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No. Consider the case $p=0$. In that case, solutions are constant along characteristics. But polynomial growth does not preclude characteristics from diverging to infinity in finite time.

The question whether appropriate weights exist is discussed in the following paper: L.R. Volevich, A problem of linear programming arising in differential equations, Uspekhi Mat. Nauk 18 (1963), No. 3 …

The standard references on parabolic PDEs assume bounded coefficients. When dealing with Fokker-Planck equations, however, the coefficients are usually unbounded. The way to get around this is to appr …

Example for first question: $\Delta^2 u=0$, say on the unit disk, with boundary condition ${\partial\over\partial n}\Delta u=0$, $\Delta u-u=0$. The boundary conditions do not satisfy the Lopatinskii …

Under reasonable boundary conditions, you will have $$\|D_3f\|_{L^2}\le C\|f\|_{H^3}\le C\|f\|_{H^4}^{1/2}\|f\|_{H^2}^{1/2}\le C\|\Delta^2f\|_{L^2}^{1/2}\|\Delta f\|_{L^2}^{1/2}\le C\lambda^{3/2}\|f\| …

This is certainly possible, and it seems rather obvious on physical grounds. Consider an initial condition where you have two hot regions connected by a thin corridor which is also hot, and the surrou …

Isn't the Laplacian with Neumann conditions a counterexample? There are no lower order terms, yet there is no inverse.

The answer is yes. Suppose $g$ is orthogonal to the image of $S$, and let $v$ be the solution of the Dirichlet problem $\Delta v=g\delta(\partial D_1)$ on $D_2$, where $\delta(\partial D_1)$ is a delt …