\documentclass[reqno]{amsart} \usepackage{amssymb,graphicx} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 42, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/42\hfil Degenerate elliptic equations] {A note on a degenerate elliptic equation with applications for lakes and seas} \author[D. Bresch, J. Lemoine, \& F. Guillen-G.\hfil EJDE-2004/42\hfilneg] {Didier Bresch, J\'er\^ome Lemoine, \& Francisco Gu\'{\i}llen-Gonzalez} % in alphabetical order \address{Didier Bresch\hfill\break Laboratoire de Mod\'elisation et de Calcul - Imag (CNRS UMR 5523), Universit\'e Joseph Fourier, 38041 Grenoble cedex, France} \email{Didier.Bresch@imag.fr} \address{J\'er\^ome Lemoine \hfill\break Laboratoire de Math\'ematiques Appliqu\'ees, Universit\'e Blaise Pascal et C.N.R.S., 63177 Aubi\`ere cedex, France} \email{Jerome.lemoine@math.univ-bpclermont.fr} \address{Francisco Guillen-Gonzalez \hfill\break Dpto. de Ecuaciones Diferenciales y An\'alisis Num\'erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain} \email{guillen@us.es} \date{} \thanks{Submitted June 15, 2002. Published March 23, 2004.} \subjclass[2000]{35Q30, 35B40, 76D05} \keywords{Regularity result, degenerate elliptic equation, geophysics, \hfill\break\indent weighted Sobolev spaces, splitting projection method} \begin{abstract} In this paper, we give an intermediate regularity result on a degenerate elliptic equation with a weight blowing up on the boundary. This kind of equations is encountoured when modelling some phenomena linked to seas or lakes. We give some examples where such regularity is useful. \end{abstract} \maketitle \newtheorem{theorem}{Theorem}[section] % theorems numbered with section # \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} \label{sec1} This paper is devoted to a degenerate elliptic equation that we can find in several models in oceanography when we consider a domain with a depth vanishing on the shore. A lot of mathematical studies in oceanography assume a domain with a strictly positive depth in order to prevent the study in weighted spaces. Few papers have been devoted to coefficients with degenerated behavior. Regularity results in weighted Sobolev spaces on degenerate elliptic equations have been studied for instance in \cite{2,3} with a vanishing weight on the boundary that implies no boundary condition on the unknown. Here we study a degenerate elliptic equation with a weight with a blowing up comportment on the shore. A $H^2$ regularity in weighted spaces is proved allowing to consider general weights. We will obtain such regularity by a careful study of the weight adapting the standard method of translation, see \cite{8}. For example, we use and adapt some results on weighted Sobolev spaces that have been studied in \cite{7}. Section \ref{sec2} is devoted to the regularity result related to the degenerate elliptic equation. Then in Section \ref{sec3}, we explain why this kind of equation is important in oceanography. In the last section we describe precise examples where such regularity is used. At first we give some examples where such regularity result is used for existence result or error estimates that means the planetary-geostrophic equations and the vertical geostrophic equations. Then we give an example where it is used in a splitting projection method. We also mention that such equation is obtained from the Great Lake equation. We remark that this degenerate elliptic equation may be found in an other field such as in electromagnetism with the maxwell's system, see \cite{18}. Similar degenerate elliptic equation may also be encountoured for a problem related to Saint-Venant's equations if we want to apply Babuska-Brezzi's Inf-Sup Lemma in weighted spaces, see \cite{1}. \section{The degenerate elliptic equation} \label{sec2} Let $\mathcal{O}$ be a two-dimensional domain. This section is devoted to a regularity result on the following degenerate elliptic equation: Given $h:\mathcal{O} \to \mathbb{R}$ and $g: \mathcal{ \theta} \to \mathbb{R}$, the problem is to find $\Psi:\mathcal{O}\to \mathbb{R}$ such that \begin{equation} -\nabla_{\rm x}\cdot (\frac 1h\nabla_{\rm x} \Psi) = g \text{in } {\mathcal{O}}, \quad \Psi= 0 \text{ on } \partial\mathcal{O}. \label{e1} \end{equation} Here, $h$ is a function data satisfying \begin{gather} h\in W^{1,\infty}(\mathcal{O}), \quad h>0 \text{ in } \mathcal{O}, \label{e2}\\ h({\rm x}) = \varphi(\delta({\rm x})) \text{in a neightbourhood of } \partial\mathcal{O} \label{e3} \end{gather} with $\delta({\rm x})=\mathop{\rm dist}({\rm x},\mathcal{O})$. Moreover we assume that \begin{gather} \varphi \text{ is a non decreasing Lipschitz function}, \quad \varphi(0) = 0, \label{e4} \\ \exists \> c>0 \text{ such that: }\ \forall s>0,\ \Bigl|\frac{\varphi'(s)}{\varphi(s)}\Bigr|\le \frac cs \label{e5} \end{gather} and for all $c_1,c_2>0$, there exist $\alpha_1,\alpha_2>0$ such that \begin{equation} \forall s,r >0,\; c_1\le\frac sr\le c_2\Longrightarrow \alpha_1\le \frac{\varphi(s)}{\varphi(r)}\le\alpha_2. \label{e6} \end{equation} \noindent{\textbf{Remark} Note that $\varphi(s)= c\,s^\alpha$, $0<\alpha<1$ and $c>0$, satisfies the previous hypothesis. \smallskip Now, we define the function space \begin{equation} H(\mathcal{O}) = \{\Psi\in L^2(\mathcal{O}) : \frac{\nabla_{\rm x} \Psi}{h^{1/2}}\in (L^2(\mathcal{O}))^2, \Psi = 0 \text{ on } \partial\mathcal{O}\} \label{e7} \end{equation} endowed with the norm $$ \|\Psi\|_{H(\mathcal{O})} =\|\frac{\nabla_{\rm x} \Psi }{ h^{1/2}}\|_{(L^2(\mathcal{O}))^2}. $$ where $h$ is defined by \eqref{e2}--\eqref{e3}. We remark that $\|\cdot\|_{H(\mathcal{O})}$ is a norm since $\frac 1h \ge c >0$ and $\Psi = 0$ on $\partial\mathcal{O}$ implies that there exists $c>0$ such that for all $\Psi \in {H(\mathcal{O})}$, $$ \|\Psi\|_{L^2(\mathcal{O})} \le c \|\frac{\nabla_{\rm x} \Psi }{ h^{1/2}}\|_{(L^2(\mathcal{O}))^2}. $$ \begin{lemma} \label{lm1} Let $H(\mathcal{O})$ be defined by \eqref{e7} with $h$ satisfying \eqref{e2}--\eqref{e6}. Then $\mathcal{D}(\mathcal{O})$ is dense in ${H(\mathcal{O})}$. \end{lemma} The proof of this lemma is similar to the proof of \cite[Theorem 11.2]{7}. Therefore, we omit it. The main result of this paper is the following \begin{theorem} \label{thm2} Let $\mathcal{O}$ be a (two-dimensional) bounded domain of class $\mathcal{C^3}$. Let $g$ be such that $\delta h^{1/2} g\in L^2(\mathcal{ O})$ with $h$ satisfying \eqref{e2}--\eqref{e6}. There exists a unique solution $\Psi$ of \eqref{e1} such that $\Psi \in {H(\mathcal{O})}$ and $$ \|\Psi\|_{H(\mathcal{O})} \le c \|\delta h^{1/2} g\|_{L^2(\mathcal{ O})}. $$ Moreover, if \begin{equation} h^{1/2} g \in L^2(\mathcal{O}) \label{e8} \end{equation} then \begin{gather} h^{1/2} \nabla_{\rm x}(\frac 1h\nabla_{\rm x} \Psi) \in (L^2(\mathcal{O}) )^4, \label{e9}\\ \|h^{1/2} \nabla_{\rm x}(\frac 1h\nabla_{\rm x} \Psi)\|_{(L^2(\mathcal{O} ))^4} \le c\|h^{1/2} g\|_{L^2(\mathcal{O})} \end{gather} with $c$ a constant depending only on the domain. \end{theorem} \begin{proof} \textbf{Weak solutions:}\quad The existence and uniqueness of weak solutions of \eqref{e1} follows from the Lax-Milgram theorem, since \begin{align*} \|\Psi\|^2_{H(\mathcal{O})} = \int_{\mathcal{O}} g \Psi & \le \|\delta h^{1/2} g \|_{L^2(\mathcal{O})} \|\frac{\Psi }{ \delta h^{1/2}}\|_{L^2(\mathcal{O})} \\ & \le c\| \delta h^{1/2} g\|_{L^2(\mathcal{O})} \|\frac{\nabla_{\rm x} \Psi }{ h^{1/2}}\|_{(L^2(\mathcal{O}))^2}. \end{align*} In the previous estimate we have used Hardy's inequality in weighted space which will be proved in Lemma \ref{lm3}. \noindent\textbf{Regularity:} We use the usual difference quotients (cf. Brezis \cite{8}). The interior regularity is well known since $h \ge c(\omega) >0$ in each $\omega \Subset \mathcal{O}$. To obtain the regularity result up to the boundary, we define a local diffeomorphism $T$ which preserves the normal direction. More precisely we define the local diffeomorphism $T: Q \to V$ by $T(x^*,r) = (x^*,\alpha(x^*)) + r\, n(x^*,\alpha(x^*))$ for all $ (x^*, r) \in Q$ where $\partial \mathcal{O}$ is locally the graph of a $\mathcal{C}^3$ function $\alpha$ (see Figure 1). This property, combined with the hypothesis \eqref{e3} of $h$, will be strongly used in the sequel. We also define a cut-off function $\theta$ such that \begin{gather*} \theta \in {\mathcal{C}}^2 \text{ in } V, \cr \theta = 0 \text{ on }\mathbb{R}^2\backslash V, \quad \theta = 1 \text{ in } \mathcal{V}_1, \quad \cr \frac{\partial\theta}{ \partial \widetilde n} = 0 \text{ on a neightbourhood of } \partial\mathcal{O}\cap V, \cr \end{gather*} where $\mathcal{V}_1\Subset V$ and the extension $\widetilde n$ of the normal $n$ is defined for all $(x,y)\in V$ by $$ \widetilde n(x,y)= n(T(x^*,0))$$ where $(x,y)= T(x^*,r)$, $(x^*,r)\in Q$. \begin{figure}[ht] \begin{center} \includegraphics[width= 0.6\textwidth]{fig1.eps} \end{center} \caption{The local diffeomorphism $T$} \end{figure} \noindent\textbf{remark} Due to the $\mathcal{C}^3$ regularity of $\partial\mathcal{O}$, $T$ is a $\mathcal{C}^2$ diffeormorphism from $Q$ onto a neightbourhood of $(x_0,y_0)$ denoted by $V$. \smallskip Multiplying \eqref{e1} by $\theta$, and denoting $\xi = \theta\Psi$, it follows that $\xi$ is the (unique) solution in $H(V\cap \mathcal{O})$ of \begin{equation} \begin{gathered} -\nabla_{\rm x}\cdot(\frac 1h\nabla_{\rm x}\xi ) = f \text{ in } V \cap \mathcal{O}, \\ \xi\vert_{\partial(V \cap\mathcal{O})} = 0, \\ \end{gathered} \label{e10} \end{equation} where $f= \theta g + {\nabla_{\rm x} h\cdot \nabla_{\rm x} \theta\over h^2}\Psi -\frac 1h \Delta_{\rm x}\theta \Psi -{2\over h}\nabla_{\rm x}\theta \cdot\nabla_{\rm x}\Psi$ (since $h^{1/2}f\in L^2(V\cap\mathcal{O})$). For this, it suffices to check that $$ {\nabla_{\rm x} h\cdot \nabla_{\rm x} \theta\over h^{3/2}}\Psi \in L^2(V\cap\mathcal{O}). $$ Indeed, since ${\partial\theta/\partial\widetilde n} = 0$ on a neightbourhood of $\partial\mathcal{O}\cap V$, then $$ \nabla_{\rm x} h\cdot \nabla_{\rm x} \theta ={\partial h \over \partial \widetilde \tau} {\partial \theta \over \partial \widetilde \tau} = 0 $$ on a neightbourhood of $\partial\mathcal{O}\cap V$. We recall that $h$ doesn't depend on $\widetilde \tau$ (where $\widetilde\tau$ is defined as $\widetilde n$) using that $h$ is given by \eqref{e3} and using the definition of $T$. Now we use the difference quotient technic on \eqref{e10} to deduce the weight regularity announced in the theorem. Let $\varphi:\mathcal{O}\to\mathbb{R}^n$ ($n\ge 1$) be a function. We denote, for all $(x^*,r) \in Q_+$, \begin{gather*} \widetilde \varphi (x^*,r) = \varphi(T(x^*,r)),\\ a_{kl} = \sum_{j= 1}^2 \partial_j T^{-1}_k (T(x^*,r))\partial_j T^{-1}_l (T(x^*,r)) |\mathop{\rm Jac} T(x^*,r)|, \quad k,l= 1,2 ,\\ \hat k (x^*,r) = \widetilde f(x^*,r) |\mathop{\rm Jac} T(x^*,r)|. \end{gather*} Then we get that $\widetilde\xi$ is the unique solution in $\widetilde H(Q_+)$ of: \begin{equation} \int_{Q_+} \sum_{k,l} {a_{kl}\over \widetilde h} \,\partial_k\widetilde \xi \,\partial_l\widetilde\varphi dx^*dr = \int_{Q_+} \hat k \widetilde \varphi d x^*dr \label{e11} \end{equation} for all $\widetilde \varphi\in \widetilde H(Q_+)$ where $$ \widetilde H(Q_+) = \{\widetilde \varphi\in L^2(Q_+) : \widetilde h^{-1/2}\nabla_{\rm x} \widetilde \varphi\in (L^2(Q_+))^2, \widetilde \varphi = 0 \text{ on } \partial Q_+\}. $$ We choose $\widetilde \varphi = D_{-\tau}(D_\tau \widetilde \xi)$ with $\tau= |\tau| e_1$, $D_{\tau}\widetilde \xi= \bigl(\widetilde \xi(x+\tau)-\widetilde \xi(x)\bigr)/|\tau|$ and $|\tau|$ small enough in order to obtain $\widetilde \varphi \in H(Q_+)$. Using $$ \|{1\over \widetilde h^{1/2}} D_{-\tau}(D_\tau \widetilde \xi) \|_{L^2(Q_+)} \le c \|{1\over \widetilde h^{1/2}} \nabla_{\rm x} (D_\tau \widetilde \xi) \|_{L^2(Q_+)} $$ and $$ \|\widetilde h^{1/2} \hat k \|_{L^2(Q_+)} \le c \| h^{1/2}g\|_{L^2(\mathcal{O})} $$ we get \begin{equation} \|\hat k D_{-\tau}(D_\tau \widetilde \xi) \|_{L^1(Q_+)} \le c\| h^{1/2}g\|_{L^2(\mathcal{O})} \|{1\over \widetilde h^{1/2}} \nabla_{\rm x} (D_\tau \widetilde\xi) \|_{ L^2(Q_+)}. \label{e12} \end{equation} Moreover, denoting $$ I = \sum_{k,l} \int_{Q_+} D_\tau ({1\over \widetilde h} a_{kl} \partial_k \widetilde \xi) \partial_l(D_\tau\widetilde\xi) $$ since $D_\tau (a_{kl}/\widetilde h)= D_\tau (a_{kl})/ \widetilde h$, (recall that $\widetilde h$ does not depend on $\tau$) and $T\in \mathcal{C}^2$, we have \begin{equation} \begin{aligned} I& \ge c\|{1\over\widetilde h^{1/2}}\nabla_{\rm x} (D_\tau\widetilde\xi)\|_{(L^2(Q_+))^2}^2 - c\|{1\over\widetilde h^{1/2}} \nabla_{\rm x} \widetilde\xi\|_{L^2} \|{1\over \widetilde h^{1/2}}\nabla_{\rm x} (D_\tau\widetilde\xi) \|_{(L^2(Q_+))^2} \\ & \ge c\|{1\over\widetilde h^{1/2}}\nabla_{\rm x} (D_\tau\widetilde\xi)\|_{(L^2(Q_+))^2}^2 - c\| h^{1/2}g\|_{L^2(\mathcal{O})}\| {1\over \widetilde h^{1/2}}\nabla_{\rm x} (D_\tau\widetilde\xi) \|_{(L^2(Q_+))^2} \end{aligned} \label{e13} \end{equation} Using the variational formulation satisfied by $\widetilde \xi$, \eqref{e12} and \eqref{e13} we get \begin{equation} \|{1\over \widetilde h^{1/2}}\nabla_{\rm x} (D_\tau\widetilde\xi) \|_{(L^2(Q_+))^2} \le c\|h^{1/2}g\|_{L^2(\mathcal{O})}. \label{e14} \end{equation} Thus, by classical arguments, \begin{equation} {\partial_1^2\widetilde \xi \over \widetilde h^{1/2}}\in L^2(Q_+) \quad\text{and}\quad {\partial_2\partial_1\widetilde \xi \over \widetilde h^{1/2}}\in L^2( Q_+), \label{e15} \end{equation} and their respective norms are bounded by $c\|h^{1/2} g\|_{L^2(\mathcal{O})}$. In particular, \begin{equation} \widetilde h^{1/2} \partial_1({\widetilde h}^{-1} \partial_1\widetilde \xi) \in L^2(Q_+), \quad \widetilde h^{1/2} \partial_1({\widetilde h}^{-1} \partial_2\widetilde\xi) \in L^2(Q_+) \label{e16} \end{equation} and their respective norms depend continuously on $h^{1/2} g$ in $L^2(\mathcal{O})$. We remark that contrary to the homogeneous case, that means the standard Laplacian operator, we have not yet the regularity $\widetilde h^{1/2}\partial_2({\widetilde h}^{-1}\partial_1\widetilde \xi) \in L^2(Q_+)$. We will obtain such regularity using the hypothesis \eqref{e5} on $h$. Indeed, in the distribution sense, \begin{equation} \widetilde h^{1/2}\partial_2({1\over \widetilde h}\partial_1\widetilde\xi ) = {-\partial_2\widetilde h\over \widetilde h^{3/2}}\partial_1\widetilde \xi +{1\over \widetilde h^{1/2}}\partial_2\partial_1\widetilde \xi. \label{e17} \end{equation} Since $\partial_1\widetilde\xi= 0$ on $\partial Q_+$ then \eqref{e15} yields $\partial_1\widetilde\xi\in H(Q_+)$. Using the Hardy's inequality \eqref{e20} and Hypothesis \eqref{e5}, we get $$ \int_{Q_+}\Bigl|{\partial_2\widetilde h\over \widetilde h^{3/2}}\partial_1\widetilde \xi\Bigr|^2 \le c\int_{Q_+}{|\partial_1\widetilde \xi|^2\over \widetilde\delta^2\widetilde h} \le c\int_{Q_+}{|\partial_2\partial_1\widetilde\xi|^2\over \widetilde h}. $$ Thus, using the regularity \eqref{e15}, we get from \eqref{e17} \begin{gather} \widetilde h^{1/2}\partial_2({1\over \widetilde h}\partial_1\widetilde \xi)\in L^2(Q_+), \label{e18} \\ \|\widetilde h^{1/2}\partial_2(\partial_1\widetilde \xi/ \widetilde h) \|_{(L^2(Q_+))^2} \le c\|h^{1/2}g\|_{L^2(\mathcal{O})}. \nonumber \end{gather} Now we use the variational formulation \eqref{e11} satisfied by $\widetilde \xi$ to obtain the regularity on $\widetilde h^{1/2}\partial_2(\partial_2\widetilde \xi/\widetilde h)$. We have $$ \bigl|\int_{Q_+} {a_{22}\over \widetilde h} \partial_2 \widetilde \xi \partial_2 \Phi \Bigr| \le c \|h^{1/2}g\|_{L^2(\mathcal{O})}\|{1\over \widetilde h^{1/2}}\Phi\|_{L^2(Q_+)}$$ for all $\Phi \in \mathcal{D}(Q_+)$. Using now the weak regularity of $\widetilde \xi$ and $a_{22} \ge c >0$ in $Q_+$, this gives \begin{equation} \widetilde h^{1/2}\partial_2({1\over \widetilde h}\partial_2\widetilde \xi ) \in L^2(Q_+). \label{e19} \end{equation} Therefore, \eqref{e16}, \eqref{e18} and \eqref{e19} give the regularity \eqref{e9}. \end{proof} \begin{lemma}[Hardy's inequality in weighted spaces] \label{lm3} Let $h$ satisfy \eqref{e2}--\eqref{e6} and let $\Psi\in H(\mathcal{O})$. Then \begin{equation} \|{\Psi\over \delta h^{1/2}}\|_{L^2(\mathcal{O})} \le c \| {\nabla_{\rm x} \Psi\over h^{1/2}}\|_{L^2(\mathcal{O})^2}\label{e20} \end{equation} where $c$ depends only on $\mathcal{O}$. \end{lemma} \begin{figure}[t] \includegraphics[width= 0.6\textwidth]{fig2.eps} \caption{The local coordinates} \end{figure} \begin{proof} The proof of this lemma is similar to the proof of the classical Hardy's inequality (see for instance \cite{14}) introducing the corresponding weight. By density, it suffices to consider $\Psi\in \mathcal{D}(\mathcal{O})$. The interior estimate is obvious. In the local coordinates (see Figure 2), we write \begin{align*} &\int_{\alpha(x)}^z {|\Psi(x,y)|^2 dy \over |y-\alpha(x)|^2 \varphi(y-\alpha(x))} \\ & \le 2\int_{\alpha(x)}^z\Bigl(\int_y^{+\infty} {dt\over |t-\alpha(x)|^2\varphi(t-\alpha(x))}\Bigr) \Psi(x,y)\partial_y\Psi(x,y) dy\\ & \le 2\int_{\alpha(x)}^z {|\Psi(x,y)||\partial_y\Psi(x,y)|\over |y-\alpha(x)|\varphi(y-\alpha(x))}dy \\ & \le 2\Bigl(\int_{\alpha(x)}^z {|\Psi(x,y)|^2\over |y-\alpha(x)|^2\varphi(y-\alpha(x))}dy\Bigr)^{1/2} \Bigl(\int_{\alpha(x)}^z {|\partial_y \Psi(x,y)|^2\over \varphi(y-\alpha(x))}dy\Bigr)^{1/2}. \end{align*} Therefore, $$ \int_{\alpha(x)}^z {|\Psi(x,y)|^2 \over |y-\alpha(x)|^2\varphi(y-\alpha(x))} \,dy \le 4 \int_{\alpha(x)}^z {|\partial_y \Psi(x,y)|^2 \over \varphi(y-\alpha(x))}\,dy. $$ Thus, integrating with respect to $x$, we get $$ \int_{V\cap\mathcal{O}}{|\Psi(x,z)|^2\over |z-\alpha(x)|^2\varphi(z-\alpha(x))}\, dz\,dx \le 4\int_{V\cap\mathcal{O}} {|\partial_y \Psi|^2\over \varphi(\xi-\alpha(x))}\,d\xi\, dx. $$ Since $\alpha$ is smooth enough, there exists $c>1$ such that, for all $(x,z)\in V\cap\mathcal{O}$, $$\delta(x,z)\le |z-\alpha(x)|\le c\delta(x,z). $$ Therefore, using \eqref{e5}--\eqref{e6}, we get $$ \int_{V\cap\mathcal{O}}{|\Psi|^2\over \delta^2\varphi(\delta)} \le c\int_{V\cap\mathcal{O}} {|\partial_y \Psi|^2\over \varphi(\delta)} $$ and the result follows. \end{proof} \section{Importance of this degenerate equation} \label{sec3} Let us introduce the three-dimensional oceanographic domain $$ \Omega =\{({\rm x},z)\in \mathbb{R} ^3 : {\rm x}= (x,y)\in \mathcal{O}, -h({\rm x}) 0$ in $\mathcal{O}$, the bottom function. Moreover, $\Gamma_s =\overline{\mathcal{O}} \times \{0\}$ is the surface boundary and $\Gamma_b =\partial \Omega \setminus\Gamma_s$ the bottom. We denote $\nabla=(\nabla_{\rm x}, \partial_z) $ the three dimensional gradient vector (with $\nabla_{\rm x}=(\partial_x , \partial_y)$ the vectorial horizontal part) and $\Delta$ is the Laplace operator. We explain, in this section, why such degenerate equation naturally appears in different models issued from oceanography when hydrostatic pressure is assumed. In all these equations, the field $u=(v,w)$ and the pressure $p$ satisfy the equation \begin{equation} \begin{aligned} Lv + \nabla_{\rm x}\, p = f, \quad \partial_z p = 0 \quad\text{ in }\Omega, \\ \partial_z w = - \mathop{\rm div}_x v \quad\text{ in } \Omega, \end{aligned} \label{e21} \end{equation} and at least one of the the boundary conditions \begin{equation} (v,w)\cdot n_{\partial\Omega}= 0, \quad \overline v \cdot n_{\partial \mathcal{O}} = 0 \label{e22} \end{equation} where we use the notation $\overline v = \int_{-h}^0 v\, dz$. We note that $L$ is a certain operator (algebraic or differential), see \eqref{e29}, \eqref{e31} or \eqref{e36} for some examples. \noindent\textbf{Remark} Of course Boundary conditions \eqref{e22} are not necessary or sufficient to solve System \eqref{e21}. We have to choose other boundary conditions following the choice of the operator $L$. \smallskip Integrating the divergence free equation with respect to the vertical coordinate and using the boundary condition \eqref{e22}, part 1, we obtain \begin{equation} \nabla_{\rm x}\cdot \overline v = 0 \text{ in } \mathcal{O}. \label{e23} \end{equation} If the domain is simply connected then, using \eqref{e23}, there exists a stream function $\Psi$ such that \begin{equation} \overline v = \nabla_{\rm x}^\bot \Psi \text{ in } \mathcal{O}, \label{e24} \end{equation} where $\nabla_x^\bot$ is the $2D$ curl operator, i.e., $(-\partial_y,\partial_x)$. The boundary condition \eqref{e22}, part 2, gives \begin{equation} \Psi = 0 \quad\text{on} \partial \mathcal{O}. \label{e25} \end{equation} We assume that $v$ maybe formally written as \begin{equation} v = A\nabla_{\rm x} p + g_1\quad \text{ in } \Omega. \label{e26} \end{equation} where $A$ is a matrix function (see the examples below). The purpose is to obtain some regularity result on $v$. Integrating \eqref{e26} with respect to $z$ (taking into account that $\partial_z p =0$ in $\Omega$), we obtain $$ \overline v = \overline A\nabla_{\rm x} p + \overline g_1, $$ where $\overline A= \int_{-h}^0 A$ and $\overline g_1= \int_{-h}^0 g_1= $. Therefore, using \eqref{e24}, we obtain $$ \nabla^\bot_{\rm x} \Psi = \overline A \nabla_{\rm x} p + \overline g_1 $$ and thus, assuming $\overline A$ invertible \begin{equation} \nabla_{\rm x} p = B(\nabla_{\rm x}^\bot \Psi - \overline g_1) \label{e27} \end{equation} where $B=(\overline A)^{-1}$. Taking the horizontal curl operator of \eqref{e27}, using that $\nabla_{\rm x}^\bot\cdot \nabla_{\rm x} = 0$, we get \begin{equation} \nabla_{\rm x}^\bot \cdot (B(\nabla_{\rm x}^\bot \Psi - \overline g_1)) = 0 \text{ in } \mathcal{O}, \quad \Psi= 0 \text{ on } \partial \mathcal{= O}. \label{e28} \end{equation} On the other-hand, \eqref{e26} and \eqref{e27} yield $$ \nabla_{\rm x} v = \nabla_{\rm x} \Bigl( A\bigl(B(\nabla_{\rm x}^\bot \Psi - \overline g_1)\bigr)\Bigr) + \nabla_{\rm x} g_1. $$ Thus the regularity of $\nabla_{\rm x} v$ depends on the regularity of $\Psi$ and $g_1$. Theorem \ref{thm2} may be extended easily to more general degenerate elliptic equations including for instance \eqref{e28}. Now assume that $A= \mathop{\rm Id}$ then we get $B= 1/h \mathop{\rm Id}$ and therefore $A\bigl(B(\nabla_{\rm x}^\bot \Psi )\bigr) = \nabla^\bot \Psi/h$. Then the regularity of $\nabla_{\rm x} v$ in $(L^2(\Omega))^4$ is given by the regularity of $\Psi$ $$ h^{1/2}\nabla_{\rm x}(h^{-1}\nabla_{\rm x}\Psi)\in (L^2(\mathcal{O}))^4 $$ deduced from Theorem \ref{thm2}.\smallskip Let us give now some applications of such regularity results on the stream function. \section{Some applications for lakes and seas} \label{sec4} We consider again, in the three first examples, the three-dimensional domain $$ \Omega =\{({\rm x},z)\in \mathbb{R}^3 : {\rm x}= (x,y)\in \mathcal{O} , \ -h({\rm x})