Existence of a global weak solution for a reaction–diffusion problem with membrane conditions

Abstract

Several problems, issued from physics, biology or the medical science, lead to parabolic equations set in two sub-domains separated by a membrane with selective permeability to specific molecules. The corresponding boundary conditions, describing the flow through the membrane, are compatible with mass conservation and energy dissipation and are called the Kedem–Katchalsky conditions. Additionally, in these models, written as reaction–diffusion systems, the reaction terms have a quadratic behaviour. M. Pierre and his collaborators have developed a complete \(L^1\) theory for reaction–diffusion systems with different diffusions. Here, we adapt this theory to the membrane boundary conditions and prove the existence of weak solutions when the initial data have only \(L^1\) regularity using the truncation method for the nonlinearities. In particular, we establish several estimates as the \(W^{1,1}\) regularity of the solutions. Also, a crucial step is to adapt the fundamental \(L^2\) (space, time) integrability lemma to our situation.

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Funding

The authors have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 740623). The work of G.C. was also partially supported by GNAMPA-INdAM.

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Correspondence to Giorgia Ciavolella.

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Appendices

Regularity

We now analyse in detail regularity in our problem referring to Lemma 1.2 that we have rewritten here below, whereas in the next Appendix, we discuss about compactness. We extend previous results for reaction–diffusion systems without membrane [2, 4, 16, 17, 21] and we refer to [24] for the general theory of parabolic equations. We also refer to [17] for a regularity lemma.

Lemma A.1

(A priori bounds). We consider w solution of the following problem in dimension \(d\ge 2\)

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t w-D\Delta w=f,&{}\quad \text{ in }\; Q_T,\\ w=0,&{}\quad \text{ in }\; \Sigma _T,\\ \partial _{{\varvec{n}}^{1}} w^{1}=\partial _{{\varvec{n}}^{1}} w^2=k(w^2-w^{1}), &{} \quad \text{ in }\; \Sigma _{T,\Gamma },\\ w(0,x)=w_0(x)\ge 0, &{} \quad \text{ in }\; \Omega , \end{array} \right. \end{aligned}$$
(30)

with \(f\in L^1(Q_T)\) and \(w_0\in L^1(\Omega )\). Then,

  • \(w \in L^{\beta }\big (0,T;W^{1,\beta }(\Omega )\big ), \; \forall \beta \in \left[ 1,\frac{d}{d-1}\right) \) and \((1+|w|)^\alpha \in L^2\big (0,T;H^1(\Omega )\big )\; \text{ for } \alpha \in \left[ 0,\frac{1}{2}\right) \).

  • The mapping \((w_0,f)\longmapsto w\) is compact from \(L^1(\Omega )\times L^1(Q_T)\) into \(L^1\big (0,T;L^{\gamma _1} (\Omega )\big )\), for all \(\gamma _1<\frac{d}{d-2}\) and \(L^{\gamma _2}(Q_T)\) for all \({\gamma _2} < \frac{2+d}{d}\).

  • The trace mapping \((w_0,f)\longmapsto Tr_\Gamma (w)\in L^\beta \big (0,T;L^\beta (\Gamma )\big ),\; \beta \in \left[ 1,\frac{d}{d-1}\right) \) is also compact.

Notice that we do not use the information \(w\in L^2(Q_T)\) here but \(w\in L^\infty (0,T; L^1(\Omega ))\). That is used in [22] and leads to the exponent \(\beta < \frac{4}{3}\).

Proof

The proof is based on manipulating nonlinear quantities and Sobolev embeddings. We divide it in several steps.

Some \(L^2\) regularity of \(\nabla w\). Multiplying the equation of w in (30) by \(\frac{w}{(1+|w|^\frac{1}{\mu })^\mu }\) and integrating on \(\Omega \), we obtain three terms which we estimate separately.

We begin with the Laplacian term. Recalling the membrane conditions and applying the Leibniz rule and the divergence theorem, arguing by a regularization and a limit technique, we gain, since \(\frac{w}{(1+|w|^\frac{1}{\mu })^\mu }\) is an increasing function,

$$\begin{aligned} \int _{\Omega } \frac{w}{(1+|w|^\frac{1}{\mu })^{\mu }}\; \Delta w&=\int _{\Gamma }\frac{w^1}{(1+|w^1|^{\frac{1}{\mu }})^{\mu }} \partial _{n_1} w^1\;+\;\int _{\Gamma }\frac{w^2}{(1+|w^2|^{\frac{1}{\mu }})^{\mu }} \partial _{n_2} w^2\nonumber \\&\quad -\int _{\Omega } \frac{|\nabla w|^2}{(1+|w|^{\frac{1}{\mu }})^{\mu +1}}\nonumber \\&=\int _{\Gamma }\left( \frac{w^1}{(1+|w^1|{^\frac{1}{\mu })^{\mu }}} - \frac{w^2}{(1+|w^2|{^\frac{1}{\mu }})^{\mu }}\right) k(w^2-w^1)\nonumber \\&\quad -\int _{\Omega } \frac{|\nabla w|^2}{(1+|w|^{\frac{1}{\mu }})^{\mu +1}}\\&\le - \int _{\Omega } \frac{|\nabla w|^2}{(1+|w|^{\frac{1}{\mu }})^{\mu +1}}. \end{aligned}$$

We analyse now the reaction term. We remark that \(0\le \frac{w}{(1+|w|^\frac{1}{\mu })^\mu }\le 1\) and, using that \(f\in L^1(Q_T)\), we conclude

$$\begin{aligned} \int _{\Omega } \left| \frac{w}{(1+|w|^\frac{1}{\mu })^{\mu }}\;f\right| \le \int _{\Omega } |f| = \Vert f\Vert _{L^1(\Omega )}. \end{aligned}$$

Next, for the time derivative, we define the anti-derivative \(0 \le \psi _\mu (w)= \int _0^w \frac{v\, dv}{(1+|v| ^\frac{1}{\mu })^{\mu }} \le w\), then

$$\begin{aligned} \frac{w}{(1+|w|^\frac{1}{\mu })^{\mu }}\;\partial _t w\;=:\; \partial _t \psi _{\mu }(w). \end{aligned}$$

Therefore, combining the previous equality and inequalities, we find

$$\begin{aligned} \int _{\Omega } \partial _t \psi _{\mu }(w)\;+\;D\int _{\Omega } \frac{|\nabla w|^2}{(1+|w|^{\frac{1}{\mu }})^{\mu +1}}\le \Vert f\Vert _{L^1(\Omega )}. \end{aligned}$$

At this point, we can integrate in time and obtain

$$\begin{aligned} D\int _{Q_T} \frac{|\nabla w|^2}{(1+|w|^{\frac{1}{\mu }})^{\mu +1}} \le \int _{\Omega } \psi _{\mu }\big (w_0(x)\big ) + \Vert f\Vert _{L^1(Q_T)} \le \Vert w_0\Vert _{L^1(\Omega )} + \Vert f\Vert _{L^1(Q_T)}. \end{aligned}$$

Since, for all \(\mu > 1\) there is a \(C_\mu \) such that

$$\begin{aligned} (1+|w|^{\frac{1}{\mu }})^{\mu +1} \le C_\mu (1+|w|)^{2(1-\alpha )}, \qquad \alpha = \frac{1}{2}\left( 1-\frac{1}{\mu }\right) , \end{aligned}$$

we conclude that

$$\begin{aligned} \int _{Q_T} (1+|w|)^{2(\alpha -1)} |\nabla w|^2 \le \frac{C_\mu }{D}\left[ \Vert w_0\Vert _{L^1(\Omega )} + \Vert f\Vert _{L^1(Q_T)}\right] , \qquad 0< \alpha < \frac{1}{2}. \end{aligned}$$

And thus, there is a constant \(C_\alpha \) which also depends on \(\Vert w_0\Vert _{L^1(\Omega )} + \Vert f\Vert _{L^1(Q_T)}\) such that

$$\begin{aligned} \int _{Q_T} |\nabla (1+|w|)^\alpha |^2 \le C_\alpha , \qquad 0< \alpha < \frac{1}{2}. \end{aligned}$$
(31)

Integrability of w. The Sobolev embedding (see Appendix C) gives

$$\begin{aligned} \left( \int _\Omega (1+|w|)^{\alpha 2^*} \right) ^{\frac{2}{2^*}} \le C\int _{\Omega } |\nabla (1+|w|)^\alpha |^2, \quad \qquad {2^*}= \frac{2d}{d-2}. \end{aligned}$$
(32)

which is only useful when \(\alpha 2^* >1\), i.e. \( \frac{d-2}{2d} <\alpha \). Then, we can interpolate between \(L^1\) and \(L^{\alpha 2^*}\) and find

$$\begin{aligned}&\left( \int _\Omega (1+|w|)^{\gamma } \right) ^{\frac{1}{\gamma }} \le C \left( \int _\Omega (1+|w|)\right) ^{\theta } \left( \int _{\Omega } |\nabla (1+|w|)^\alpha |^2\right) ^{\frac{1-\theta }{2\alpha }}, \\&\quad \frac{1}{\gamma }= \theta +\frac{1-\theta }{\alpha 2^*} . \end{aligned}$$

We may choose \(\frac{1-\theta }{2\alpha }=1\), and, recalling that \(\alpha < \frac{1}{2}\), we find the integrability

$$\begin{aligned} w\in L^1\big (0,T; L^{\gamma _1}(\Omega )\big ) \qquad \text {with} \qquad {\gamma _1} = \frac{d}{2\big ( d(1-\alpha ) -1\big ) } < \frac{d}{d-2} . \end{aligned}$$

We may also choose \(\frac{\gamma (1-\theta )}{2\alpha }=1\), \(\alpha < \frac{1}{2}\) and find the integrability

$$\begin{aligned} w\in L^{\gamma _2}(Q_T) \qquad \text {with} \qquad {\gamma _2} = \frac{2\big (1+\alpha d\big )}{d} < \frac{2+d}{d} . \end{aligned}$$

Regularity of \(\nabla w\). On the other hand, Hölder inequality gives

$$\begin{aligned} \begin{aligned} \displaystyle \int _{\Omega } |\nabla w|^\beta&=\int _{\Omega } \frac{|\nabla w|^\beta }{(1+|w|)^\eta }(1+|w|)^\eta \displaystyle \le \left( \int _{\Omega } \frac{|\nabla w|^{\beta r}}{(1+|w|)^{\eta r}}\right) ^\frac{1}{r}\left( \int _{\Omega }(1+|w|)^{\eta p}\right) ^\frac{1}{p}\\&\displaystyle \le C\left( \int _{\Omega } |\nabla (1+|w|)^\alpha |^2 \right) ^{\frac{1}{r}} \left( \int _{\Omega }(1+|w|)^{\eta p}\right) ^\frac{1}{p} \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \frac{1}{r}+\frac{1}{p}=1, \qquad \beta =\frac{2}{r} \le 2, \quad \eta r=2 (1-\alpha ). \end{aligned}$$

We can choose \(\eta p = \gamma _1\) from above, which requires \(\eta \left( \frac{1}{2(1-\alpha )} + \frac{1}{\gamma _1} \right) = 1\), \(\beta = \frac{\eta }{1-\alpha }= \frac{2 \gamma _1}{\gamma _1+2(1-\alpha )}\) and we find, thanks to the estimate (31),

$$\begin{aligned} \int _{\Omega } |\nabla w|^\beta \in L^1(0,T) \qquad \text {with} \qquad \beta < \frac{d}{d-1}. \end{aligned}$$

This concludes the proof of the gradient estimate. Moreover, considering that \(\beta <\gamma _2\), thanks to Sobolev embeddings, we can infer that \(w \in L^\beta (0,T;L^\beta (\Omega ))\).

The trace. The regularity of the trace derives from its continuity property [5] (p. 315), i.e.

$$\begin{aligned} \int _0^T\Vert \text{ Tr }(w)\Vert ^\beta _{W^{1-\frac{1}{\beta },\beta }(\Gamma )}\le \int _0^T\Vert w\Vert ^\beta _{W^{1,\beta }(\Omega )}, \quad 1\le \beta <\frac{d}{d-1}. \end{aligned}$$
(33)

Compactness

In order to conclude the proof of Lemma A.1, it remains to adapt compactness arguments to the case of the membrane problem. A proof based on a dual approach, see [2, 4], could be used. We rather go to a direct proof.

Compactness in space. It can be obtained using the Rellich–Kondrachov theorem [1], since we know the approximate family is bounded in the spaces \(W^{1,\beta }(\Omega ^\lambda )\), \(\lambda =1, \; 2\) which are compactly embedded in \(L^{\gamma _1}(\Omega ^\lambda )\), with \(\gamma _1<\frac{d}{d-2}\).

Compactness in time. We use the Fréchet–Kolmogorov criteria, see [5] for instance. Let \(\varphi (x)\) be a non-negative, radially symmetric, \(C^\infty _c(\mathbb {R}^d)\) standard mollifier with mass 1. We define the family \((\varphi _\delta )_{\delta >0}\) by

$$\begin{aligned} \varphi _\delta (x)=\frac{1}{\delta ^d}\,\varphi \left( \frac{x}{\delta }\right) , \qquad \Vert \varphi _\delta \Vert _{L^1(\Omega )}=1. \end{aligned}$$
(34)

Moreover, we have

$$\begin{aligned} \Vert g*\varphi _\delta \Vert _{L^p(\Omega )}\le \Vert \varphi _\delta \Vert _{L^1(\Omega )}\Vert g\Vert _{L^p(\Omega )} , \end{aligned}$$
(35)

and it holds ([13], p. 273) that for any function \(g\in W^{1,p}(\Omega )\),

$$\begin{aligned} \Vert g*\varphi _\delta -g\Vert _{L^p(\Omega )}\le \delta \Vert \nabla g\Vert _{L^p(\Omega )}. \end{aligned}$$
(36)

About the derivative of order k of \(\varphi _\delta \), we know that

$$\begin{aligned} \nabla ^k \varphi _\delta (x)=\frac{1}{\delta ^{d+k}}\nabla ^k \varphi \left( \frac{x}{\delta }\right) , \qquad \Vert \nabla ^k \varphi _\delta \Vert _{L^1(\Omega )}\le \frac{C}{\delta ^{k}}. \end{aligned}$$
(37)

Proof

To complete the proof of time compactness, we shall prove that, as \(h \rightarrow 0\),

$$\begin{aligned} \int _{0}^{T-h}\int _{\Omega }|w(t+h,x)-w(t,x)|\hbox {d}x \hbox {d}t \rightarrow 0. \end{aligned}$$
(38)

By comparison with the mollified versions, the triangular equality yields

$$\begin{aligned}&\int _{0}^{T-h}\int _{\Omega }|w(t+h,x)-w(t,x)|\hbox {d}x\hbox {d}t \nonumber \\&\quad \le \int _{0}^{T-h}\int _{\Omega }|w(t,x)- w(t,\cdot )*\varphi _\delta (x)|\hbox {d}x\hbox {d}t\nonumber \\&\qquad + \int _{0}^{T-h}\int _{\Omega }|w(t+h,x)- w(t+h,\cdot )*\varphi _\delta (x)|\hbox {d}x\hbox {d}t \nonumber \\&\qquad +\int _{0}^{T-h}\int _{\Omega }|w(t+h,\cdot )*\varphi _\delta (x)-w(t,\cdot )*\varphi _\delta (x)|\hbox {d}x \hbox {d}t \end{aligned}$$

Here, \(\delta \) depends on h (to be specified later on) and converges to zero. It suffices to prove that each integral converges to zero as \(h\rightarrow 0\).

First term. We analyse the first term in the right-hand side. It holds that

$$\begin{aligned} \int _{0}^{T-h}\int _{\Omega }|w(t,x)- w(t,\cdot )*\varphi _\delta (x)|\hbox {d}x \hbox {d}t\le \delta \int _0^{T-h} \Vert \nabla w(t,x)\Vert _{L^1(\Omega )} \hbox {d}t \le C\delta (h),\nonumber \\ \end{aligned}$$
(39)

thanks to w regularity and to (36), which proves that it converges to zero as \(h\rightarrow 0\).

Second term. For the second integral, we can proceed as for the first one obtaining

$$\begin{aligned} \int _{0}^{T-h}\int _{\Omega }|w(t+h,x)- w(t+h,\cdot )*\varphi _\delta (x)|\hbox {d}x \hbox {d}t \le C \delta (h). \end{aligned}$$
(40)

Third term. Remembering (30), the last term can be written as

$$\begin{aligned}&\int _{0}^{T-h}\int _{\Omega }|w(t+h,\cdot )*\varphi _\delta (x)-w(t,\cdot )*\varphi _\delta (x)|\hbox {d}x\, \hbox {d}t\\&\quad =\int _{0}^{T-h}\int _{\Omega }\left| \int _{t}^{t+h} \frac{\partial w}{\partial s}(s,x)*\varphi _\delta (x)\hbox { d}s\right| \hbox {d}x\,\hbox {d}t\\&\quad =\int _{0}^{T-h}\int _{\Omega }\left| \int _{t}^{t+h} \left[ D\Delta w + f\right] *\varphi _\delta \hbox {d}s\right| \hbox {d}x\,\hbox {d}t \\&\quad = \int _{0}^{T-h}\int _{\Omega }\left| \int _{t}^{t+h} D w *\Delta \varphi _\delta + f*\varphi _\delta \, \hbox {d}s \right| \hbox {d}x\,\hbox {d}t \end{aligned}$$

after exchanging derivatives in the convolution. From (35) we deduce

$$\begin{aligned}&\int _{0}^{T-h}\int _{\Omega }|w(t+h,\cdot )*\varphi _\delta (x)-w(t,\cdot )*\varphi _\delta (x)|\hbox {d}x\, \hbox {d}t \\&\quad \le \int _{0}^{T-h}\int _t^{t+h} D\Vert w\Vert _{L^1(\Omega )} \Vert \Delta \varphi _\delta \Vert _{L^1(\Omega )}\nonumber \\&\qquad + \int _{0}^{T-h}\int _t^{t+h}\Vert f\Vert _{L^1(\Omega )} \Vert \varphi _\delta \Vert _{L^1(\Omega )}. \end{aligned}$$

Finally, thanks to (34) and (37), we obtain choosing \(\delta =h^{1/4}\)

$$\begin{aligned} \int _{0}^{T-h}\int _{\Omega }|w(t+h,\cdot )*\varphi _\delta (x)-w(t,\cdot )*\varphi _\delta (x)|\hbox {d}x\, \hbox {d}t \le C\left[ \frac{h}{\delta ^2} +h\right] \le C \sqrt{h} \end{aligned}$$

and (38) follows combining this estimate with (39) and (40). \(\square \)

Applying the Fréchet–Kolmogorov theorem [5], we conclude that the set of functions \(w \in L^1(Q_T)\) under consideration is compact in \(L^1(Q_T)\). Consequently, we claim compactness in \(L^1\big (0,T; L^{\gamma _1}(\Omega )\big )\) with \({\gamma _1}< \frac{d}{d-2}\) and in \(L^{\gamma _2}(Q_T)\) with \({\gamma _2}< \frac{2+d}{d} \). In fact, since we have \(L^1\)-convergence of \(L^p\)-functions, we deduce convergence in the space \(L^q\), for \(q<p\).

Compactness of traces in \(L^\beta \big (0,T;L^\beta (\Gamma )\big )\). Space compactness can be deduced, in each \(\Omega ^\lambda \), from trace continuity and a compactness result for the boundary ([11], p. 167) such that \(W^{1-\frac{1}{\beta },\beta }(\Gamma ) \subset \subset L^{\beta }(\Gamma )\). Time compactness is again achieved through the Fréchet–Kolmogorov theorem. Following the same proof as before and W changing the order of the time integrals, we need to recall Kedem–Katchalsky membrane conditions from which we can infer that \(\partial _tTr_\Gamma (w) \in L^1(0,T;L^1(\Gamma ))\) and so we can conclude the proof. \(\square \)

Sobolev and Poincaré inequalities with membrane

For completeness, we explain why the Sobolev embeddings can be extended to the membrane problem, leading to (31) and (32). More precisely, we explain how to arrive to

$$\begin{aligned} \Vert \phi _{\alpha }(w^1)\Vert ^2_{L^{2^*}(\Omega ^1)} + \Vert \phi _{\alpha }(w^2)\Vert ^2_{L^{2^*}(\Omega ^2)}\; \le C \left( \Vert \nabla \phi _{\alpha }(w^1)\Vert ^2_{L^2(\Omega ^1)} + \Vert \nabla \phi _{\alpha }(w^2)\Vert ^2_{L^2(\Omega ^2)} \right) . \end{aligned}$$

There are two difficulties. First, the boundary condition is not Dirichlet everywhere. Second, we are dealing with a singular domain \(\Omega \) and so we cannot use directly the Sobolev or Poincaré inequalities in \(\Omega \), but only some easy generalizations that we explain now.

We are going to prove the

Theorem C.1

(Gagliardo–Nirenberg–Sobolev inequality with membrane). We consider the bounded domain \(\Omega = \Omega ^1\, \cup \,\Omega ^2 \subset \mathbb {R}^d, \; d\ge 2\), with piecewise \(C^1\) sub-domains \(\Omega ^1\) and \(\Omega ^2\) and a \(C^1\) membrane \(\Gamma =\partial \Omega ^1\, \cap \,\partial \Omega ^2\) which decomposes \(\Omega \) in the two parts. We take the function \(v=(v^1,v^2) \in {\mathbf{H}^\mathbf{1}}\) (see Definition 1.3), then, for \(\lambda =1,2\),

$$\begin{aligned} \Vert v^\lambda \Vert _{L^{2^*}(\Omega ^\lambda )}\;\le \;C(\Omega ^\lambda )\;\;\Vert \nabla v^\lambda \Vert _{{L^{2}(\Omega ^\lambda )}^d}, \end{aligned}$$
(41)

and consequently

$$\begin{aligned} {[}\;\Vert v^1\Vert _{L^{2^*}(\Omega ^1)}\;+\;\Vert v^2\Vert _{L^{2^*}(\Omega ^2)} \;]\;\le \;C(\Omega ^1,\Omega ^2)\;[\;\Vert \nabla v^1\Vert _{{L^{2}(\Omega ^1)}^d}\;+\;\Vert \nabla v^2\Vert _{{L^{2}(\Omega ^2)}^d}\;].\nonumber \\ \end{aligned}$$
(42)

The reason why we want to prove this theorem is that the domain \(\Omega \) described above is not enough regular to use the usual Gagliardo–Nirenberg–Sobolev inequality ([5], p. 284). Consequently, we need to build smoother domains containing each \(\Omega ^i\) in which we can apply known results and then, with a restriction to \(\Omega \), we can find (41) and (42). The construction is made considering an extension of \(\Gamma \) and a domain with the same internal structure as \(\Omega \) such that it contains \(\Omega \) and each extension of the \(\Omega ^i\) is of class \(C^1\).

We first recall the standard Sobolev inequality ([5], p. 284) in a bounded open set.

Theorem C.2

(Sobolev embedding). Let Q be a bounded open subset of class \(C^1\) in \(\mathbb {R}^d\). There is a constant \( C_Q\) such that for all \(v\in H^1(Q),\) we have

$$\begin{aligned} v\in L^{2^*}(Q) \quad \text{ and } \quad \Vert v\Vert _{L^{2^*}(Q)}\;\le \;C_Q\;\left[ \;\Vert v\Vert _{L^{2}(Q)}\;+\;\Vert \nabla v\Vert _{{L^{2}(Q)}^d}\;\right] . \end{aligned}$$

Proof

We recall how to prove Theorem C.2 departing from the case of the full space. We use the regularity of the domain which assures us the existence of a linear and continuous extension operator \(T: H^1(Q) \rightarrow H^1(R^d)\), which is also the extension from \(L^2(Q)\) into \(L^2(\mathbb {R}^d)\) ([5], p. 272). So, we obtain that:

$$\begin{aligned}&\bullet \; \text{ taken } v\in H^1(Q), \quad T(v)\in H^1(\mathbb {R}^d) \text{ and } T(v)=v \text{ on } Q; \end{aligned}$$
(43)
$$\begin{aligned}&\bullet \; \Vert T(v)\Vert ^2_{L^2(\mathbb {R}^d)} \le C^2_{\small {\text{ exten }}L^2}(Q)\;\Vert v\Vert ^2_{L^2(Q)};\end{aligned}$$
(44)
$$\begin{aligned}&\bullet \; \Vert \nabla T(v)\Vert ^2_{{L^2(\mathbb {R}^d)}^d} \le C^2_{\small {\text{ exten }}H^1}(Q)\;\Vert v\Vert ^2_{H^1(Q)}. \end{aligned}$$
(45)

Moreover, for construction (see the proof of the extension theorem [5], p. 272), this operator is in \(H^1_0(R^d)\). Consequently, using a corollary of the Sobolev inequality ([13], p. 265), we get that

$$\begin{aligned} T(v)\in L^{2^*}(\mathbb {R}^d) \text{ and } \Vert T(v)\Vert _{L^{2^*}(\mathbb {R}^d)}\le C_{\small {\text{ sob }}}(d,2)\; \Vert \nabla T(v)\Vert _{{L^2(\mathbb {R}^d)}^d}. \end{aligned}$$

We proceed with some estimates due to the application of (43), (44), (45). First of all, we deduce

$$\begin{aligned} \Vert \nabla v\Vert ^2_{{L^2(Q)}^d}= & {} \Vert \nabla T(v)\Vert _{{L^2(Q)}^d}\le \Vert \nabla T(v)\Vert _{{L^2(\mathbb {R}^d)}^d}\le C_{\small {\text{ exten }}H^1}(Q)\;\Vert v\Vert _{H^1(Q)}^2\\= & {} C_{\small {\text{ exten }}H^1}(Q)\;\left[ \Vert v\Vert _{L^2(Q)}+\Vert \nabla v\Vert _{{L^2(Q)}^d}\right] . \end{aligned}$$

Since \(T(v)\in L^{2^*}(\mathbb {R}^d)\) and \(T(v)=v\) on Q, we get \(v\in L^{2^*}(Q)\) and

$$\begin{aligned} \Vert v\Vert _{L^{2^{*}}(Q)}^2= & {} \Vert T(v)\Vert _{L^{2^{*}}(Q)}^2\le \Vert T(v)\Vert _{L^{2^{*}}(\mathbb {R}^d)}^2\le (C_{\small {\text{ sob }}}(d,2))^2 \;\Vert \nabla T(v)\Vert _{{L^2(\mathbb {R}^d)}^d}^2\\\le & {} (C_{\small {\text{ sob }}}(d,2))^2\; C^2_{\small {\text{ exten }}H^1}(Q)\;\left[ \Vert v\Vert _{L^2(Q)}^2+\Vert \nabla v\Vert _{{L^2(U)}^d}^2\right] . \end{aligned}$$

The proof of Theorem C.2 is complete. \(\square \)

Since we do not impose Dirichlet conditions on the full boundary, we need the following generalized Poincaré inequality ([20] p. 82).

Theorem C.3

(Poincaré inequality). Suppose Q a bounded and connected open subset of \(\mathbb {R}^d\) of class \(C^1\) and consider a portion of its boundary \(\Sigma _0 \subset \partial Q \) such that \(|\Sigma _0|>0\). Then, there exists a constant \(C(Q,\Sigma _0)\) such that

$$\begin{aligned} \forall v\in H^1(Q) \ \mathrm{such \ that}\ Tr_{\Sigma _0}(v)=0, \quad \Vert v\Vert _{L^2(Q)}^2\le C(Q,\Sigma _0) \Vert \nabla v\Vert _{{L^2(Q)}^d}^2. \end{aligned}$$
(46)

Proof

If the statement is not true, we can find a sequence \({v_n}\) such that each \(v_n\in H^1(Q)\) and

$$\begin{aligned} \Vert v_n\Vert _{L^2(Q)}^2>n\;\left[ \Vert \nabla v_n\Vert _{{L^2(Q)}^d}^2\;+\;\left( \int _{\Sigma _0}|v_n|\hbox {d}S\right) ^2\right] . \end{aligned}$$

On account of the homogeneity (normalizing), we may assume that \(\Vert v_n\Vert _{L^2(Q)}=1\), for each n. So, we infer that

$$\begin{aligned} n\;\left[ \Vert \nabla v_n\Vert _{{L^2(Q)}^d}^2\;+\;\left( \int _{\Sigma _0}|v_n|\hbox {d}S\right) ^2\right] <1, \end{aligned}$$
(47)

which implies that

$$\begin{aligned} \Vert \nabla v_n\Vert _{{L^2(Q)}^d}^2<\frac{1}{n}. \end{aligned}$$

Therefore, \(\nabla v_n \rightarrow 0\) in \(L^2(Q)\). Moreover, \(v_n\) is bounded in \(H^1(Q)\), so, up to a sub-sequence, it converges weakly in \(H^1(Q)\) to some v. So, \(\nabla v_n \rightharpoonup \nabla v\), that means \(\nabla v=0\). This shows that v is a constant (since Q is connected). For the continuity of the trace operator and (47), we deduce

$$\begin{aligned} 0=\lim _{n\rightarrow +\infty }\int _{\Sigma _0} |v_n| \hbox {d}S=\int _{\Gamma _0} |v| \hbox {d}S=|c||\Gamma _0|, \end{aligned}$$

and so \(v=0\).

At the same time, thanks to the Rellich–Kondrachov compactness theorem [1, 5, 13], up to a sub-sequence, \(v_n\) converges strongly in \(L^2(Q)\) to \(v=0\). Hence, since \(\Vert v_n\Vert _{L^2(Q)}=1\), we arrive to a contradiction. \(\square \)

At this point we are able to give the proof of Theorem C.1.

Proof

We apply Theorems C.2 and C.3. First of all, we consider the extension of \(\Gamma \) into the space \(\mathbb {R}^d\) such that now \(\Gamma \) separates the space into two pieces \(P^\lambda \) with \(\lambda =1,2\). Since we have Dirichlet boundary conditions on \(\Gamma ^\lambda \), we can extend the function to zero in the whole \(P^\lambda \). So now, considering \(Q^\lambda \) a domain of class \(C^1\) such that \(\Omega ^\lambda \subset Q^\lambda \subset P^\lambda \) and for \(\lambda , \sigma =1,2\), \(\;Q^\lambda \cap P^\sigma \) is a portion of \(\Gamma \), we can apply Theorems C.2 and C.3 to

$$\begin{aligned} {\tilde{v}}^\lambda = \left\{ \begin{array}{ll} v^\lambda , &{} \quad \text{ in } \Omega ^\lambda , \\ 0, &{} \quad \text{ in } \Gamma ^\lambda \cup \{Q^\lambda \setminus \Omega ^\lambda \}. \end{array}\right. \end{aligned}$$

This proves Theorem C.1 in \(Q^\lambda \) and, so, in \(\Omega ^\lambda \). \(\square \)

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Ciavolella, G., Perthame, B. Existence of a global weak solution for a reaction–diffusion problem with membrane conditions. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00633-7

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Keywords

  • Kedem–Katchalsky conditions
  • Membrane boundary conditions
  • Reaction–diffusion equations
  • Mathematical biology

Mathematics Subject Classification

  • 35K57
  • 35D30
  • 35Q92