient conditions in order that there exists a unique solution u in $W^{1,p}(0, T; L^p(\Omega )) \cap L^p(0, T; W^{2,p}(\Omega ))$, with $\gamma u$ in $W^{1,p}(0, T; L^p(\partial \Omega )) \cap L^p(0, T; W^{2,p}(\partial \Omega ))$($p \in (1, \infty ) \setminus \{\frac{3}{2}\}$). So we prove a maximal regularity result in a class of functions which is larger that the one considered in [4]. As in [11], E is essentially an arbitrary linear partial differential operator of order not exceeding one. The argument of the proof is quite simple: we begin by studying the case $E = 0$ and employ the results of Sect. 2, together with classical results for mixed parabolic problems with Dirichlet boundary conditions (see [16]). The general case can be treated by a perturbation argument.

In Sect. 4, we show that, for any p in $(1, \infty )$, the unbounded operator $G_p$ defined as follows:

$$\begin{aligned} \begin{array}{l} D(G_p) := \{(u, \gamma u) : u \in W^{2,p}(\Omega ), \gamma u \in W^{2,p}(\partial \Omega )\}, \\ G_p(u, \gamma u) := (\mathcal{A}u, L \gamma u + \gamma Eu). \end{array} \end{aligned}$$

is the infinitesimal generator of an analytic semigroup in $L^p(\Omega ) \times L^p(\partial \Omega )$.

Finally, in Sect. 5 we establish the following precise relation between problems (1.1) and (1.3). We introduce the operator $M_p$ defined as follows:

$$\begin{aligned} \begin{array}{l} D(M_p) := \{(u, \gamma u) : u \in C^2(\overline{\Omega }), \gamma \mathcal{A}u - L \gamma u - \gamma Eu = 0\}, \\ M_p(u, \gamma u) = (\mathcal{A}u, \gamma \mathcal{A}u) = (Au, L \gamma u + \gamma Eu ). \end{array} \end{aligned}$$

and show that, if the coefficients and the boundary of $\Omega$ are suitably regular, $M_p$ is closable in $X_p = L^p(\Omega ) \times L^p(\partial \Omega )$ and its closure coincides with $G_p$. The closure of $M_p$ is precisely the main operator studied in [7] and [8], as we explain more in detail in Sect. 5.

In conclusion of this introduction, we precise some notation. $\mathbb {N}$ will indicate the set pf positive integers; BC(A) is the class of complex valued continuous and bounded functions with domain A; if $A \subseteq \mathbb {R}^n$, BUC(A) will be the class of complex valued uniformly continuous and bounded functions with domain A.

Given the Banach spaces $X_0, X_1, X$, with $X_1 \hookrightarrow X \hookrightarrow X_0$, and $\alpha \in (0, 1)$, we shall write $X \in J^\alpha (X_0, X_1)$ to indicate that there exists M positive, such that, for any x in $X_1$,

$$\begin{aligned} \Vert x\Vert _X \le M \Vert x\Vert _{X_0}^{1-\alpha } \Vert x\Vert _{X_1}^{\alpha }. \end{aligned}$$

The symbol $\gamma$ will be employed to indicate the trace operator.

2 Elliptic problems depending on a parameter and parabolic problems in a differentiable manifold

We introduce the following assumptions:

(A1)$\Gamma$is a compact, smooth differentiable manifold of class$C^2$and dimensionm($m \in \mathbb {N}$).

(A2)Lis a second-order, partial differential operator in$\Gamma$. More precisely: for every local chart$(U, \Phi )$, withUopen in$\Gamma$and$\Phi$$C^{2}-$diffeomorphism betweenUand$\Phi (U)$, with$\Phi (U)$open in$\mathbb {R}^{m}$, for any$v \in C^{2}(\Gamma )$, if$x' \in U$,

$$\begin{aligned} L v(x') = \sum _{|\alpha | \le 2} l_{\alpha , \Phi }(x') D_y^\alpha (v \circ \Phi ^{-1})(\Phi (x')); \end{aligned}$$

(2.1)

we suppose, moreover, that, if$|\alpha | \le 2$, $l_{\alpha , \Phi } \in L^\infty _{\mathrm{loc}}(U)$, if$|\alpha | = 2$, $l_{\alpha , \Phi } \in C(U)$and is real valued, for any$x' \in U$there exists$\nu (x') > 0$such that,$\forall \eta \in \mathbb {R}^m$,

$$\begin{aligned} \sum _{|\alpha | = 2} l_{\alpha , \Phi } (x') \eta ^\alpha \ge \nu (x') |\eta |^2. \end{aligned}$$

We consider the elliptic system depending on the parameter $\lambda \in \mathbb {C}$

$$\begin{aligned} \lambda g(x') - Lg(x') = h(x'), \quad x' \in \Gamma . \end{aligned}$$

(2.2)

We prove the following

Theorem 2.1

Suppose that (A1) and (A2) hold. Let $p \in (1, \infty )$. Then:

• (I) there exists $\omega$ in $\mathbb {R}$ such that, if $\lambda \in \mathbb {C}$, $Re(\lambda ) \ge \omega$ and $h \in L^p(\Gamma )$, (2.2) has a unique solution g in $W^{2,p}(\Gamma )$; moreover, there exists $C_0 > 0$ such that

  $$\begin{aligned} |\lambda | \Vert g\Vert _{L^p(\Gamma )} + \Vert g\Vert _{W^{2,p}(\Gamma )} \le C_0 \Vert h\Vert _{L^p(\Gamma )}. \end{aligned}$$

• (II) As a consequence, the operator $L_p: W^{2,p}(\Gamma ) \rightarrow L^{p}(\Gamma )$, $L_p u = Lu$ is the infinitesimal generator of an analytic semigroup in $L^{p}(\Gamma )$.

Proof

We follow the argument in [11], proof of Theorem 2.1.

We take an arbitrary $x^0 \in \Gamma$ and consider a local chart $(U, \Phi )$ around $x^0$, with U open subset of $\Gamma$ and $\Phi$ diffeomorphism between U and $\Phi (U)$, open subset in $\mathbb {R}^m$. We introduce in $\Phi (U)$ the strongly elliptic operator $L^\sharp$,

$$\begin{aligned} L^\sharp v(y) := L(v \circ \Phi )(\Phi ^{-1}(y)), \quad y \in \Phi (U). \end{aligned}$$

(2.3)

By shrinking U (if necessary), we may assume that the coefficients of $L^\sharp$ are in $BC(\overline{\Phi (U)})$ and are extensible to elements $l_\beta$ in $BUC(\mathbb {R}^m)$, in such a way that the operator which we continue to call $L^\sharp = \sum _{|\alpha | \le 2} l_\beta (y) D_y^\beta$ is uniformly strongly elliptic in $\mathbb {R}^m$. Now we consider the problem

$$\begin{aligned} \lambda v(y) - L^\sharp v(y) = k(y), \quad y \in \mathbb {R}^m, \end{aligned}$$

(2.4)

with $k \in L^p(\mathbb {R}^m)$. Then, (see [17, Chapter 3.1.2]), there exists $\omega (x^0) \in \mathbb {R}$, such that, if $\lambda \in \mathbb {C}$ and $Re(\lambda ) \ge \omega (x^0)$, then (2.4) has a unique solution v in $W^{2,p}(\mathbb {R}^m)$; moreover, there exists $C(x^0) > 0$ such that

$$\begin{aligned} \sum _{j=0}^2 |\lambda |^{1-j/2} \Vert v\Vert _{W^{j,p}(\mathbb {R}^m)} \le C(x^0) \Vert k\Vert _{L^p(\mathbb {R}^m)}. \end{aligned}$$

Now we fix $U_1$ open subset of U, with $\overline{U_1}$ contained in U, $x^0 \in U_1$ and $\phi \in C^{2}(\Gamma )$, with compact support in U, $\phi (x) = 1$ for any $x \in U_1$. Given $h \in L^p(\Gamma )$, we indicate with k the trivial extension of $(\phi h) \circ \Phi ^{-1}$ to $\mathbb {R}^m$. If $\lambda$ is such that (2.4) is uniquely solvable for every k in $L^p(\mathbb {R}^m)$, we set

$$\begin{aligned}{}[S(x^0, \lambda ) h](x) := \phi (x) v(\Phi (x)), \quad x \in \Gamma , \end{aligned}$$

(2.5)

with v solving (2.4). We observe that

• $(\alpha _1)$$S(x^0, \lambda ) h \in W^{2,p}(\Gamma )$;

• $(\alpha _2)$

  $$\begin{aligned} \sum _{j=0}^2 |\lambda |^{1-j/2} \Vert S(x^0, \lambda ) h\Vert _{W^{j,p}(\Gamma )} \le C_1(x^0) \Vert h\Vert _{L^p(\Gamma )}; \end{aligned}$$

• $(\alpha _3)$$(\lambda - L) S(x^0, \lambda ) h = h$ in $U_1$;

• $(\alpha _4)$ if (2.2) is satisfied, for $h \in L^p(\Gamma )$, by some $g \in W^{2,p}(\Gamma )$ and g vanishes outside $U_1$, then $g = S(x^0, \lambda ) h$;

in fact, the trivial extension of $g \circ \Phi ^{-1}$ solves (2.4), with k trivial extension of $h \circ \Phi ^{-1}$.

Now we fix, for every $x \in \Gamma$, neighbourhoods U(x), $U_1(x)$ of x as before. As $\Gamma$ is compact, there exist $x_1, \dots , x_N$ in $\Gamma$ such that $\Gamma = \cup _{j=1}^N U_1(x_j)$.

Let $\lambda \in \mathbb {C}$. We show that, if $g \in W^{2,p}(\Gamma )$, it solves (2.2) with $h \equiv 0$ and $Re(\lambda )$ sufficiently large, then $g \equiv 0$. In fact, let $(\phi _j)_{j=1}^N$ be a $C^{2}-$ partition of unity in $\Gamma$, with $supp(\phi _j) \subseteq U_1(x_j)$, for each $j \in \{1, \dots , N\}$. Observe that

$$\begin{aligned} (\lambda - L) (\phi _j g) = [\phi _j; L] g, \end{aligned}$$

where we have indicated with $[\phi _j; L]$ the commutator $\phi _j L - L (\phi _j \cdot )$, which is a differential operator of order one. As $(\phi _j g)(x) = 0$ outside $U_1(x_j)$, we deduce from $(\alpha _4)$, if $Re(\lambda )$ is sufficiently large,

$$\begin{aligned} \phi _j g = S(x_j, \lambda )([\phi _j; L] g). \end{aligned}$$

So, from $(\alpha _2)$,

$$\begin{aligned} \Vert g\Vert _{W^{1,p}(\Gamma )} \le \sum _{j=1}^N \Vert \phi _j g\Vert _{W^{1,p}(\Gamma )} \le C_1 |\lambda |^{-1/2} \sum _{j=1}^N \Vert [\phi _j; L] g\Vert _{L^p(\Gamma )} \le C_2 |\lambda |^{-1/2} \Vert g\Vert _{W^{1,p}(\Gamma )}, \end{aligned}$$

implying $g \equiv 0$ if $Re(\lambda )$ is sufficiently large.

Next, we show that, if $|\lambda |$ is large enough, then (2.2) is solvable for every $h \in L^p(\Gamma )$. This time we fix, for each $j \in \{1, \dots , N\}$, $\psi _j \in C^{2} (\Gamma )$, vanishing outside $U_1(x_j)$ and such that $\sum _{j=1}^N \psi _j(x)^2 = 1$ for any x in $\Gamma$. We look for g in the form

$$\begin{aligned} g = \sum _{j=1}^N \psi _j S(x_j, \lambda )(\psi _j {\tilde{h}}), \end{aligned}$$

for some ${\tilde{h}} \in L^p(\Gamma )$. Again observing that $\psi _j S(x_j, \lambda )(\psi _j {\tilde{h}})$ vanishes outside $U_1(x_j)$ and that

$$\begin{aligned} (\lambda - L) [\psi _j S(x_j, \lambda )(\psi _j {\tilde{h}})] = \psi _j^2 {\tilde{h}} + [\psi _j; L] [S(x_j, \lambda )(\psi _j {\tilde{h}})], \end{aligned}$$

we deduce

$$\begin{aligned} (\lambda - L) g = {\tilde{h}} + \sum _{j=1}^N [\psi _j; L] [S(x_j, \lambda )(\psi _j {\tilde{h}})]. \end{aligned}$$

So, we have to choose ${\tilde{h}}$ in such a way that

$$\begin{aligned} {\tilde{h}} + \sum _{j=1}^N [\psi _j; L] [S(x_j, \lambda )(\psi _j {\tilde{h}})] = h. \end{aligned}$$

(2.6)

This is uniquely possible if $Re(\lambda )$ is sufficiently large, because

$$\begin{aligned}&\left\| \sum _{j=1}^N [\psi _j; L] [S(x_j, \lambda )(\psi _j {\tilde{h}})]\right\| _{L^p(\Gamma )} \le C_0 \sum _{j=1}^N \Vert S(x_j, \lambda )(\psi _j {\tilde{h}})]\Vert _{W^{1,p}(\Gamma )} \\&\quad \le C_1 |\lambda |^{-1/2} \Vert {\tilde{h}}\Vert _{L^p(\Gamma )}. \end{aligned}$$

So, if $C_1 |\lambda |^{-1/2} \le \frac{1}{2}$, we deduce from (2.6)

$$\begin{aligned} \Vert {\tilde{h}}\Vert _{L^p(\Gamma )} \le 2 \Vert h\Vert _{L^p(\Gamma )}, \end{aligned}$$

which, together with $(\alpha _2)$, implies (I).

(II) follows from (I). Observe also that, as $W^{2,p}(\Gamma )$ is dense in $L^p(\Gamma )$, the domain of $L_p$ is dense in $L^p(\Gamma )$. $\square$

Corollary 2.2

Suppose that (A1)–(A2) are satisfied. Let $1< p < \infty$, $\epsilon \in \mathbb {R}^+$, $g_0 \in W^{2,p}(\Gamma )$, $T \in \mathbb {R}^+$, $f \in C^\epsilon ([0, T]; L^p(\Gamma ))$. Then the problem

$$\begin{aligned} \begin{array}{ll} u'(t) - L_p u(t) = f(t), &{} t \in [0, T], \\ u(0) = g_0. \end{array} \end{aligned}$$

(2.7)

has a unique solution u in $C^1([0, T]; L^p(\Gamma )) \cap C([0, T]; W^{2,p}(\Gamma ))$ and

$$\begin{aligned} u(t) = e^{tL_p} u_0 + \int _0^t e^{(t-s) L_p} f(s) \mathrm{d}s, \end{aligned}$$

(2.8)

with $(e^{tL_p})_{t \ge 0}$ analytic semigroup generated by $L_p$.

The following “maximal regularity” result holds also:

Proposition 2.3

Let $p \in (1, \infty )$. Consider the problem (2.7). Then the following conditions are necessary and sufficient in order that there exists a unique solution u in $W^{1,p}(0, T; L^p(\Gamma )) \cap L^p(0, T; W^{2,p}(\Gamma ))$:

• (a) $f \in L^p(0, T; L^p(\Gamma ))$;

• (b) $g_0 \in W^{2-2/p,p}(\Gamma )$

If (a)–(b) hold, this unique solution is given by (2.8).

Proof

(a) is obviously necessary. The necessity of (b) follows from the fact that

$$\begin{aligned}&\{v(0) : v \in W^{1,p}(0, T; L^p(\Gamma )) \cap L^{p}(0, T; W^{2,p}(\Gamma ))\} \nonumber \\&\quad = (L^p(\Gamma ), W^{2,p}(\Gamma ))_{1-1/p,p} = W^{2-2/p,p}(\Gamma ) \end{aligned}$$

(2.9)

(see [17], Chapter 2.2.1 and Theorem 3.2.3).

On the other hand, suppose that (a)–(b) hold. It is well known that the only possible solution of (2.7) is (2.8). So the solution with the desired properties is, if it exists, unique. It is known that, if $v(t) = e^{tL_p} u_0$, $v \in W^{1,p}(0, T; L^p(\Gamma )) \cap L^p(0, T; W^{2,p}(\Gamma ))$ (see [17], Chapter 2.2.1). Assume that $u_0 = 0$. In this case, we deduce, for any $t \in [0, T]$, as $W^{1,p}(\Gamma ) \in J^{1/2} (L^p(\Gamma ); W^{2,p}(\Gamma ))$, if u is given by (2.8),

$$\begin{aligned} \Vert u(t)\Vert _{W^{1,p}(\Gamma )} \le C_0 \int _0^t (t-s)^{-1/2} \Vert f(s)\Vert _{L^p(\Gamma )} \mathrm{d}s \end{aligned}$$

so that, by Young’s inequality,

$$\begin{aligned} \Vert u\Vert _{L^p(0, T; W^{1,p}(\Gamma ))} \le C_1 \Vert f\Vert _{L^p(0, T; L^p(\Gamma ))}. \end{aligned}$$

(2.10)

Suppose now that $f \in C^\epsilon ([0, T]; L^p(\Gamma ))$. Then u really solves (2.7) (by Corollary 2.2). We fix a local chart $(U, \Phi )$ and take $\phi \in C^2(\Gamma ))$, with support in U. Then, if

$$\begin{aligned}u_\phi (t,x):= \phi (x) u(t,x), \end{aligned}$$

we get

$$\begin{aligned} \begin{array}{ll} D_t(u_\phi )(t,x) - L_p(u_\phi ) (t,x) = \phi (x) f(t,x) + ([\phi ; L_p]u)(t,x), &{} (t,x) \in [0, T] \times \Gamma , \\ u_\phi (0,x) = 0, &{} x \in \Gamma . \end{array} \end{aligned}$$

Setting

$$\begin{aligned} v(t,y) := u_\phi (t, \Phi ^{-1}(y)), \quad (t,y) \in [0, T] \times \Phi (U), \end{aligned}$$

and identifying v with its trivial extension to $[0, T] \times \mathbb {R}^m$, we get

$$\begin{aligned}&D_tv(t,y) - L^\sharp v(t,y) = \phi (\Phi ^{-1}(y)) f(t,\Phi ^{-1}(y)) \\&\quad + ([\phi ; L_p]u)(t,\Phi ^{-1}(y)), \quad (t,y) \in [0, T] \times \mathbb {R}^m, \\&v(0,y) = 0, \quad y \in \mathbb {R}^m, \end{aligned}$$

where we have employed again the operator $L^\sharp$ introduced in (2.3). From well-known maximal regularity results in $\mathbb {R}^m$ (which can be deduced, for example, from [15], Theorem 6.8), we obtain

$$\begin{aligned}&\Vert u_\phi \Vert _{W^{1,p}(0, T; L^p(\Gamma ))} + \Vert u_\phi \Vert _{L^{p}(0, T; W^{2,p}(\Gamma ))} \\&\quad \le C_1 (\Vert v\Vert _{W^{1,p}(0, T; L^p(\mathbb {R}^m))} + \Vert v\Vert _{L^{p}(0, T; W^{2,p}(\mathbb {R}^m))}) \\&\quad \le C_2 (\Vert f\Vert _{L^{p}(0, T; L^{p}(\Gamma )} + \Vert u\Vert _{L^{p}(0, T; W^{1,p}(\Gamma ))}) \\&\quad \le C_3 \Vert f\Vert _{L^{p}(0, T; L^{p}(\Gamma ))}, \end{aligned}$$

by (2.10). From this estimate, it follows immediately that

$$\begin{aligned} \Vert u\Vert _{W^{1,p}(0, T; L^p(\Gamma ))} + \Vert u\Vert _{L^{p}(0, T; W^{2,p}(\Gamma ))} \le C \Vert f\Vert _{L^{p}(0, T; L^{p}(\Gamma ))}. \end{aligned}$$

This implies the conclusion, taking a sequence $(f_k)_{k \in \mathbb {N}}$ in (say) $C^1([0, T]; L^p(\Gamma ))$ and converging to f in $L^p(0, T; L^p(\Gamma ))$. $\square$

Example 2.4

We show an example of an operator fulfilling conditions (A1)–(A2). Let $\Gamma$ be a smooth compact Riemannian manifold with dimension m and class $C^2$. For every x in $\Gamma$, we indicate with $T_x(\Gamma )$ the tangent space and with $T_x(\Gamma ) + i T_x(\Gamma )$ its complexification. The real scalar product $(\cdot , \cdot )_x$ in $T_x(\Gamma )$ can be extended in a natural way to a complex scalar product, which we continue to indicate with $(\cdot , \cdot )_x$ (for these elementary facts, see [19], Chapter 6.5). We shall indicate with $T(\Gamma ) + iT(\Gamma )$ the disjoint union of the spaces $T_x(\Gamma ) + i T_x(\Gamma )$ ($x \in \Gamma$), which is naturally equipped with a structure of $m-$dimensional complex vector bundle on $\Gamma$.

If $f : \Gamma \rightarrow \mathbb {C}$ is of class $C^1$, we indicate with $\nabla f(x)$ the gradient of f in x, which belongs to $T_x(\Gamma ) + i T_x(\Gamma )$. $\nabla$ is a first-order differential operator, mapping smooth complex valued functions defined in $\Gamma$ into sections of $T(\Gamma )+ iT(\Gamma )$. We recall that $\nabla f(x)$ is the element of $T_x(\Gamma ) + i T_x(\Gamma )$ such that, for every $v \in T_x(\Gamma )$,

$$\begin{aligned} (\nabla f(x), v)_x = v(f) \end{aligned}$$

(see, for example, [2], Chapter V). Suppose that we fix a local chart $(U, \Phi )$ in $\Gamma$. We indicate with $\frac{\partial }{\partial x_j}$ ($1 \le j \le m$) the field in U such that

$$\begin{aligned} \frac{\partial f}{\partial x_j} (x) = \frac{\partial (f \circ \Phi ^{-1})}{\partial y_j}(\Phi (x)), \quad x \in U, \end{aligned}$$

where we have indicated by $y_1, \dots , y_m$ the standard coordinates in $\mathbb {R}^m$. Moreover, we set

$$\begin{aligned} g(x) = \left( \left( \frac{\partial }{\partial x_i} (x), \frac{\partial }{\partial x_j} (x)\right) _x\right) _{1\le i,j \le m}. \end{aligned}$$

It is easily seen that the matrix g(x) is symmetric and positive definite. We introduce also its inverse

$$\begin{aligned} G(x) := g(x)^{-1}, \end{aligned}$$

again symmetric and positive definite. Then it is not difficult to check that, in local coordinates,

$$\begin{aligned} \nabla f(x) = \sum _{i=1}^m \sum _{j=1}^m G_{ij}(x) \frac{\partial f}{\partial x_j}(x) \frac{\partial }{\partial x_i}(x). \end{aligned}$$

(2.11)

Now we assume that, for any $x \in \Gamma$, B(x) is a linear operator from $T_x(\Gamma )$ into itself, Hermitian and positive definite with respect to $(\cdot , \cdot )_x$, that is, $\forall \xi , \eta \in T_x(\Gamma )$,

$$\begin{aligned} (B(x) \xi , \eta )_x = (\xi , B(x)\eta )_x \end{aligned}$$

and, if $v \in T_x(\Gamma ) \setminus \{0\}$,

$$\begin{aligned} (B(x) v, v)_x > 0. \end{aligned}$$

We suppose also that B(x) depends smoothly on x. This is equivalent to prescribe that, for every local chart $(U, \Phi )$ the following conditions are satisfied:

• (a) for each $i \in \{1, \dots , m\}$, $B(x)(\frac{\partial }{\partial x_i}(x)) = \sum _{j=1}^m B_{ij}(x) \frac{\partial }{\partial x_j}(x)$, with $B_{ij} \in C^1(U)$;

• (b) if we set, for any x in U, ${\mathcal {B}}(x):= (B_{ij}(x))_{1 \le i,j \le m}$, the product ${\mathcal {B}}(x) g(x)$ is symmetric and positive definite.

Observe that (a)–(b) imply that, for any x in U, even $G(x) {\mathcal {B}}(x)$ is symmetric and positive definite. In fact,

$$\begin{aligned}&(G(x) {\mathcal {B}}(x))^T = {\mathcal {B}}(x)^T G(x) = G(x) (g(x) {\mathcal {B}}(x)^T) G(x) \\&\quad = G(x) ({\mathcal {B}}(x) g(x))^T G(x) = G(x) {\mathcal {B}}(x) g(x) G(x) = G(x) {\mathcal {B}}(x). \end{aligned}$$

Moreover, if $\xi \in \mathbb {R}^m \setminus \{0\}$,

$$\begin{aligned} (G(x) {\mathcal {B}}(x) \xi ) \cdot \xi = ({\mathcal {B}}(x) g(x) G(x) \xi ) \cdot G(x) \xi > 0. \end{aligned}$$

We indicate by $\sigma$ the measure induced by the Riemannian metric in $\Gamma$ and by $-div$ the adjoint operator of $\nabla$. So, if $u : \Gamma \rightarrow \mathbb {C}$ and v is a smooth vector field,

$$\begin{aligned} \int _\Gamma (\nabla u(x), v(x))_x \mathrm{d}\sigma = - \int _\Gamma u(x) \overline{\mathrm{div}(v)(x)} \mathrm{d}\sigma . \end{aligned}$$

It is not difficult to check that, if $(U, \phi )$ is the usual chart, and if $\rho : \Phi (U) \rightarrow \mathbb {R}^+$ is such that, for every measurable subset A of U

$$\begin{aligned} \sigma (A) = \int _{\Phi (A)} \rho (y) \mathrm{d}y, \end{aligned}$$

for every smooth vector field $X = \sum _{k=1}^m X_k \frac{\partial }{\partial x_k}$ in U, one has

$$\begin{aligned} \mathrm{div}(X)(x) = (\nabla \cdot X)(x) = \sum _{k=1}^m \frac{\partial }{\partial x_k} ((\rho \circ \Phi ) X_k)(x). \end{aligned}$$

(2.12)

We introduce now the operator

$$\begin{aligned} L u (x) := \mathrm{div}(B(x) \nabla _x u) \end{aligned}$$

(2.13)

Observe that, if $B(x) = I_{T_x(\Gamma )}$ for any x in $\Gamma$, ${\mathcal {B}}$ is nothing but the Laplace-Beltrami operator. We show that it satisfies the conditions (A1)–(A2). In fact, if $f : U \rightarrow \mathbb {C}$ is sufficiently smooth and $x \in U$, we have, on account of (2.11),

$$\begin{aligned}&B(x) \nabla f(x) = \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^m G_{ij}(x) \frac{\partial f}{\partial x_j}(x) B(x) \left( \frac{\partial f}{\partial x_i}(x)\right) \\&\quad = \sum \nolimits _{i=1}^m \sum \nolimits _{j=1}^m \sum \nolimits _{k=1}^m G_{ij}(x) \frac{\partial f}{\partial x_j}(x) B_{ik}(x) \frac{\partial f}{\partial x_k}(x) \\&\quad = \sum \nolimits _{j=1}^m \sum \nolimits _{k=1}^m (G(x) {\mathcal {B}}(x))_{jk} \frac{\partial f}{\partial x_j}(x) \frac{\partial f}{\partial x_k}(x), \end{aligned}$$

so that, by (2.13),

$$\begin{aligned} Lf(x) = \sum _{k=1}^m \sum _{j=1}^m \frac{\partial }{\partial x_k} \left[ (\rho \circ \Phi )(x) (G(x) {\mathcal {B}}(x))_{jk} \frac{\partial f}{\partial x_j}(x)\right] . \end{aligned}$$

or

$$\begin{aligned} Lf(x) = \sum _{k=1}^m \sum _{j=1}^m \frac{\partial }{\partial y_k}\left[ \rho (y) (G {\mathcal {B}})_{jk}(\Phi ^{-1} (y)) \frac{\partial (f \circ \Phi ^{-1})}{\partial y_j}(y)\right] (\Phi (x)). \end{aligned}$$

The principal part of the operator is

$$\begin{aligned} \sum _{k=1}^m \sum _{j=1}^m \rho (\Phi (x)) (G {\mathcal {B}})_{jk}(x) \frac{\partial ^2 (f \circ \Phi ^{-1})}{\partial y_k \partial y_j}(\Phi (x)). \end{aligned}$$

and the matrix $\rho (\Phi (x)) (G {\mathcal {B}})(x)$ is symmetric and positive definite.

So L, defined in (2.13), satisfies the conditions (A1)–(A2).

3 Maximal regularity

Now we consider the following classical Cauchy–Dirichlet parabolic problem

$$\begin{aligned} \begin{array}{ll} D_t u(t,x) = \mathcal{A}u(t,x) + f(t,x), &{} (t,x) \in (0, T) \times \Omega , \\ \gamma u(t,\cdot ) = g(t,\cdot ) , &{} t \in (0, T), \\ u(0,x) = u_0(x), &{} x \in \Omega , \end{array} \end{aligned}$$

(3.1)

with the following conditions:

(B1)$\Omega$is an open, bounded subset of$\mathbb {R}^n$, lying on one side of its boundary$\Gamma$, which is a submanifold of$\mathbb {R}^n$of class$C^2$.

(B2)$\mathcal{A}= \sum _{i,j= 1}^n a_{ij}(x) D_{x_i x_j} + \sum _{j=1}^n b_j(x) D_{x_j} + c(x)$, with$a_{ij}, b_j, c \in C(\overline{\Omega })$ ($1 \le i,j \le n$); the functions$a_{ij}$are real valued and there exists$\nu \in \mathbb {R}^+$such that$\sum _{i,j= 1}^n a_{ij}(x) \xi _i \xi _j \ge \nu |\xi |^2$, for any$x \in \overline{\Omega }$, $\xi = (\xi _1, \dots , \xi _n) \in \mathbb {R}^n$.

The following classical result holds (see [16], Theorem 9.1):

Theorem 3.1

Suppose that (B1)–(B2) hold. Let $p \in (1, \infty ) \setminus \{\frac{3}{2}\}$. Then the following conditions are necessary and sufficient, in order that (3.1) has a unique solution u in $W^{1,p}(0, T; L^p(\Omega )) \cap L^p(0, T; W^{2,p}(\Omega ))$:

• (I) $f \in L^p(0, T; L^p(\Omega ))$;

• (II) $g \in W^{1- 1/(2p), p}(0, T; L^p(\Gamma )) \cap L^p(0, T; W^{2-1/p,p}(\Gamma ))$;

• (III) $u_0 \in W^{2-2/p,p}(\Omega )$;

• (IV) in case $p > \frac{3}{2}$, $\gamma u_0 = g(0)$.

Remark 3.2

Observe that, as $u \in L^p(0, T; W^{2,p}(\Omega ))$, the second equation in (3.1) is assumed to be satisfied only almost everywhere in (0, T).

However, the identity (2.9) and the analogous identity obtained by replacing $\Gamma$ with $\Omega$ imply that

$$\begin{aligned}&W^{1,p}(0, T; L^p(\Omega )) \cap L^{p}(0, T; L^p(\Omega )) \subseteq C([0, T]; W^{2-2/p,p}(\Omega )),\\&W^{1,p}(0, T; L^p(\Gamma )) \cap L^{p}(0, T; L^p(\Gamma )) \subseteq C([0, T]; W^{2-2/p,p}(\Gamma )). \end{aligned}$$

If $p > \frac{3}{2}$, then $2 - \frac{2}{p} > \frac{1}{p}$, so that $\gamma u \in C([0, T]; L^p(\Gamma ))$ and the second equation in (3.1) can be assumed to be satisfied for every $t \in [0, T]$. This explains the necessity of (IV) in this case. Observe also that, as $1 - \frac{1}{2p} > \frac{1}{p}$, (II) implies that $g \in C([0, T]; L^p(\Gamma ))$.

Now we consider the problem

$$\begin{aligned} \begin{array}{ll} D_t u(t,x) = \mathcal{A}u(t,x) + f(t,x), &{} (t,x) \in (0, T) \times \Omega , \\ D_t \gamma u(t,\cdot ) = L \gamma u(t,\cdot ) + h(t,\cdot ), &{} t \in (0, T), \\ u(0,x) = u_0(x), &{} x \in \Omega , \end{array} \end{aligned}$$

(3.2)

with L as in (2.1).

We consider first the case $p > \frac{3}{2}$:

Proposition 3.3

Let $p \in (\frac{3}{2}, \infty )$. Consider problem (3.2). Suppose that (B1)–(B2) hold and L is as in (2.1). Then the following conditions are necessary and sufficient in order that (3.2) has a unique solution u in $W^{1,p}(0, T; L^p(\Omega )) \cap L^p(0, T; W^{2,p}(\Omega ))$ with $\gamma u \in W^{1,p}(0, T; L^p(\Gamma )) \cap L^p(0, T; W^{2,p}(\Gamma ))$:

• (I) $f \in L^p(0, T; L^p(\Omega ))$;

• (II) $h \in L^p(0, T; L^p(\Gamma ))$;

• (III) $u_0 \in W^{2-2/p,p}(\Omega )$, $\gamma u_{0} \in W^{2-2/p,p}(\Gamma )$.

Proof

(I)–(II) are obviously necessary. The belonging of $u_0$ to $W^{2-2/p,p}(\Omega )$ follows from Theorem 3.1. From what we have observed in Remark 3.2, if we set $v:= \gamma u$, the identity $v(t) = \gamma [u(t)]$ can be intended pointwise. We deduce that v(0) must coincide with $\gamma u_0$. So from Proposition 2.3, we deduce the necessity of (III).

On the other hand, suppose that (I)–(III) hold. We consider the system

$$\begin{aligned} \begin{array}{ll} D_t v(t,\cdot ) = L v(t,\cdot ) + h(t,\cdot ), &{} t \in (0, T) \\ v(0,\cdot ) = \gamma u_0. \end{array} \end{aligned}$$

(3.3)

Then, by Proposition 2.3, (3.3) has a unique solution v in $W^{1,p}(0, T; L^p(\Gamma )) \cap L^{p}(0, T; W^{2,p}(\Gamma ))$. Now we consider the solution u to

$$\begin{aligned} \begin{array}{ll} D_t u(t,x) = \mathcal{A}u(t,x) + f(t,x), &{} (t,x) \in (0, T) \times \Omega , \\ \gamma u(t,\cdot ) = v(t,\cdot ) , &{} t \in (0, T), \\ u(0,x) = u_0(x), &{} x \in \Omega , \end{array} \end{aligned}$$

By Theorem 3.1, such u is the unique solution to (3.2).

$\square$

Now we consider the case $p < \frac{3}{2}$. In this case, (3.2) is underdetermined. It is more convenient to consider the problem

$$\begin{aligned} \begin{array}{ll} D_t u(t,x) = \mathcal{A}u(t,x) + f(t,x), &{} (t,x) \in (0, T) \times \Omega , \\ D_t \gamma u(t,\cdot ) = L \gamma u(t,\cdot ) + h(t,\cdot ), &{} t \in (0, T) \\ u(0,x) = u_0(x), &{} x \in \Omega , \\ (\gamma u)(0) = v_0. \end{array} \end{aligned}$$

(3.4)

The following result holds:

Proposition 3.4

Let $p \in (1, \frac{3}{2})$. Consider problem (3.4). Suppose that (B1)–(B2) hold and L is as in (2.1). Then the following conditions are necessary and sufficient in order that (3.4) has a unique solution u in $W^{1,p}(0, T; L^p(\Omega )) \cap L^p(0, T; W^{2,p}(\Omega ))$ with $\gamma u \in W^{1,p}(0, T; L^p(\Gamma )) \cap L^p(0, T; W^{2,p}(\Gamma ))$:

• (I) $f \in L^p(0, T; L^p(\Omega ))$;

• (II) $h \in L^p(0, T; L^p(\Gamma ))$;

• (III) $u_0 \in W^{2-2/p,p}(\Omega )$, $v_0 \in W^{2-2/p,p}(\Gamma )$.

Proof

The necessity of (I)–(III) follows immediately from Proposition 2.3 and Theorem 3.1. The proof of the sufficiency is the same as in Proposition 3.3. $\square$

Remark 3.5

As already observed in Remark 3.2, if $v(t) = \gamma u (t)$, the identity should be intended to be satisfied only for almost every t. In our case v should be extensible to an element of $C([0, T]; L^p(\Gamma ))$, but v(0) should not necessarily coincide with $\gamma u_0$; by the way, as $u_0 \in W^{2-2/p,p}(\Omega )$ and $2 - \frac{2}{p} < \frac{1}{p}$ if $p < \frac{3}{2}$, $u_0$ does not necessarily admit a trace on $\Gamma$.

It is convenient to reformulate together the results of Propositions 3.3 and 3.4:

Proposition 3.6

Let $p \in (1, \infty ) \setminus \{\frac{3}{2}\}$. Consider problem (3.4). Suppose that (B1)–(B2) hold and L is as in (2.1). Then the following conditions are necessary and sufficient in order that (3.4) has a unique solution u in $W^{1,p}(0, T; L^p(\Omega )) \cap L^p(0, T; W^{2,p}(\Omega ))$ with $\gamma u \in W^{1,p}(0, T; L^p(\Gamma )) \cap L^p(0, T; W^{2,p}(\Gamma ))$:

• (I) $f \in L^p(0, T; L^p(\Omega ))$;

• (II) $h \in L^p(0, T; L^p(\Gamma ))$;

• (III) $u_0 \in W^{2-2/p,p}(\Omega )$, $v_0 \in W^{2-2/p,p}(\Gamma )$ and, in case $p > \frac{3}{2}$, $\gamma u_0 = v_0$.

We proceed with some useful estimates.

Lemma 3.7

Consider problem (3.4). Suppose that (B1)–(B2) hold and L is as in (2.1). Let $p \in (1, \infty ) \setminus \{\frac{3}{2}\}$, $T_0 \in \mathbb {R}^+$, $0 < T \le T_0$. Suppose that $f \in L^p(0, T; L^p(\Omega ))$, $h \in L^p(0, T; L^p(\Gamma ))$, $u_0 \in W^{2-2/p,p}(\Omega )$, $v_0 \in W^{2-2/p,p}(\Gamma )$ and, in case $p > \frac{3}{2}$, $\gamma u_0 = v_0$. Then there exists $C(T_0)$ in $\mathbb {R}^+$ such that

$$\begin{aligned}&\Vert D_t u\Vert _{L^p(0, T; L^p(\Omega ))} + \Vert u\Vert _{L^p(0, T; W^{2,p}(\Omega ))} + \Vert D_t \gamma u\Vert _{L^p(0, T; L^p(\Gamma ))} + \Vert \gamma u\Vert _{L^p(0, T; W^{2,p}(\Gamma ))} \\&\quad \le C(T_0) (\Vert f\Vert _{L^p(0, T; L^p(\Omega ))} + \Vert h\Vert _{L^p(0, T; L^p(\Gamma ))} + \Vert u_0\Vert _{W^{2-2/p,p}(\Omega )} + \Vert v_0\Vert _{W^{2-2/p,p}(\Gamma )}). \end{aligned}$$

Proof

We set, for $t \in (0, T_0)$,

$$\begin{aligned} F(t, \cdot )= & {} \left\{ \begin{array}{lll} f(t, \cdot ) &{} {\mathrm{if }} &{} t \in (0, T), \\ 0 &{} {\mathrm{if }} &{} t \in [T, T_0), \end{array} \right. \\ H(t, \cdot )= & {} \left\{ \begin{array}{lll} h(t, \cdot ) &{} {\mathrm{if }} &{} t \in (0, T), \\ 0 &{} {\mathrm{if }} &{} t \in [T, T_0), \end{array} \right. \end{aligned}$$

and consider the problem

$$\begin{aligned} \begin{array}{ll} D_t U(t,x) = \mathcal{A}U(t,x) + F(t,x), &{} (t,x) \in (0, T_0) \times \Omega , \\ D_t \gamma U(t,\cdot ) = L \gamma U(t,\cdot ) + H(t,\cdot ), &{} t \in (0, T_0) \\ U(0,x) = u_0(x), &{} x \in \Omega , \\ (\gamma U)(0) = v_0. \end{array} \end{aligned}$$

(3.5)

By Proposition 3.6, (3.5) has a unique solution U in $W^{1,p}(0, T_0; L^p(\Omega )) \cap L^p(0, T_0; W^{2,p}(\Omega ))$ with $\gamma U \in W^{1,p}(0, T_0; L^p(\Gamma )) \cap L^p(0, T_0; W^{2,p}(\Gamma ))$, which is clearly an extension of u. We deduce

$$\begin{aligned}&\Vert D_t u\Vert _{L^p(0, T; L^p(\Omega ))} + \Vert u\Vert _{L^p(0, T; W^{2,p}(\Omega ))} + \Vert D_t \gamma u\Vert _{L^p(0, T; L^p(\Gamma ))} + \Vert \gamma u\Vert _{L^p(0, T; W^{2,p}(\Gamma ))} \\&\quad \le \Vert D_t U\Vert _{L^p(0, T_0; L^p(\Omega ))} + \Vert U\Vert _{L^p(0, T_0; W^{2,p}(\Omega ))} + \Vert D_t \gamma U\Vert _{L^p(0, T_0; L^p(\Gamma ))} \\&\qquad + \Vert \gamma U\Vert _{L^p(0, T_0; W^{2,p}(\Gamma ))} \\&\quad \le C(T_0) (\Vert F\Vert _{L^p(0, T; L^p(\Omega ))} + \Vert H\Vert _{L^p(0, T; L^p(\Gamma ))} + \Vert u_0\Vert _{W^{2-2/p,p}(\Omega )} + \Vert v_0\Vert _{W^{2-2/p,p}(\Gamma )}) \\&\quad = C(T_0) (\Vert f\Vert _{L^p(0, T; L^p(\Omega ))} + \Vert h\Vert _{L^p(0, T; L^p(\Gamma ))} + \Vert u_0\Vert _{W^{2-2/p,p}(\Omega )} + \Vert v_0\Vert _{W^{2-2/p,p}(\Gamma )}). \end{aligned}$$

$\square$

Lemma 3.8

Suppose that the assumptions of Lemma 3.7 are fulfilled. Suppose that $u_0 = 0$ and let $\theta \in [0, 2]$. Then there exists $C(T_0,\theta ) > 0$ such that

$$\begin{aligned}&\Vert u\Vert _{L^p(0, T; W^{\theta ,p}(\Omega ))} \\&\quad \le C(T_0) T^{1-\theta /2} (\Vert f\Vert _{L^p(0, T; L^p(\Omega ))} + \Vert h\Vert _{L^p(0, T; L^p(\Gamma ))} + \Vert v_0\Vert _{W^{2-2/p,p}(\Gamma )}). \end{aligned}$$

Proof

Consider first the case $\theta = 0$. Then, as $u_0 = 0$, $u = 1 *D_tu$. It follows from Young’s inequality and Lemma 3.7 that

$$\begin{aligned}&\Vert u\Vert _{L^p(0, T; L^p(\Omega ))} \le T \Vert D_t u\Vert _{L^p(0, T; L^p(\Omega ))}\\&\quad \le C(T_0) T (\Vert f\Vert _{L^p(0, T; L^p(\Omega ))} + \Vert h\Vert _{L^p(0, T; L^p(\Gamma ))} + \Vert v_0\Vert _{W^{2-2/p,p}(\Gamma )}). \end{aligned}$$

In general, there exists $C(\theta )>$ such that, for any $z \in W^{2,p}(\Omega )$,

$$\begin{aligned} \Vert z\Vert _{W^{\theta ,p}(\Omega )} \le C(\theta ) \Vert z\Vert _{L^p(\Omega )}^{1-\theta /2} \Vert z\Vert _{W^{2,p}(\Omega )}^{\theta /2}, \end{aligned}$$

as $W^{\theta ,p}(\Omega )$ coincides with the real interpolation space $(L^p(\Omega ), W^{2,p}(\Omega ))_{\theta /2, p}$ in case $\theta \ne 1$, with the complex interpolation space $(L^p(\Omega ), W^{2,p}(\Omega ))_{[\frac{1}{2}]}$ in case $\theta = \frac{1}{2}$ (see [21]). We deduce that

$$\begin{aligned}&\Vert u\Vert _{L^p(0, T; W^{\theta ,p}(\Omega ))} \le C(\theta ) \left( \int _0^T \Vert u(t)\Vert _{L^p(\Omega )}^{p(1-\theta /2)} \Vert u(t)\Vert _{W^{2,p}(\Omega )}^{p\theta /2} dt\right) ^{1/p} \\&\quad \le C(\theta ) \Vert u\Vert _{L^p(0, T; L^{p}(\Omega ))}^{1-\theta /2} \Vert u\Vert _{L^p(0, T; W^{2,p}(\Omega ))}^{\theta /2}. \end{aligned}$$

So the conclusion follows from the case $\theta = 0$ and Lemma 3.7. $\square$

Now we introduce an operator E of order not exceeding one, with coefficients in $C^1(\overline{\Omega })$:

$$\begin{aligned} Eu(x) = \sum _{j=1}^n e_j(x) D_{x_j}u(x) + e_0(x) u(x) \end{aligned}$$

(3.6)

and the following system:

$$\begin{aligned} \begin{array}{ll} D_t u(t,x) = \mathcal{A}u(t,x) + f(t,x), &{} (t,x) \in (0, T) \times \Omega , \\ D_t \gamma u(t,\cdot ) = L \gamma u(t,\cdot ) + \gamma Eu(t,\cdot ) + h(t,\cdot ), &{} t \in (0, T) \\ u(0,x) = u_0(x), &{} x \in \Omega , \\ (\gamma u)(0) = v_0. \end{array} \end{aligned}$$

(3.7)

We show the following

Theorem 3.9

Let $p \in (1, \infty ) \setminus \{\frac{3}{2}\}$. Consider problem (3.7). Suppose that (B1)–(B2) hold, L is as in (2.1) and E is as in (3.6) with coefficients in $C^1(\overline{\Omega })$. Then the following conditions are necessary and sufficient in order that (3.7) has a unique solution u in $W^{1,p}(0, T; L^p(\Omega )) \cap L^p(0, T; W^{2,p}(\Omega ))$ with $\gamma u \in W^{1,p}(0, T; L^p(\Gamma )) \cap L^p(0, T; W^{2,p}(\Gamma ))$:

• (I) $f \in L^p(0, T; L^p(\Omega ))$;

• (II) $h \in L^p(0, T; L^p(\Gamma ))$;

• (III) $u_0 \in W^{2-2/p,p}(\Omega )$, $v_0 \in W^{2-2/p,p}(\Gamma )$ and, in case $p > \frac{3}{2}$, $\gamma u_0 = v_0$.

Proof

The fact that (I)–(III) are necessary can be shown with the same arguments as in the proofs of Propositions 3.3 and 3.4.

We show that they are also sufficient. We fix $\theta \in (1+ \frac{1}{p}, 2)$. We observe that, by classical trace theorems, $u \rightarrow \gamma Eu$ belongs to $\mathcal{L}(W^{\theta ,p}(\Omega ), L^p(\Gamma ))$. We take $\tau \in (0, T]$ and consider the system

$$\begin{aligned} \begin{array}{ll} D_t u(t,x) = \mathcal{A}u(t,x) + f(t,x), &{} (t,x) \in (0, \tau ) \times \Omega , \\ D_t \gamma u(t,\cdot ) = L \gamma u(t,\cdot ) + \gamma EU(t,\cdot ) + h(t,\cdot ), &{} t \in (0, \tau ), \\ u(0,x) = u_0(x), &{} x \in \Omega , \\ (\gamma u)(0) = v_0, \end{array} \end{aligned}$$

(3.8)

with $U \in L^p(0, \tau ; W^{\theta ,p}(\Omega ))$. By Proposition 3.6, (3.8) has a unique solution $u = S(U)$ in $W^{1,p}(0, \tau ; L^p(\Omega )) \cap L^p(0, \tau ; W^{2,p}(\Omega ))$ with $\gamma u \in W^{1,p}(0, \tau ; L^{p}(\Gamma )) \cap L^p(0, \tau ; W^{2,p}(\Gamma ))$. If $U_j \in L^p(0, \tau ; W^{\theta ,p}(\Omega ))$ ($j \in \{1,2\}$), we set $u_j:= S(U_j)$. Then $u_1 - u_2$ solves the system

$$\begin{aligned} \begin{array}{ll} D_t (u_1 - u_2)(t,x) = \mathcal{A}(u_1 - u_2) (t,x), &{} (t,x) \in (0, \tau ) \times \Omega , \\ D_t \gamma (u_1 - u_2) (t,\cdot ) = L \gamma (u_1 - u_2) (t,\cdot ) + \gamma E(U_1 - U_2) (t,\cdot ), &{} t \in (0, \tau ), \\ (u_1 - u_2) (0,x) = 0, &{} x \in \Omega , \\ \gamma (u_1 - u_2)(0) = 0. \end{array} \end{aligned}$$

(3.9)

We deduce from Lemma 3.8 the estimate

$$\begin{aligned}&\Vert u_1 - u_2\Vert _{L^p(0, \tau ; W^{\theta ,p}(\Omega ))} \\&\quad \le C(T) \tau ^{1-\theta /2} \Vert \gamma E(U_1 - U_2)\Vert _{L^p(0, \tau ; L^p(\Gamma ))} \le C_1(T) \tau ^{1-\theta /2} \Vert U_1 - U_2\Vert _{L^p(0, \tau ; W^{\theta ,p}(\Omega ))}. \end{aligned}$$

So, if we choose $\tau$ so small that $C_1(T) \tau ^{1-\theta /2} < 1$, S has a unique fixed point in $L^p(0, \tau ; W^{\theta ,p}(\Omega ))$. We deduce that (3.8) has a unique solution u in $W^{1,p}(0, \tau ; L^p(\Omega )) \cap L^p(0, \tau ; W^{2,p}(\Omega ))$ with $\gamma u$ in $W^{1,p}(0, \tau ; L^p(\Gamma )) \cap L^p(0, \tau ; W^{2,p}(\Gamma ))$. Observe that $\tau$ can be chosen independently of $f, h, u_0, v_0$.

Now we show that, in case $f \equiv 0$, $h \equiv 0$, $u_0 = 0$, $v_0 = 0$, the unique solution u in $W^{1,p}(0, T; L^p(\Omega )) \cap L^p(0, T; W^{2,p}(\Omega ))$ with $\gamma u$ in $W^{1,p}(0, T; L^p(\Gamma )) \cap L^p(0, T; W^{2,p}(\Gamma ))$ is $u \equiv 0$. This is true (by the uniqueness of the fixed point for S), if we replace T by $\tau$ sufficiently small. Assume that there exists a nontrivial solution u in (0, T). We set

$$\begin{aligned} \sigma := \inf \{t \in [0, T]: u(t,\cdot ) \ne 0\}. \end{aligned}$$

As $u \in C([0, T]; W^{2-2/p,p}(\Omega ))$ and $u(0,\cdot ) = 0$, $\sigma \in [0, T)$ and $u(\sigma ,\cdot ) = 0$. Moreover, $\gamma u(t,\cdot ) = 0$ for almost every t in $[0, \sigma )$. As $\gamma u \in C([0, T]; W^{2-2/p,p}(\Gamma ))$, we deduce that $(\gamma u) (\sigma , \cdot ) = 0$. So, if $\tau > 0$, and $\sigma + \tau \le T$, $w(t):= u(\sigma + t)$ solves the system

$$\begin{aligned} \begin{array}{ll} D_t w(t,x) = \mathcal{A}w (t,x), &{} (t,x) \in (0, \tau ) \times \Omega , \\ D_t \gamma w (t,\cdot ) = L \gamma w (t,\cdot ) + \gamma E w (t,\cdot ), &{} t \in (0, \tau ), \\ w (0,x) = 0, &{} x \in \Omega , \\ (\gamma w)(0) = 0. \end{array} \end{aligned}$$

If $\tau$ is sufficiently small, we deduce $w(t,\cdot ) = 0$ for any $t \in [0, \tau ]$, so that $u(t,\cdot ) = 0$ for any $t \in [0, \sigma + \tau ]$, in contradiction with the definition of $\sigma$.

Finally, we show the existence of a global solution. We have already proved the existence of a solution z in some interval $[0, \tau ]$, independent of the data. Suppose that $\tau < T$. We extend the solution to $[0, (2\tau ) \wedge T]$. We have that $z(\tau ,\cdot ) \in W^{2-2/p,p}(\Omega )$, $(\gamma z)(\tau ) \in W^{2-2/p,p}(\Gamma )$. In case $p > \frac{3}{2}$ we have also

$$\begin{aligned} \gamma [z(\tau )] = (\gamma z)(\tau ). \end{aligned}$$

So we consider the system

$$\begin{aligned} \begin{array}{ll} D_t w(t,x) = \mathcal{A}w(t,x) + f(\tau + t,x), &{} (t,x) \in (0, \tau \wedge (T-\tau )) \times \Omega , \\ D_t \gamma w(t,\cdot ) = L \gamma w(t,\cdot ) + \gamma Ew(t,\cdot ) + h(\tau + t,\cdot ), &{} t \in (0, \tau \wedge (T-\tau )) \\ w(0,x) = z(\tau ,x), &{} x \in \Omega , \\ (\gamma w)(0) = (\gamma z)(\tau ). \end{array} \end{aligned}$$

(3.10)

(3.10) has a unique solution w in $W^{1,p}(0, \tau \wedge (T-\tau ); L^p(\Omega )) \cap L^p(0, \tau \wedge (T-\tau ); W^{2,p}(\Omega ))$, with $\gamma w$ in $W^{1,p}(0, \tau \wedge (T-\tau ); L^p(\Gamma )) \cap L^p(0, \tau \wedge (T-\tau ); W^{2,p}(\Gamma ))$. If we set

$$\begin{aligned} u(t,\cdot ) := \left\{ \begin{array}{lll} z(t,\cdot ) &{} {\mathrm{if }} &{} t \in (0, \tau ], \\ w(t - \tau ,\cdot ) &{} {\mathrm{if }} &{} t \in (\tau , \tau \wedge (T-\tau )], \end{array} \right. \end{aligned}$$

it is easily seen that $u \in W^{1,p}(0, (2\tau )\wedge T; L^p(\Omega )) \cap L^p(0, (2\tau ) \wedge T; W^{2,p}(\Omega ))$, with $\gamma u$ in $W^{1,p}(0, (2\tau ) \wedge T ; L^p(\Gamma )) \cap L^p(0, (2\tau ) \wedge T; W^{2,p}(\Gamma ))$ and solves (3.7) , if we replace T with $(2\tau ) \wedge T$. In case $2\tau < T$, we iterate the argument extending the solution to $(3\tau ) \wedge T$. It is clear that in a finite number of steps we reach the conclusion. $\square$

Remark 3.10

It is easily seen that the conclusion of Theorem 3.9 still holds if we replace $\gamma E$ with an arbitrary operator F which is bounded from $W^{\theta ,p}(\Omega )$ to $L^p(\Gamma )$, for some $\theta$ in [0, 2).

4 Generation of an analytic semigroup

Now we prove a result of generation of an analytic semigroup.

Theorem 4.1

Suppose that the conditions (B1)–(B2) hold, L is as in (2.1) and E is as in (3.6), with coefficients $e_j$ in $C^1(\overline{\Omega })$ ($0 \le j \le n$). Let $p \in (1, \infty )$. Consider the space $X_p:= L^p(\Omega ) \times L^p(\Gamma )$ and define the following operator $G_p$ acting on $X_p$:

$$\begin{aligned} \begin{array}{ll} D(G_p) := \{(u, \gamma u) : u \in W^{2,p}(\Omega ), \gamma u \in W^{2,p}(\Gamma )\}, \\ G_p(u, \gamma u) := (\mathcal{A}u, L \gamma u + \gamma Eu). \end{array} \end{aligned}$$

(4.1)

Then $G_p$ is the infinitesimal generator of an analytic semigroup in $X_p$.

In the proof, we shall employ the following

Lemma 4.2

For any $p \in [1, \infty ]$, there exists a linear operator $P : W^{2,p}(\Gamma ) \rightarrow W^{2,p}(\Omega )$ such that $\gamma Pg = g$ for any $g \in W^{2,p}(\Gamma )$ and, for some $C > 0$, independent of g,

$$\begin{aligned} \Vert Pg\Vert _{L^p(\Omega )} \le C \Vert g\Vert _{L^p(\Gamma )}, \quad \Vert Pg\Vert _{W^{2,p}(\Omega )} \le C \Vert g\Vert _{W^{2,p}(\Gamma )}. \end{aligned}$$

Proof

Firstly, P can be constructed in the particular case $\Omega = \mathbb {R}^{n-1} \times \mathbb {R}^+$, $\Gamma = \mathbb {R}^{n-1} \times \{0\}$, setting, for $g \in W^{2,p}(\Gamma )$,

$$\begin{aligned} Pg(x',x_n) := g(x',0) \phi (x_n), \end{aligned}$$

with $\phi \in C^2([0, \infty ))$, $\phi (t) = 1$ if $0 \le t \le 1$, $\phi (t) = 0$ if $t \ge 2$. The general case can be reduced to this one, employing partitions of unity and changes of variable. $\square$

Remark 4.3

It can be easily seen that P can be extended to a linear bounded operator from $L^p(\Gamma )$ to $L^p(\Omega )$, for any p in $[1, \infty ]$, and from $C^\alpha (\Gamma )$ to $C^\alpha (\overline{\Omega })$ for any $\alpha$ in [0, 2].

Proof of Theorem 4.1

Let $\lambda \in \mathbb {C}$, $Re(\lambda ) \ge 0$. We shall show that the problem

$$\begin{aligned} \lambda (u, \gamma u) - G_p (u, \gamma u) = (f,h) \end{aligned}$$

(4.2)

has a unique solution $(u, \gamma u)$ in $D(G_p)$ if $|\lambda |$ is sufficiently large. Moreover, there exists $C > 0$, independent of $\lambda$ and (f, h), such that

$$\begin{aligned} \Vert (u, \gamma u)\Vert _{X_p} \le C |\lambda |^{-1} \Vert (f,h)\Vert _{X_p}. \end{aligned}$$

Observe that (4.2) is equivalent to

$$\begin{aligned} \left\{ \begin{array}{l} \lambda u(x) - Au(x) = f(x), \quad x \in \Omega , \\ \lambda \gamma u(x') - L \gamma u(x') -\gamma Eu(x') = h(x'), \quad x ' \in \Gamma . \end{array} \right. \end{aligned}$$

(4.3)

We begin by considering the particular case $E = 0$, that is,

$$\begin{aligned} \left\{ \begin{array}{l} \lambda u(x) - Au(x) = f(x), \quad x \in \Omega , \\ \lambda \gamma u(x') - L \gamma u(x') = h(x'), \quad x ' \in \Gamma . \end{array} \right. \end{aligned}$$

(4.4)

By Theorem 2.1, there exists $R_1$ positive such that, if $|\lambda |\ge R_1$, the equation

$$\begin{aligned} \lambda v(x') - L v(x') = h(x'), \quad x ' \in \Gamma \end{aligned}$$

has a unique solution v in $W^{2,p}(\Gamma )$. Moreover, for some $C_1$ positive, independent of $\lambda$ and h,

$$\begin{aligned} |\lambda | \Vert v\Vert _{L^p(\Gamma )} + \Vert v\Vert _{W^{2,p}(\Gamma )} \le C_1 \Vert h\Vert _{L^p(\Gamma )}. \end{aligned}$$

Now we consider the system

$$\begin{aligned} \begin{array}{l} \lambda u(x) - Au(x) = f(x), \quad x \in \Omega , \\ \gamma u(x') = v(x'), \quad x ' \in \Gamma . \end{array} \end{aligned}$$

(4.5)

By [20], Chapter 3.8, there exists $R \ge R_1$ such that (4.5) has a unique solution u in $W^{2,p}(\Omega )$. Moreover, for some $C_2 > 0$ independent of $\lambda$ and f, for any $V \in W^{2,p}(\Omega )$ such that $\gamma V = v$,

$$\begin{aligned} |\lambda | \Vert u\Vert _{L^p(\Omega )} + \Vert u\Vert _{W^{2,p}(\Omega )} \le C_2 (\Vert f\Vert _{L^p(\Omega )} + \Vert V\Vert _{W^{2,p}(\Omega )} + |\lambda | \Vert V\Vert _{L^{p}(\Omega )}). \end{aligned}$$

Choosing $V = Pv$, with P as in Lemma 4.2, we deduce

$$\begin{aligned}&|\lambda | \Vert u\Vert _{L^p(\Omega )} + \Vert u\Vert _{W^{2,p}(\Omega )} \le C_2 (\Vert f\Vert _{L^p(\Omega )} + \Vert Pv\Vert _{W^{2,p}(\Omega )} + |\lambda | \Vert Pv\Vert _{L^{p}(\Omega )}) \nonumber \\&\quad \le C_3 (\Vert f\Vert _{L^p(\Omega )} + \Vert v\Vert _{W^{2,p}(\Gamma )} + |\lambda | \Vert v\Vert _{L^{p}(\Gamma )}) \nonumber \\&\quad \le C_4 (\Vert f\Vert _{L^p(\Omega )} + \Vert h\Vert _{L^p(\Gamma )}). \end{aligned}$$

(4.6)

Now we consider the general case $E \ne 0$. For any $\theta \in [0, 2]$, it follows from (4.6) that

$$\begin{aligned} \Vert u\Vert _{W^{\theta ,p}(\Omega )} \le C(\theta ) |\lambda |^{\theta /2 - 1} (\Vert f\Vert _{L^p(\Omega )} + \Vert h\Vert _{L^p(\Gamma )}). \end{aligned}$$

(4.7)

Now we fix $\theta \in (1+ \frac{1}{p}, 2)$ and, for $U \in W^{\theta ,p}(\Omega )$, we consider the system

$$\begin{aligned} \begin{array}{l} \lambda u(x) - Au(x) = f(x), \quad x \in \Omega , \\ \lambda \gamma u(x') - L \gamma u(x') = \gamma EU(x') + h(x'), \quad x ' \in \Gamma . \end{array} \end{aligned}$$

(4.8)

If $|\lambda |$ is sufficiently large, there exists a unique solution $u = u(U)$ in $W^{2,p}(\Omega )$. We shall think of $U \rightarrow u(U)$ as an operator from $W^{\theta ,p}(\Omega )$ into itself. If $u_j = u(U_j)$ ($j \in \{1,2\}$), we have

$$\begin{aligned} \begin{array}{l} \lambda (u_1 - u_2) (x) - A(u_1 - u_2)(x) = 0, \quad x \in \Omega , \\ \lambda \gamma (u_1 - u_2)(x') - L \gamma (u_1 - u_2)(x') = \gamma E(U_1 - U_2)(x'), \quad x ' \in \Gamma , \end{array} \end{aligned}$$

so that, by (4.7),

$$\begin{aligned}&\Vert u_1 - u_2\Vert _{W^{\theta ,p}(\Omega )} \le C(\theta ) |\lambda |^{\theta /2 - 1} \Vert \gamma E (U_1 - U_2)\Vert _{L^p(\Gamma )})\\&\quad \le C_1(\theta ) |\lambda |^{\theta /2 - 1} \Vert U_1 - U_2\Vert _{W^{\theta ,p}(\Omega )}. \end{aligned}$$

We deduce that $U \rightarrow u(U)$ is a contraction if $|\lambda |$ is sufficiently large. We conclude that, for such choice of $\lambda$, (4.3) has a unique solution u. Moreover, from (4.7),

$$\begin{aligned}&\Vert u\Vert _{W^{\theta ,p}(\Omega )} \le C(\theta ) |\lambda |^{\theta /2 - 1} (\Vert f\Vert _{L^p(\Omega )} + \Vert h\Vert _{L^p(\Gamma )} + \Vert \gamma Eu\Vert _{L^p(\Gamma )}) \\&\quad \le C_1 |\lambda |^{\theta /2 - 1} (\Vert f\Vert _{L^p(\Omega )} + \Vert h\Vert _{L^p(\Gamma )} + \Vert u\Vert _{W^{\theta ,p}(\Omega )}), \end{aligned}$$

implying

$$\begin{aligned} \Vert u\Vert _{W^{\theta ,p}(\Omega )} \le C_2 (\Vert f\Vert _{L^p(\Omega )} + \Vert h\Vert _{L^p(\Gamma )}) \end{aligned}$$

if $|\lambda |$ is sufficiently large. We deduce that

$$\begin{aligned}&|\lambda | \Vert u\Vert _{L^p(\Omega )} + \Vert u\Vert _{W^{2,p}(\Omega )} + |\lambda | \Vert \gamma u\Vert _{L^p(\Gamma )} + \Vert \gamma u\Vert _{W^{2,p}(\Gamma )} \\&\quad \le C_3 (\Vert f\Vert _{L^p(\Omega )} + \Vert h\Vert _{L^p(\Gamma )} + \Vert u\Vert _{W^{\theta ,p}(\Omega )}) \\&\quad \le C_4 (\Vert f\Vert _{L^p(\Omega )} + \Vert h\Vert _{L^p(\Gamma )}). \end{aligned}$$

The proof is complete. $\square$

Remark 4.4

Here also the assertion of Theorem 4.1 holds replacing $\gamma E$ with any operator F which is bounded from $W^{\theta ,p}(\Omega )$ to $L^p(\Omega )$, for some $\theta$ in [0, 2).

Remark 4.5

We have chosen to prove Theorem 4.1 estimating directly the resolvent $(\lambda - G_p)^{-1}$. In fact, the result can be obtained quite quickly, applying Theorem 3.1 together with a nice theorem by G. Dore (see [5]).

5 General Wentzell boundary conditions

In [7] and [8], the authors considered the problem

$$\begin{aligned} \left\{ \begin{array}{ll} D_t u(t,x) = M u(t,x), &{} \quad (t,x) \in (0, T) \times \Omega , \\ M u(t,x') + \beta (x') \partial _\nu ^a u(t,x') - q \beta (x') L_\partial \gamma u(t,x')\\ + q \tilde{a} (x') \cdot \nabla _\tau \gamma u(t,x') + \tilde{r}(x') \gamma u(t,x') = 0, &{} (t,x') \in (0, T) \times \Gamma , \\ u(0,x) = u_0(x), &{} \quad x \in \Omega . \end{array}\right. \end{aligned}$$

(5.1)

Here

$$\begin{aligned} Mu = \sum _{i,j=1}^n \partial _i(a_{ij}(\cdot ) \partial _j u) + \sum _{i=1}^n c_i \partial _i u + ru, \\ \end{aligned}$$

with $a_{ij}$ real valued, $a_{ij} = a_{ji}$, $\sum _{i,j=1}^n a_{ij}(x) \xi _i \xi _j \ge \alpha _0 |\xi |^2$ for any $(x,\xi ) \in \overline{\Omega }\times \mathbb {R}^n$ for some $\alpha _0$ positive, $\beta$ positive, $\partial _\nu ^a = \sum _{i,j=1}^n a_{ij}(\cdot ) \nu _i \partial _j u$, $q \in \mathbb {R}^+$, $a_{ij}$, $c_i$, r defined and sufficiently regular on $\overline{\Omega }$, $\tilde{a}$ and $\tilde{r}$ defined and sufficiently regular on $\Gamma$. $\nabla _\tau$ stands for the gradient operator in $\Gamma$ and

$$\begin{aligned} L_\partial \gamma u = \mathrm{div} (B(x) \nabla _\tau \gamma u) \end{aligned}$$

is an operator of the form considered in Example 2.4. Of course, the Riemannian structure in $\Gamma$ is that inherited as an embedded submanifold of $\mathbb {R}^n$. The open set $\Omega$ is not assumed to be bounded. System (5.1) is studied in the following way: it is introduced the following operator $\tilde{M}_p$:

$$\begin{aligned} \begin{array}{l} D(\tilde{M}_p) := \{(u, \gamma u) : u \in C_c^2(\overline{\Omega }), \gamma M u + \beta \partial _\nu ^a u - q \beta L_\partial \gamma u + q \tilde{a} \cdot \nabla _\tau u + \tilde{r} \gamma u = 0\}, \\ \tilde{M}_p(u, \gamma u) = (Mu, \gamma Mu) = (Au, - \beta \partial _\nu ^a u + q \beta L_\partial \gamma u - q \tilde{a} \cdot \nabla _\tau u - \tilde{r} \gamma u ). \end{array} \end{aligned}$$

(5.2)

Then it is proved that the closure of ${\hat{M_p}}$ in $L^p(\Omega ) \times L^p(\Gamma )$ generates an analytic semigroup. It follows that, for every $u_0$ belonging to the domain of ${\hat{M_p}}$, (5.1) has a solution (in some generalised sense).

Following this idea, we can consider the problem

$$\begin{aligned} \begin{array}{ll} D_t u(t,x) = \mathcal{A}u(t,x), &{} (t,x) \in (0, T) \times \Omega , \\ \gamma \mathcal{A}u(t,\cdot ) - L \gamma u(t,\cdot ) - \gamma Eu(t,\cdot ) = 0, &{} t \in (0, T) \\ u(0,x) = u_0(x), &{} x \in \Omega , \end{array} \end{aligned}$$

(5.3)

with the assumptions of Theorem 3.9: we introduce the following operator $M_p$, for $p \in (1, \infty )$:

$$\begin{aligned} \begin{array}{l} D(M_p) := \{(u, \gamma u) : u \in C^2(\overline{\Omega }), \gamma \mathcal{A}u - L \gamma u - \gamma Eu = 0\}, \\ M_p(u, \gamma u) = (\mathcal{A}u, \gamma \mathcal{A}u) = (Au, L \gamma u + \gamma Eu ). \end{array} \end{aligned}$$

(5.4)

We show the following

Theorem 5.1

Suppose that (B1)–(B2) hold, L is as in (2.1) and E is as in (3.6) with coefficients in $C^1(\overline{\Omega })$. Moreover,

• (a) $\Gamma = \partial \Omega$ is of class $C^{2+\alpha }$, for some $\alpha \in (0, 1)$;

• (b) the coefficients $a_{ij}$, $b_j$, c of $\mathcal{A}$ ($1 \le i, j \le n$) are of class $C^\alpha (\overline{\Omega })$;

• (c) the coefficients $l_{\alpha , \Phi }$ in (2.1) are in $C^\alpha (U)$;

• (d) the coefficients $e_j$ ($0 \le j \le n$) of E (see (3.6) are in $C^\alpha (\overline{\Omega }))$.

Then, if $1< p < \infty$, $M_p$ is closable in $X_p = L^p(\Omega ) \times L^p(\partial \Omega )$ and its closure coincides with $G_p$ (defined in (4.1)).

Proof

We have to prove the following:

$\forall (u, \gamma u) \in D(G_p)$there exists a sequence$((u_k, \gamma u_k))_{k \in \mathbb {N}}$in$D(M_p)$such that

$$\begin{aligned} \Vert (u_k, \gamma u_k) - (u, \gamma u)\Vert _{X_p} + \Vert M_p(u_k, \gamma u_k) - G_p(u, \gamma u)\Vert _{X_p} \rightarrow 0 \quad (k \rightarrow \infty ). \end{aligned}$$

We start by proving three steps.

Step 1: Let$(u, \gamma u) \in D(G_p)$be such that, for some$\lambda \in \mathbb {C}$, $(\lambda - G_p)(u, \gamma u) \in C^\alpha (\overline{\Omega }) \times C^\alpha (\Gamma )$. Then$(u, \gamma u) \in C^{2+\alpha }(\overline{\Omega }) \times C^{2+\alpha }(\Gamma )$

We start by considering the case $E = 0$. Then, $\lambda \gamma u - L \gamma u = h \in C^\alpha (\Gamma )$ and so $\gamma u \in C^{2+\alpha }(\Gamma )$ (see [11], Theorem 2.1). So $u \in W^{2,p}(\Omega )$ and solves the system

$$\begin{aligned} \begin{array}{ll} (\lambda - \mathcal{A})u = f \in C^\alpha (\overline{\Omega }), \\ \gamma u \in C^{2+\alpha }(\Gamma ), \end{array} \end{aligned}$$

again implying $u \in C^{2+\alpha } (\overline{\Omega })$.

Now we consider the case $E \ne 0$, employing a bootstrap argument. Suppose that we have shown that $(u, \gamma u) \in W^{2,q}(\Omega ) \times W^{2,q}(\Gamma )$ for some $q \ge p$. Then $\gamma Eu \in W^{1-1/q,q}(\Gamma )$. Assume that

$$\begin{aligned} q \ge \frac{n}{1-\alpha }. \end{aligned}$$

Then $W^{1-1/q,q}(\Gamma ) \hookrightarrow C^\alpha (\Gamma )$, so that $(\lambda u - \mathcal{A}u, \lambda \gamma u - L \gamma u) \in C^\alpha (\overline{\Omega }) \times C^\alpha (\Gamma )$ and the conclusion follows.

Suppose $q > n$. Then $\gamma Eu \in C^{\alpha '}(\Gamma )$, for some $\alpha ' \in (0, 1)$. It follows that $(\lambda u - \mathcal{A}u, \lambda \gamma u - L \gamma u) \in C^{\alpha '} (\overline{\Omega }) \times C^{\alpha '} (\Gamma )$. This implies $u \in C^{2+\alpha '}(\overline{\Omega })$, so that $\gamma Eu \in C^{1+\alpha '}(\Gamma ) \hookrightarrow C^\alpha (\Gamma )$, and we have again the conclusion.

Suppose $q < n$. Then $\gamma Eu \in W^{1-1/q,q}(\Gamma ) \hookrightarrow L^{\frac{n-1}{n-q}q}(\Gamma )$. We deduce $(\lambda u - \mathcal{A}u, \lambda \gamma u - L \gamma u) \in C^\alpha (\overline{\Omega }) \times L^{\frac{n-1}{n-q}q}(\Gamma )$, implying $(u, \gamma u) \in W^{2,q_1}(\Omega ) \times W^{2,q_1}(\Gamma )$, with $q_1 = \frac{n-1}{n-q}q > q$. If $q_1 > n$, we can conclude. Otherwise, we deduce that $(u, \gamma u) \in W^{2,q_2}(\Omega ) \times W^{2,q_2}(\Gamma )$, with $q_2 = \frac{n-1}{n-q_1}q_1 > q_1$. We can iterate the process until we get the belonging of $(u, \gamma u)$ to $W^{2,r}(\Omega ) \times W^{2,r}(\Gamma )$ for some $r > n$. This can be necessarily achieved in a finite number of steps. Otherwise, we should obtain the belonging of $(u, \gamma u)$ to $W^{2,q_k}(\Omega ) \times W^{2,q_k}(\Gamma )$ with $q< q_1< \dots< q_k< q_{k+1} \le \dots < n$ for a certain sequence $(q_k)_{k \in \mathbb {N}}$. But this is not possible, because

$$\begin{aligned} q_k = \frac{n-1}{n-q_{k-1}}q_{k-1} \ge \left( \frac{n-1}{n-q}\right) ^k q \rightarrow \infty \quad (k \rightarrow \infty ), \end{aligned}$$

a contradiction.

Step 2: Let$(u, \gamma u) \in D(G_p)$be such that, for some$\lambda \in \mathbb {C}$, $(\lambda - G_p)(u, \gamma u) = (f,h) \in C^\alpha (\overline{\Omega }) \times C^\alpha (\Gamma )$, with$\alpha \in (0, 1)$and$h = \gamma f$. Then$(u,\gamma u) \in D(M_p)$.

In fact, by Step 1, $(u, \gamma u) \in C^{2+\alpha }(\overline{\Omega }) \times C^{2+\alpha }(\Gamma )$. Moreover,

$$\begin{aligned} \gamma \mathcal{A}u - L \gamma u - \gamma Eu = \lambda \gamma u - \gamma f - \lambda \gamma u + h = 0. \end{aligned}$$

Step 3:$\{(\psi , \gamma \psi ): \psi \in C^\alpha (\overline{\Omega })\}$is dense in$X_p$.

In fact, let $(f,h) \in X_p$. We begin by considering a sequence $(h_k)_{k \in \mathbb {N}}$ with values in $C^\alpha (\Gamma )$, such that $\Vert h_k - h\Vert _{L^p(\Gamma )} \rightarrow 0$$(k \rightarrow \infty )$. Let P be the extension operator described in Lemma 4.2. By Remark 4.3, it can be extended to a linear bounded operator from $C^\alpha (\Gamma )$ to $C^\alpha (\overline{\Omega })$ and from $L^p(\Gamma )$ to $L^p(\Omega )$. So $Ph_k \in C^\alpha (\overline{\Omega })$ for every $k \in \mathbb {N}$ and $(Ph_k)_{k \in \mathbb {N}}$ converges to Ph in $L^p(\Omega )$. Now we consider a sequence $(\phi _k)_{k \in \mathbb {N}}$ in $C_0^\infty (\Omega )$ converging to $f - Ph$ in $L^p(\Omega )$. We set $\psi _k:= Ph_k +\phi _k$. Then $\psi _k \in C^\alpha (\overline{\Omega })$, $(\psi _k)_{k \in \mathbb {N}}$ converges to f in $L^p (\Omega )$ and $(\gamma \psi _k)_{k \in \mathbb {N}} = (h_k)_{k \in \mathbb {N}}$ converges to h in $L^p(\Gamma )$.

Now let us consider $(u, \gamma u) \in D(G_p)$. We fix $\lambda \in \rho (G_p)$ and set $(f,h) := \lambda (u, \gamma u) - G_p((u, \gamma u)) \in X_p$. We take a sequence $((\psi _k, \gamma \psi _k))_{k \in \mathbb {N}}$ with $\psi _k \in C^{\alpha }(\overline{\Omega })$, converging to (f, h) in $X_p$. We set $(u_k, \gamma u_k):= (\lambda - G_p)^{-1}(\psi _k, \gamma \psi _k)$$(k \in \mathbb {N})$. Then $(u_k, \gamma u_k) \in D(M_p)$, the sequence $((u_k, \gamma u_k))_{k \in \mathbb {N}}$ converges to $(u, \gamma u)$ in $W^{2,p}(\Omega ) \times W^{2,p}(\Gamma )$, so that $(M_p(u_k, \gamma u_k))_{k \in \mathbb {N}}$ converges to $G_p(u, \gamma u)$ in $X_p$. $\square$