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I would like some references concerning the following subject.

Suppose that $\Omega$ is a bounded subset in $\mathbb{R}^n$ with smooth boundary and consider the following PDE there stated $$L(f)(x) = g(x),~ x\in \Omega,~~f(y) = h(y),~y\in \partial \Omega,$$ where $L : W^{2,p}(\overline \Omega) \to L^p(\overline \Omega)$ is an elliptic operator in $\Omega$ but such that its principal symbol vanishes for points in $\partial \Omega$.

Which are some classical references concerning this kind of problem, such as, existence of solutions and eventual regularity theory?

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    $\begingroup$ These are degenerate elliptic equations and there is no general theory but many specail cases. have a look at M. K. V. MURTHY and G. STAMPACCHIA, Boundary value problems for some degenerate elliptic operators, Annali di Mat. Pura ed Appl. (IV) LXXX (1968), 1-122. You will have to work with weighted Sobolev space $\endgroup$
    – Liviu Nicolaescu
    Commented Oct 22, 2021 at 22:08

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