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3 added 109 characters in body

Hello all,

could someone point me to a reference that ties the smoothness of the solution $u$ to the classical elliptic problem

$\nabla \cdot ( q \nabla u ) = f \;,\; x \in \Omega$

$u = g \;,\; x \in \Gamma = \partial \Omega$

to the smoothness of $f$, $q$ and $g$?

$\Omega$ is a convex polygonal domain in $\Re^d$ with $d \in {2,3}$. The boundary $\Gamma$ is piecewise linear (can have corners, e.g., if $\Omega$ is the unit square).

I am particularly interested in the (minimal) smoothness requirements for the forcing and boundary data $f$ and $g$, such that $u \in {\cal H}^2(\Omega)$ (not just locally).

I went through Evans' book on PDEs but he assumes homogeneous boundaries and proves only local smoothness $u \in {\cal H}_{\rm loc}^s(\Omega)$ based on assumptions on the forcing $f$. My $g$ is generally nonzero.

Also, would the smoothness theory for the BVP above extend to a Helmholtz problem with a pure Neumann BC?

Thanks for any good pointers!

Kind regards, -- Mihai

2 added 196 characters in body

Hello all,

could someone point me to a reference that ties the smoothness of the solution $u$ to the classical elliptic problem

$\nabla \cdot ( q \nabla u ) = f \;,\; x \in \Omega$

$u = g \;,\; x \in \Gamma$ Gamma = \partial \Omega$to the smoothness of$f$,$q$and$g$?$\Omega$is a convex polygonal domain in$\Re^d$with$d \in {2,3}$. The boundary$\Gamma$is piecewise linear (can have corners, e.g., if$\Omega$is the unit square). I am particularly interested in the (minimal) smoothness requirements for the forcing and boundary data$f$and$g$, such that$u \in {\cal H}^2(\Omega)$(not just locally). I went through Evans' book on PDEs but he assumes homogeneous boundaries and proves only local smoothness$u \in {\cal H}_{\rm loc}^s(\Omega)$based on assumptions on the forcing$f$. My$g$is generally nonzero. Thanks for any good pointers! Kind regards, -- Mihai 1 # smoothness of solution for second order elliptic problem Hello all, could someone point me to a reference that ties the smoothness of the solution$u$to the classical elliptic problem$\nabla \cdot ( q \nabla u ) = f \;,\; x \in \Omegau = g \;,\; x \in \Gamma$to the smoothness of$f$,$q$and$g$? I am particularly interested in the (minimal) smoothness requirements for the forcing and boundary data$f$and$g$, such that$u \in {\cal H}^2(\Omega)$(not just locally). I went through Evans' book on PDEs but he assumes homogeneous boundaries and proves only local smoothness$u \in {\cal H}_{\rm loc}^s(\Omega)$based on assumptions on the forcing$f$. My$g\$ is generally nonzero.

Thanks for any good pointers!

Kind regards, -- Mihai