Analyticity of solutions of a second order elliptic PDE with smooth coefficients.

Question

Under which conditions the solutions of a second order elliptic PDE with smooth coefficients on a bounded domain with analytic boundary is analytic?

Any reference will be extremely appreciated!

Answer 1

The only reference I know is the book of Morrey, Multiple Integrals in the Calculus of Variations. Section 6.6 studies analyticity of solutions both in the interior and on the boundary. Unfortunately it is not an easy book to read. Sorry.

EDIT This is the most general result from Morrey's book. Let $\Omega\subset\mathbb{R}^{N}$ be an open set and set $ D=\left( D_{1},\ldots,D_{N}\right)$, where $D_{j}=\frac{1}{i}\frac{\partial }{\partial y_{j}}$, $1\leq j\leq N$. Let $$L_{kj}\left( y,D\right)=\sum_{\left| \alpha\right| \leq s_{k}+t_{j}% }a_{kj}^{\alpha}\left( y\right) D^{\alpha} ,\quad 1\leq j,k\leq n,$$ be linear differential operators with continuous complex valued coefficients and consider the system of partial differential equations in the dependent variables $u^{1},\ldots ,u^{n}$ \begin{equation} \sum_{j=1}^{n}L_{kj}\left( y,D\right) u^{j}\left( y\right) =f_{k}\left( y\right) \quad \text{ in }\Omega,\quad 1\leq k\leq n. \label{elliptic system}% \end{equation} To each equation assign an integer weight $s_{k}\leq0$ and to each dependent variable an integer weight $t_{k}\geq0$ such that \begin{align*} \text{order }L_{kj}\left( y,D\right) & \leq s_{k}+t_{j}\quad\text{in }\Omega,\quad 1\leq k\leq n,\\ \max_{k}s_{k} & =0, \end{align*} where $L_{kj}\left( y,D\right) \equiv0$ if $s_{k}+t_{j}<0.$ The principal part of $L_{kj}\left( y,D\right) $ is defined by $$ L_{kj}^{\prime}\left( y,D\right) =\sum_{\left| \alpha\right| =s_{k}+t_{j}% }a_{kj}^{\alpha}\left( y\right) D^{\alpha}. $$ The system is elliptic if \begin{equation} \text{rank }(L_{kj}^{\prime}(y,\xi))=n\text{ for each }\xi\in\mathbb{R}% ^{N}\setminus\left\{ 0\right\} \text{ and }y\in\Omega, \label{rank condition}% \end{equation} and for each pair of independent vectors $\xi,\eta\in\mathbb{R}^{N}$ and $y\in\Omega$ the polynomial \begin{equation} p(z)=\det L_{kj}^{\prime}(y,\xi+z\eta) \label{even solutions}% \end{equation} has exactly $\mu=\frac{1}{2}\deg p$ roots with positive imaginary part and $\mu=\frac{1}{2}\deg p$ roots with negative imaginary part.

A general system of equations \begin{equation} F_{k}\left( y,\mathbf{u}(y),D\mathbf{u}(y),\ldots,D^{\ell}\mathbf{u}% (y)\right) =0\quad\text{in }\Omega,\quad 1\leq k\leq n, \label{nonlinear}% \end{equation} where $\mathbf{u=}\left( u_{1},\ldots,u_{n}\right) $ and $D^{m}$ stands for the set of all partial derivatives of order $m$, is elliptic along the solution $\mathbf{u}$ if the equations \begin{equation} \sum_{j=1}^{n}L_{kj}\left( y,D\right) \overline{u}^{j}\left( y\right) :=\left. \frac{d}{dt}F_{k}\left( y,\mathbf{u}(y)+t\overline{\mathbf{u}% },D(\mathbf{u}(y)+t\overline{\mathbf{u}}),\ldots,D^{\ell}(\mathbf{u}% (y)+t\overline{\mathbf{u}})\right) \right\vert _{t=0}=0 \label{linearized}% \end{equation} are an elliptic system as defined above.

Let $B_{hj}\left( y,D\right) $, $1\leq h\leq\mu$, $1\leq j\leq n$, be linear differential operator with continuous coefficients and assume that a portion of the boundary $\partial\Omega$ is contained in the hyperplane $y_{N\text{ }% }=0.$ We say that the set of boundary conditions $$ \sum_{j=1}^{n}B_{hj}\left( y,D\right) u^{j}\left( y\right) =g_{h}(y)\text{ on }S\subset\partial\Omega\cap\left\{ y_{N\text{ }}=0\right\} ,\text{ }1\leq h\leq\mu $$ is coercive for the system if

• the system is elliptic and $ 2\mu=\sum_{j=1}^{n}(s_{j}+t_{j})\geq0$;
• there exist integers $r_{h},$ $1\leq h\leq\mu,$ such that the order of $B_{hj}\left( y,D\right) \leq r_{h}+t_{j}$ on $S$;
• for every $y_{0}\in S$ the homogeneous boundary value problem \begin{align*} \sum_{j=1}^{n}L_{kj}^{\prime}\left( y_{0},D\right) u^{j}\left( y\right) & =0\text{ in }\mathbb{R}_{+}^{N},\text{ }1\leq k\leq n,\\ \sum_{j=1}^{n}B_{hj}^{\prime}\left( y_{0},D\right) u^{j}\left( y\right) & =0\text{ on }y_{N\text{ }}=0,\text{ }1\leq h\leq\mu, \end{align*} where $B_{hj}^{\prime}$ is the part of $B_{hj}$ of order $r_{h}+t_{j},$ admits no nontrivial bounded exponential solutions of the form $$ u^{j}\left( y\right) =e^{i\xi^{\prime}y^{\prime}}\varphi_{j}(y_{N\text{ }% }),\text{ }1\leq j\leq n\text{, }\xi^{\prime}\in\mathbb{R}^{N-1}, $$ where $y^{\prime}=\left( y_{1},\ldots,y_{N-1}\right) .$

A set of (nonlinear) boundary conditions $$ \Psi_{h}\left( y,\mathbf{u}(y),D\mathbf{u}(y),\ldots,D^{s}\mathbf{u}% (y)\right) =0\quad\text{on }S,\quad 1\leq h\leq\mu, $$ is coercive for the system along the solution $\mathbf{u}$ if there exist weights $r_{1},\ldots,r_{\mu}$ such that the set of linearized boundary conditions \begin{equation} \sum_{j=1}^{n}B_{hj}\left( y,D\right) \overline{u}^{j}\left( y\right) :=\left. \frac{d}{dt}\Psi_{k}\left( y,\mathbf{u}(y)+t\overline{\mathbf{u}% },D(\mathbf{u}(y)+t\overline{\mathbf{u}}),\ldots,D^{s}(\mathbf{u}% (y)+t\overline{\mathbf{u}})\right) \right\vert _{t=0}=0 \label{linearized boundary}% \end{equation} in $S$ is coercive for the linearized system on $S.$

Theorem Let $U$ be a neighborhood of $0$ in $\mathbb{R}% _{+}^{N}$ and $S=\partial U\cap\left\{ y_{N\text{ }}=0\right\} .$ Assume that $\mathbf{u}$ is a solution of the elliptic and coercive system \begin{align*} F_{k}\left( y,\mathbf{u}(y),D\mathbf{u}(y),\ldots,D^{\ell}\mathbf{u}% (y)\right) & =0\quad\text{in }U,\quad 1\leq k\leq n,\\ \Psi_{h}\left( y,\mathbf{u}(y),D\mathbf{u}(y),\ldots,D^{s}\mathbf{u}% (y)\right) & =0\quad\text{on }S,\quad 1\leq h\leq\mu, \end{align*} with weights $s_{k},$ $t_{j},$ $r_{h},$ $1\leq j,k\leq n,$ $1\leq h\leq\mu.$

Suppose also that $F_{k}$ and $\Psi_{h}$ are analytic. If $u^{j}\mathbf{\in C}^{t_{h}+r_{0},\alpha}\left( U\cup S\right) $, for some $\alpha>0$ and where $r_{0}=\max_{h}\left( 0,1+r_{h}\right) ,$ then the $u^{j}$ are analytic in $U\cup S,$ $1\leq j\leq n.$

Answer 2

Are you interested in real analytic, complex analytic solutions? I don't have a direct answer for you, but maybe some of these will be interesting to you nonetheless. I've been studying for a PDE qualifying exam, so here we go...

If you are not interested strictly in analytic solutions, I'd look at Evans PDE chapter 6, section 3. Chapter 6 is on general second order elliptic PDEs, and section 3 is on regularity. He discusses the regularity of $u$ in the case with smooth $C^\infty$ coefficients, $C^k$ coefficients, and bounded coefficients.

For a more applied treatise on the same topics as Evans, in sense of potential theory, see chapter 8 section 6 of Guenther and Lee's dover text PDE of mathematical physics and integral equations.

If you are interested in elliptic equations with nonlinear lower order terms, the last chapter of Paul Garabedian's (older but classic) book Partial Differential Equations discusses the analyticity of solutions to such equations.

Fritz John doesn't talk much about regularity but he does give a detailed discussion on real analytic functions and the Cauchy-Kowalevski theorem.

It does seem that many books are missing discussion on analytic solutions to general second order elliptic equations... it seems like the general interest has moved in the opposite direction, toward studying weak solutions.