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Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.

4 votes

Intuition for Agmon-Douglis-Nirenberg ellipticity

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 …
Michael Renardy's user avatar
0 votes

Density of traces of solutions to an elliptic equation

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 …
Michael Renardy's user avatar
5 votes

Linear transport equation with unbounded coefficients

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.
Michael Renardy's user avatar
3 votes

Well-posedness of Fokker-Planck equation

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 …
Michael Renardy's user avatar
1 vote
Accepted

Strong maximum principle for the heat equation in non-cylindrical domains

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 …
Michael Renardy's user avatar
2 votes

Failure of Fredholm property of elliptic PDE systems

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 …
Michael Renardy's user avatar
2 votes

Bound deg 3 partial differential operator on Laplace eigenfunction?

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\| …
Michael Renardy's user avatar
1 vote

Is the left regularizer for elliptic BVP a left inverse for the principal part?

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