Abstract
Linear reaction–diffusion equations with inhomogeneous boundary and transmission conditions are shown to possess the property of maximal \(L_\mathrm{p}\) regularity. The new feature is the fact that the transmission interface is allowed to intersect the boundary of the domain transversally.
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1 Introduction
The emerging and understanding of the theory of maximal regularity for parabolic differential equations, which took place within the last three or so decades, has provided a firm basis for a successful handling of many challenging nonlinear problems. Among them, phase transition issues play a particularly prominent role. The impressive progress which has been made in this field with the help of maximal regularity techniques is well documented in the book by Prüss and Simonett [32]. The reader may also consult the extensive list of references and the ‘Bibliographic Comments’ in [32] for works of other authors and historical developments.
The relevant mathematical setup is usually placed in the framework of parabolic equations in bounded Euclidean domains, the interface being modeled as a hypersurface. In most works known to the author, it is assumed that the interface lies in the interior of the domain. Noteworthy exceptions are the papers by Wilke [37], Prüss et al. [33], Abels et al. [1], and Rauchecker [34] who study various important parabolic free boundary problems, presupposing that the membrane makes a ninety-degree boundary contact. In addition, in all of them, except for [1], a capillary (i.e., cylindrical) geometry is being studied. The same ninety-degree condition is employed by Garcke and Rauchecker [25] who carry out a linearized stability computation at a stationary solution of a Mullins–Sekerka flow in a two-dimensional bounded domain.
The assumption of the ninety-degree contact considerably simplifies the analysis since it allows to use reflection arguments. This does not apply in the case of general transversal intersection.
The only paper, we are aware of, in which a general contact angle is being considered is the one by Laurençot and Walker [28]. These authors establish the unique solvability in the strong \(L_2\) sense of a two-dimensional stationary transmission problem taking advantage of a particularly favorable geometric setting.
Elliptic problems with boundary and transmission conditions have also been investigated in a series of papers by Nistor et al. [21, 29,30,31]. The motivation for these works stems from the desire to get optimal convergence rates for approximations used for numerical computations. Although these authors employ weighted \(L_2\) Sobolev spaces, their methods and results are quite different from the ones presented here.
In this paper, we establish the maximal regularity of linear inhomogeneous parabolic transmission boundary value problems for the case where the interface intersects the boundary transversally. This is achieved by allowing the equations to degenerate near the intersection manifold and working in suitable weighted Sobolev spaces. We restrict ourselves to the simplest case of a fixed membrane and a single reaction–diffusion equation.
In a forthcoming publication, we shall use our present result to establish the local well-posedness of quasilinear equations with nonlinear boundary and transmission conditions.
The author is deeply grateful to G. Simonett for carefully reading the first draft of this paper, valuable suggestions, and pointing out misprints, errors, and the above references to related moving boundary problems.
2 The main result
Now we outline—in a slightly sketchy way—the main result of this paper. Precise definitions of notions, facts, and function spaces which we use here without further explanation are given in the subsequent sections.
Let \(\varOmega \) be a bounded domain in \({{\mathbb {R}}}^m\), \(m\ge 2\), with a smooth boundary \(\varGamma \) lying locally on one side of \(\varOmega \). By a membrane in \(\overline{\varOmega }\), we mean a smooth oriented hypersurface S of (the manifold) \(\overline{\varOmega }\) with a (possibly empty) boundary \(\varSigma \) such that \(S\cap \varGamma =\varSigma \). Thus, S lies in \(\varOmega \) if \(\varSigma =\emptyset \). Otherwise, \(\varSigma \) is an \((m-2)\)-dimensional oriented smooth submanifold of \(\varGamma \). In this case, it is assumed that S and \(\varGamma \) intersect transversally. Note that we do not require that S be connected. Hence, even if \(\varSigma \ne \emptyset \), there may exist interior membranes. However, the focus in this paper is on membranes with boundary. Thus, we assume until further notice that \(\varSigma \ne \emptyset \).
We denote by \(\nu \) the inner (unit) normal (vector field) on \(\varGamma \) and by \(\nu _S\) the positive normal on S. (Thus, \(\nu _S(x)\in T_x\overline{\varOmega }=T_x{{\mathbb {R}}}^m=\{x\}\times {{\mathbb {R}}}^m\), the latter being also identified with \(x+{{\mathbb {R}}}^m\subset {{\mathbb {R}}}^m\) for \(x\in S\)). As usual, \(\llbracket \cdot \rrbracket =\llbracket \cdot \rrbracket _S\) is the jump across S. We fix any \(T\in (0,\infty )\) and set \(J=J_T:=[0,T]\).
Of concern in this paper are linear reaction–diffusion equations with nonhomogeneous boundary and transmission conditions of the following form.
Set
with \(\gamma \) being the trace operator on \(\varGamma \). We assume (for the moment) that \(a\in {{\bar{C}}}^1\bigl ((\overline{\varOmega }\setminus S)\times J\bigr )\) and \(a>0\). A bar over a symbol for a standard function space means that its elements may undergo jumps across S. (The usual definitions based on decompositions of \(\overline{\varOmega }\setminus S\) in ‘inner’ and ‘outer’ domains cannot be used since \(\overline{\varOmega }\setminus S\) may be connected). Then, the problem under investigation reads:
where \(\gamma _0\) is the trace operator at \(t=0\).
We are interested in the strong \(L_\mathrm{p}\) solvability of (2.1), that is, in solutions possessing second order space derivatives in \(L_\mathrm{p}\). However, since S intersects \(\varGamma \), we cannot hope to get solutions which possess this regularity up to \(\varSigma \). Instead, it is to be expected that the derivatives of u blow up as we approach \(\varSigma \). For this reason, we set up our problem in weighted Sobolev spaces where the weights control the behavior of \(\partial ^\alpha u\) for \(0\le |\alpha |\le 2\) in relation to the distance from \(\varSigma \). This requires that the differential operator is adapted to such a setting, which means that the adapted ‘diffusion coefficient’ tends to zero near \(\varSigma \). In other words, we will have to deal with parabolic problems which degenerate near \(\varSigma \). To describe the situation precisely, we introduce curvilinear coordinates near \(\varSigma \) as follows.
Since \(\varSigma \) is an oriented hypersurface in \(\varGamma \), there exists a unique positive normal vector field \(\mu \) on \(\varSigma \) in \(\varGamma \). Given \(\sigma \in \varSigma \), we write \(\mu (\cdot ,\sigma )\) for the unique geodesic in \(\varGamma \) satisfying \(\mu (0,\sigma )=\sigma \) and \({{\dot{\mu }}}(0,\sigma )=\mu (\sigma )\). Similarly, for each \(y\in \varGamma \), we set \(\nu (\xi ,y):=y+\xi \nu (y)\) for \(\xi \ge 0\). Then, we can choose \(\varepsilon \in (0,1)\) and a neighborhood \(\tilde{U}(\varepsilon )\) of \(\varSigma \) in \(\overline{\varOmega }\) with the following properties: for each \(x\in \tilde{U}(\varepsilon )\), there exists a unique triple
such that
Thus, \(x\in \varGamma \cap \tilde{U}(\varepsilon )\) iff \((\xi ,\eta ,\sigma )\in \{0\}\times (-\varepsilon ,\varepsilon )\times \varSigma \).
Now we define curvilinear derivatives for \(u\in C^2\bigl (\tilde{U}(\varepsilon )\bigr )\) by
for \(x\in \tilde{U}(\varepsilon )\). It follows thatFootnote 1
on \(\tilde{U}(\varepsilon )\), where \(\mathop {{\mathrm{div}}}\nolimits _\varSigma \) and \(\mathop {{\mathrm{grad}}}\nolimits _\varSigma \) denote the divergence and the gradient, respectively, in \(\varSigma \) (with respect to the Riemannian metric \(g_\varSigma \) induced by the one of \(\varGamma \) which, in turn, is induced by the Euclidean metric on \(\overline{\varOmega }\)).
For x given by (2.2), we set
which is the geodesic distance in \(\overline{\varOmega }\) from x to \(\varSigma \) (and not, in general, the distance in the ambient space \({{\mathbb {R}}}^m\)). We fix \(\omega \in C^\infty \bigl (N(\varepsilon ),[0,1]\bigr )\), depending only on r, such that \(\omega |N(\varepsilon /3)=1\) and \(\mathop {{\mathrm{supp}}}\nolimits (\omega )\subset N(2\varepsilon /3)\) and set
Then, we define on
a singular linear reaction–diffusion operator \({{\mathcal {A}}}_U\) by
for \(u\in {{\bar{C}}}^2(U\setminus S)\). The corresponding singular boundary operator is given by
Since S intersects \(\varGamma \) transversally, it follows that there exists a smooth function \(s:[0,\varepsilon )\times \varSigma \rightarrow (-\varepsilon ,\varepsilon )\) such that \(s(0,\sigma )=0\) for \(\sigma \in \varSigma \) and
Using this, we associate with \({{\mathcal {A}}}_U\) a transmission operator \({{\mathcal {C}}}_U\) on U by setting
for \(u\in {{\bar{C}}}^2(U\setminus S)\), where \((\cdot |\cdot )_\varSigma =g_\varSigma \) and
Now we define a singular transmission boundary value problem on \(\overline{\varOmega }\setminus S\) by putting \(V:=\overline{\varOmega }\,\big \backslash \tilde{U}(2\varepsilon /3)\) and
It follows from (2.4) and the properties of \(\rho \) that this definition is unambiguous.
To introduce weighted Sobolev spaces on \(U\setminus S\), we put
where \(\nabla _{\varSigma }\) is the Levi–Civita connection on \(\varSigma \) for the metric \(g_\varSigma \). Also, \(1<p<\infty \) and
Then, \({{\bar{W}}}_{p}^2(U\setminus S;r)\) is the completion of \({{\bar{C}}}^2(U\setminus S)\) in \(L_{1,\mathrm{loc}}(U\setminus S)\) with respect to the norm \(\Vert \cdot \Vert _{{{\bar{W}}}_{p}^2(U\setminus S;r)}\).
The (global) weighted Sobolev space
consists of all \(u\in L_{1,\mathrm{loc}}(\overline{\varOmega }\setminus S)\) with \(u\big |U\in {{\bar{W}}}_{p}^2(U\setminus S;r)\) and \(u\big |V\in {{\bar{W}}}_{p}^2(V\setminus S)\). It is a Banach space with the norm
whose topology is independent of the specific choice of \(\varepsilon \) and \(\omega \). Similarly, the Lebesgue space
is obtained by replacing \(\langle u\rangle \) in (2.12) by |u|. Moreover,
where \((\cdot |\cdot )_{\theta ,p}\) is the real interpolation functor of exponent \(\theta \).
We also need time-dependent anisotropic spaces. For this, we use the notation \(s/\varvec{2}:=(s,\,s/2)\), \(0\le s\le 2\). Then,
and \({{\mathcal {X}}}_p^{0/\varvec{2}}:=L_p(J,{{\mathcal {X}}}_p^0)\). If \(X\in \{\varGamma ,S\}\) and \(s\in \{1-1/p,\ 2-1/p\}\), then
Here, the \({{\bar{W}}}_{p}^s(X\setminus \varSigma ;r)\) are trace spaces of \({{\mathcal {X}}}_p^2\) (cf. (14.11) and (14.12)). Moreover,
By \({\bar{BC}}(\overline{\varOmega }\setminus S)\), we mean the space of bounded and continuous functions (with possible jumps across S), endowed with the maximum norm. Then, \({\bar{BC}}^1(\overline{\varOmega }\setminus S;r)\) is the Banach space of all \(u\in {\bar{BC}}(\overline{\varOmega }\setminus S)\) with \(\partial _ju\in {\bar{BC}}(V\setminus S)\), \(1\le j\le m\), and
Furthermore,
To indicate the nonautonomous structure of (2.1), we write \(a(t):=a(\cdot ,t)\) and, correspondingly, \({{\mathcal {A}}}(t)\), \({{\mathcal {B}}}(t)\), and \({{\mathcal {C}}}(t)\).
Now we are ready to formulate the main result of this paper, the optimal solvability of linear reaction–diffusion transmission boundary value problems:
Theorem 2.1
Let \(1<p<\infty \) with \(p\notin \{3/2,\,3\}\) and
for some \(\underline{\alpha }\in (0,1)\). Suppose
and that the following compatibility conditions are satisfied:
where \(\psi :=(\psi ^0,\psi ^1)\). Then,
has a unique solution \(u\in {{\mathcal {X}}}_p^{2/\varvec{2}}\). It depends continuously on the data.
Corollary 2.2
Suppose a is independent of t, that is, \(a\in {\bar{BC}}^1(\overline{\varOmega }\setminus S;r)\). Set
and \(A_r:={{\mathcal {A}}}_r|{{\mathcal {X}}}_{p,0}^2\). Then, \(-A_r\), considered as a linear operator in \({{\mathcal {X}}}_p^0\) with domain \({{\mathcal {X}}}_{p,0}^2\), generates on \({{\mathcal {X}}}_p^0\) a strongly continuous analytic semigroup.
Proof
The theorem implies that \(A_r\) has the property of maximal \({{\mathcal {X}}}_p^0\) regularity. This fact is well known to imply the claim (e.g., [7, Chapter III] or [23]).\(\square \)
Theorem 2.1 is a special case of the much more general Theorems 7.1 and 14.1. They also include Dirichlet boundary conditions and apply to transmission problems in general Riemannian manifolds with boundary and bounded geometry.
The situation is considerably simpler if \(\varSigma =\emptyset \), that is, if only interior transmission hypersurfaces are present. Of course, if \(S=\emptyset \), then (2.14) reduces to a linear reaction–diffusion equation with inhomogeneous boundary conditions. In these cases, no degenerations do occur.
We refrain from considering operators \(({{\mathcal {A}}}_r,{{\mathcal {B}}}_r)\) with lower order terms. This case will be covered by the forthcoming quasilinear result.
In the case of an interior transmission surface (that is, \(\varSigma =\emptyset \)) and if a is independent of t, Theorem 2.1 is a special case of Theorem 6.5.1 in [32]. The latter theorem applies to systems and provides an \(L_p\)-\(L_q\) theory.
If \(\varSigma \ne \emptyset \), then the basic difficulty in proving Theorem 2.1 stems from the fact that \(\overline{\varOmega }\setminus \varSigma \) and, consequently, \(S\setminus \varSigma \) and \(\varGamma \setminus \varSigma \), are no longer compact. The fundamental observation which makes the proofs work is the fact that we can consider \(\overline{\varOmega }\setminus \varSigma \) as a (noncompact) Riemannian manifold with a metric g which coincides on \(U(\varepsilon /3)\) with the singular metric \(r^{-2}\mathrm{d}\nu \otimes \mathrm{d}\mu +g_\varSigma \) and on V with the Euclidean metric. With respect to this metric, \({{\mathcal {A}}}_r\) is a uniformly elliptic operator.
Theorems 4.4 and 5.1 show that \((\overline{\varOmega }\setminus \varSigma ,\,g)\) is a uniformly regular Riemannian manifold in the sense of [10]. Thus, we are led to consider linear parabolic equations with boundary and transmission conditions on such manifolds. As in the compact case, by means of local coordinates the problem is reduced to Euclidean settings. However, since we have to deal with noncompact manifolds, we have to handle simultaneously infinitely many model problems. In order for this technique to work, we have to establish uniform estimates which are in a suitable sense independent of the specific local coordinates. In addition, special care has to be taken in ‘gluing together the local model problems.’ These are no points to worry about in the compact case.
In our earlier paper [12], we have established an optimal existence theory for linear parabolic equations on uniformly regular Riemannian manifolds without boundary. The present proof extends those arguments to the case of manifolds with boundary. The presence of boundary and transmission conditions adds considerably to the complexity of the problem and makes the paper rather heavy.
In Sect. 3, we collect the needed background information. In the subsequent two sections, we establish the differential geometric foundation of transmission surfaces in uniformly regular and singular Riemannian manifolds.
After having introduced the relevant function spaces in Sect. 6, we present in Sect. 7 the basic maximal regularity theorem in anisotropic Sobolev spaces for linear nonautonomous reaction–diffusion equations with nonhomogeneous boundary and transmission conditions on general uniformly regular Riemannian manifolds. Its rather complex proof occupies the next five sections. Finally, in the last section, it is shown that our general results apply to the Euclidean setting presented here.
3 Uniformly regular Riemannian manifolds
In this section, we recall the definition of uniformly regular Riemannian manifolds and collect those properties of which we will make use. Details can be found in [9,10,11], and in the comprehensive presentation [15]. Thus, we shall be rather brief.
We use standard notation from differential geometry and function space theory. In particular, an upper, resp. lower, asterisk on a symbol for a diffeomorphism denominates the corresponding pullback, resp. push-forward (of tensors). By c, resp. \(c(\alpha )\) etc., we denote constants \(\ge 1\) which can vary from occurrence to occurrence. Assume S is a nonempty set. On the cone of nonnegative functions on S, we define an equivalence relation \({}\sim {}\) by \(f\sim g\) iff \(f(s)/c\le g(s)\le cf(s)\), \(s\in S\).
An m-dimensional manifold is a separable metrizable space equipped with an m-dimensional smooth structure. We always work in the smooth category.
Let M be an m-dimensional manifold with or without boundary. If \(\kappa \) is a local chart, then we use \(U_{\kappa }\) for its domain, the coordinate patch associated with \(\kappa \). The chart is normalized if \(\kappa (U_{\kappa })=Q_\kappa ^m\), where \(Q_\kappa ^m=(-1,1)^m\) if , the interior of M, and \(Q_\kappa ^m=[0,1)\times (-1,1)^{m-1}\) otherwise. An atlas \({{\mathfrak {K}}}\) is normalized if it consists of normalized charts. It is shrinkable if it normalized and there exists \(r\in (0,1)\) such that \(\bigl \{\,\kappa ^{-1}(rQ_\kappa ^m)\ ;\ \kappa \in {{\mathfrak {K}}}\,\bigr \}\) is a covering of M. It has finite multiplicity if there exists \(k\in {{\mathbb {N}}}\) such that any intersection of more than k coordinate patches is empty.
The atlas \({{\mathfrak {K}}}\) is uniformly regular (ur) if
Two ur atlases \({{\mathfrak {K}}}\) and \(\tilde{{{\mathfrak {K}}}}\) are equivalent if
This defines an equivalence relation in the class of all ur atlases. An equivalence class thereof is a ur structure. By a ur manifold, we mean a manifold equipped with a ur structure. A Riemannian metric g on a ur manifold M is ur if, given a ur atlas \({{\mathfrak {K}}}\),
Here, \(g_m:=(\cdot |\cdot )=\mathrm{d}x^2\) is the Euclidean metricFootnote 2 on \({{\mathbb {R}}}^m\) and (i) is understood in the sense of quadratic forms. This concept is well-defined, independently of the specific \({{\mathfrak {K}}}\). A uniformly regular Riemannian (urR) manifold is a ur manifold, endowed with a urR metric.
Remarks 3.1
-
(a)
Given a (nonempty) subset S of M and an atlas \({{\mathfrak {K}}}\),
$$\begin{aligned} {{\mathfrak {K}}}_S:=\{\,\kappa \in {{\mathfrak {K}}}\ ;\ U_{\kappa }\cap S\ne \emptyset \,\}\ . \end{aligned}$$We say that \({{\mathfrak {K}}}\) is normalized on S, resp. has finite multiplicity on S, resp. is shrinkable on S if \({{\mathfrak {K}}}_S\) possesses the respective properties. Moreover, \({{\mathfrak {K}}}\) is ur on S if (3.1) applies with \({{\mathfrak {K}}}\) replaced by \({{\mathfrak {K}}}_S\). Similarly, two atlases \({{\mathfrak {K}}}\) and \(\tilde{{{\mathfrak {K}}}}\), which are ur on S, are equivalent on S if (3.2) holds with \({{\mathfrak {K}}}\) and \(\tilde{{{\mathfrak {K}}}}\) replaced by \({{\mathfrak {K}}}_S\) and \(\tilde{{{\mathfrak {K}}}}_S\), respectively. This induces a ur structure on S. Finally, M is ur on S if it is equipped with a ur structure on S.
-
(b)
Suppose \({{\mathfrak {K}}}\) is a ur atlas for M on S. Given any \(\varepsilon >0\), there exists a ur atlas \({{\mathfrak {K}}}'\) on S such that \(\mathop {{\mathrm{diam}}}\nolimits _g(U_{\kappa })<\varepsilon \) for \(\kappa \in {{\mathfrak {K}}}'\), where \(\mathop {{\mathrm{diam}}}\nolimits _g\) is the diameter with respect to the Riemannian distance \(d_g\). \(\square \)
In the following examples, we use the natural ur structure (e.g., the product ur structure in Example 3.2(c)) if nothing else is mentioned.
Example 3.2
-
(a)
Each compact Riemannian manifold is a urR manifold, and its ur structure is unique.
-
(b)
Let \(\varOmega \) be a bounded domain in \({{\mathbb {R}}}^m\) with a smooth boundary such that \(\varOmega \) lies locally on one side of it. Then, \((\overline{\varOmega },g_m)\) is a urR manifold.
More generally, suppose that \(\varOmega \) is an unbounded open subset of \({{\mathbb {R}}}^m\) whose boundary is ur in the sense of Browder [20] (also see [27, IV.§4]). Then, \((\overline{\varOmega },g_m)\) is a urR manifold. In particular, \(({{\mathbb {R}}}^m,g_m)\) and \(({{\mathbb {H}}}^m,g_m)\) are urR manifolds, where \({{\mathbb {H}}}^m:={{\mathbb {R}}}_+\times {{\mathbb {R}}}^{m-1}\) is the closed right half-space in \({{\mathbb {R}}}^m\).
-
(c)
If \((M_i,g_i)\), \(i=1,2\), are urR manifolds and at most one of them has a nonempty boundary, then \((M_1\times M_2,\ g_1+g_2)\) is a urR manifold.
-
(d)
Assume M is a manifold and N a topological space. Let \(f:N\rightarrow M\) be a homeomorphism. If \({{\mathfrak {K}}}\) is an atlas for M, then \(f^*{{\mathfrak {K}}}:=\{\,f^*\kappa \ ;\ \kappa \in {{\mathfrak {K}}}\,\}\) is an atlas for N which induces the smooth ‘pullback’ structure on N. If \({{\mathfrak {K}}}\) is ur, then \(f^*{{\mathfrak {K}}}\) also is ur.
Let \((M,g)\) be a urR manifold. Then, \(f^*(M,g):=(N,f^*g)\) is a urR manifold and the map \(f:(N,f^*g)\rightarrow (M,g)\) is an isometric diffeomorphism. \(\square \)
It follows from these examples, for instance, that the cylinders \({{\mathbb {R}}}\times M_1\) or \({{\mathbb {R}}}_+\times M_2\), where \(M_i\) are compact Riemannian manifolds with \(\partial M_2=\emptyset \), are urR manifolds. More generally, Riemannian manifolds with cylindrical ends are urR manifolds (see [11] where more examples are discussed).
Without going into detail, we mention that a Riemannian manifold without boundary is a urR manifold iff it has bounded geometry (see [10] for one half of this assertion and [24] for the other half). Thus, for example, is not a urR manifold. A Riemannian manifold with boundary is a urR manifold iff it has bounded geometry in the sense of Schick [35] (also see [17,18,19, 26] for related definitions). Detailed proofs of these equivalences can be found in [15].
4 Uniformly regular hypersurfaces
Let \((M,g)\) be an oriented Riemannian manifold with (possibly empty) boundary \(\varGamma \). If it is not empty, then there exists a unique inner (unit) normal vector field \(\nu =\nu _\varGamma \) on \(\varGamma \), that is, a smooth section of \(T_\varGamma M\), the restriction of the tangent bundle TM of M to \(\varGamma \). Furthermore, \(\varGamma \) is oriented by the inner normal in the usual sense.
Suppose that S is an oriented hypersurface in , an embedded submanifold of codimension 1. Then, there is a unique positive (unit) normal vector field \(\nu _S\) on S, where ‘positive’ means that \(\bigl [\nu _S(p),\beta _1,\ldots ,\beta _{m-1}\bigr ]\) is a positive basis for \(T_pM\) if \([\beta _1,\ldots ,\beta _{m-1}]\) is one for \(T_pS\).
Let \(Z\in \{\varGamma ,S\}\). Then, we write
This means that, given \(p\in Z\), \(\gamma _p^Z\) is the unique geodesic in M satisfying \(\gamma _p^Z(0)=p\) and \({{\dot{\gamma }}}_p^Z(0)=\nu _Z(p)\) and being defined (at least) on \(I_Z\bigl (\varepsilon (p)\bigr )\), where \(I_\varGamma \bigl (\varepsilon (p)\bigr )=\bigl [0,\varepsilon (p)\bigr )\) and \(I_S\bigl (\varepsilon (p)\bigr )=\bigl (-\varepsilon (p),\varepsilon (p)\bigr )\) for some \(\varepsilon (p)>0\). Note that for \(t>0\).
We say that Z has a uniform normal geodesic tubular neighborhood of width \(\varepsilon \) if the following is true: there exist \(\varepsilon >0\) and an open neighborhood \(Z(\varepsilon )\) of Z in M such that
is a diffeomorphism satisfying \(\varphi _Z(Z)=\{0\}\times Z\). If \(Z=\varGamma \), then a uniform tubular neighborhood is a uniform collar.
Given any embedded submanifold C of M, with or without boundary, we denote by \(g_C\) the pullback metric \(\iota ^*g\), where \(\iota :C\hookrightarrow M\) is the natural embedding.
Now we suppose that
This means that there exists an oriented ur atlas for M.
Let S be a hypersurface with boundary \(\varSigma \) such that \(\varSigma =S\cap \varGamma \). Thus, if \(\varSigma =\emptyset \). An atlas \({{\mathfrak {K}}}\) for M is S-adapted if for each \(\kappa \in {{\mathfrak {K}}}_S\) one of the following alternatives applies:
Then, S is a regularly embedded hypersurface in M, a membrane for short, if there exists an oriented ur atlas \({{\mathfrak {K}}}\) for M which is S-adapted.
Let S be a membrane. Each S-adapted atlas for M induces (by restriction) a ur structure and a (natural) orientation on S. Moreover, the ur structure and the orientation of S are independent of the specific choice of \({{\mathfrak {K}}}\).
For the proof of all this and the following theorem, we refer to [15].
Theorem 4.1
Let (4.2) be satisfied and suppose S is a membrane in M. Assume \(Z\in \{\varGamma ,S\}\). Then,
-
(i)
\((Z,g_Z)\) is an (\(m-1\))-dimensional oriented urR manifold.
-
(ii)
If \(\varSigma =\partial S\ne \emptyset \), then \(\varSigma \) is a membrane in \(\varGamma \) without boundary.
-
(iii)
Let \(\varSigma =\emptyset \) if \(Z=S\). Then, Z has a uniform tubular neighborhood
$$\begin{aligned} \varphi _Z:Z(\varepsilon )\rightarrow I_Z(\varepsilon )\times Z \end{aligned}$$and \(\varphi _{Z*}g\sim \mathrm{d}s^2+g_Z\). Moreover, \(\varphi _Z\) is an orientation-preserving diffeomorphism.
-
(iv)
Suppose \(\varSigma \ne \emptyset \). Then, given \(\rho >0\), there exists \(\varepsilon (\rho )>0\) such that
$$\begin{aligned} S\cap \bigl \{\,q\in M\ ;\ d_g(q,\varGamma )>\rho \,\bigr \} \end{aligned}$$has a uniform tubular neighborhood of width \(\varepsilon (\rho )\) in .
Now we suppose that S is a membrane with \(\varSigma \ne \emptyset \). It follows from (ii) and (iii) that \(\varSigma \) has a uniform tubular neighborhood \(\psi :\varSigma ^\varGamma (\varepsilon )\rightarrow (-\varepsilon ,\varepsilon )\times \varSigma \) in \(\varGamma \) for some \(\varepsilon >0\). By part (iii), \(\varGamma \) has a uniform collar \(\varphi :\varGamma (\varepsilon )\rightarrow [0,\varepsilon )\times \varGamma \) in M, where we assume without loss of generality that \(\varphi \) and \(\psi \) are of the same width. Then,
is an open neighborhood of \(\varSigma \) in M and
is an orientation-preserving diffeomorphism, a tubular neighborhood of \(\varSigma \) in M of width \(\varepsilon \).
We refer once more to [15] for the proof of the next theorem. Henceforth, \(h:=g_\varSigma \) and \(N(\varepsilon ):=[0,\varepsilon )\times (-\varepsilon ,\varepsilon )\).
Theorem 4.2
Assume (4.2) and S is a membrane with nonempty boundary \(\varSigma \). Then,
It follows that
is for each \(\sigma \in \varSigma \) a two-dimensional submanifold of \(\varSigma (\varepsilon )\) and \(S\cap \varSigma _\sigma (\varepsilon )\) is a one-dimensional submanifold of \(\varSigma _\sigma (\varepsilon )\).
Remark 4.3
(The two-dimensional case) Suppose \(\dim (M)=2\) and \(\varSigma \ne \emptyset \). It follows from Theorem 4.1(ii) and the fact that M has a countable base that \(\varSigma \) is a countable discrete subspace of M. Thus, we can find \(\varepsilon >0\) with the following properties: If we denote by \(\psi ^{-1}(\cdot ,\sigma ):(-\varepsilon ,\varepsilon )\rightarrow \varGamma \) the local arc-length parametrization of \(\varGamma \) with \(\psi ^{-1}(0,\sigma )=\sigma \) for \(\sigma \in \varSigma \), then the above definition of \(\chi \) applies and defines a tubular neighborhood of \(\varSigma \) in M of width \(\varepsilon \).
Note that \(\varSigma (\varepsilon )\) is the countable pairwise disjoint union of \(\varSigma _\sigma (\varepsilon )\), \(\sigma \in \varSigma \). The term \(+h\) in (4.3) (and everywhere else) has to be disregarded, and the volume measure of \(\varSigma \) is the counting measure. Thus, in this case, integration with respect to \(\mathrm{d}\mathop {{\mathrm{vol}}}\nolimits _\varSigma \) reduces to summation over \(\sigma \in \varSigma \). \(\square \)
Now we restrict the class of membranes under consideration by requiring that S intersects \(\varGamma \) uniformly transversally. This means the following: there exists \(f\in C^\infty \bigl ([0,\varepsilon )\times \varSigma ,\,(-\varepsilon ,\varepsilon )\bigr )\) such that, setting \(f_\sigma :=f(\cdot ,\sigma )\),
In general, a submanifold C of a manifold B intersects \(\partial B\) transversally if
The following theorem furnishes an important large class of urR manifolds and membranes intersecting the boundary uniformly transversally.
Theorem 4.4
Let \((M,g)\) be a compact oriented Riemannian manifold with boundary \(\varGamma \). Assume S is an oriented hypersurface in M with nonempty boundary \(\varSigma \subset \varGamma \) and S intersects \(\varGamma \) transversally. Then, \((M,g)\) is a urR manifold and S is a membrane intersecting \(\varGamma \) uniformly transversally.
Proof
Example 3.2(a) guarantees that \((M,g)\) is an oriented urR manifold. Hence, \((\varGamma ,g_\varGamma )\) is an oriented urR manifold by Theorem 4.1(i). Since S intersects \(\varGamma \) transversally, it is a well-known consequence of the implicit function theorem that \(\varSigma \) is a compact hypersurface in \(\varGamma \) without boundary. It is oriented, being the boundary of the oriented manifold S. Hence, invoking Example 3.2(a) once more, \((\varSigma ,g_\varSigma )\) is an oriented urR manifold. As it is compact, it has a uniform tubular neighborhood in \(\varGamma \). Thus, \(\varGamma \) having a uniform collar, \(\varSigma \) has a uniform tubular neighborhood \(\chi \) in M of some width \(\varepsilon \).
Since S intersects \(\varGamma \) transversally, \(\chi \bigl (S\cap \varSigma _\sigma (\varepsilon )\bigr )\) can be represented as the graph of a smooth function \(f_\sigma :[0,\varepsilon )\rightarrow (-\varepsilon ,\varepsilon )\) with \(f_\sigma (0)=0\), and \(f_\sigma \) depends smoothly on \(\sigma \in \varSigma \). The compactness of \(\varSigma \) implies that (4.4) is true. Hence, S intersects \(\varGamma \) uniformly transversally. Now, due to the compactness of S, it is not difficult to see that S is a regularly embedded submanifold of M. The theorem is proved.\(\square \)
Remarks 4.5
-
(a)
This theorem applies to the case \((M,g)=(\overline{\varOmega },g_m)\) considered in Sect. 2.
-
(b)
Suppose \((M,g)\) is an oriented urR manifold and S a membrane without boundary. Then, the fact that S has a uniform tubular neighborhood in prevents S from either reaching \(\varGamma \) or ‘collapsing’ at infinity.\(\square \)
5 The singular manifold
In this section,
By Theorem 4.1 and the considerations following it, we can choose a uniform tubular neighborhood
We write \(D(\varepsilon ):=\bigl \{\,(x,y)\in {{\mathbb {R}}}^2\ ;\ x^2+y^2<\varepsilon ^2,\ x\ge 0\,\bigr \}\). Then,
is an open neighborhood of \(\varSigma \) in M contained in \(\varSigma (\varepsilon )\). We put
Furthermore, r and \(\rho \) are given by (2.5) and (2.6), respectively. Then, we define a Riemannian metric \(\hat{g}\) on \(\hat{M}\) by
Note that, see Theorem 4.2,
and
Hence, \(({{\hat{M}}},{{\hat{g}}})\) is a Riemannian manifold with a wedge singularity near \(\varSigma \).
The following theorem is the basis for our approach. It implies that it suffices to study transmission problems for membranes without boundary on urR manifolds.
Theorem 5.1
\(({{\hat{M}}},{{\hat{g}}})\) is an oriented urR manifold, and \(\hat{S}:=S\setminus \varSigma \) is a membrane in \(\hat{M}\) without boundary.
Proof
(1) We set \(\dot{D}(\varepsilon ):=D(\varepsilon )\setminus \{0,0\}\) and define \(\delta \in C^\infty [0,\varepsilon )\) by
Then, we fix \(\hat{\varepsilon }\in (2\varepsilon /3,\,\varepsilon )\), define a diffeomorphism
and set \(t:=s^{-1}\). It follows, see [14, Lemma 5.1], that
We also consider the polar coordinate diffeomorphism
Then,
that is,
Hence,
is a diffeomorphism satisfying
where \(\beta :=t^*(r/\delta )\). By (2.6), \(r/\delta =r/(1-\omega +r\omega )\) for \(0<r\le \hat{\varepsilon }\). Hence, \(\beta \) is smooth and \(\beta \sim 1\). Thus, \(\gamma \) is a metric on \(N:=(0,\infty )\times [-\pi /2,\,\pi /2]\) which is uniformly equivalent to \(g_2=\mathrm{d}s^2+\mathrm{d}\alpha ^2\). By Examples 3.2 (a)–(c),
is a urR manifold. From this, we deduce, see Remark 3.1(a), that \((N,\gamma )\) is a urR manifold on \((\overline{s},\infty )\) for each \(\overline{s}>0\).
It follows that
is an isometric isomorphism. Hence, we derive from Example 3.2(d) and Remark 3.1(a) that \(U(\hat{\varepsilon })\) is a urR manifold on \(U(\overline{\varepsilon })\), where \(\overline{\varepsilon }:=5\varepsilon /6\). Since \((M,g)\) is a urR manifold, it is a urR manifold on \(M\big \backslash \tilde{U}(\varepsilon /3)\). Thus, it is a consequence of (5.5) that \(({{\hat{M}}},{{\hat{g}}})\) is a urR manifold. The first assertion is now clear.
(2) Fix \(\overline{\varepsilon }\in (\varepsilon /3,\,\hat{\varepsilon })\). It can be assumed that (4.4) applies with this choice of \(\overline{\varepsilon }\). Set \(\tilde{f}_\sigma :=t^*f_\sigma \). Note that (4.4)(ii) implies
Also note that \(t(s)=ce^{-s}\) for \(s\ge s(\varepsilon /3)\) and some \(c>0\). Hence,
Thus, it follows from (4.4)(iii) that
that is, uniformly with respect to \(\sigma \in \varSigma \).
We write \(\tilde{G}_\sigma \) for the graph of \(\tilde{f}_\sigma \) in \(\bigl [s(\overline{\varepsilon }),\infty \bigr )\times [-\pi /2,\,\pi /2]\). We can assume that
is the positive normal for \(\tilde{G}_\sigma \) at \(\bigl (s,\tilde{f}_\sigma (s)\bigr )\) (otherwise replace \(\tilde{\nu }_\sigma (s)\) by \(-\tilde{\nu }_\sigma (s)\)). It follows from (5.8) that
From this and (5.7), we deduce that \(\tilde{G}_\sigma \) has a uniform tubular neighborhood in \(\bigl ([s(\overline{\varepsilon }),\infty )\times [-\pi /2,\,\pi /2],\,\mathrm{d}s^2+\mathrm{d}\alpha ^2\bigr )\) whose width is independent of \(\sigma \in \varSigma \). It follows from step (1) that its pullback by w is a uniform tubular neighborhood of \(\hat{S}\) in \(U(\overline{\varepsilon })\). Now the second part of the assertion is a consequence of Theorem 4.1(iv), since, given any \(\delta >0\), \(\hat{g}\) and g are equivalent on \(M\big \backslash \tilde{U}(\delta )\).\(\square \)
6 Function spaces
Let \((M,g)\) be a Riemannian manifold. We consider the tensor bundles
the tangent, cotangent, and trivial bundle, respectively, and
endow \(T_\tau ^\sigma M\) with the tensor bundle metric \(g_\sigma ^\tau :=g^{\otimes \sigma }\otimes g^{*\,\otimes \tau }\), \(\sigma ,\tau \in {{\mathbb {N}}}\), and setFootnote 3
By \(\nabla =\nabla _{g}\), we denote the Levi–Civita connection and interpret it as covariant derivative. Then, given a smooth function u on M, \(\nabla ^ku\in C^\infty (T_k^0M)\) is defined by \(\nabla ^0u:=u\), \(\nabla ^1u=\nabla u:=\mathrm{d}u\), and \(\nabla ^{k+1}u:=\nabla (\nabla ^ku)\) for \(k\in {{\mathbb {N}}}\).
Let \(\kappa =(x^1,\ldots ,x^m)\) be a local coordinate system and set \(\partial _i:=\partial /\partial x^i\). Then,
where
are the Christoffel symbols. It follows that
and
As usual, \(\mathrm{d}\mathop {{\mathrm{vol}}}\nolimits _g=\sqrt{g}\,\mathrm{d}x\) is the Riemann–Lebesgue volume element on \(U_{\kappa }\).
Let \(\sigma ,\tau \in {{\mathbb {N}}}\), put \(V:=T_\tau ^\sigma M\), and write \(\vert \cdot \vert _V:=\vert \cdot \vert _{g_\sigma ^\tau }\). Then, \({{\mathcal {D}}}(V)\) is the linear subspace of \(C^\infty (V)\) of compactly supported sections.
For \(1\le q<\infty \), we set
Then,
is the usual Lebesgue space of \(L_q\) sections of V. Hence, \(L_q(M,g)=L_q(V,g)\) for \(V=T_0^0M=M\times {{\mathbb {R}}}\). If \(k\in {{\mathbb {N}}}\), then
and
The Sobolev space \(W_{q}^k(V)=W_{q}^k(V,g)\) is the completion of \({{\mathcal {D}}}(V)\) in \(L_q(V)\) with respect to the norm \(\Vert \cdot \Vert _{W_{q}^k(V)}\). If \(k<s<k+1\), the Slobodeckii space \(W_{q}^s(V)\) is obtained by real interpolation:
We also need the time-dependent function spaces
Thus, \(W_{q}^{0/\varvec{2}}(M\times J)\doteq L_q\bigl (J,L_q(M)\bigr )\), where \({}\doteq {}\) means ‘equivalent norms.’
By \(BC^k(V)=BC^k(V,g)\), we denote the Banach space of all \(u\in C^k(V)\) for which \(\Vert u\Vert _{BC^k(V)}\) is finite, and \(BC:=BC^0\). Then,
with the usual Hölder space \(C^{1/2}\).
The following lemma shows that in the Euclidean setting these definitions return the classical spaces.
Lemma 6.1
Suppose that \({{\mathbb {X}}}\in \{{{\mathbb {R}}}^m,{{\mathbb {H}}}^m\}\), \((M,g):=({{\mathbb {X}}},g_m)\), and \(V:={{\mathbb {X}}}\times F\) with \(F={{\mathbb {R}}}^{m^\sigma \times m^\tau }\simeq T_\tau ^\sigma {{\mathbb {X}}}\). Then,
the standard Sobolev–Slobodeckii spaces, and
Proof
The second assertion is obvious.
If \(k\in {{\mathbb {N}}}\), then the above definition of \(W_{q}^k(V)\) coincides with the one in [13, (VII.1.2.2)]. Now the first assertion follows from (6.4), Theorems VII.2.7.4 and VII.2.8.3 in [13], and the fact that the Besov space \(B_q^s=B_{qq}^s\) coincides with \(W_{q}^s\) for \(s\notin {{\mathbb {N}}}\).\(\square \)
Now we suppose that
By Theorem 4.1(iii), we can choose a uniform tubular neighborhood
in . We set
By \(V_\pm :=V_{M_\pm }\) and \(V_0:=V_{M_0}\), we denote the restrictions of V to \(M_\pm \) and \(M_0\), respectively. Then, \({{\bar{W}}}_{q}^s(M\setminus S,\,V)\), \(s\in {{\mathbb {R}}}_+\), resp. \({\bar{BC}}^k(M\setminus S,\,V)\), \(k\in {{\mathbb {N}}}\), is the closed linear subspace of
consisting of all \(u=u_0\oplus u_+\oplus u_-\) satisfying \((u_0-u_\pm )|M_0\cap M_\pm =0\). Definitions analogous to (6.5) and (6.6) give the Banach spaces \({{\bar{W}}}_{q}^{s/\varvec{2}}\bigl ((M\setminus S)\times J\bigr )\) and \({\bar{BC}}^{1/\varvec{2}}\bigl ((M\setminus S)\times J\bigr )\), respectively. Note that \({{\bar{W}}}_{p}^0(M\setminus S,\,V)=L_p(M,V)\), since \(\mathop {{\mathrm{vol}}}\nolimits _g(S)=0\).
Remark 6.2
Let \((M,g):=({{\mathbb {R}}}^m,g_m)\) and \(S:=\partial {{\mathbb {H}}}^m\). We can set \(\varepsilon =\infty \) in (6.8) to get \(M_+={{\mathbb {H}}}^m\), \(M_-=-{{\mathbb {H}}}^m\), and \(M_0=\emptyset \). Then,
and
as follows from Lemma 6.1. \(\square \)
Assume \(a\in {\bar{BC}}(M\setminus S,\,V)\). Then, the one-sided limits
exist and \(a_\pm \in BC(S)\). Hence, the jump across S,
is well-defined. Note that \(a_\pm \) is the trace of a on S ‘from the positive/negative side of S.’
Let \(u\in {\bar{BC}}^1(M\setminus S)\). Then, \(u\circ \gamma _p^S\in {\bar{BC}}^1\bigl ((-\varepsilon ,\varepsilon )\setminus \{0\}\bigr )\). We set
for \(\varphi (q)=(\tau ,p)\in (-\varepsilon ,\varepsilon )\times S\) with \(\tau \ne 0\). Thus, \(\partial u/\partial \nu _S\) is the normal derivative of u in \((M_+\cup M_-)\setminus S\), that is, the derivative along the normal geodesic \(\gamma _p^S\). Hence,
Consequently, the jump of the normal derivative
is also well-defined.
7 The parabolic problem on manifolds
We presuppose (6.7) and assume
where \(\underline{\alpha }\le 1\). Then,
Fix \(\delta \in C\bigl (\varGamma ,\{0,1\}\bigr )\). Then, \(\varGamma _j:=\delta ^{-1}(j)\), \(j=0,1\), is open and closed in \(\varGamma \) and \(\varGamma _0\cup \varGamma _1=\varGamma \). Either \(\varGamma \), \(\varGamma _0\) or \(\varGamma _1\) may be empty. In such a case, all references to these empty sets have to be disregarded. Recall that \(\gamma \) denotes the trace operator on \(\varGamma \) (for any manifold).
We introduce an operator \({{\mathcal {B}}}=({{\mathcal {B}}}^0,{{\mathcal {B}}}^1)\) on \(\varGamma \) by \({{\mathcal {B}}}^0u=\gamma _{\varGamma _0}u\), the Dirichlet boundary operator, on \(\varGamma _0\), and a Neumann boundary operator
On S, we consider the transmission operator \({{\mathcal {C}}}=({{\mathcal {C}}}^0,{{\mathcal {C}}}^1)\), where
Note that equals \(\llbracket a\partial _{\nu _S}u \rrbracket \) and not .
Of concern in this paper is the inhomogeneous linear transmission problem
We assume that
For abbreviation, we set, for \(1<q<\infty \),
and introduce the trace spaces
and
As a rule, we often drop the index q if \(q=p\). Thus, \({{\bar{W}}}^{2/\varvec{2}}={{\bar{W}}}_{p}^{2/\varvec{2}}\), \(\partial W=\partial W_{p}\), etc. Finally,
is the closed linear subspace of \(\partial W\oplus \partial _SW\oplus \gamma _0{{\bar{W}}}\) consisting of all \((\varphi ,\psi ,u_0)\) satisfying the compatibility conditions
where \(\varphi =(\varphi ^0,\varphi ^1)\) and \(\psi =(\psi ^0,\psi ^1)\). It follows from the anisotropic trace theorem ([13, Example VIII.1.8.6]) that \(\partial _{{{\mathcal {B}}},{{\mathcal {C}}}}W\) is well-defined.
Given Banach spaces E and F, we write \({{\mathcal {L}}}{{\mathrm{is}}}(E,F)\) for the set of all isomorphisms in \({{\mathcal {L}}}(E,F)\), the Banach space of continuous linear maps from E into F.
Now we can formulate the following maximal regularity theorem for problem (7.2). Its proof, which needs considerable preparation, is found in Sect. 13.
Theorem 7.1
Let (7.3) be satisfied. Then,
8 The uniform Lopatinskii–Shapiro condition
In the proof of Theorem 7.1, we need to consider systems of elliptic boundary value problems. For this, we have to be precise on the concept of uniform ellipticity.
Let \((M,g)\) be any Riemannian manifold. We consider a general second-order linear differential operator on M,
with \(u=(u^1,\ldots ,u^n)\) and
Here, \(\nabla ^iu=(\nabla ^iu^1,\ldots ,\nabla ^iu^n)\) so that, for example,
where s is summed from 1 to n and denotes complete contraction, that is, summation over all twice occurring indices in any local coordinate representation.
The (principal) symbol \({{\mathfrak {s}}}{{\mathcal {A}}}\) of \({{\mathcal {A}}}\) is the \((n\times n)\)-matrix-valued function defined by
Then, \({{\mathcal {A}}}\) is uniformly normally elliptic if there exists an ‘ellipticity constant’ \(\underline{\alpha }\in (0,1)\) such that
for all \(p\in M\) and \(\xi \in T_p^*M\) with \(|\xi |_{g^*(p)}^2=1\), where \(\sigma (\cdot )\) denotes the spectrum.
Suppose \(\varGamma \ne \emptyset \) and \({{\mathcal {B}}}=({{\mathcal {B}}}^1,\ldots ,{{\mathcal {B}}}^n)\) is a linear boundary operator of order at most 1. More precisely, we assume that there is \(k\in \{0,\ldots ,n\}\) such that
with
Then, the (principal) symbol of \({{\mathcal {B}}}\) is the \((n\times n)\)-matrix-valued function \({{\mathfrak {s}}}{{\mathcal {B}}}\) given by
Observe that if X is a vector, \(\omega \) a covector field, and \(\langle {}\cdot {},{}\cdot {}\rangle \) the duality pairing.
We denote by \(\nu ^\flat \in T_\varGamma ^*M\) the inner conormal on \(\varGamma \) defined in local coordinates by \(\nu ^\flat =g_{ij}\nu ^j\mathrm{d}x^i\). Given \(q\in \varGamma \), we write \({{\mathbb {B}}}(q)\) for the set of all
Then, if \((\xi ,\lambda )\in {{\mathbb {B}}}(q)\), we introduce linear differential operators on \({{\mathbb {R}}}\) by
where \(i=\sqrt{-1}\).
As usual, \(C_0({{\mathbb {R}}}_+,{{\mathbb {C}}}^n)\) is the closed linear subspace of \(BC({{\mathbb {R}}}_+,{{\mathbb {C}}}^n)\) consisting of the functions that vanish at infinity.
Suppose \({{\mathcal {A}}}\) is uniformly normally elliptic. Then, it follows that the homogeneous problem
has for each \(q\in \varGamma \) and \((\xi ,\lambda )\in {{\mathbb {B}}}(q)\) precisely n linearly independent solutions whose restrictions to \({{\mathbb {R}}}_+\) belong to \(C_0({{\mathbb {R}}}_+,{{\mathbb {C}}}^n)\). We denote their span by \(C_0(q,\xi ,\lambda )\). It is an n-dimensional linear subspace of \(C_0({{\mathbb {R}}}_+,{{\mathbb {C}}}^n)\).
Now we consider the initial value problem on the half-line:
Then, \(({{\mathcal {A}}},{{\mathcal {B}}})\) satisfies the uniform parameter-dependent Lopatinskii–Shapiro (LS) conditions if problem (8.5) has for each \(\eta \in {{\mathbb {C}}}^n\) a unique solution
and
unif. w.r.t. \(q\in \varGamma \) and \((\xi ,\lambda )\in {{\mathbb {B}}}(q)\).
The basic feature, which distinguishes the above definition from the usual form of the LS condition, is the requirement of the uniform bound (8.7). Without this requirement, the LS condition is much simpler to formulate (e.g., [5, 6, 22, 23, 32], for example) and to verify.
It is known that the LS condition is equivalent to the parameter-dependent version of the complementing condition of Agmon et al. [2] (see, for example, [27, VII§9] or [38, Sect. 10.1]). Using this version, it is possible to define a uniform complementing condition which is equivalent to (8.7) (see [3] and [4]). However, that condition is even more difficult to verify in concrete situations. We refer to [15] for a detailed exposition of all these facts. It should be noted that the uniformity condition (8.7) is fundamental for the following, since we will have to work with infinitely many linear model problems.
9 Model cases
For the proof of Theorem 7.1, we have to understand the model cases to which problem (7.2) reduces in local coordinates.
Until further notice, it is assumed that
By Remark 3.1(b), we can assume that \(\mathop {{\mathrm{diam}}}\nolimits (U_{\kappa })<\varepsilon /2\) for \(\kappa \in {{\mathfrak {K}}}_S\) where \(\varepsilon \) is the width of the tubular neighborhood of S.
We can choose a family \(\{\,\pi _\kappa ,\chi \ ;\ \kappa \in {{\mathfrak {K}}}\,\}\) with the following properties:
(See Lemma 3.2 in [10] or [15].) We fix an \(\tilde{\omega }\in {{\mathcal {D}}}\bigl ((-1,1)^m,[0,1]\bigr )\) which is equal to 1 on \(\mathop {{\mathrm{supp}}}\nolimits (\chi )\). Then,
is a Riemannian metric on \({{\mathbb {R}}}^m\) such that
and
This follows from (3.3). Furthermore,
Note that
We write \(\mathop {{\mathrm{grad}}}\nolimits _\kappa :=\mathop {{\mathrm{grad}}}\nolimits _{g_\kappa }\) and \(\mathop {{\mathrm{div}}}\nolimits _\kappa :=\mathop {{\mathrm{div}}}\nolimits _{g_\kappa }\) for \(\kappa \in {{\mathfrak {K}}}\). Then,
Let \(\delta _\kappa :=\kappa _*\delta \). Then,
where \(\nu _\kappa \) is the inner normal on \(\partial {{\mathbb {H}}}^m\) with respect to \(({{\mathbb {H}}}^m,g_\kappa )\), and \((\cdot |\cdot )_\kappa =g_\kappa \). If \(\kappa \in {{\mathfrak {K}}}_S\), then
Using these notations, we consider the three model problems:
and
and
In the following two sections, we prove that each one of them, complemented by appropriate initial and compatibility conditions, enjoys a maximal regularity result, unif. w.r.t. \(\kappa \).
10 Continuity
First, we note that
and, given \(k\in {{\mathbb {N}}}\),
with \(\nabla _{\kappa }u:=\kappa _*\nabla \kappa ^*u\) (cf. [10, Lemma 3.1] or [15]).
We set
Then,
and
where \(0\le s\le 2\). For the sake of a unified presentation,
If \({{\mathbb {X}}}\in \{{{\mathbb {R}}}^m,{{\mathbb {H}}}^m\}\), then \(W_{p}^s({{\mathbb {X}}}):=W_{p}^s({{\mathbb {X}}},g_m)\). It is a consequence of (10.1) and (10.2) that
where \({}\doteq {}\) stands for ‘equal except for equivalent norms.’
Since
(cf. [13, Theorems VII.2.7.4 and VII.2.8.3, as well as (VII.3.6.3)]), it follows from definition (6.4) and from (10.4) that
Due to (10.1) and (10.2), we get, with an analogous definition of \({\mathsf {BC}}\),
Using this, (6.5), and (6.6), we infer that
First, we note that (3.1), (7.1), (9.4), and (10.2) imply
In local coordinates, \(\mathop {{\mathrm{grad}}}\nolimits u=g^{ij}\partial _ju\,\partial /\partial x^i\). Using this, (3.3), and (10.8), we deduce that
Given a vector field \(Y=Y^i\partial /\partial x^i\), it holds \(\mathop {{\mathrm{div}}}\nolimits Y=\sqrt{g}^{-1}\partial _i\bigl (\sqrt{g}\,Y^i\bigr )\). By this and the above, it is verified that
Now we consider Sobolev–Slobodeckii spaces on \(\partial {{\mathbb {H}}}^m\simeq {{\mathbb {R}}}^{m-1}\). We set for \(\kappa \in {{\mathfrak {K}}}_\varGamma \). Then,
and
Suppose \(0<\sigma<s<1\). Then,
where the \(BUC^\rho \) are the usual Hölder spaces and \(b_\infty ^{\sigma /\varvec{2}}\) is an anisotropic little Besov space. Indeed, the first embedding follows from the mean value theorem and by using the localized Hölder norm (cf. [13, (VII.3.7.1)]). For the norm equivalence, we refer to definition (VII.3.6.4) and Remark VII.3.6.4. The last embedding is implied by Lemma VII.2.2.3 and Theorem VII.7.3.4. By Theorem VII.2.7.4 in [13],
We deduce from (3.3) and (10.2) that
\({{\mathfrak {K}}}_\varGamma \)-unif. Now it follows from (10.12) and (10.13) that
Since, trivially, \(\gamma \in {{\mathcal {L}}}\bigl (BC^{1/\varvec{2}}({{\mathbb {H}}}^m\times J), \,BC^{1/\varvec{2}}(\partial {{\mathbb {H}}}^m\times J)\bigr )\), it is now clear that
In local coordinates,
Hence, \(\delta _\kappa {{\mathcal {B}}}_\kappa u=\delta _\kappa b_\kappa ^i\gamma \partial _iu =\delta _\kappa \gamma a_\kappa \nu _\kappa ^i\gamma \partial _iu_\kappa \), where it follows from (3.3), (9.5), and \(\Vert a_\kappa \Vert _\infty \le c\) that
and, from (10.14),
for \(\kappa \in {{\mathfrak {K}}}_\varGamma \). Thus, it is a consequence of (10.14), (10.16), and the boundary operator retraction theorem [13, Theorem VIII.2.2.1] thatFootnote 4
Clearly, ‘\({{\mathfrak {K}}}_\varGamma \)-uniform’ means that there exists a coretraction \({{\mathcal {B}}}_\kappa ^c\) such that \(\Vert {{\mathcal {B}}}_\kappa \Vert \) and \(\Vert {{\mathcal {B}}}_\kappa ^c\Vert \) are \({{\mathfrak {K}}}_\varGamma \)-uniformly bounded.
Obviously,
If we replace in the preceding arguments the boundary operator retraction argument by Theorem VIII.2.3.3 of [13], we find that
It follows from (10.5) that
The anisotropic trace theorem ( [13, Corollary VII.4.6.2, Theorems VIII.1.2.9 and VIII.1.3.1]) implies that
is a retraction. Using Theorems VII.2.7.4 and VII.2.8.3, definition VII.3.6.3 and Remark VII.3.6.4 of [13], we get
Now we infer from (10.7), (10.20)–(10.22), and (10.5) that
11 Maximal regularity
First, we rewrite \(({{\mathcal {A}}},{{\mathcal {B}}})\) in terms of covariant derivatives. For this, we define
in local coordinates by
Then, we get
where \(\varDelta \) is the Laplace–Beltrami operator (e.g., [36, (2.4.10)]). Consequently,
For the boundary operator, we find
Hence,
Clearly, these formulas apply to any oriented Riemannian manifold, thus to \((\partial {{\mathbb {H}}}^m,g_\kappa )\), \(\kappa \in {{\mathfrak {K}}}_\varGamma \).
It follows from (9.5) and (11.2) that
for \(x\in {{\mathbb {X}}}_\kappa \), \(\xi \in T_x^*{{\mathbb {X}}}_\kappa \), \(t\in J\), and \(\kappa \in {{\mathfrak {K}}}\). Hence,
We begin with the full-space problem.
Proposition 11.1
It holds
that is,
Proof
It is obvious from (10.10) and (10.23) that
Due to (11.5), the assertion now follows from Corollary 9.7 in [16] and Theorem III.4.10.8 in [7] and (the proof of) Theorem 7.1 in [8]. (See [15] for a different demonstration).\(\square \)
Next, we study the case where \(\kappa \in {{\mathfrak {K}}}_\varGamma \). For this, we first establish the validity of the uniform LS condition. Henceforth, it is always assumed that
We fix any \(t\in J\) and omit it from the notation. The reader will easily check that all estimates are uniform with respect to \(t\in J\). From (11.2), we see that the first equation in (8.4) has the form
where
with the principal value of the square root.
Suppose \(|\xi |_{g_\kappa ^*(x)}^2\le 1/2\). Then, \(\zeta \in {{\mathbb {B}}}(x)\) implies \(\rho _\kappa ^2(\zeta )\ge 1/2a_\kappa (x)\). Otherwise, \(\rho _\kappa ^2(\zeta )\ge 1/2\). Thus, since \(\Vert a_\kappa \Vert _\infty \le c\), we find a \(\beta >0\) such that
From \(a_\kappa \ge \underline{\alpha }\), we infer that \(|\rho _\kappa (\zeta )|\le c\) for \(\kappa \in {{\mathfrak {K}}}_\varGamma \). Set
Then, \({{\mathbb {C}}}v_\kappa (\zeta )\) is the subspace of \(C_0({{\mathbb {R}}},{{\mathbb {C}}})\) of decaying solutions of (11.7).
Let \(\kappa \in {{\mathfrak {K}}}_{\varGamma _0}\) so that \({{\mathcal {B}}}_\kappa =\gamma \), the Dirichlet operator on \(\partial {{\mathbb {H}}}^m\). Then (recall (8.6)), \(R_\kappa (\zeta )\eta =\eta v_\kappa (\zeta )\). Thus, by (11.9),
Now assume \(\kappa \in {{\mathfrak {K}}}_{\varGamma _1}\). Then, we see from (11.3) and (11.10) that
Consequently,
This proves that \(({{\mathcal {A}}}_\kappa ,{{\mathcal {B}}}_\kappa )\) satisfies the uniform parameter-dependent LS condition, unif. w.r.t. \(\kappa \in {{\mathfrak {K}}}_\varGamma \) and \(t\in J\).
Proposition 11.2
It holds
Proof
We deduce from (10.17), (10.23), and [13, Example VIII.1.8.6] that \([\partial W_{\kappa }\oplus \gamma _0W_{\kappa }]_{{{\mathcal {B}}}_\kappa }\) is a well-defined closed linear subspace of \(\partial W_{\kappa }\oplus \gamma _0W_{\kappa }\) and, using also (10.10),
The uniform LS condition implies now the remaining assertions. For this, we refer to [15].\(\square \)
Nonhomogeneous linear parabolic boundary value problems (of arbitrary order and in a Banach-space-valued setting) on Euclidean domains have been studied in [23]. It follows, in particular from Proposition 6.4 therein, that the isomorphism assertion is true for each \(\kappa \in {{\mathfrak {K}}}_\varGamma \). However, it is not obvious whether the \({{\mathfrak {K}}}_\varGamma \)-uniformity statement does also follow from the results in [23]. For this, one would have to check carefully the dependence of all relevant estimates on the various parameters involved, which would be no easy task. (The same observation applies to Proposition 11.1). In [15], we present an alternative proof which takes care of the needed uniform estimates.
Lastly, we assume that \(\kappa \in {{\mathfrak {K}}}_S\). We set, once more suppressing a fixed \(t\in J\),
and
Then,
Furthermore, \({{\mathfrak {B}}}_\kappa =({{\mathfrak {B}}}_\kappa ^0,{{\mathfrak {B}}}_\kappa ^1)\), where
on \(\partial {{\mathbb {H}}}^m\).
Clearly,
Thus, \({{\mathfrak {A}}}_\kappa \) is uniformly normally elliptic on \({{\mathbb {H}}}^m\), \({{\mathfrak {K}}}_S\)-unif.
We define \(\rho _\kappa ^i\), \(1=1,2\), by replacing \(a_\kappa \) in (11.8) by \(a_\kappa ^i\) and introduce \(v_\kappa ^i\) by changing \(\rho _\kappa \) in(11.10) to \(\rho _\kappa ^i\). Then,
is the subspace of \(C_0({{\mathbb {R}}}_+,{{\mathbb {C}}}^2)\) of decaying solutions of
From (11.11) and (11.12), we see that the initial conditions in (8.5) are in the present case
Omitting \(x\in \partial {{\mathbb {H}}}^m\), the solution of this system is
From this, the uniform boundedness of \(a_\kappa \), and \(\mathop {{\mathrm{Re}}}\nolimits (a_\kappa ^1\rho _\kappa ^1+a_\kappa ^2\rho _\kappa ^2)\ge 1/\underline{\alpha }\beta \), it follows that \(({{\mathfrak {A}}}_\kappa ,{{\mathfrak {B}}}_\kappa )\) satisfies the uniform parameter-dependent LS condition, unif. w.r.t. \(\kappa \in {{\mathfrak {K}}}_S\) and \(t\in J\).
Proposition 11.3
It holds
unif. w.r.t. \(\kappa \in {{\mathfrak {K}}}_S\) and \(t\in J\).
Proof
Set \({{\mathfrak {u}}}(x):=\bigl (u(x),u(-x)\bigr )\) for \(x\in {{\mathbb {H}}}^m\) and \({{\bar{{{\mathfrak {W}}}}}}_\kappa ^s:={{\bar{W}}}_\kappa ^s\oplus {{\bar{W}}}_\kappa ^s\), etc. Then, the assertion is true iff
By the preceding considerations, the proof of Proposition 11.2 applies verbatim to the system for \({{\mathfrak {u}}}\). This proves the claim.\(\square \)
12 Localizations
We presuppose (7.3) and fix an S-adapted atlas for M with \(\mathop {{\mathrm{diam}}}\nolimits (U_{\kappa })<\varepsilon /2\) for \(\kappa \in {{\mathfrak {K}}}_S\). Then,
\({{\mathfrak {N}}}_\varGamma (\kappa ):={{\mathfrak {N}}}(\kappa )\cap {{\mathfrak {K}}}_\varGamma \), and \({{\mathfrak {N}}}_S(\kappa ):={{\mathfrak {N}}}(\kappa )\cap {{\mathfrak {K}}}_S\). By the finite multiplicity of \({{\mathfrak {K}}}\),
We set for \(\kappa \in {{\mathfrak {K}}}\) and \({\tilde{\kappa }}\in {{\mathfrak {N}}}(\kappa )\)
It follows from (3.1)(ii) that, given \(s\in [0,2]\),
Interpreting \({{\mathfrak {K}}}\) as an index set, we put
endowed with the product topology. For \(\alpha \in \{0,\varGamma \}\), we set
Since \({{\mathfrak {K}}}\) is the disjoint union of \({{\mathfrak {K}}}_0\), \({{\mathfrak {K}}}_\varGamma \), and \({{\mathfrak {K}}}_S\),
A similar definition and direct sum decomposition applies to \(\gamma _0{\varvec{{{\mathsf {W}}}}}\). We also set
We define linear operators \({{\mathcal {R}}}\) and \({{\mathcal {R}}}^c\) by
for \({\varvec{u}}=(u_\kappa )\in {\varvec{{{\mathsf {W}}}}}^{0/\varvec{2}}\) and \(u\in L_1\bigl (J,L_{1,\mathrm{loc}}(M\setminus S)\bigr )\), respectively. Note that the sum is locally finite and \(\pi _\kappa \) is identified with the multiplication operator \(v\mapsto \pi _\kappa v\).
We want to evaluate \({{\mathcal {A}}}\circ {{\mathcal {R}}}{\varvec{u}}\) for \({\varvec{u}}\in {\varvec{{{\mathsf {W}}}}}^{2/\varvec{2}}\). Observe
the commutator being given by
Thus, we get from (12.5)
By \({{\mathcal {A}}}(\kappa ^*u_\kappa )=\kappa ^*{{\mathcal {A}}}_\kappa u_\kappa \), the first sum equals \({{\mathcal {R}}}{\varvec{{{\mathsf {A}}}}}{\varvec{u}}\), where \({\varvec{{{\mathsf {A}}}}}:={\mathop {{\mathrm{diag}}}\nolimits }[{{\mathcal {A}}}_\kappa ]\). Using \(1=\sum _{\tilde{\kappa }}\pi _{\tilde{\kappa }}^2\), we find
Set
Then, (12.2), (12.6), (9.1), (10.8), and \(\kappa _*\pi _\kappa =(\kappa _*\pi _\kappa )\chi \) imply
We define \({\varvec{{{\mathsf {A}}}}}_\kappa ^0:{\varvec{{{\mathsf {W}}}}}^{1/\varvec{2}}\rightarrow {{\mathsf {W}}}_{\kappa }^{0/\varvec{2}}\) by
Then, we deduce from (12.1) and (12.9) that
Moreover, \({\varvec{{{\mathsf {A}}}}}^0:=({\varvec{{{\mathsf {A}}}}}_\kappa ^0)_{\kappa \in {{\mathfrak {K}}}}\).
Now we sum (12.8) over \(\kappa \in {{\mathfrak {K}}}\) and interchange the order of summation to find that the second sum in (12.7) equals \({{\mathcal {R}}}{\varvec{{{\mathsf {A}}}}}^0{\varvec{u}}\). Thus, in total,
Similar considerations lead to
Here, \(\tilde{{\varvec{{{\mathsf {A}}}}}}{}^0=(\tilde{{\varvec{{{\mathsf {A}}}}}}{}_\kappa ^0)_{\kappa \in {{\mathfrak {K}}}}\) with \(\tilde{{\varvec{{{\mathsf {A}}}}}}{}_\kappa ^0\in {{\mathcal {L}}}({\varvec{{{\mathsf {W}}}}}^{1/\varvec{2}},{{\mathsf {W}}}_{\kappa }^{0/\varvec{2}})\) satisfying
We turn to \(\varGamma \) and define for \(\kappa \in {{\mathfrak {K}}}_\varGamma \). Then,
and
Observe that
and
Similarly as above, we find
where \({\varvec{B}}:={\mathop {{\mathrm{diag}}}\nolimits }[{{\mathcal {B}}}_\kappa ]\) and \({\varvec{B}}^0=({\varvec{B}}_\kappa ^0)_{\kappa \in {{\mathfrak {K}}}_\varGamma }\) with \({\varvec{B}}_\kappa ^0:{\varvec{W}}^{1/\varvec{2}}[\varGamma ]\rightarrow \{0\}\oplus \partial _1W_{\kappa }\) satisfying
Analogously,
where \(\tilde{{\varvec{B}}}{}^0=(\tilde{{\varvec{B}}}{}_\kappa ^0)\) with \(\tilde{{\varvec{B}}}{}_\kappa ^0 :{\varvec{W}}^{1/\varvec{2}}[\varGamma ]\rightarrow \{0\}\oplus \partial _1W_{\kappa }\) is such that
Concerning the transmission interface S, we define \({{\mathcal {R}}}_S\) and \({{\mathcal {R}}}_S^c\) analogously to (12.14) and (12.15). Observe that
From this, it is now clear that
where \({\varvec{C}}:={\mathop {{\mathrm{diag}}}\nolimits }[{{\mathcal {C}}}_\kappa ]\), \({\varvec{C}}^0=({\varvec{C}}_\kappa ^0)\), and \(\tilde{{\varvec{C}}}{}^0=(\tilde{{\varvec{C}}}{}_\kappa ^0)\) with
for \(\kappa \in {{\mathfrak {K}}}_S\) and \({\varvec{u}}\in {{\bar{{\varvec{W}}}}}^{2/\varvec{2}}\).
The following consequences of the preceding results are needed to establish Theorem 7.1.
Lemma 12.1
Fix \(s\in (1,2)\) and \(q>p\) with \(1/p-1/q<(2-s)/(m+2)\).Then,
Proof
(1) Set \({{\mathbb {X}}}:={{\mathbb {R}}}^m\) and \(s_1:=s+(m+2)(1/p-1/q)<2\). By (VII.3.6.3) and Example VII.3.6.5 of [13]
where \(W_{q}^{s/\varvec{2}}=W_{q}^{s/\varvec{2}}({{\mathbb {X}}}\times J)\), etc. Hence, see [13, Theorem VII.2.2.4(iv)],
Consequently, invoking (10.5),
(2) Since \(\mathop {{\mathrm{supp}}}\nolimits (\chi )\subset (-1,1)^m\), Hölder’s inequality implies
Hence, by interpolating and then using (10.5) once more,
From this, we get
and
Now the assertion follows in this case from step (1).
(3) The preceding arguments also hold if we replace \({{\mathbb {X}}}={{\mathbb {R}}}^m\) by \({{\mathbb {X}}}={{\mathbb {H}}}^m\). Then (see Remark 6.2), it also applies to \({{\mathbb {X}}}={{\mathbb {R}}}^m\setminus \partial {{\mathbb {H}}}^m\). Thus, the lemma is proved.\(\square \)
Given a function space \({{\mathfrak {F}}}\) defined on J, we write \({{\mathfrak {F}}}(\tau )\) for its restriction to \(J_\tau \), \(0<\tau \le T\).
Lemma 12.2
Let \(\hat{{\varvec{{{\mathsf {A}}}}}}_\kappa \in \{{\varvec{{{\mathsf {A}}}}}_\kappa ^0,\tilde{{\varvec{{{\mathsf {A}}}}}}_\kappa \}\). There exists \(\varepsilon >0\) such that
unif. w.r.t. \(\kappa \in {{\mathfrak {K}}}\) and \(0<\tau \le T\).
Proof
Fix s and q as in Lemma 12.1 and set \(\varepsilon :=1/p-1/q\). We get from Hölder’s inequality, Lemma 12.1, (12.10), and (12.13)
for \(0<\tau \le T\), \({{\mathfrak {K}}}\)-unif., where \(\Vert \cdot \Vert _{L_q\cap {{\mathsf {W}}}_{\kappa }^{s/2}}\) is the norm in the image space occurring in Lemma 12.1.\(\square \)
The next lemma provides analogous estimates for the boundary and transmission operators.
Lemma 12.3
Let \(\hat{{\varvec{B}}}_\kappa \in \{{\varvec{B}}_\kappa ^0,\tilde{{\varvec{B}}}{}_\kappa ^0\}\) and \(\hat{{\varvec{C}}}_\kappa \in \{{\varvec{C}}_\kappa ^0,\tilde{{\varvec{C}}}{}_\kappa ^0\}\). There exists \(\varepsilon >0\) such that
and
for \(0<\tau \le T\).
Proof
Let s, q, and \(\varepsilon \) be as in the preceding proof.
Given any Banach space E, Hölder’s inequality gives
Hence, by interpolation (see [13, Theorems VII.2.7.4 and VII.7.3.4]),
Set
Then, we get from (12.23)
It is clear from the structure of \(\hat{{\varvec{B}}}_\kappa \) and the mapping properties of \(\gamma \) that
Since \({{\mathcal {L}}}_\kappa ^s(\tau )\hookrightarrow {{\mathcal {L}}}_\kappa ^1(\tau )\) \({{\mathfrak {K}}}_\varGamma \)-unif. and unif. w.r.t. \(\tau \), the first assertion now follows from (12.24), Lemma 12.1, and the fact that \(\hat{\varvec{B}}_\kappa \) has its image in the closed linear subspace \(\partial _1W_{\kappa }(\tau )\) of \(\partial W_{\kappa }(\tau )\).
The proof of the second claim is similar.\(\square \)
13 Proof of Theorem 7.1
Let \({\varvec{E}}=\prod _{\alpha \in {{\mathsf {A}}}}E_\alpha \), where each \(E_\alpha \) is a Banach space and \({{\mathsf {A}}}\) is a countable index set. Then, \(\ell _p({\varvec{E}})\) is the Banach space of p-summable sequences in \({\varvec{E}}\).
We put
and
for \(i=0,2\). Then,
Moreover, \({{\mathbb {F}}}_\varGamma :=\ell _p(\partial {\varvec{W}})\), \({{\mathbb {F}}}_S:=\ell _p(\partial _S{{\bar{{\varvec{W}}}}})\).
Recall the definitions of \(({{\mathcal {R}}},{{\mathcal {R}}}^c)\), \(({{\mathcal {R}}}_\varGamma ,{{\mathcal {R}}}_\varGamma ^c)\), and \(({{\mathcal {R}}}_S,{{\mathcal {R}}}_S^c)\) in Sect. 12.
Proposition 13.1
\(({{\mathcal {R}}},{{\mathcal {R}}}^c)\) is an r-c pair for \(({{\mathbb {E}}}^k,{{\bar{W}}}^{k/\varvec{2}})\), \(k=0,2\), and
is an r-c pair for
Proof
This is obtained from [10, Theorem 6.1] (also see [9, Theorem 9.3] or [15]).\(\square \)
Lemma 13.2
\(\gamma _0\) is a retraction from \({{\bar{W}}}^{2/\varvec{2}}\) onto \(\gamma _0{{\bar{W}}}\) and from \({{\mathbb {E}}}^2\) onto \(\gamma _0{{\bar{{{\mathbb {E}}}}}}\). Furthermore, \(\gamma _0{{\mathcal {R}}}={{\mathcal {R}}}\gamma _0\).
Proof
Due to (10.23), there exists \(\gamma _0^c\in {{\mathcal {L}}}(\gamma _0{{\bar{{{\mathbb {E}}}}}},{{\mathbb {E}}}^2)\) such that \((\gamma _0,\gamma _0^c)\) is an r-c pair for \(({{\mathbb {E}}}^2,\gamma _0{{\bar{{{\mathbb {E}}}}}})\). For the moment, we denote it by \(({{\bar{\gamma }}}_0,{{\bar{\gamma }}}_0^c)\). Then, it follows from Proposition 13.1 that
Since \(\gamma _0\) is the evaluation at \(t=0\) and \(({{\mathcal {R}}},{{\mathcal {R}}}^c)\) is independent of \(t\in J\), we see that \(\gamma _0{{\mathcal {R}}}={{\mathcal {R}}}{{\bar{\gamma }}}_0\). Hence, \(\gamma _0\in {{\mathcal {L}}}({{\bar{W}}}^{2/\varvec{2}},\gamma _0{{\bar{W}}})\) and \(\gamma _0^c:={{\mathcal {R}}}{{\bar{\gamma }}}_0^c{{\mathcal {R}}}^c\) is a continuous right inverse for \(\gamma _0\).\(\square \)
We write
and, using (12.4), set \({\varvec{u}}=({\varvec{v}},{\varvec{w}},{\varvec{z}})\in {\varvec{{{\mathsf {W}}}}}^{2/\varvec{2}}\) and, analogously,
Moreover, \({{\mathbb {G}}}:={{\mathbb {F}}}_\varGamma \oplus {{\mathbb {F}}}_S\oplus \gamma _0{{\bar{{{\mathbb {E}}}}}}\) and \([{{\mathbb {G}}}]_{{\varvec{B}},{\varvec{C}}}\) is the linear subspace consisting of all \(({\varvec{\varphi }},\varvec{\psi },{\varvec{u}}_0)\) satisfying the compatibility conditions
if \(p>3\), with the corresponding modifications if \(p<3\). Analogous definitions apply to \([{{\mathbb {G}}}]_{\hat{{{\mathbb {B}}}},\hat{{{\mathbb {C}}}}}\).
Proposition 13.3
\(\bigl (\partial _t+\hat{{{\mathbb {A}}}},\,(\hat{{{\mathbb {B}}}},\hat{{{\mathbb {C}}}},\gamma _0)\bigr ) \in {{\mathcal {L}}}{{\mathrm{is}}}\bigl ({{\mathbb {E}}}^2,\,{{\mathbb {E}}}^0\oplus [{{\mathbb {G}}}]_{\hat{{{\mathbb {B}}}},\hat{{{\mathbb {C}}}}}\bigr )\).
Proof
(1) First, we prove that
Since \({{\mathbb {L}}}\) has diagonal structure, the claim is a direct consequence of Propositions 11.1–11.3.
(2) Set \({{\mathbb {L}}}_0:=(\hat{{\varvec{{{\mathsf {A}}}}}},\hat{{\varvec{B}}},\hat{{\varvec{C}}},0)\). Then,
It follows from (12.1) and Lemmas 12.2 and 12.3 that \(\hat{{{\mathbb {L}}}}_0\in {{\mathcal {L}}}({{\mathbb {E}}}^2,\,{{\mathbb {E}}}^0\oplus {{\mathbb {G}}})\) and there exists \(\varepsilon >0\) such that
From this and step (1), we see that
Write
By Lemma 13.2, we can choose a coretraction \(\gamma _0^c\) for \(\gamma _0\in {{\mathcal {L}}}({{\mathbb {E}}}^2,\gamma _0{{\bar{{{\mathbb {E}}}}}})\). Let \(({\varvec{f}},{\varvec{g}})\in {{\mathbb {E}}}^0\oplus [{{\mathbb {G}}}]_{\hat{{{\mathbb {B}}}},\hat{{{\mathbb {C}}}}}\) with \({\varvec{g}}=({\varvec{\varphi }},{\varvec{\psi }},{\varvec{u}}_0)\). Set \(\overline{{\varvec{u}}}:=\gamma _0^c{\varvec{u}}_0\). Then, \({\varvec{u}}\in {{\mathbb {E}}}^2\) satisfies \(\hat{{{\mathbb {L}}}}{\varvec{u}}=({\varvec{f}},{\varvec{g}})\) iff \({\varvec{v}}:={\varvec{u}}-\overline{{\varvec{u}}}\) is such that
Note that \({\varvec{v}}\in {{\mathbb {E}}}_0^2\) and \({\varvec{g}}_0\in [{{\mathbb {G}}}_0]_{\hat{{{\mathbb {B}}}},\hat{{{\mathbb {C}}}}}\). Suppose \(p>3\). Given any \({\varvec{w}}\in {{\mathbb {E}}}_0^2\), we get
From this, the analogous relation for \({\varvec{C}}\) and \(\hat{{{\mathbb {C}}}}\), and (13.4), we infer that it suffices to prove that \(\hat{{{\mathbb {L}}}}:{{\mathbb {E}}}_0^2\rightarrow {{\mathbb {E}}}^0\oplus [{{\mathbb {G}}}_0]_{{\varvec{B}},{\varvec{C}}}\) is surjective and has a continuous inverse. Obvious modifications apply to \(p<3\).
(3) Let \({{\mathbb {F}}}:={{\mathbb {E}}}^0\oplus [{{\mathbb {G}}}_0]_{{\varvec{B}},{\varvec{C}}}\) and \({\varvec{h}}\in {{\mathbb {F}}}\). Suppose \({\varvec{u}}\in {{\mathbb {E}}}_0^2\) and set \({\varvec{v}}:={{\mathbb {L}}}{\varvec{u}}\in {{\mathbb {F}}}\). By step (1) and (13.2), the equation \(\hat{{{\mathbb {L}}}}u={\varvec{h}}\) is equivalent to \({\varvec{v}}+\hat{{{\mathbb {L}}}}_0{{\mathbb {L}}}^{-1}{\varvec{v}}={\varvec{h}}\). Observe \(\hat{{{\mathbb {L}}}}_0{{\mathbb {L}}}^{-1}{\varvec{v}}\in {{\mathbb {F}}}\).
Due to (13.3), we can fix \(\overline{\tau }\in (0,T]\) such that \(\Vert \hat{{{\mathbb {L}}}}_0{{\mathbb {L}}}^{-1}\Vert _{{{\mathcal {L}}}({{\mathbb {F}}}(\tau ))}\le 1/2\). As is well-known (e.g., [12, Lemma 12.2]), this implies that \(\hat{{{\mathbb {L}}}}\in {{\mathcal {L}}}{{\mathrm{is}}}\bigl ({{\mathbb {F}}}(\overline{\tau }),{{\mathbb {F}}}(\overline{\tau })\bigr )\).
(4) If \(\overline{\tau }=T\), then we are done. Otherwise, we repeat this argument for the problem in \([0,\,T-\overline{\tau }]\) obtained by the time shift \(t\mapsto t-\overline{\tau }\) and with the initial value \({\varvec{u}}(\overline{\tau })\). After finitely many such steps, we reach T. The proposition is proved.\(\square \)
Proof of Theorem 7.1
Now we write \(({{\mathbb {A}}},{{\mathbb {B}}},{{\mathbb {C}}})\) for \((\hat{{{\mathbb {A}}}},\hat{{{\mathbb {B}}}},\hat{{{\mathbb {C}}}})\) if \((\hat{{\varvec{{{\mathsf {A}}}}}},\hat{{\varvec{B}}},\hat{{\varvec{C}}})\) equals \(({\varvec{{{\mathsf {A}}}}}^0,{\varvec{B}}^0,{\varvec{C}}^0)\), and \((\tilde{{{\mathbb {A}}}},\tilde{{{\mathbb {B}}}},\tilde{{{\mathbb {C}}}})\) otherwise. We also set \(G:=\partial W\oplus \partial _SW\oplus \gamma _0{{\bar{W}}}\). Then, Proposition 13.1 implies
and
for \(k=0,2\).
Let \(\bigl ({\varvec{u}},({\varvec{\varphi }},{\varvec{\psi }},{\varvec{u}}_0)\bigr )\in {{\mathbb {E}}}^2\oplus {{\mathbb {G}}}\) and write \(\bigl (u,(\varphi ,\psi ,u_0)\bigr )\) for its image under \({\vec {{{\mathcal {R}}}}}\). Then, we obtain from (12.16) and (12.20)
Suppose \(p>3\), \({\varvec{u}}_0=\gamma _0{\varvec{u}}\), and \({{\mathbb {B}}}(0){\varvec{u}}_0={\varvec{\varphi }}(0)\). Then,
Since Lemma 13.2 and \({\varvec{u}}_0=\gamma _0{\varvec{u}}\) imply \(u_0=\gamma _0u\), we see that \({{\mathcal {B}}}(0)u_0=\varphi _0\). Similarly, we find that it follows from \({{\mathbb {C}}}(0){\varvec{u}}_0={\varvec{\psi }}(0)\) that \({{\mathcal {C}}}(0)u_0=\psi (0)\). Thus, letting \({{\mathbb {F}}}:=[{{\mathbb {G}}}]_{{{\mathbb {B}}},{{\mathbb {C}}}}\) and \(F:=[G]_{{{\mathcal {B}}},{{\mathcal {C}}}}\), we have shown that
Consequently,
We find analogously that
This holds for \(p>3\). The case \(p<3\) is similar.
Now we set
Define \(\tilde{{{\mathbb {L}}}}:=\bigl (\partial +\tilde{{{\mathbb {A}}}},\,(\tilde{{{\mathbb {B}}}},\tilde{{{\mathbb {C}}}},\gamma _0)\bigr )\). It is a consequence of (12.11), (12.12), (12.16), (12.18), (12.20), and the fact that \(({{\mathcal {R}}},{{\mathcal {R}}}^c)\) is independent of t that
Proposition 13.3 guarantees
Suppose \(Lu=0\). Then, (13.7) and (13.8) imply \({{\mathcal {R}}}^cu=0\). Thus, \(u={{\mathcal {R}}}{{\mathcal {R}}}^cu=0\). This shows that L is injective.
Let \((f,g)\in {{\bar{W}}}^{0/\varvec{2}}\oplus F\). By (13.8), we find \({\varvec{u}}\in {{\mathbb {E}}}^2\) satisfying \({{{\mathbb {L}}}{\varvec{u}}=\vec {{{\mathcal {R}}}}^c(f,g)}\). Put
Hence, L is surjective and, by (13.5) and (13.6),
The proof is accomplished.\(\square \)
Remarks 13.4
-
(a)
Recall that either some of \(\varGamma _0\), \(\varGamma _1\), and S, or all of them, can be empty. If \((\varGamma ,S)\ne \{\emptyset ,\emptyset \}\), then the result is new. Otherwise, it is a special case of the more general Theorem 1.23(ii) of [12].
-
(b)
Theorem 7.1 is true for systems \(u=(u^1,\ldots ,u^n)\), provided the uniform Lopatinskii–Shapiro conditions apply. This is trivially the case if a is a diagonal matrix. \(\square \)
14 Membranes with boundary
Now we turn to the case of membranes intersecting \(\varGamma \) transversally. This case is handled by reducing it to the situation studied in the preceding section.
Theorem 14.1
Let (5.1) be satisfied. Theorem 7.1 applies with \((M,g)\) replaced by \((\hat{M},\hat{g})\).
Proof
Theorem 5.1.\(\square \)
In preparation for the proof of Theorem 2.1, we derive rather explicit representations of \(({{\mathcal {A}}},{{\mathcal {B}}},{{\mathcal {C}}})\) and the relevant function spaces in a tubular neighborhood of \(\varSigma =\partial S\) in \(\hat{M}\).
We use the notations of Sects. 4 and 5 and set
The Christoffel symbols \(\tilde{\varGamma }_{ij}^k\) for the metric \(\tilde{g}\) turn out to be
We set \(D:=(D_1,D_2)\) with \(D_i:=\rho \partial _i\) and \(\tilde{\nabla }:=\nabla _{\tilde{g}}\). Then, for \(u\in {{\bar{C}}}^2(\dot{N}\setminus G_\sigma )\) with \(G_\sigma :=\mathop {{\mathrm{graph}}}\nolimits (f_\sigma )\),
and
Let
and
We write
for the space of all \(u\in L_{1,\mathrm{loc}}(\dot{N}\times \varSigma )\) for which the norm
is finite. Moreover, \({{\bar{{{\mathcal {W}}}}}}_{p}^0\) is obtained by replacing \(\langle \cdot \rangle _\rho \) by \(\vert \cdot \vert \). It is then clear how to define the anisotropic spaces \({{\bar{{{\mathcal {W}}}}}}_{p}^{k/\varvec{2}}\), \(k=0,2\).
Proposition 14.2
\(u\in {{\bar{W}}}_{p}^2(U\setminus S;\hat{g})\) iff \(\chi _*u\in {{\bar{{{\mathcal {W}}}}}}_{p}^2\).
Proof
First, we note that
It follows from (14.3) that
Consequently, since \(\nabla =\chi ^*(\tilde{\nabla }\oplus \nabla _{\varSigma })\chi _*\),
If \((u_j)\) is a converging sequence in \({{\bar{W}}}_{p}^2:={{\bar{W}}}_{p}^2(U\setminus S;\hat{g})\), then we infer from (14.1)–(14.3) that \((\chi _*u_j)\) converges in \({{\bar{{{\mathcal {W}}}}}}_{p}^2\). Hence, \(\chi _*u\in {{\bar{{{\mathcal {W}}}}}}_{p}^2\) if \(u\in {{\bar{W}}}_{p}^2\). From this, (14.5), and Banach’s homomorphism theorem, we obtain
This proves the claim.\(\square \)
Let \(\lambda :V_\lambda \rightarrow (-1,1)^{m-2}\), \(\sigma \mapsto z\) be a local chart for \(\varSigma \). Then,
is a local chart for \(\hat{M}\) with \(U_{\kappa }\subset U\) and \(\kappa (U_{\kappa })=\dot{N}\times (-1,1)^{m-2}\). Set \(\tilde{h}:=\lambda _*h\). It follows from (5.4) that \(\kappa _*\hat{g}=\tilde{g}+\tilde{h}\).
Assume \(u\in C^2(\hat{M})\), put \(v:=\kappa _*u\in C^2(Q_\kappa ^m)\), and denote by \(e_1,\ldots ,e_m\) the standard basis of \({{\mathbb {R}}}^m\). Then,
Similarly, let \(X=X^i\partial /\partial x^i\in C^1(TU_{\kappa })\) and set
where \(\tilde{Y}=Y^\alpha e_\alpha \) with \(\alpha \) running from 3 to m. Observe that v and Y depend on (x, y, z). It follows
Since \(\sqrt{\kappa _*\hat{g}}=\sqrt{\tilde{g}}\,\sqrt{\tilde{h}} =\rho ^{-2}\sqrt{\tilde{h}}\), we see that
From this and (14.6), we obtain, letting \(\tilde{a}:=\kappa _*a\),
As in (2.3), we define curvilinear derivatives by
Then, it follows from (14.7) that
where \({{\mathcal {A}}}_U\) is the restriction of \({{\mathcal {A}}}\) to U. Due to (6.2), the regularity assumption for a stipulated in Theorem 2.1 means that \(a\in {\bar{BC}}^{1/\varvec{2}}(\hat{M}\setminus S\times J,\ \hat{g}+dt^2)\).
Analogously,
From (10.15) and \((\kappa _*\hat{g})^{1i}=0\) for \(i\ge 2\), we deduce that \(\kappa _*\nu (0,y,z)=\rho (0,y)e_1\). Hence,
This implies
Next, we determine the first-order transmission operator on \(({{\hat{M}}},{{\hat{g}}})\). Recalling definition (4.4), we set \(s:=\chi ^*f\in C^\infty (U)\).
Proposition 14.3
Define
Then,
Proof
(1) It follows from (4.4) that
We write \(f(x,z):=f_{\lambda ^{-1}(z)}(x)\). Then, \((x,z)\mapsto F(x,z):=\bigl (x,f(x,z),z\bigr )\) is a local parametrization of \(\tilde{S}\), and
Hence, given \(({{\bar{x}}},{{\bar{z}}})\in (0,\varepsilon )\times (-1,1)^{m-2}\),
For \(\varXi :=(\xi ,\zeta )\in {{\mathbb {R}}}\times {{\mathbb {R}}}^{m-2}\) and \(\hat{\varXi }:=(\hat{\xi },\hat{\eta },\hat{\zeta })\in {{\mathbb {R}}}\times {{\mathbb {R}}}\times {{\mathbb {R}}}^{m-2}\) we find
where \(\alpha ({{\bar{x}}},{{\bar{z}}}):=\rho ^{-2}\bigl ({{\bar{x}}},f({{\bar{x}}},{{\bar{z}}})\bigr )\). Choose \(\hat{\xi }:=\partial _xf({{\bar{x}}},{{\bar{z}}})\), \(\hat{\eta }:=-1\), and \(\hat{\zeta }\) in \({{\mathbb {R}}}^{m-2}\) such that \((\hat{\zeta }|\tilde{\zeta })_{\tilde{h}} =\bigl \langle \partial _zf({{\bar{x}}},{{\bar{z}}}),\tilde{\zeta }\bigr \rangle _{{{\mathbb {R}}}^{m-2}}\) for all \(\tilde{\zeta }\in {{\mathbb {R}}}^{m-2}\), that is, \(\hat{\zeta }\) equals \(\mathop {{\mathrm{grad}}}\nolimits _{\tilde{h}}f({{\bar{x}}},{{\bar{z}}})\). Then \(\bigl (({{\bar{x}}},{{\bar{z}}}),\hat{\varXi }\bigr )\perp T_{({{\bar{x}}},{{\bar{z}}})}\tilde{S}\). Now we define
Then,
is the positiveFootnote 5 normal on \(\tilde{S}\), since \(\tilde{\nu }^1({{\bar{x}}},{{\bar{z}}})e_1+\tilde{\nu }^2({{\bar{x}}},{{\bar{z}}})e_2\) is a positive multiple of the positive normal of the graph of \(f_{{{\bar{\sigma }}}}\) at \(\bigl ({{\bar{x}}},f_{{{\bar{\sigma }}}}({{\bar{x}}})\bigr )\), where \({{\bar{\sigma }}}:=\lambda ^{-1}({{\bar{z}}})\) (cf. (5.9)).
(2) Using (14.6) and \(\kappa _*\nu _S=\tilde{\nu }\), we obtain
the \(\nu _S^i\) being given by (14.10). From this, the assertion is now clear.\(\square \)
Note that, see (14.4),
Since, by (4.4),
uniformly with respect to \(\tau =\bigl (x,f_\sigma (x)\bigr )\in G_\sigma \) and \(\sigma \in \varSigma \), we see that
On the basis of (14.11) and (14.12) it is possible to represent the trace spaces on \(\varGamma \setminus \varSigma \) and \(S\setminus \varSigma \) analogously to \({{\bar{{{\mathcal {W}}}}}}_{p}^{2/\varvec{2}}\). Details are left to the reader.
Proof of Theorem 2.1
The claim is an easy consequence of Theorems 4.4 and 7.1, (14.2), (14.3), Proposition 14.2, (14.8), (14.9), and Proposition 14.3.\(\square \)
Notes
If \(m=2\), then the last term must be disregarded. It is understood that similar interpretations and adaptions are to be made throughout this paper.
We use the same symbol for a Riemannian metric and its restrictions to submanifolds of the same dimension.
If V is a vector bundle over M, then \(C^k(V)\) denotes the vector space of \(C^k\) sections of V.
An operator \(r\in {{\mathcal {L}}}(E,F)\) is a retraction if it has a continuous right inverse, a coretraction \(r^c\). Then, \((r,r^c)\) is an r-c pair for \((E,F)\).
By the conventions employed in (5.9).
References
H. Abels, M. Rauchecker, M. Wilke. Well-posedness and qualitative behaviour of the Mullins–Sekerka problem with ninety-degree angle boundary contact (2019). arXiv:1902.03611.
S. Agmon, A. Douglis, L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math., 17 (1964), 35–92.
H. Amann. Existence and regularity for semilinear parabolic evolution equations. Ann. Scuola Norm. Sup. Pisa, Ser. IV, 11 (1984), 593–676.
H. Amann. Global existence for semilinear parabolic systems. J. reine angew. Math., 360 (1985), 47–83.
H. Amann. Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems. Differential Integral Equations, 3(1) (1990), 13–75.
H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), pages 9–126. Teubner-Texte Math., 133, Stuttgart, 1993.
H. Amann. Linear and quasilinear parabolic problems. Vol. I Abstract linear theory. Birkhäuser, Basel, 1995.
H. Amann. Maximal regularity for nonautonomous evolution equations. Adv. Nonl. Studies, 4 (2004), 417–430.
H. Amann. Anisotropic function spaces on singular manifolds. (2012). arXiv:1204.0606.
H. Amann. Function spaces on singular manifolds. Math. Nachr., 286 (2012), 436–475.
H. Amann. Uniformly regular and singular Riemannian manifolds. In Elliptic and parabolic equations, volume 119 of Springer Proc. Math. Stat., pages 1–43. Springer, Cham, 2015.
H. Amann. Cauchy problems for parabolic equations in Sobolev-Slobodeckii and Hölder spaces on uniformly regular Riemannian manifolds. J. Evol. Equ., 17(1) (2017), 51–100.
H. Amann. Linear and quasilinear parabolic problems. Vol. II Function spaces. Birkhäuser, Basel, 2019.
H. Amann. Linear parabolic equations with strong boundary degenerations. J. Elliptic Parabolic Equ., 6 (2020), 123–144.
H. Amann. Linear and quasilinear parabolic problems. Vol. III Differential equations. Birkhäuser, Basel, 2021. In preparation.
H. Amann, M. Hieber, G. Simonett. Bounded \({H}_\infty \)-calculus for elliptic operators. Diff. Int. Equ., 7 (1994), 613–653.
B. Ammann, N. Große, V. Nistor. Analysis and boundary value problems on singular domains: an approach via bounded geometry. C. R. Math. Acad. Sci. Paris, 357(6) (2019), 487–493.
B. Ammann, N. Große, V. Nistor. The strong Legendre condition and the well-posedness of mixed Robin problems on manifolds with bounded geometry. Rev. Roumaine Math. Pures Appl., 64(2-3) (2019), 85–111.
B. Ammann, N. Große, V. Nistor. Well-posedness of the Laplacian on manifolds with boundary and bounded geometry. Math. Nachr., 292(6) (2019), 1213–1237.
F.E. Browder. Estimates and existence theorems for elliptic boundary value problems. Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 365–372.
C. Băcuţă, A.L. Mazzucato, V. Nistor, L. Zikatanov. Interface and mixed boundary value problems on \(n\)-dimensional polyhedral domains. Doc. Math., 15 (2010), 687–745.
R. Denk, M. Hieber, J. Prüss. \({\cal{R}}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc., 166(788) (2003).
R. Denk, M. Hieber, J. Prüss. Optimal \(L^p\)-\(L^q\)-estimates for parabolic boundary value problems with inhomogeneous data. Math. Z., 257(1) (2007), 193–224.
M. Disconzi, Y. Shao, G. Simonett. Remarks on uniformly regular Riemannian manifolds. Math. Nachr., 289 (2016), 232–242.
H. Garcke, M. Rauchecker. Stability analysis for stationary solutions of the Mullins–Sekerka flow with boundary contact. (2019). arXiv:1907.00833.
N. Große, C. Schneider. Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces. Math. Nachr., 286(16) (2013), 1586–1613.
O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’ceva. Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Transl. Math. Monographs, Providence, R.I., 1968.
Ph. Laurençot, Ch. Walker. Shape Derivative of the Dirichlet Energy for a Transmission Problem. Arch. Ration. Mech. Anal., 237(1) (2020), 447–496.
H. Li, A.L. Mazzucato, V. Nistor. Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains. Electron. Trans. Numer. Anal., 37 (2010), 41–69.
H. Li, V. Nistor, Y. Qiao. Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric finite element method. In Numerical analysis and its applications, volume 8236 of Lecture Notes in Comput. Sci., pages 12–23. Springer, Heidelberg, 2013.
A.L. Mazzucato, V. Nistor. Well-posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks. Arch. Ration. Mech. Anal., 195(1) (2010), 25–73.
J. Prüss, G. Simonett. Moving Interfaces and Quasilinear Parabolic Evolution Equations, volume 105 of Monographs in Mathematics. Birkhäuser, Basel, 2016.
J. Prüss, G. Simonett, M. Wilke. The Rayleigh–Taylor instability for the Verigin problem with and without phase transition. NoDEA Nonlinear Differential Equations Appl., 26(3) (2019), Paper No. 18, 35.
M. Rauchecker. Strong solutions to the Stefan problem with Gibbs–Thomson correction and boundary contact. (2020). arXiv:2001.06438.
Th. Schick. Manifolds with boundary and of bounded geometry. Math. Nachr., 223 (2001), 103–120.
M.E. Taylor. Partial differential equations I. Basic theory. Springer-Verlag, New York, 1996.
M. Wilke. Rayleigh–Taylor instability for the two-phase Navier–Stokes equations with surface tension in cylindrical domains. (2017). arXiv:1703.05214.
J.T. Wloka, B. Rowley, B. Lawruk. Boundary Value Problems for Elliptic Systems. Cambridge University Press, Cambridge, 1995.
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Dedicated to Matthias Hieber, a pioneer of maximal regularity, on the occasion of his sixtieth birthday
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Amann, H. Maximal regularity of parabolic transmission problems. J. Evol. Equ. 21, 3375–3420 (2021). https://doi.org/10.1007/s00028-020-00612-y
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DOI: https://doi.org/10.1007/s00028-020-00612-y
Keywords
- Linear reaction–diffusion equations
- Inhomogeneous boundary and transmission conditions
- Interfaces with boundary intersection
- Maximal regularity
- Riemannian manifolds with bounded geometry and singularities
- Weighted Sobolev spaces